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1309.2765	# Enhancements of Multi-class Support Vector Machine Construction from Binary Learners using Generalization Performance Patoomsiri Songsiri [email protected] Thimaporn Phetkaew [email protected] Boonserm Kijsirikul [email protected] Department of Computer Engineering, Chulalongkorn University, Pathumwan, Bangkok Thailand. 10330 School of Informatics, Walailak University, Thasala District, Nakhon Si Thammarat Thailand. 80161 ###### Abstract We propose several novel methods for enhancing the multi-class SVMs by applying the generalization performance of binary classifiers as the core idea. This concept will be applied on the existing algorithms, i.e., the Decision Directed Acyclic Graph (DDAG), the Adaptive Directed Acyclic Graphs (ADAG), and Max Wins. Although in the previous approaches there have been many attempts to use some information such as the margin size and the number of support vectors as performance estimators for binary SVMs, they may not accurately reflect the actual performance of the binary SVMs. We show that the generalization ability evaluated via a cross-validation mechanism is more suitable to directly extract the actual performance of binary SVMs. Our methods are built around this performance measure, and each of them is crafted to overcome the weakness of the previous algorithm. The proposed methods include the Reordering Adaptive Directed Acyclic Graph (RADAG), Strong Elimination of the classifiers (SE), Weak Elimination of the classifiers (WE), and Voting based Candidate Filtering (VCF). Experimental results demonstrate that our methods give significantly higher accuracy than all of the traditional ones. Especially, WE provides significantly superior results compared to Max Wins which is recognized as the state of the art algorithm in terms of both accuracy and classification speed with two times faster in average. ###### keywords: support vector machine , multi-class classification , generalization performance ††journal: Neurocomputing ## 1 Introduction The support vector machine (SVM) [1, 2] is a high performance learning algorithm constructing a hyperplane to separate two-class data by maximizing the margin between them. There are two approaches for extending SVMs to multi- class problems, i.e., solving the problem by formulating all classes of data under a single optimization, and combining several two-class subproblems. However, the difficulty and complexity to solve the problem with the first method are due to the increase of the number of classes and the size of training data, so the second method is more suitable for practical use. In this paper, we focus on the second approach. For constructing a multi-class classifier from binary ones, the method called one-against-one trains each binary classifier on only two out of $N$ classes, and builds $N(N-1)/2$ possible classifiers. Several strategies have been proposed for combining the trained classifiers to make the final classification for an unseen data. Friedman [3] suggested the combination strategy called Max Wins. In the classification process of Max Wins, every binary classifier provides one vote for its preferred class and the class with the largest vote will be set to be the final output. Chang and Lee [4] investigated an adaptive framework to manage a nuisance vote which is a vote for an unrelated class by allowing a classifier to make a non-vote for data of unrelated class. Instead of a binary classifier, they employed a ternary classifier that consists of two particular classes and the rest of the classes fused as the third class for solving this problem. Vapnik [1] proposed the one-against-the-rest approach working by constructing a set of $N$ binary classifiers in which each $i^{th}$ classifier is learned from all examples in the $i^{th}$ class, and the remaining classes labeled with the positive and negative classes, respectively. The class corresponding to the classifier with the highest output value is used to make the final output. Moreover, Manikandan and Venkataramani [5] adapted the traditional one-against-the-rest to work as a sequential classifier. All classifiers will be ordered corresponding to their misclassification. This method needs a lower number of classifiers on avearge compared with the traditional one-against- the-rest, but both algorithms have the same problem in the training phase because of the difficulty for calculating the absolutely separating hyperplane between a class and all of the other classes. Dietterich and Bakiri [6] introduced the Error Correcting Output Code (ECOC) based on the fundamental of information theory. For a given code matrix with $N$ rows and $L$ columns, each element contains either $\textquoteleft$1’, or $\textquoteleft$-1’. Each column denotes the bit string showing the combination of positive and negative classes for constructing a binary classifier, and each row of the code matrix indicates the unique bit string for representing a specific class (each bit string is called a codeword). Allwein et al. [7] extended the coding method by adding the third symbol $\textquoteleft$0’ as “ don’t care bit” to allow the binary model learned without considering some particular classes. Unlike the previous method, the number of classes for training a binary classifier can be varied from 2 to $N$ classes. Based on these two systems for an $N$ classes problem, the maximum numbers of different binary classifiers are $2^{N-1}-1$ [6], and $\frac{3^{N}-2^{N+1}+1}{2}$ [8], respectively. Design of code matrices with different subsets of binary classifiers gives different abilities for separating classes, and the problem of selecting a suitable subset of binary classifiers is complicated with a large size of $N$. To obtain the suitable code matrix, some techniques using the Genetic Algorithm have been proposed [9, 10]. In the classification phase, a test example is classified by all classifiers corresponding to the column of the code matrix, and then the class with the closet codeword is assigned to the final output class. Platt et al. [11] proposed the Decision Directed Acyclic Graph (DDAG) in order to reduce evaluation time [12]. In each round, a binary model will be randomly selected from all $N(N-1)/2$ classifiers. The binary classification result is employed to eliminate the candidate output classes, and to ignore all binary classifiers related to the defeated class. It guarantees that the number of classifications (applied classifiers) of the DDAG is always $N-1$. This recursive task will be applied until there is only one last candidate class. However, the misclassification of the DDAG can be occurred at the time when selected binary classifiers related to the target class (hence forth BCRT) give the wrong answer. The more times the number of BCRTs are applied, the more chance the misclassification is produced by the DDAG. In order to reduce this risk, Kijsirikul and Ussivakul [13] proposed the Adaptive Directed Acyclic Graphs (ADAG) that has a reversed triangular structure of the DDAG. It requires only $\lceil log_{2}N\rceil$ times or less that the target class is tested against the other classes, while the DDAG possibly requires at most $N-1$ times. In addition, there have been many attempts that apply some information such as the margin size [11], the number of support vectors [14], and the separability measures between classes [15, 16], to improve the performance of the multi- class classification. The margin size and the number fo support vectors were applied for selecting the suitable two-class classifiers in the DDAG [11, 14]. The separability measure was employed for automatically constructing a binary tree of multi-class classification based on the concept of the minimum spanning tree [15]. Li, et al. [16] used similar information to vote the preferred class for data in unclassifiable region for both the one-against-one and the one-against-the-rest techniques. In this research, we investigate the framework for enhancing three well-known methods, which are the DDAG, the ADAG, and Max Wins. Max Wins is currently recognized as the-state-of-the-art combining algorithm and it is also the most powerful technique among all of our focused works with a need of $N(N-1)/2$ number of classifications for an $N$-class problem, while the other two approaches reduce the number of classifications to $N-1$. We study the characteristics of these methods that lead to wrong classification results. The first two techniques have the same hierarchical structure and have the same weak point that they “trust on individual opinion” for making decision to discard the candidate classes. Intuitively, if only one of BCRTs makes a mistake, the whole system will give the wrong output. The last technique as the high performance one, Max Wins is based on the concept of “trust on most popular opinion” for making decision to select the output class. If all of $N-1$ BCRTs give the correct answer, Max Wins will always provide the correct output class. However, if there exists only one of BCRTs give the wrong answer, it may lead to misclassification due to equal voting, or other non- target classes reaching the largest vote as shown later in the paper. Examples which are incorrectly classified in this scenario can be recovered by our proposed strategies. In this paper, we demonstrate that the above traditional methods can be improved based on the same idea that if we access further important information of generalization performance of all binary classifiers and properly estimate it, it can be employed for enhancing the performance of the methods. Based on this idea, we propose four novel approaches including (1) the Reordering Adaptive Directed Acyclic Graph (RADAG), (2) Strong Elimination of the classifiers (SE), (3) Weak Elimination of the classifiers (WE), and (4) Voting based Candidate Filtering (VCF). The first approach, the next two approaches, and the last approach are improved from the ADAG, the DDAG, and Max Wins, respectively. We also empirically evaluate our methods by comparing them with the traditional methods on the sixteen datasets from the UCI Machine Learning Repository [17]. This paper is organized as follows. Section 2 reviews the traditional multi- class classification frameworks. Section 3 describes how to properly estimate the generalization performance of binary classifiers. Section 4 presents our proposed methodologies. Section 5 performs experiments and explains the results and discussions. Section 6 concludes the research. ## 2 Multi-class Support Vector Machines ### 2.1 Max Wins For an $N$-class problem, all possible pairs of two-class data are learned for constructing $N(N-1)/2$ classifiers. All binary classifiers are applied for voting the preferred class. A class with maximum vote will be assigned as the final output class. This method is called Max Wins [3]. However, in case that there exists more than one class giving the same maximum vote, the final output class can be obtained by random selection from candidate classes with the equal maximum-vote. An example of the classification using this technique for a four-class problem is shown in Fig. 1. Each class will be voted (solid- line) or ignored (dash-line) by all related binary models. For example, class $1$ has three related classifiers, i.e., 1 vs 2, 1 vs 3, and 1 vs 4. The voting result of class $1$, class $2$, class $3$ and class $4$ are three, zero, one, and, two, respectively. In this case, class $1$ has the largest score, and therefore it is assigned as the final output class. Figure 1: An example of a four-class classification with Max Wins. ### 2.2 Decision Directed Acyclic Graphs Platt et al. [11] introduced a learning algorithm using the Directed Acyclic Graph (DAG) to represent the classification task, called the Decision Directed Acyclic Graph (DDAG). This architecture represents a set of nodes connected by edges with no cycles. Each edge has an orientation and each node has either 0 or 2 edges. Among these nodes, there exists a root node which is the unique node with no edge pointing into it. In a DDAG, the nodes are arranged with a triangular shape in which each node is labeled with an element of a boolean function. There exists a single root node at the top, two nodes in the second layer, and so on until the final layer of $N$ leaves for an $N$-class problem. To make a classification, an example with an unknown class label is evaluated by the nodes as binary decision functions. The binary output result in each layer is applied to eliminate the candidate output classes and the binary classifiers related to the defeated class are removed. At the first layer (see Fig. 2), the root node can be randomly selected from all possible $N(N-1)/2$ classifiers and there are $N$ candidate output classes. After the root node is tested, its binary result is employed to eliminate the candidate output classes and the binary classifiers corresponding to the defeated class are discarded. In the next layer, the remaining binary classifiers are randomly selected to continue the same process in which some classes are eliminated from the remaining candidate classes. The process is repeated until there is only one class remained which is then assigned as the final output class. This algorithm requires only $N-1$ decision nodes in order to obtain the final answer. Figure 2: The DDAG finding the best class out of four classes [11]. One disadvantage of the DDAG is that its classification result is affected by the sequence of binary classifiers randomly selected in the evaluation process. Platt et al. also proposed the other method that prefers the binary decision function with the higher generalization performance measured by its margin sizes, called the large margin DAGs [11]. The margin size ($\Delta$) is a parameter for bounding the generalization ability of the binary SVM as shown in terms of the VC-dimension in Eq. (2). It illustrates that the generalization performance of the binary model is proportional to the size of the margin. A binary classifier with the larger margin size will be firstly applied in each round of the evaluation step. Moreover, Takahashi and Abe [14] proposed a similar framework that employed the number of support vectors as a performance measure. In this method, the generalization error ($\epsilon_{ij}$) for classes $i$ and $j$ was bounded by Eq. (1) [18]: $\epsilon_{ij}=\frac{SV_{ij}}{M_{ij}},$ (1) where $SV_{ij}$ is the number of support vectors for classes $i$ and $j$ and $M_{ij}$ is the number of training data for classes $i$ and $j$. ### 2.3 Adaptive Directed Acyclic Graphs Figure 3: The structure of an adaptive DAG for an 8-class problem. In the DDAG, binary classification result of a previously employed binary classifier is used to eliminate a candidate output classe, and there are only current remaining candidate classes that can be possibly assigned as the final output class. Therefore, the misclassification of a selected BCRT is the crucial point. The ADAG was originally designed to reduce this risk of the DDAG by using reversed triangular structure [13]. In an $N$-class problem, there are $\lceil\frac{N}{2}\rceil$ nodes at the top, $N/2^{2}$ nodes in the second layer and so on until the lowest layer of the final node, as illustrated in Fig. 3. Like the DDAG, binary output results of the ADAG in each layer are applied to discard candidate output classes and the binary classifiers related to the defeated classes are also ignored. Therefore, the ADAG also evaluates only $N-1$ nodes to obtain the final answer. According to the critical issue of misclassification mentioned above, even only one selected classifier related to the target class provides a wrong answer, the misclassification on the final output class cannot be avoided. Hence, the number of times the target class is tested against other classes indicates the risk of misclassification. The DDAG requires at most $N-1$ times that the target class is tested against other classes, while the ADAG requires only $\lceil log_{2}N\rceil$ times or less. This shows that the opportunities of the target class tested against other classes on the ADAG is much lower than the DDAG. ## 3 An Estimation of the Generalization Performan-ces of Binary Support Vector Machines The generalization performance of a learning model is the actual performance evaluated on unseen data. For support vector machines, a model is trained by using the concept of the Structure Risk Minimization principle [19] in which the generalization performance of the model is estimated based on both terms of the complexity of model (the VC dimension of approximating functions) and the quality of fitting training data (empirical error). Consider the problem of binary classification where dataset $X$ of $m$ samples in real $n$-dimensional space is randomly independent identically distributed observations drawn according to $P(x,y)=P(x)P(y\|x)$. The expected risk ($R(\alpha)$) with probability at least $1-\delta$ can be bounded by the following equation [20, 21]: $R(\alpha)\leq\frac{l}{m}+\sqrt{\frac{c}{m}({\frac{R^{2}}{\Delta^{2}}{log^{2}m}+log{\frac{1}{\delta}}})},$ (2) where there is a corresponding constant $c$ for all probability distributions, $l$ is the number of labeled examples in $z$ with margin less than $\Delta$, $R$ indicates the radius of the smallest sphere that contains all the data points, and $\Delta$ is the distance between the hyperplane and the closest points of the training set (margin size). The first and second terms of inequality in Eq. (2) denote the bound of the empirical error, and the VC dimension, respectively. In our frameworks, the generalization ability will be applied to improve the multi-class classification. Although there have been many attempts to use some performance measures such as the margin size [11], the number of support vectors [14], they may not accurately reflect the actual performance of each binary SVM. Consider a two-class problem where hyperplanes $h_{1}$ and $h_{2}$ are learning models created to separate the positive and the negative examples. Suppose that they provide different margin sizes of $\Delta_{1}$ and $\Delta_{2}$, and the different numbers, $l_{1}$ and $l_{2}$, of labeled examples in $z$ with the margin less than their margin sizes, respectively. In case that the parameters $c$ and $\delta$ are fixed, there are only two parameters including $\Delta$ and $l$ that affect the performance of the learning model (as the parameters $m_{1}$ and $m_{2}$, as well as $R_{1}$ and $R_{2}$ are the same for the same pair of a two-class problem). Now consider two-learning models learned from different pairs of a two-class problem. In case that the parameters $c$, and $\delta$ are fixed, according to inequality in Eq. (2), obviously, if we use only $\Delta$, $\l$, or combination of them, they are not sufficient to represent the whole term of their generalization abilities. This shows that a binary model with the larger margin size may not provide more accurate result of classification. The use of only the number of support vectors is also shown in [21] that it is not predictive for generalization ability. (a) (b) (c) (d) Figure 4: Generalization errors of 325 classifiers of the Letter dataset based on $k$-fold cross-validation (CV Bound), the number of support vectors (shown in term of the ratio between the number of support vectors and the number of training data: SV Bound), the margin size (shown in term of its inverse value normalized to be in [0,1]: Normalized Margin Bound), and their actual risks on test data (unseen data) by applying the polynomial kernel of $d=4$. Figure (a) compares generalization errors calculated by all techniques where classifiers are sorted in the ascending order by their actual generalization performances (actual risk), and figures (b)-(d) show the comparisons between the actual risks, and the estimated generalization errors with different measures, i.e., CV Bound, SV Bound, and Normalized Margin Bound, respectively (the classifiers will be sorted in ascending order by the estimated generalization errors, and for ease of visualization we show only $10\%$ of classifiers by sampling every ten classifers from the sorted list of the classifiers. Figure 5: Classification process of the RADAG. As described above, the generalization ability can be employed to enhance the performance of multi-class classification, by carefully design algorithms which utilize this information as a selection measure for good classifiers. We believe that the generalization performance of binary SVMs can be directly estimated by $k$-fold cross-validation [22] (see Algorithm 1), and it can be used to fairly compare the performances of binary SVMs on different two-class problems. Below we give an example which demonstrates that $k$-fold cross- validation is more suitable for estimating the generalizaiton performance of the classifiers than the other measures used by the previous methods, i.e. the number of support vectors, the margin size. Algorithm 1 An estimation of the generalization error of a classifier by using $k$-fold cross-validation. 1:procedure Cross Validation 2: Set of training data $T$ is partitioned into $k$ disjoint equal-sized subsets 3: Initial the classification error of round $i$: $\epsilon_{i}\leftarrow 0$ 4: for $i$=1 to $k$ do 5: validate set $\leftarrow$ $i^{th}$ subset 6: training set $\leftarrow$ all remaining subsets 7: Learn model based on training set 8: $\epsilon_{i}$ $\leftarrow$ Evaluate the learned model by validate set, and find the number of examples with misclassification 9: end for 10: generalization error $\leftarrow$ $\sum_{i=1}^{k}\epsilon_{i}\times\frac{1}{\|T\|}$ 11: return generalization error 12:end procedure Fig. 4 shows the generalization performance measured by the previous methods [11, 14], and $k$-fold cross-validation, which we propose to use as the performance measure, for the Letter dataset with 26 classes, by applying the polynomial kernel of $d=4$. Fig. 4 (a) illustrates that the trend of estimated generalization error by $k$-fold cross-validation is very closed to the actual risk, while the other two techniques give high variation. To further investigation in more details, we select about $10\%$ of all classifiers to show in Fig. 4 (b-d); these figures illustrate the comparisons between the actual risk and the estimated generalization errors with different measures, i.e., CV Bound, SV Bound, and Normalized Margin Bound, respectively. In each figure, classifiers are sorted in ascending order by the estimated generalization errors. It is expected that if a specific measure is a good estimator for generalization error, its value should be in the same trend as the actual risk (its value shoud increase with the increase of the actual risk). A good trend is found in Fig. 4 (b), while the other two methods give no clear trend and contain confusing patterns. In order to evaluate the efficiency of these estimating methods, we apply the correlation analysis between two variables [23], i.e., the actual risks and these three estimated generalization errors. These evaluations are based on 325 classifiers as in Fig. 4 (a), and the statistical r-values of them are 0.805, 0.372, and -0.230 as shown in Fig. 4 (b-d), respectively. The r-values also confirm that CV Bound and actual risk have high correlation, while the other two methods give very low correlation. They show that $k$-fold cross-validation is more suitable to be the measure for the performance of binary classifiers. According to the above reason, we apply this measure in our research. ## 4 The proposed methods The combination of binary SVMs with high generalization performance directly affects the accuracy of the multi-class classification. In this section, we introduce four enhanced approaches based on the previous techniques i.e., the ADAG, the DDAG, and Max Wins by applying the generalization abilities in order to select suitable binary classifiers. An improvement of the ADAG is called the Reordering Adaptive Directed Acyclic Graph (RADAG). There are two improved versions for the DDAG i.e., Strong Elimination of the classifiers (SE) and Weak Elimination of the classifiers (WE). The last technique is Voting-based Candidate Filtering (VCF) enhanced from Max Wins. To increase the classification accuracy, the generalization estimated by $k$-fold cross- validation is utilized as the goodness measure of classifiers in our frameworks. ### 4.1 Reordering Adaptive Directed Acyclic Graph The ADAG is designed to reduce the number of times the binary classifiers related to the target class are applied, from at most $N-1$ times required by the DDAG, to $\lceil log_{2}N\rceil$ times or less. However, binary classifiers in the first level of the ADAG are still randomly selected, and its misclassification can be produced at the time even when only one BCRT gives a wrong answer. In this section, we introduce a more effective method which uses the minimum weight perfect matching to select the optimal pairs of classes in each level with minimum generalization error. We called the method the Reordering Adaptive Directed Acyclic Graph (RADAG). The structure of the RADAG is similar to the ADAG, but they are different in the initialization of the binary classifiers in the top level and the order of classes in lower levels (see Fig. 4.1). The reordering algorithm with minimum weight perfect matching is described in Algorithm 2. The algorithm selects the optimal order of classes in each level. It is different from the ADAG in that the initial order of classes in the ADAG is obtained randomly, and the matching of classes in successive levels depends on the classification results of nodes from the previous level. For the RADAG, the reordering process will be applied to the remaining candidate classes in all levels for determining the optimal sequence of them. Figure 6: (a) A graph for an 8-class problem (b) An example of the output of the reordering algorithm. To select the optimal set of classifiers, the generalization measure in Section 3 is used as a criterion. This scheme provides less chance to predict the wrong class from all possible $\frac{N!}{2^{\lfloor{N/2}\rfloor}{\lfloor{N/2}\rfloor}!}$ orders. Among $N(N-1)/2$ classifiers, $N/2$ classifiers which have the smallest sum of generalization errors will be used in the classification. Algorithm 2 Reordering Adaptive Directed Acyclic Graph (RADAG). 1:procedure RADAG 2: Initial set of candidate output classes $C=\\{1,2,3,...,N\\}$, and set of discarded classes $D=\emptyset$ 3: Calculate generalization errors of all possible pairs of classes on $C$ as described in Section 3 4: Create the binary SVMs from all possible pairs of classes on $C$ 5: while $\|C\|>1$ do 6: Apply the minimum weight perfect matching [24] to find the optimal $\lfloor\frac{\|C\|}{2}\rfloor$ pairs of classes from all possible pairs on $C$ to obtain the optimal binary models with minimum generalization error 7: $D$ $\leftarrow$ Classify the example by the optimal binary models, and find the defeated classes 8: $C$ $\leftarrow$ $C-D$ 9: end while 10: final output class $\leftarrow$ the last remaining candidate class 11: return final output class 12:end procedure Let $G=(V,E)$ be a graph with node set $V$ and edge set $E$. Each node in $G$ denotes one class and each edge indicates one binary classifier of which generalization error is estimated from Section 3 (see Fig. 6(a)). The output of the reordering algorithm for graph $G$ is a subset of edges with the minimum sum of generalization errors of all edges and each node in $G$ is met by exactly one edge in the subset (see Fig. 6(b)). Given a real weight $\epsilon_{e}$ being generalization error for each edge $e$ of $G$, the problem of reordering algorithm can be solved by the minimum weight perfect matching [24] that finds a perfect matching $M$ of minimum weight $\sum(\epsilon_{e}:e\in M)$. For $U\subseteq V$, let $E(U)=\\{(i,j):(i,j)\in E,i\in U,j\in U\\}$. $E(U)$ is the set of edges with both endpoints in $U$. The set of edges incident to node $i$ in the node-edge incidence matrix is denoted by $\delta(i)$. The convex hull of perfect matchings on a graph $G=(V,E)$ with $\|V\|$ even is given by a) $x\in\\{0,1\\}^{m}$ b) $\sum_{e\in\delta(v)}x_{e}=1$ for $v\in V$ c) $\sum_{e\in E(U)}x_{e}\leq\lfloor\frac{\|U\|}{2}\rfloor$ for all odd sets $U\subseteq V$ with $\|U\|\geq 3$ or by (a),(b) and d) $\sum_{e\in\delta(U)}x_{e}\geq 1$ for all odd sets $U\subseteq V$ with $\|U\|\geq 3$ where $\|E\|=m$, and $x_{e}=1$ ($x_{e}=0$) means that $e$ is (is not) in the matching. Hence, the minimum weight of a perfect matching is at least as large as the value of $min\sum_{e\in E}\epsilon_{e}x_{e}$ (3) where $x$ satisfies “(a), (b), and (c)” or “(a), (b) and, (d)”. Therefore, the reordering problem can be solved by the integer program in Eq. (3). ### 4.2 Strong & Weak Elimination of Classifiers for Enhancing Decision Directed Acyclic Graph Algorithm 3 Strong Elimination of the classifiers (SE). 1:procedure SE 2: Initial set of candidate output classes $C=\\{1,2,3,...,N\\}$, and set of discarded classes $D=\emptyset$ 3: Calculate generalization errors of all possible pairs of classes on $C$ as described in Section 3 4: Create the binary models from all possible pairs of classes on $C$ 5: Sort the list of the binary models in ascending order by the generalization errors 6: current classifier $\leftarrow$ the first element on the sorted list 7: while $\|C\|>1$ do 8: $D$ $\leftarrow$ Classify the example by the current classifier, and find the defeated class 9: $C$ $\leftarrow$ $C-D$ 10: current classifier $\leftarrow$ the next element on the sorted list where it is not related to any classes discarded from $C$ 11: end while 12: final output class $\leftarrow$ the last remaining candidate class 13: return final output class 14:end procedure Algorithm 4 Weak Elimination of the classifiers (WE). 1:procedure WE 2: Initial set of candidate output classes $C=\\{1,2,3,...,N\\}$, and set of discarded classes $D=\emptyset$ 3: Calculate generalization errors of all possible pairs of classes on $C$ as elaborated in section 3 4: Create the binary models from all possible pairs of classes on $C$ 5: Sort the list of the binary models in ascending order by the generalization errors 6: current classifier $\leftarrow$ the first element on the sorted list 7: while $\|C\|>1$ do 8: $D$ $\leftarrow$ Classify the example by current classifier, and find the defeated class 9: $C$ $\leftarrow$ $C-D$ 10: current classifier $\leftarrow$ the next element on the sorted list where it does not include all two classes discarded from $C$ 11: end while 12: final output class $\leftarrow$ the last remaining candidate class 13: return final output class 14:end procedure According to the characteristic of the DDAG, binary classification results of the previously employed binary classifiers are used to eliminate the candidate output classes, and thus the final output class will be assigned with one of the remaining candidate classes. By using the random technique for selecting a binary classifier, the DDAG produces mis-classification at the time when a BCRT with very low performance is selected and provides the wrong answer, as the target class will be discarded from the remaining candidate classes, and it is not possible to reach the correct output class. In this section, we propose the framework to enhance the performance of the DDAG to select the binary classifier with high performance based on the generalization abilities of binary classifiers as described in Section 3. We propose two methods that are Strong Elimination of the classifiers (SE) and Weak Elimination of the classifiers (WE). Both algorithms are described in Algorithm 3 and Algorithm 4. We also show a classification process of SE and WE for an $N$-class problem in Fig. 7 and Fig. 8, respectively. For both of the DDAG and SE, in each round, a defeated class will be removed from candidate output classes, and all binary classifiers related to the defeated class are ignored. Due to this reason, they guarantee $N-1$ number of classifications for an $N$-class problem. However, these ignored classifiers may have high generalization abilities and thus are helpful to eliminate the other remaining candidate classes. Therefore, we then propose WE to make use of binary classifiers with high generalization abilities. According to the classifier elimination of WE, the number of classifications is bounded with the best case of $N-1$, and the worst case of $N(N-1)/2$. However, WE provides the opportunities to employ better classifiers as shown in the Fig. 8. At round $r$, suppose that classifier $A_{i}$ vs $A_{j}$ has lower generalization error than classifier $A_{i}$ vs $A_{k}$ , and both of them are active classifiers. In this case, it is possible that classifier $A_{i}$ vs $A_{j}$ can remove the class $A_{i}$ from the list of two remaining candidate classes, and can avoid using classifier $A_{i}$ vs $A_{k}$ with lower reliability that is unavoidable for SE as shown in Fig. 7. Figure 7: Classification process of SE for an $N$-class problem. Figure 8: Classification process of WE for an $N$-class problem. ### 4.3 Voting Based Candidate Filtering Max Wins is one of high performance techniques that work based on the concept of “trust on the most popular opinion” for making decision to select the output class. If all of $N-1$ BCRTs give the correct answer, Max Wins will always provide the correct output class. It does not depend on the answers of the other binary classifiers. However, if only one of BCRTs gives a wrong answer, it may lead to misclassification due to equal voting, or another non- target class reaching the largest vote. Fig. 9 shows an example of such cases, taken from our experiment on the Letter dataset (see Section for more details); Fig. 9(a) and (b) show the cases of equal voting and another non- target class having the largest vote, respectively. We propose a novel multi-class classification approach that alleviates the above problem of Max Wins, and uses the same concept “trust on the most popular opinion” for filtering out the low competitive classes. On the other hand, high competitive classes will be voted to be candidate output classes, though there exist some BCRTs providing the wrong answer. If there is more than one remaining class, the output class will be selected via the mechanism of WE. Our proposed technique aims to combine the strong point of both Max Wins and WE, and is called Voting based Candidate Filtering (VCF). The details of our algorithm are shown in Algorithm 5. Algorithm 5 Voting based Candidate Filtering (VCF). 1:procedure VCF 2: Initial set of candidate output classes $C=\\{1,2,3,...,N\\}$, and score of class $i$: $s_{i\in N}\leftarrow 0$ 3: Create the binary models from all possible pairs of classes on $C$ 4: for $j$=1 to $N(N-1)/2$ do 5: $w$ $\leftarrow$ Classify the example by classifier $j^{th}$, and find the winner class 6: $s_{w}$ $\leftarrow$ $s_{w}+1$ 7: end for 8: $s_{top}$ $\leftarrow$ Find the top voting score of all $s_{i\in N}$ 9: for $i$=1 to $N$ do 10: $dp_{i}\leftarrow\frac{(s_{top}-s_{i})\times 100}{s_{top}}$ 11: if $dp_{i}\leq$ $threshold\\_value$ then 12: Add class $i$ into the set of candidate output classes $C$ 13: end if 14: end for 15: if $\|C\|>1$ then 16: final output class $\leftarrow$ Call the WE procedure 17: else 18: final output class $\leftarrow$ the last remaining candidate class 19: end if 20: return final output class 21:end procedure Let $s_{top}$, and $s_{i}$ indicate the maximum of scores for all $N$ classes, and the score of class $i\in[N]$ for a test data, respectively. Also let $dp_{i}$ denotes the percentage of the difference between $s_{top}$ and $s_{i}$. An example of the calculation of $dp_{i}$ is shown in Fig. 9 (a), where $i=\textquoteleft$E’, the score of class $\textquoteleft$E’ = 23 points, and the score of class $\textquoteleft$C’ = 24 points (as the top score). Then $dp_{i}$ value can be calculated by $\frac{(24-23)\times 100}{24}=4.17$. We also define $threshold\\_value$ to be the threshold of $dp_{i}$ for considering class $i$ as a candidate for the target class; class $i$ will be accepted into the set of high competitive candidate classes if and only if its $dp_{i}$ is less than or equal to $threshold\\_value$. We want to keep the size of the filtered candidate classes as small as possible while still containing the target class. (a) (b) Figure 9: An example of high risk of misclassification of Max Wins together with score distribution of all classes: two cases of misclassification of class $\textquoteleft$C’ due to only one BCRT giving the wrong answer in the Letter problem having 26 classes (25 possible BCRTs, and 25 points as the largest possible score), a) three classes, including $\textquoteleft$C’, $\textquoteleft$G’, and $\textquoteleft$L’, with equal score (only one BCRT $\textquoteleft$C vs G’ giving the wrong class), and b) the non-target class $\textquoteleft$E’ with the highest score (only one BCRT $\textquoteleft$C vs E’ providing the wrong class). Figure 10: A case study of the $461$ examples with high risk of misclassification in the Letter problem. The maximum voting scores of these examples are reached (1) by both of the target class and the non$-$target class (equal vote: $1^{st}$ rank), or (2) by a non$-$target class (absolutely wrong: $2^{nd}-8^{th}$ rank). The figure shows the target class score of these examples by observing between the $dp_{t}$ and the rank of the target class. A case study of high risk of misclassification in the Letter dataset including 4,010 examples where Max Wins provides 3,549 examples with the correct result, and 461 examples with high risk of misclassification. By a high-risk example, we mean (1) the example with an equal vote (the score of the target class is equal to those of other non-target classes) and (2) the example with a vote less than the maximum vote that is then mis-classified by Max Wins. These high risk examples will be hopefully recovered with the correct class label by our proposed algorithm. In our experiment, the high-risk examples includes 24 examples (around 5$\%$) with an equal vote, and 437 examples (around 95$\%$) with a vote less than the maximum as shown in Fig. 10, where $dp_{t}$ represents the percentage of the difference between $s_{top}$ and the score of the target class. For each example, we calculate the rank of the voting score of the target class compared to the other non-target classes, and consider only the first eight ranks. There are 24 examples (around $5\%$) in the first rank, while in the second to the eighth ranks, the numbers of examples are 171, 77, 31, 30, 15, 14, and 10 (around $39\%,18\%,7\%,7\%,3\%,3\%$, and $2\%$), respectively. The examples with the different ranks have different ranges of $dp_{t}$ values, such as, in the second rank, the $dp_{t}$ values are varied from $4.0$ to $12.0$, in the third rank, the $dp_{t}$ values are varied from $4.2$ to $20.0$, in the fourth rank, the $dp_{t}$ values are varied from $8.7$ to $20.0$, and so on. According to this case study, there can be at most $5\%$ of examples that will be correctly classified with the correct class label by random selection of Max Wins, while the other $95\%$ of examples will be absolutely misclassified. We want to recover an example that is not correctly classified by Max Wins, as its actual target class is not in the first rank or its target class has equal vote with some other output classes. If $threshold\\_value$ is set as $1$ in the VCF algorithm, it will guarantee that all high-risk misclassified examples with $dp_{t}$ values no greater than can be filtered into the set of the candidate output classes; in this case only the examples in the first rank ($5\%$ of examples) will be selected. When we apply a bigger threshold, e.g. $threshold\\_value$ $=10$, it covers all misclassified examples in the first and the second ranks ($5\%+39\%$), almost of the third rank ($18\%$), and some parts of the fourth rank ($7\%$). It shows that the increase of $threshold\\_value$ covers more candidate classes, while the larger size of $threshold\\_value$ creates a higher risk to employ an unnecessarily large number of binary classifiers. On the other hand, if $threshold\\_value$ is too low, the target class may be removed. However, a suitable $threshold\\_value$ can be obtained by general tuning techniques. For our experiment, we just define $threshold\\_value$ to be $10$ for all of datasets without fine-tuning which is good enough to demonstrate the effectiveness of the VCF algorithm. ## 5 Experiments In this section, we design the experimental setting to evaluate the performance of the proposed methods. We compare our methods with the traditional algorithms, i.e., the DDAG, the ADAG, and Max Wins. We divide this section into two parts as experimental protocols, and results & discussions. ### 5.1 Experimental Protocol We run experiments on sixteen datasets from the UCI Machine Learning Repository [17] including Page Block, Glass, Segment, Arrhyth, Mfeat-factor, Mfeat-fourier, Mfeat-karhunen, Mfeat-zernike, Optdigit, Pendigit, Primary tumor, Libras Movement, Abalone, Krkopt, Spectrometer, and Letter (see Table 1). For the datasets containing both training data and test data, we added up both of them into one set, and used 5-fold cross validation for evaluating the classification accuracy. Table 1: Description of the datasets used in the experiments. Datasets \| #Cases \| #Classes \| #Features ---\|---\|---\|--- Page Block \| 5,473 \| 5 \| 10 Glass \| 214 \| 6 \| 9 Segment \| 2,310 \| 7 \| 18 Arrhyth \| 438 \| 9 \| 255 Mfeat-factor \| 2,000 \| 10 \| 216 Mfeat-fourier \| 2,000 \| 10 \| 76 Mfeat-karhunen \| 2,000 \| 10 \| 64 Mfeat-zernike \| 2,000 \| 10 \| 47 Optdigit \| 5,620 \| 10 \| 62 Pendigit \| 10,992 \| 10 \| 16 Primary tumor \| 315 \| 13 \| 15 Libras Movement \| 360 \| 15 \| 90 Abalone \| 4,098 \| 16 \| 8 Krkopt \| 28,056 \| 18 \| 6 Spectrometer \| 475 \| 21 \| 101 Letter \| 20,052 \| 26 \| 16 In these experiments, we scaled data to be in [-1,1] and employed two kernel functions i.e., the Polynomial kernel $K(x_{i},x_{j})\equiv\|({\bf x_{i}}\cdot{\bf x_{j}}+1)\|^{d}$, and the RBF kernel $K(x_{i},x_{j})\equiv e^{-\gamma\|\|{\bf x_{i}}-{\bf x_{j}}\|\|^{2}}$. For the polynomial kernel we applied the same set of degrees $d=\\{2,3,4,5\\}$ to all datasets, and for the RBF kernel we applied the set of degrees $\gamma_{1}=\\{1,0.5,0.1,0.05\\}$ to Page Block, Glass, Segment, Mfeat-zernike, Pendigit, Libras Movement, Abalone, Krkopt, and Letter, and applied the set of degrees $\gamma_{2}=\\{0.1,0.05,0.01,\\\ 0.005\\}$ to the other datasets. The default parameter of regularization parameter $C$ was used for model construction; this parameter is used to trade off between error of the SVM on training data and margin maximization. In the training phase, we used software package $SVM^{light}$ version 6.02 [25, 26] to create the $N(N-1)/2$ binary classifiers. For the DDAG and the ADAG, we examined all possible orders of classes for datasets having not more than 8 classes, whereas we randomly selected 50,000 orders for datasets having more than 8 classes, and we then calculated the average of accuracy of these orders. ### 5.2 Results & Discussions We compare the original methods with their enhanced techniques in three tasks including: (1) the ADAG with the RADAG, (2) the DDAG with two improved approaches, i.e., SE and WE, and (3) Max Wins with VCF. We also selected the best techniques from (1) and (2), i.e., the RADAG and WE, respectively, and compared them with Max Wins as the state of the art technique. These comparison results are shown in Table 2 to Table 5. Moreover, paired comparison among all of three traditional methods (the DDAG, the ADAG, and Max Wins), and all proposed techniques (SE, the RADAG, WE, and VCF) are concluded in Table 6. The best accuracy among these methods is represented in bold-face. In addition, we used the one-tailed paired t-test technique to analyze the significant difference between the accuracies of the traditional algorithms and the proposed algorithms. To estimate the difference between accuracies, we use a $k$-fold cross-validation method [22]. To indicate the level of the confidence interval using a one-tailed paired t-test in the Table 2 to Table 5, the symbol ‘$+$’ and ‘$-$’ are used to represent that the corresponding method has higher accuracy, and lower accuracy compared to a baseline method, respectively. The number of symbols shows the level of confidence interval for estimating the difference between accuracies of two algorithms i.e., one symbol, two symbols, and three symbols represent 90%, 95%, and 99% respectively. Table 2: A comparison of the classification accuracy of the ADAG and the RADAG. \| Polynomial \| \| RBF ---\|---\|---\|--- Data sets \| ADAG \| RADAG \| \| ADAG \| RADAG Page Block \| 93.597 \| 93.541 $---$ \| \| 93.562 \| 93.555 $-$ Glass \| 63.879 \| 64.019 \| \| 63.084 \| 63.318 Segment \| 93.207 \| 93.236$+{\color[rgb]{1,1,1}}$ \| \| 93.348 \| 93.366 Arrhyth \| 63.489 \| 63.470 \| \| 58.049 \| 57.991 Mfeat-factor \| 97.238 \| 97.225 $--{\color[rgb]{1,1,1}}$ \| \| 96.921 \| 96.938 Mfeat-fourier \| 82.839 \| 82.863 \| \| 82.456 \| 82.513$+$ Mfeat-karhunen \| 96.864 \| 96.863 \| \| 96.890 \| 96.900 Mfeat-zernike \| 82.368 \| 82.413$+++{\color[rgb]{1,1,1}}$ \| \| 81.867 \| 81.888 Optdigit \| 98.995 \| 98.999 \| \| 98.620 \| 98.630 Pendigit \| 99.400 \| 99.402 \| \| 99.313 \| 99.320$+$ Primary tumor \| 47.266 \| 47.619$++{\color[rgb]{1,1,1}}$ \| \| 46.089 \| 46.429$+++$ Libras Movement \| 73.218 \| 73.194 \| \| 72.289 \| 72.569$+$ Abalone \| 27.603 \| 27.648 \| \| 27.353 \| 27.337 Krkopt \| 53.102 \| 53.239$+++$ \| \| 53.088 \| 53.173$+++$ Spectrometer \| 54.445 \| 54.842$++{\color[rgb]{1,1,1}}$ \| \| 50.808 \| 51.579$+{\color[rgb]{1,1,1}}$ Letter \| 88.668 \| 88.787$+++$ \| \| 89.989 \| 90.090$+++$ The experimental results in Table 2 uses the ADAG as the baseline algorithm. It shows that the RADAG yields highest accuracy in several datasets. The results also show that, at $95$% confidence interval, the RADAG performs statistically better than the ADAG in five datasets using the Polynomial kernel and better in three datasets using the RBF kernel. As shown in the table, the RADAG performs better when the number of classes is comparatively large, and does not perform well in the datasets with the small number of classes, i.e., the Page Block, and the Mfeat-factor with 5 and 10 classes, respectively. We believe that in case of datasets with the large number of classes, the variety of generalization errors of classifiers in consideration is rich and the RADAG is able to choose good classifiers freely, whereas the RADAG may be forced to select ineffective classifiers in case of the small number of classes, and it could lead to an incorrect output class. Table 3: A comparison of the classification accuracy between the DDAG, and our methods, i.e, SE, and WE. \| Polynomial \| \| RBF ---\|---\|---\|--- Data sets \| DDAG \| SE \| WE \| \| DDAG \| SE \| WE Page Block \| 93.597 \| \| 93.541 $---$ \| 93.623$+$ \| \| 93.562 \| \| 93.555 $-$ \| 93.582$++{\color[rgb]{1,1,1}}$ Glass \| 63.892 \| \| 64.019 \| 64.019 \| \| 63.084 \| \| 63.201 \| 63.201 Segment \| 93.207 \| \| 93.236 $+$ \| 93.247$++$ \| \| 93.350 \| \| 93.344 \| 93.366 Arrhyth \| 63.490 \| \| 63.527 \| 63.527 \| \| 58.048 \| \| 57.991 \| 58.162 Mfeat-factor \| 97.238 \| \| 97.250 \| 97.238 \| \| 96.923 \| \| 96.975$+$ \| 96.975$++{\color[rgb]{1,1,1}}$ Mfeat-fourier \| 82.837 \| \| 82.863 \| 82.863 \| \| 82.443 \| \| 82.475 \| 82.538 Mfeat-karhunen \| 96.863 \| \| 96.875 \| 96.875 \| \| 96.861 \| \| 96.850 \| 96.988$+{\color[rgb]{1,1,1}}$ Mfeat-zernike \| 82.362 \| \| 82.400$++{\color[rgb]{1,1,1}}$ \| 82.350 \| \| 81.869 \| \| 81.888 \| 81.863 Optdigit \| 98.994 \| \| 99.013$++{\color[rgb]{1,1,1}}$ \| 99.008 $+{\color[rgb]{1,1,1}}$ \| \| 98.618 \| \| 98.643$+$ \| 98.630 Pendigit \| 99.399 \| \| 99.404 \| 99.402 \| \| 99.312 \| \| 99.318 \| 99.320$+$ Primary tumor \| 47.227 \| \| 47.064 \| 47.460$++$ \| \| 46.019 \| \| 46.032 \| 46.111 Libras Movement \| 73.142 \| \| 73.264 \| 73.472$++{\color[rgb]{1,1,1}}$ \| \| 72.283 \| \| 72.569$+$ \| 72.431 Abalone \| 27.611 \| \| 27.648 \| 27.672$+$ \| \| 27.354 \| \| 27.330 \| 27.398 Krkopt \| 53.101 \| \| 53.263 $++$ \| 53.472$+++$ \| \| 53.088 \| \| 53.212 $+++$ \| 53.320$+++$ Spectrometer \| 54.373 \| \| 54.632 \| 54.421 \| \| 50.821 \| \| 51.316 \| 51.842$++$ Letter \| 88.609 \| \| 88.707 $+++$ \| 88.835$+++$ \| \| 89.903 \| \| 89.977 $++$ \| 90.294$+++$ Table 3 shows the experimental results of SE and WE compared with the DDAG as the baseline algorithm. Both WE and SE have higher accuracy than the traditional DDAG in almost all datasets. The results also show that at $95$% confidence interval, SE performs statistically significantly better than the DDAG in four datasets using the Polynomial kernel and significantly better than the DDAG in two datasets using the RBF kernel. It is similar to the previous comparison between the ADAG and the RADAG that in datasets with the small number of classes, the classifier manipulation of SE may be forced to select inaccurate classifiers and it possibly leads to the misclassification. The results also show that WE performs statistically significantly better than the DDAG in five datasets in both cases of the Polynomial kernel and the RBF kernel. These results illustrate that WE can reduce the risk of selecting inaccurate classifiers compared to SE. Figure 11: An example of generalization errors of binary SVMs used by WE and SE in the Letter dataset. Table 4: A comparison of the classification accuracy of Max Wins and VCF. Data sets \| Polynomial \| \| RBF ---\|---\|---\|--- \| Max Wins \| VCF \| \| Max wins \| VCF Page Block \| 93.600 \| 93.623 \| \| 93.567 \| 93.582$++{\color[rgb]{1,1,1}}$ Glass \| 63.863 \| 64.019 \| \| 63.143 \| 63.201 Segment \| 93.209 \| 93.247$++{\color[rgb]{1,1,1}}$ \| \| 93.351 \| 93.366 Arrhyth \| 63.489 \| 63.527 \| \| 58.048 \| 58.162 Mfeat-factor \| 97.242 \| 97.238 \| \| 96.927 \| 96.975$++{\color[rgb]{1,1,1}}$ Mfeat-fourier \| 82.852 \| 82.825 \| \| 82.454 \| 82.525 Mfeat-karhunen \| 96.879 \| 96.875 \| \| 96.952 \| 96.963 Mfeat-zernike \| 82.338 \| 82.350 \| \| 81.825 \| 81.863 Optdigit \| 99.004 \| 99.013 \| \| 98.631 \| 98.630 Pendigit \| 99.402 \| 99.402 \| \| 99.315 \| 99.320 Primary tumor \| 47.394 \| 47.460 \| \| 46.508 \| 46.191 Libras Movement \| 73.194 \| 73.472$++{\color[rgb]{1,1,1}}$ \| \| 72.373 \| 72.431 Abalone \| 27.614 \| 27.672$+$ \| \| 27.375 \| 27.398 Krkopt \| 53.149 \| 53.475$+++$ \| \| 53.146 \| 53.328$+++$ Spectrometer \| 54.263 \| 54.421$+{\color[rgb]{1,1,1}}$ \| \| 51.026 \| 51.842$++$ Letter \| 88.706 \| 88.869$+++$ \| \| 90.112 \| 90.316$+++$ Table 5: A comparison of the classification accuracy between Max Wins and the RADAG, and WE. \| Polynomial \| \| RBF ---\|---\|---\|--- Data sets \| Max Wins \| RADAG \| WE \| \| Max Wins \| RADAG \| WE Page Block \| 93.600 \| 93.541 $---$ \| 93.623 \| \| 93.567 \| 93.555 $--{\color[rgb]{1,1,1}}$ \| 93.582$++{\color[rgb]{1,1,1}}$ Glass \| 63.863 \| 64.019 \| 64.019 \| \| 63.143 \| 63.318 \| 63.201 Segment \| 93.209 \| 93.236 $+{\color[rgb]{1,1,1}}$ \| 93.247$++$ \| \| 93.351 \| 93.366 \| 93.366 Arrhyth \| 63.489 \| 63.470 \| 63.527 \| \| 58.048 \| 57.991 \| 58.162 Mfeat-factor \| 97.242 \| 97.225 $---$ \| 97.238 \| \| 96.927 \| 96.938 \| 96.975$++{\color[rgb]{1,1,1}}$ Mfeat-fourier \| 82.852 \| 82.863 \| 82.863 \| \| 82.454 \| 82.513 \| 82.538 Mfeat-karhunen \| 96.879 \| 96.863 \| 96.875 \| \| 96.952 \| 96.900 \| 96.988 Mfeat-zernike \| 82.338 \| 82.413$++{\color[rgb]{1,1,1}}$ \| 82.350 \| \| 81.825 \| 81.888$++{\color[rgb]{1,1,1}}$ \| 81.863 Optdigit \| 99.004 \| 98.999 \| 99.008 \| \| 98.631 \| 98.630 \| 98.630 Pendigit \| 99.402 \| 99.402 \| 99.402 \| \| 99.315 \| 99.320 \| 99.320 Primary tumor \| 47.394 \| 47.619$++$ \| 47.460 \| \| 46.508 \| 46.429 \| 46.111 Libras Movement \| 73.194 \| 73.194 \| 73.472$++{\color[rgb]{1,1,1}}$ \| \| 72.373 \| 72.569 \| 72.431 Abalone \| 27.614 \| 27.648 \| 27.672$+$ \| \| 27.375 \| 27.337 \| 27.398 Krkopt \| 53.149 \| 53.239 $++{\color[rgb]{1,1,1}}$ \| 53.472$+++$ \| \| 53.146 \| 53.173 \| 53.320$+++$ Spectrometer \| 54.263 \| 54.842$++$ \| 54.421 $+$ \| \| 51.026 \| 51.579 \| 51.842$++$ Letter \| 88.706 \| 88.787 $++{\color[rgb]{1,1,1}}$ \| 88.835$+++$ \| \| 90.112 \| 90.090 \| 90.294$+++$ We further analyze the results comparing WE and SE on the Letter dataset which consists of $26$ classes and 325 binary learners, as shown in Fig. 11. These 325 classifiers in the figure are sorted in ascending order by the generalization error, and this sequence of classifiers is maintained in the classification phase. SE requires $25$ classifiers and WE requires $93$ classifiers in this case, and the generalization error of the worst binary classifier in WE is almost five times lower than in SE (the largest generalization errors of all binary SVMs used in SE and WE are 0.015, 0.073, respectively). As a result, the average performance of the binary classifiers in WE is higher than SE. As shown in Table 4 with Max Wins as the baseline method, VCF yields higher accuracy than Max Wins in almost all of datasets. The results show that, at $95$% confidence interval, in the Polynomial kernel VCF performs statistically significantly better than Max Wins in four datasets, and in the RBF kernel VCF performs statistically significantly better than Max Wins in five datasets. The previous three tables show that our proposed methods improve the accuracy of the ADAG, the DDAG, and Max Wins significantly. Next, we select the best algorithm in each table from the first two tables, i.e, the RADAG, and WE, and then compare them to Max Wins. According to experimental result in Table 5, at $95$% confidence interval, the RADAG performs statistically significantly better than Max Wins in five datasets using the Polynomial kernel, and significantly higher than Max Wins in one dataset using the RBF kernel. In case of the small number of classes, it is possible that the RADAG will have the effect mentioned above. For WE, the results show that it performs statistically significantly better than Max Wins in four datasets in case of the Polynomial kernel and significantly better than Max Wins in five datasets in case of the RBF kernel. There is no any dataset in which Max Wins has significantly higher accuracy than WE. Table 6: Paired comparisons among all techniques including of three traditional techniques (DDAG, ADAG, and Max Wins), and four proposed techniques (RADAG, SE, WE, and VCF). \| \| Traditional Methods \| \| Proposed Methods ---\|---\|---\|---\|--- Kernel Function \| Algorithms \| DDAG \| ADAG \| Max Wins \| \| SE \| RADAG \| WE \| VCF Polynomial \| DDAG \| \| 1-15-0 \| 2-14-0 \| \| 4-11-1 \| 5-9-2 \| 5-11-0 \| 6-10-0 \| ADAG \| \| \| 2-13-1 \| \| 4-11-1 \| 5-9-2 \| 4-12-0 \| 5-11-0 \| Max Wins \| \| \| \| \| 3-12-1 \| 5-9-2 \| 4-12-0 \| 4-12-0 \| SE \| \| \| \| \| \| 2-14-0 \| 4-11-1 \| 4-11-1 \| RADAG \| \| \| \| \| \| \| 2-12-2 \| 4-10-2 \| WE \| \| \| \| \| \| \| \| 1-15-0 RBF \| DDAG \| \| 2-14-0 \| 2-14-0 \| \| 2-14-0 \| 3-13-0 \| 5-11-0 \| 5-11-0 \| ADAG \| \| \| 2-14-0 \| \| 1-15-0 \| 3-13-0 \| 5-11-0 \| 5-11-0 \| Max Wins \| \| \| \| \| 2-12-2 \| 1-14-1 \| 5-11-0 \| 5-11-0 \| SE \| \| \| \| \| \| 2-13-1 \| 4-12-0 \| 3-13-0 \| RADAG \| \| \| \| \| \| \| 3-13-0 \| 3-13-0 \| WE \| \| \| \| \| \| \| \| 1-15-0 Table 6 summarizes paired comparisons of all algorithms including the traditional techniques, and the proposed works based on both of the Polynomial kernel and the RBF kernel. We show the win-draw-loss record (s) of the algorithm in the column against the algorithm in the row. A win-draw-loss record reports how many datasets the method in the column is better than the method in the row (win), is equal (draw), or is worse (loss) at $95$% confidence interval. As summarized in the table, our proposed methods are better than all previous works i.e., the DDAG, the ADAG, and Max Wins. WE and VCF give the highest accuracy among all of our methods. The result also shows that VCF gives a little better results compared to WE. However, as mentioned before in Section 4.3, the accuracies of VCF are the ones without fine-tuning, and higher accuracies can be expected if fine-tuning is performed to find the optimal $threshold\\_value$ for VCF. ### 5.3 Computational Time The computational times of all methods are shown in Fig. 12 and Fig. 13. We can classify algorithms according to the time requirement into three groups, for an $N$-class problem: 1) $N-1$ times i.e., the DDAG, the ADAG, SE, and the RADAG, 2) average about half of time of $N(N-1)/2$ i.e., WE, 3) $N(N-1)/2$ i.e., Max Wins, and VCF. The results show that algorithms in the first and the second groups require comparatively low running time in all datasets, especially when the number of classes is relatively large, while the larger the number of classes, the more running time the algorithms in the third group requires. WE in the second group requires $N-1$ classifiers in the best case and $N(N-1)/2$ classifiers in the worst case; however, in our experimental results WE takes approximately half of time required by the algorithms in the third group. For the RADAG, though the number of classes affects the running time for reordering process, it takes a little time even when there are many classes. The algorithms in the third group need $O(N^{2})$ comparisons for a problem with $N$ classes. VCF needs more time to choose the final class from the set of candidate classes which can be obtained by re-using the previous results of binary classification. The DDAG reduces the number of comparisons down to $O(N)$. SE spends a little time more than the DDAG for sorting the classifiers in the training phase. By reducing the depth of the path, the ADAG and SE require $O(N)$ comparisons of binary classifiers. WE consumes more time than SE due to each round of classification can reduce only one classifier while SE can eliminate all classifiers built from the discarded class. The number of testing classifiers for WE is equal to that for Max Wins in the worst case; fortunately, the experimental results show that WE actually spends only half of Max Wins’ times in the average case. The RADAG needs a little time more than the ADAG for reordering the order of classes. Note that, the minimum weight perfect matching algorithm, which is used in the reordering algorithm, runs in time bounded by $O(N(M+NlogN))$ [24], where $N$ is the number of nodes (classes) in the graph and $M=N(N-1)/2$ is the number of edges (binary classifiers). The RADAG will reorder the order of classes in every level, except for the last level. The order of classes in the top level is reordered only once and we use the order to evaluate every test example. Hence for classifying each test data, we need $log_{2}N-2$ times of reordering, where each time the number of classes is reduced by half. Therefore, the running time of the RADAG is bounded by $O(c_{1}N)+O(c_{2}N^{3}log_{2}N)$, where $c_{1}$ is much larger than $c_{2}$. Figure 12: A comparison of the computational time using the Polynomial kernel. Figure 13: A comparison of the computational time using the RBF kernel. ## 6 Conclusion Max Wins is a powerful combining technique with a need of $N(N-1)/2$ number of classifications for an $N$-class problem, while the DDAG and the ADAG reduce the number of classifications to $N-1$. We study the characteristics of these previous methods that lead to wrong classification results. We believe that the performances of them depend on the BCRTs. In case of Max Wins, if there exists only one BCRT giving an incorrect answer, it may convey misclassification due to equal voting or another non-target class reaching the largest vote, while in cases of the DDAG and the ADAG, if only one of BCRTs in the sequence of selected classifiers makes a mistake, the whole system will give the wrong output. We investigate the well-organized combination of the binary models including BCRTs in classification process to provide a more precise final result. In this research, we propose four methods for overcoming the above weakness of the previous works. All our proposed methods are based on the same principle that if the information about genearalization ability is accurately measured, then it is able to be employed for enhancing the performance of the classification. In this paper, the generalization performance is estimated by $k$-fold cross-validation technique, and we show that it is more suitable than previously used measures in other frameworks, such as the margin size and the number of support vectors. Our proposed methods are the Reordering Adaptive Directed Acyclic Graph (RADAG), Strong Elimination of the classifiers (SE), Weak Elimination of the classifiers (WE), and Voting based Candidate (VCF). The RADAG is an enhanced version for the ADAG by using the minimum weight perfect matching for selecting the optimal pair of classes in each level with minimum generalization error. Compared to the ADAG, the RADAG is not only superior in terms of accuracy, but also maintains the same testing time ($N-1$). Next, We propose two improved algorithms for the DDAG, i.e. SE and WE. In SE, a sequence of binary classifiers selected by minimum generalization error is applied to eliminate the candidate classes until only one class remained and assigned as the final output class. SE provides better accuracy than the DDAG. The testing time of SE is the same as the traditional DDAG and the RADAG, with a number of applied classifiers equal to $N-1$. We also propose the other enhanced version for the DDAG, called WE. This approach aims to efficiently use as many as possible of the classifiers with low generalization errors. This is different from the process of the DDAG and SE in which all binary classifiers related to a defeated class are ignored when the defeated class is removed from the candidate classes. In WE, however, a classifier will be ignored only if all of two related classed of that classifier are discarded from the candidate output classes, and this process enables WE to efficiently employ good classifiers. WE gives significantly higher performance compared to the DDAG, and requires the number of classifications on average about half of the number of all possible binary classifiers. Additionally, we propose VCF by applying the voting technique to carefully select the high competitive classes with high confidence. The remaining candidate classes are recursively eliminated by using WE. Although the number of classifications of VCF is equal to that of Max Wins, it shows the highest accuracy compared to all the other algorithms. Finally, more experiments were conducted to compare our proposed algorithms and Max Wins in order to find the suitable scenario for using each of them. The RADAG should be chosen when the number of classes is large and the classification time is the most concern. VCF shows the highest accuracy among our proposed algorithms, and it should be selected when the time constraints is not the main concern. In a general case, WE is the most suitable method because it is superior to Max Wins in terms of accuracy and time. All of our techniques apply the generalization performance for organizing the use of the binary classifiers. This measure can be optimally estimated by the mechanism of $k$-fold cross-validation that is independent of base learners. Consequently, all our proposed methods can be also applied to other base classifiers such as logistic regression, perceptron, linear discriminant analysis, etc. The estimation of generalization errors using k-fold cross validation requires additional computation, and this can be thought of as a drawback of our methods. However, the estimation is done in the offline training phase, and thus it does not affect the performance in the classification phase. ## 7 Acknowledgment The authors would like to thank Dr.Peerapon Vateekul for his valuable comments on an earlier version of this paper. This research is partially supported by the Thailand Research Fund, and the Graduate School, Chulalongkorn University. ## References [1] V.Vapnik , Statistical Learning Theory, New York, Wiley, 1998. * [2] V.Vapnik, An overview of statistical learning theory, IEEE Transactions on Neural Networks. 10 (1999) 988-999. * [3] J.H.Friedman, Another approach to polychotomous classification, Technical report, Stanford University, Department of Statistics, 1996. * [4] C.C.Chang, L.Chien, and Y.Lee, A novel framework for multi-class classification via ternary smooth support vector machine, Pattern Recognition 44 (2011) 1235-1244. * [5] J.Manikandan, and B.Venkataramani, Study and evaluation of a multi-class SVM classifier using diminishing learning technique, Neurocomputing 73 (2010) 1676-1685. * [6] T.G.Dietterich, and G.Bakiri, Solving multiclass learning problems via error-correcting output codes, Journal of Artificial Intelligence Research 2 (1995) 263-286. * [7] E.L.Allwein, R.E.Schapire, and Y.Singer, Reducing Multiclass to Binary: A Unifying Approach for Margin Classifiers, Journal of Machine Learning Research 1 (2000) 113-141. * [8] M.A.Bagheri, G.Montazer, and E.Kabir, A subspace approach to error correcting output codes, Pattern Recognition Letters 34 (2013) 176-184. * [9] L.I.Kuncheva, Using diversity measures for generating error-correcting output codes in classifier ensembles, Pattern Recognition Letters 26 (2005) 83-90. * [10] A.C.Lorena, and A.C.P.L.F.Carvalho Evaluation functions for the evolutionary design of multiclass Support Vector Macines, International Journal of Computational Intelligence and Applications 8 (2009) 53-68. * [11] J.Platt, N.Cristianini, and J.Shawe-Taylor, Large margin DAGs for multiclass classification, Proceedings of Neural Information Processing Systems, MIT Press (2000) 547-553. * [12] C.Hsu, and C.Lin, A comparison of methods for multiclass support vector machines, IEEE Transactions on Neural Networks 13 (2002) 415-425. * [13] B.Kijsirikul,and N.Ussivakul, Multiclass support vector machines using adaptive directed acyclic graph., Proceedings of International Joint Conference on Neural Networks (IJCNN) (2002) 980-985. * [14] F.Takahashi, and S.Abe, Optimizing directed acyclic graph support vector machines, Proceedings of Artificial Neural Networks in pattern recognition (2003) 166-170. * [15] A.C.Lorena, and A.C.P.L.F.Carvalho, Building binary-tree-based multiclass classifiers using separability measures, Neurocomputing 73 (2010) 2837-2845. * [16] R.Li, A.Li, T.Wang, and L.Li, Vector projection method for unclassifiable region of support vector machine, Expert Systems with Applications 38 (2011) 856-861. * [17] C.Blake, E.Keogh, and C.Merz, UCI repository of machine learning databases, Department of Information and Computer Science, University of California, Irvine, 1998. * [18] V.Vapnik, The Nature of Statistical Learning Theory, London, UK, Springer-Verlag, 1995. * [19] V. N.Vapnik, and A.Y.Chervonenkis, Teoriya Raspoznavaniya Obrazov: Statisticheskie Problemy Obucheniya. (Russian) [Theory of Pattern Recognition: Statistical Problems of Learning]., Moscow: Nauka, 1974. * [20] P.L.Bartlett, and J.Shawe-Taylor, Generalization performance of support vector machines and other pattern classifiers, Advances in Kernel Methods - Support Vector Learning, MIT Press, Cambridge, USA, (1999) 43-54. * [21] C.Burges, A tutorial on support vector machines for pattern recognition, Data Mining and Knowledge Discovery 2 (1998) 121-167. * [22] T.Mitchell, Machine Learning, McGraw Hill, 1997. * [23] R.Johnson and, G.Bhattacharyya, Statistics: principles and methods, New York, Wiley, 2001. * [24] W.Cook, and A.Rohe, Computing minimum-weight perfect matchings, Technical Report 97863, Forschungsinstitut f$\ddot{u}$r Diskrete Mathematik, Universit$\ddot{a}$t Bonn, (1999). * [25] T.Joachims, Making large-scale SVM learning practical, Advances in Kernel Methods - Support Vector Learning, MIT Press (1998). * [26] T.Joachims, SVMlight, http://ais.gmd.de/~thorsten/svm_light, (1999). Patoomsiri Songsiri received the B.Sc. degree in Computer Science (First class honor) from Prince of Songkla University, Thailand, in 2001 and the M.Sc. degree in Computer Science from Chulalongkorn University, Thailand, in 2006. She is currently working toward the Ph.D. degree in Computer Engineering at Chulalongkorn University. Her research interests include Pattern Recognition and Machine Learning. Thimaporn Phetkaew received her B.Sc. degree in Applied Mathematics and she also received her M.Sc. degree in Computer Science from Prince of Songkla University, Thailand in 1997 and 2000, respectively. In 2004, she received her Ph.D. degree in Computer Engineering from Chulalongkorn University, Thailand. Since 2004, as a lecturer, she has been with the School of Informatics, Walailak University. She is also a member of Informatics Innovation Research Unit at Walailak University. Her research interests include Data Mining, Machine Learning, and Software Testing. Boonserm Kijsirikul received the B.Eng. degree in Electronic and Electrical Engineering, the M.Sc. degree in Computer Science, and the Ph.D. in Computer Science from Tokyo Institute of Technology, Japan, in 1986, 1990, and 1993, respectively. He is currently a Professor at the Department of Computer Engineering, Chulalongkorn University, Thailand. His current research interests include Machine Learning, Artificial Intelligence, Natural Language Processing, and Speech Recognition.	arxiv-papers	2013-09-11T08:59:07	2024-09-04T02:49:50.690866	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Patoomsiri Songsiri, Thimaporn Phetkaew, Boonserm Kijsirikul", "submitter": "Patoomsiri Songsiri Ms.", "url": "https://arxiv.org/abs/1309.2765" }
1309.2780	# $Z^{0}$-Boson Decays in a Strong Electromagnetic Field 111Published in: Yad. Fiz. 72 (2009) 1078 , [Phys. At. Nucl. 72 (2009) 1034]. A.V.Kurilin222E-mail address: [email protected] Moscow Technological Institute 333on leave from Moscow State Open Pedagogical University, Leninsky Prospect, 38a, Moscow, 119334, Russia The probability of $Z^{0}$-boson decay to a pair of charged fermions in a strong electromagnetic field, $Z^{0}\rightarrow\bar{f}f$, is calculated. On the basis of a method that employs exact solutions to relativistic wave equations for charged particles, an analytic expression for the partial decay width $\Gamma(\varkappa)=\Gamma(Z^{0}\rightarrow\bar{f}f)$ is obtained at an arbitrary value of the parameter $\varkappa=eM_{Z}^{-3}\sqrt{-(F_{\mu\nu}q^{\nu})^{2}}$, which characterizes the external-field strength. The total $Z^{0}$-boson decay width in an intense electromagnetic field, $\Gamma_{Z}(\varkappa)$, is calculated by summing these results over all known generations of charged leptons and quarks. It is found that, in the region of relatively weak fields ($\varkappa<0,06$), the field- induced corrections to the standard $Z^{0}$-boson decay width in a vacuum do not exceed 2%. As $\varkappa$ increases, the total decay width $\Gamma_{Z}(\varkappa)$ develops oscillations against the background of its gradual decrease to the absolute-minimum point. At $\varkappa_{\rm min}=0,445$, the total $Z^{0}$-boson decay width reaches the minimum value of $\Gamma_{Z}(\varkappa_{\rm min})=2,164$ GeV, which is smaller than the $Z^{0}$-boson decay width in a vacuum by more than 10%. In the region of superstrong fields ($\varkappa>1$), $\Gamma_{Z}(\varkappa)$ grows monotonically with increasing external-field strength. In the region $\varkappa>5$, the $t$-quark-production process $Z^{0}\rightarrow\bar{t}t$, which is forbidden in the absence of an external field, begins contributing significantly to the total decay width of the $Z^{0}$\- boson. ## 1\. INTRODUCTION It is not an exaggeration to say that $Z^{0}$-bosons along with their charged partners, $W^{\pm}$-bosons, have been at the focus of attention in high-energy physics over more than the past two decades. The properties of these particles, which are mediators of weak interaction between leptons and quarks, were studied in detail in experiments at the LEP and SLC electron-positron colliders. In a series of experiments performed between 1989 and 1995 at CERN [1], enormous statistics of observations of $Z^{0}$-boson production and decays were collected. In all, approximately $2\cdot 10^{6}$ events associated with leptonic modes of $Z^{0}$-boson decay and more than $1.5\cdot 10^{7}$ events of transformation of these unstable particles into hadrons were detected. The results of these investigations made it possible to determine, among other things, the mass of the $Z^{0}$-boson, $M_{Z}=91,1876\pm 0,0021\ \mbox{GeV}$, and its total decay width, $\Gamma_{Z}=2,4952\pm 0,0023\ \mbox{GeV}$, to an extremely high degree of precision [2]. Many other parameters of the process $e^{+}e^{-}\rightarrow Z^{0}\rightarrow\bar{f}f$ were also analyzed both in the vicinity of the $Z$-resonance and far from it [3]. The precision of experimental data reached in recent years has given a motivation to theoretical physicists for performing formidable work on calculating radiative corrections to the processes involving $Z^{0}$-boson production and decay (see, for example, [4, 5, 6, 7] and references therein). Not only does $Z^{0}$-boson physics provide a precision test for Standard Model predictions concerning particle interactions, but it is also a unique tool for seeking manifestations of new physics (beyond the Standard Model), which will replace sooner or later the generally accepted $SU(3)\times SU(2)\times U(1)$ scheme. Despite the impressive successes of the Standard Model in adequately describing a formidable set of experimental data, there is no doubt among the physics community that the modern model of fundamental interactions cannot be conclusive because of a number of fundamental theoretical drawbacks. These include, first, an enormous number (more than 19!) of independent input parameters — coupling constants, the masses of quark and leptons, the parameters of the Cabibbo-Kobayashi-Maskawa mixing matrix, etc; second, the instability of the mass of the as-yet-undiscovered Higgs boson with respect to quadratically divergent radiative corrections and the allied hierarchy problem; and, third, the isolated character of gravitational interactions, which do not fit in the existing quantum model. This is not the whole story, however. Paradoxically as it is, the main drawback of the Standard Model at the present time is that it is in perfect agreement with all experimental data accumulated thus far, providing no hint as to the nature of new physics that will have to replace it. This kind of ”recess” in the experimental development of physics theory caused a flash of creative activity in theoretical physics in attempts at guessing vague outlines of a new theory of everything. By no means do Grand Unification theories, supersymmetry, supergravity, superstrings, supermembranes, and $M$-theory exhaust the list of ideas that require an experimental verification. The questions that a future theory should answer include that of the origin of particle masses, that of why our space-time is four-dimensional, that of why several generations of leptons exist, that of how many flavors quarks have, and that of how many neutrino flavors exist. Even today, physicists try to find answers to many of these questions by analyzing available experimental data. For example, the total invisible width of $Z^{0}$-bosons, $\Gamma_{\rm inv}=499,0\pm 1,5\ \mbox{MeV}$, with respect to the decays of these particles through channels that are rather difficult to detect plays an important role in $Z^{0}$-boson physics. In the experiments performed at the LEP accelerator, the invisible width $\Gamma_{\rm inv}$ was determined indirectly as the difference of the total decay width of the $Z^{0}$-boson and its total width with respect to all observed decays. In the Standard Model, the invisible decay width of the $Z^{0}$-boson is interpreted as the partial width with respect to decays to neutrinos of various flavors: $\Gamma_{\rm inv}=\sum_{i}^{N_{\nu}}\Gamma(Z^{0}\rightarrow\nu_{i}\tilde{\nu}_{i})=N_{\nu}\Gamma_{\nu\tilde{\nu}}$ This quantity can be considered as some kind of a counter of neutrino generations, and its precise measurement is of particular interest for new physics (see, for example, [8]). It is intriguing that the present-day experimental value of the number of neutrinos from the invisible decay width of the $Z^{0}$-boson, $N_{\nu}=2.92\pm 0,06$ [2], differs somewhat from the result expected in the Standard Model, $N_{\nu}=3$. Exotic channels of $Z^{0}$-boson decay frequently provide unique tests for some fundamental principles of modern physics. For example, analysis of the forbidden (in a vacuum) $Z^{0}$-boson decays to photons ($Z^{0}\rightarrow\gamma\gamma$) and gluons ($Z^{0}\rightarrow gg$) makes it possible to establish a quantitative measure of the principle of quantum indistinguishability of integer-spin particles — Bose symmetry for photons and gluons [9]. According to the Landau-Yang theorem [10], a massive vector particle cannot decay to massless vector states. In an external electromagnetic field, however, the reactions $Z^{0}\rightarrow\gamma\gamma$ and $Z^{0}\rightarrow gg$ become possible [11]. The problem of searches for new possible manifestations of supersymmetry is also tightly related to $Z^{0}$-boson physics. There is still a hope for finding relatively light superparticles produced in rare channels of $Z^{0}$-boson decay. As an example, one can consider the reaction $Z^{0}\rightarrow\tilde{g}\tilde{g}$, in which a pair of light gluinos, $\tilde{g}$, having a mass of about $m_{\tilde{g}}=12\div 16$ GeV appear. This process, which is extensively discussed in the literature (see [12] and references therein), is of special interest in view of prospects for observing new-physics manifestations at next-generation $p\bar{p}$-colliders. Projects of experiments with $Z^{0}$-bosons at the Large Hadron Collider(LHC) also caused interest in studying the properties of these particles in the presence of a quark-gluon plasma, which is quite an unusual environment of [13]. In view of the aforesaid, it is of particular interest to discuss some other new phenomena in $Z^{0}$-boson physics that manifest themselves under unusual conditions. In this study, we investigate changes in partial decay widths of the $Z^{0}$-boson in the presence of strong electromagnetic fields. Interest in such investigations is explained both by astrophysical applications and by applications in the physics of relativistic particles channeling through single crystals (so-called channeling of particles [14]). It is well known that the strength of electric fields generated by the axes and planes of single crystals may reach formidable values of ($E\geq 10^{10}\;$V/m) [15] over macroscopic distances. At the same time, it is well known that astrophysics studies exotic objects (neutron stars, white dwarfs) that, at the latest stages of their evolution, can undergo a strong compression, and this leads to a significant increase in the magnetic field strength inside them ($H\geq 10^{8}\div 10^{13}\;$ G [16]). At such enormous values of the external field strength, the physics of quantum processes changes significantly. External fields frequently remove the forbiddance of specific reactions whose occurrence is impossible in a vacuum. The first investigations into the physics of relativistic particles in the presence of a strong external field were performed within quantum electrodynamics (see, for example, [17, 18, 19]). Later, the technique of calculations that was developed there was also used in non-Abelian gauge theories [20]. Respective calculations rely on the method of exact solutions to the wave equations for charged particles, which makes it possible to take into account the interaction with the electromagnetic field beyond standard perturbation theory. In this case, the wave functions for all charged particles and their propagators are modified in quite a unwieldy manner, depending on the configuration of an external field [21]. Although the resulting expressions are cumbersome and although the calculations are quite involved, the ultimate results, which are valid at arbitrarily high values of the external-electromagnetic-field strength, carry information that cannot be obtained by perturbative methods, and this is an obvious advantage of the method. From this point of view, investigation of quantum processes in superstrong fields provides a unique possibility of analyzing the self- consistency of the physical theory globally in the case where an expansion in a small parameter is impossible. The method of calculations that is used below is based on the crossed-field model, which was successfully applied in the previous studies of the present author to studying photino-pair production [22] and to analyzing the decays of $W$-bosons in an external field [23]. ## 2\. PROBABILITY FOR $Z^{0}$-BOSON DECAY IN EXTERNAL FIELDS In the leading order of perturbation theory in the coupling constants $g_{V}$ and $g_{A}$, the matrix element of $Z^{0}$-boson decay to a pair of charged leptons, $\ell^{\pm}$, is given by $S_{fi}=i\int d^{4}x\overline{\Psi}_{\ell^{-}}(x,p)\gamma^{\mu}(g_{V}+\gamma^{5}g_{A})\Psi_{\ell^{+}}^{c}(x,p^{\prime})Z_{\mu}(x,q).$ (1) The presence of an external electromagnetic field is taken into account via choosing specific wave functions $\Psi_{\ell^{-}}(x,p)$ and $\Psi_{\ell^{+}}(x,p^{\prime})$ for the charged leptons $\ell^{\pm}$. This approach, which, in particle physics, is referred to as the Furry representation, yields, for the probability of $Z^{0}$-boson decay, an expression in which electromagnetic interactions are taken into account exactly, beyond standard perturbation theory in the electromagnetic coupling constant $\alpha=e^{2}/4\pi\simeq 1/137$. The explicit form of the wave functions for charged particles in an external electromagnetic field depends on the field configuration. In the present study, we restrict ourselves to the crossed-field configuration, in which case the strength tensor $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ has the form $F_{\mu\nu}=C(k_{\mu}a_{\nu}-k_{\nu}a_{\mu}).$ (2) It is assumed in this case that the external-electromagnetic-field potential $A_{\mu}(x)$ is chosen in the gauge $A_{\mu}(x)=a_{\mu}C(k_{\nu}x^{\nu}),$ (3) Here, the unit constant $4$-vector $a_{\mu}$ determines the spatial field configuration and satisfies the conditions $a_{\mu}a^{\mu}=-1,\qquad F_{\mu\nu}=(a_{\mu}F_{\nu\lambda}-a_{\nu}F_{\mu\lambda})a^{\lambda}.$ (4) A crossed field is a particular case of the field of a plane electromagnetic wave having an isotropic wave $4$-vector $k_{\mu}\ (k^{2}=0)$. The invariant parameter $C$ characterizes the crossed-field strength and has the dimensions of energy. Recall that, in an arbitrary reference frame, a crossed field is a superposition of constant electric and magnetic fields whose strength vectors are orthogonal and are equal in absolute value: $E=H=k_{0}C$. Moreover, the two relativistic invariants of the electromagnetic field are zero: $F_{\mu\nu}F^{\mu\nu}=F_{\mu\nu}\tilde{F}^{\mu\nu}=0.$ (5) In the crossed-field model, the choice of wave $4$-vector $k_{\mu}$ is quite arbitrarily. It is only necessary that this vector be isotropic, have dimensions of energy, and satisfy the conditions $k_{\mu}a^{\mu}=0,\qquad F^{\mu\lambda}F_{\lambda\nu}=C^{2}k^{\mu}k_{\nu}.$ (6) In particular, the wave $4$-vector $k^{\mu}$ can be expressed in terms of the electromagnetic-field-strength tensor and the constant electromagnetic- potential $4$-vector $a_{\lambda}$ as $k^{\mu}=-\frac{1}{C}\cdot F^{\mu\lambda}a_{\lambda}.$ (7) In dealing with the problem of $Z^{0}$-boson decay, it is advisable to express the wave vector $k^{\mu}$ in terms of the $Z^{0}$-boson 4-momentum $q^{\nu}$ and the electromagnetic-field-strength tensor $F^{\mu\lambda}$ (see next section) as $k^{\mu}=\frac{e^{2}\Delta}{2M_{Z}^{6}\varkappa^{2}}\cdot F^{\mu\lambda}F_{\lambda\nu}q^{\nu}.$ (8) In an arbitrary constant uniform electromagnetic field, the probability of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ is a function of three invariant dimensionless parameters, $P(Z^{0}\rightarrow\ell^{+}\ell^{-})=P(\varkappa,a,b)$. These parameters are determined by the strength tensor of an external macroscopic field as $\displaystyle\varkappa=\frac{e}{M_{Z}^{3}}\sqrt{-(F_{\mu\nu}q^{\nu})^{2}},$ (9) $\displaystyle a=-\frac{e^{2}}{4M_{Z}^{4}}F_{\mu\nu}\tilde{F}^{\mu\nu},$ (10) $\displaystyle b=\frac{e^{2}}{4M_{Z}^{4}}F_{\mu\nu}F^{\mu\nu}.$ (11) In the crossed-field model, the parameter $\varkappa$ has the simplest form $\varkappa=\frac{eC}{M_{Z}^{3}}(q_{\mu}k^{\mu}),$ (12) the square of the wave vector (8) having the form $k^{2}=\frac{\Delta^{2}}{4M_{Z}^{2}}\left({a^{2}\over\varkappa^{4}}-{2b\over\varkappa^{2}}\right).$ (13) In a crossed field, the two parameters (10) and (11) of the external-field strength vanish by virtue of the equality in (5), $a=b=0$; therefore, the right-hand side of Eq.(13) also vanishes. For a constant uniform electromagnetic field of general form, the condition that the wave vector in (8) is isotropic is not satisfied: $k^{2}\neq 0$. Even in this case, however, we can use the semiclassical crossed-field model, which ensures a satisfactory description of the decay probability in an electromagnetic field of general form, provided that $a\ll\varkappa^{2}$ and $b\ll\varkappa^{2}$. The closer to zero the right-hand side of Eq.(13) and the higher the precision to which the condition of isotropicity is satisfied, the more precise the results that are obtained in this approximation. The above constraints are obviously satisfied in the region of relatively weak electromagnetic fields for $a\ll 1$ and $b\ll 1$. Thus, the decay probability $P(Z^{0}\rightarrow\ell^{+}\ell^{-})$ in a crossed field will correspond to the first leading term $P_{0}$ in the semiclassical expansion of the total decay probability $P(\varkappa,a,b)$ in a power series in the vectors of the external-field strengths at small values of the parameters $a$ and $b$; that is, $P(\varkappa,a,b)=P_{0}(\varkappa,0,0)+a\frac{\partial P}{\partial a}(\varkappa,0,0)+b\frac{\partial P}{\partial b}(\varkappa,0,0)+\ldots$ (14) This relation between the probabilities of quantum processes in a crossed field and a constant electromagnetic field of general form was obtained for the first time within quantum electrodynamics [18]. The wave functions for charged fermions in the field of a plane electromagnetic wave were obtained as far back as 1937 by D.M.Volkov (see,for example, [24]). In the case of a crossed field, these exact solutions to the Dirac equations for leptons $\ell^{\pm}$ in an external field have the form $\displaystyle\Psi_{\ell^{-}}(x,p)=\exp\biggl{[}-ipx+\frac{ieC(pa)}{2(pk)}(kx)^{2}-\frac{ie^{2}C^{2}}{6(pk)}(kx)^{3}\biggr{]}\times$ $\displaystyle\times\biggl{\\{}1-\frac{e(kx)}{4(pk)}(F_{\mu\lambda}\gamma^{\mu}\gamma^{\lambda})\biggr{\\}}\frac{u(p)}{\sqrt{2p_{0}V}};$ (15) $\displaystyle\Psi_{\ell^{+}}^{c}(x,p^{\prime})=\exp\biggl{[}ip^{\prime}x+\frac{ieC(p^{\prime}a)}{2(p^{\prime}k)}(kx)^{2}-\frac{ie^{2}C^{2}}{6(p^{\prime}k)}(kx)^{3}\biggr{]}\times$ $\displaystyle\times\biggl{\\{}1+\frac{e(kx)}{4(p^{\prime}k)}(F_{\mu\lambda}\gamma^{\mu}\gamma^{\lambda})\biggr{\\}}\frac{u^{c}(p^{\prime})}{\sqrt{2p^{\prime}_{0}V}};$ (16) The spin part of these wave functions normalized to a three-dimensional spatial volume $V$ is determined by the constant Dirac bispinors $u(p)$ and $u^{c}(p^{\prime})$. The antilepton wave function $\Psi_{\ell^{+}}(x,p^{\prime})$ differs from expression (2.) only by the reversal of the sign of the electric charge $e$: $e\rightarrow-e$. In the expression for the matrix element of the decay in (1), the antilepton wave function appears in the charge-conjugate form (2.), which can be derived on the basis of the Dirac-conjugate bispinor $\bar{\Psi}_{\ell^{+}}(x,p^{\prime})$ with the aid of the charge-conjugation matrix $U_{C}=-i\gamma^{0}\gamma^{2}$; that is, $\Psi^{c}_{\ell^{+}}(x,p^{\prime})=U_{C}\bar{\Psi}^{T}_{\ell^{+}}(x,p^{\prime})$ (17) As for the $Z^{0}$-boson, its wave function does not change in an electromagnetic field because it is electrically neutral: $Z_{\mu}(x,q)=\exp\biggl{(}-iqx\biggr{)}\frac{v_{\mu}(q)}{\sqrt{2q_{0}V}}.$ (18) The spin states of the $Z^{0}$-boson are characterized by the complex polarization $4$-vector $v_{\mu}(q)$ satisfying standard conditions for massive vector fields; that is, $\displaystyle v_{\mu}(q)q^{\mu}=0,\quad v^{}_{\mu}(q)v^{\mu}(q)=-1,$ (19) $\displaystyle\sum_{\sigma=1}^{3}v_{\mu}(q,\sigma)v^{}_{\nu}(q,\sigma)=-g_{\mu\nu}+q_{\mu}q_{\nu}/M_{Z}^{2}.$ We substitute the particle wave functions (2.), (2.), and (18) into expression (1) for the $S$-matrix element and perform integration of $\mid S_{fi}\mid^{2}$ over the phase space of final leptons. Upon summation over the spin states of the lepton-antilepton pair and averaging over polarizations of the $Z^{0}$-boson, we arrive at $\displaystyle P(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)=\frac{(g_{V}^{2}+g_{A}^{2})M_{Z}^{2}}{12\pi^{2}q_{0}}\int\limits_{0}^{1}du\biggl{\\{}\biggl{[}1-(1+3\lambda)\frac{m_{\ell}^{2}}{M_{Z}^{2}}\biggr{]}\Phi_{1}(z)-$ $\displaystyle-\frac{2\varkappa^{2/3}}{\left[u(1-u)\right]^{1/3}}\biggl{[}1-2u+2u^{2}+(1+\lambda)\frac{m_{\ell}^{2}}{M_{Z}^{2}}\biggr{]}\Phi^{\prime}(z)\biggr{\\}}.$ (20) The probability of $Z^{0}$-boson decay is expressed in terms of the Airy functions $\Phi^{\prime}(z)$ and $\Phi_{1}(z)$ (for necessary details concerning these special mathematical functions, see the Appendix) depending on the argument $z=\frac{m_{\ell}^{2}-M_{Z}^{2}u(1-u)}{M_{Z}^{2}[\varkappa u(1-u)]^{2/3}}.$ (21) The couplings of the $Z^{0}$-boson to the charged leptons of all three generations, $e^{\pm},\mu^{\pm},$ and $\tau^{\pm}$, can be expressed in terms of the Weinberg angle $\theta_{\rm W}$ and the electroweak coupling constant $g=e/\sin\theta_{\rm W}$ as $g_{V}=-\frac{g(1-4\sin^{2}\theta_{\rm W})}{4\cos\theta_{\rm W}},\qquad g_{A}=-\frac{g}{4\cos\theta_{\rm W}}.$ (22) The dimensionless parameter $\lambda$ in expression (2.) is given by $\lambda=\frac{g_{A}^{2}-g_{V}^{2}}{g_{A}^{2}+g_{V}^{2}}=\frac{1-(1-4\sin^{2}\theta_{\rm W})^{2}}{1+(1-4\sin^{2}\theta_{\rm W})^{2}},$ (23) at $\sin^{2}\theta_{\rm W}=0,23$, it is very close to unity ($\lambda\simeq 0,987$). ## 3\. KINEMATICS OF $Z^{0}$-BOSON DECAY IN A CROSSED FIELD The fact that the law of energy-momentum conservation for particles involved in the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ does not have a conventional form, $q_{\mu}+k_{\mu}=p_{\mu}+p^{\prime}_{\mu}.$ (24) is a distinctive feature of processes occurring in an external electromagnetic field. Along with the $Z^{0}$-boson and final-lepton energy-momentum $4$-vectors $q_{\mu}$, $p_{\mu}(\ell^{-})$, and $p^{\prime}_{\mu}(\ell^{+})$, expression (24) also involves the wave vector (8) determining the energy contribution of the external field. It should be noted that, in this case, we are dealing with asymptotic momenta that charged particles possess at rather large distances, where the effect of the external field is negligible, rather than with their dynamical $4$-momenta in a crossed field. This is the meaning that the parameters $p_{\mu}$ and $p^{\prime}_{\mu}$ appearing in the Volkov solutions (2.) and (2.) to the Dirac equation in a crossed field have. It is indeed extremely difficult to observe the kinematics of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in the field itself because the charged- lepton $4$-momenta are not integrals of the motion. As soon as the leptons escape from the region of the external field, their energies and momenta do not change any longer, so that one can measure them experimentally. In this case, the $4$-momenta of all particles satisfy the ordinary kinematical conditions $p^{2}=p^{\prime 2}=m_{\ell}^{2},\quad q^{2}=M_{Z}^{2},\quad k^{2}=0,$ (25) and this makes it possible to represent the isotropic vector $k^{\mu}$ in the form $k^{\mu}=\frac{\Delta}{2(q_{\alpha}F^{\alpha\beta}F_{\beta\sigma}q^{\sigma})}\cdot F^{\mu\lambda}F_{\lambda\nu}q^{\nu},$ (26) where the parameter $\Delta$ characterizes the energy imbalance of the reaction $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in the external field, $\Delta=2(qk)=(p+p^{\prime})^{2}-q^{2}=2m_{\ell}^{2}+2(pp^{\prime})-M_{Z}^{2}$ (27) Expression (24) arises in a natural way as the argument of the four- dimensional Dirac delta function in calculating the $S$-matrix element (1). In performing integration over the phase space of final leptons, there also occurs summation of the external-field contributions characterized by various values of the parameter $\Delta$. This is the circumstance that distinguishes the kinematics of $Z^{0}$-boson decay in an external field from a similar process involving a real photon, $\gamma+Z^{0}\rightarrow\ell^{+}\ell^{-}$, where the $4$-momentum $k_{\mu}$ is fixed and is independent of the final- lepton momenta. The physical meaning of the above formulas can most easily be understood in the $Z^{0}$-boson rest frame, where the law of energy-momentum conservation in (24) can be expressed in terms of the following $4$-vectors: $q^{\mu}=(M_{Z};{\bf 0}),\quad p^{\mu}=(\varepsilon_{1};{\bf p}_{1}),\quad p^{\prime\mu}=(\varepsilon_{2};{\bf p}_{2}),\quad k^{\mu}=(k^{0};{\bf k}).$ (28) We chose the axes of the three-dimensional Cartesian coordinate system in such a way that the vectors of the electric- and magnetic-field strength, ${\bf E}$ and ${\bf H}$, are directed along the $x$ and $y$ axes, respectively. The direction of the $3$-momentum ${\bf k}$ formed by the spatial components of the wave $4$-vector $k^{\mu}$ (26) then coincides with the positive direction of the $z$ axis, which is parallel to the vector product ${\bf[E\times H]}$. In the Cartesian coordinate system chosen above, the orientation of the lepton ($\ell^{-}$) and antilepton ($\ell^{+}$) $3$-momenta ${\bf p}_{1}$ and ${\bf p}_{2}$ with respect to the $z$ axis can be described by two azimuthal angles $\vartheta_{1}$ and $\vartheta_{2}$ ($0\leq\vartheta_{1,2}\leq\pi$). Figure 1 shows schematically the three-dimensional configuration formed by all of the aforementioned vectors. Figure 1: Kinematics of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a crossed field in the $Z^{0}$-boson rest frame. If the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ occurs in the vacuum and if the external electromagnetic field is inoperative, the lepton and antilepton have momenta that are equal in absolute value but are oppositely directed and which can be expressed in terms of their masses and the $Z^{0}$-boson mass. The total energy taken away by the lepton-antilepton pair is shared between the particles in equal parts and its total amount is equal to the rest energy of the $Z^{0}$-boson, $p_{1}=p_{2}=\frac{M_{Z}}{2}\left(1-{4m_{\ell}^{2}\over M_{Z}^{2}}\right)^{1/2},\quad\varepsilon_{1}=\varepsilon_{2}=M_{Z}/2,\quad\vartheta_{1}+\vartheta_{2}=\pi.$ (29) In a crossed field, each of the equalities in (29) is in general violated, and the energy $\varepsilon_{1}$ that is taken away in the decay by the lepton is not equal to the antilepton energy $\varepsilon_{2}$. Only fulfillment of the following conservation laws is guaranteed: $\displaystyle\varepsilon_{1}+\varepsilon_{2}-M_{Z}=p_{1}\cos\vartheta_{1}+p_{2}\cos\vartheta_{2},$ (30) $\displaystyle p_{1}\sin\vartheta_{1}-p_{2}\sin\vartheta_{2}=0,$ (31) The vectors ${\bf p}_{1}$, ${\bf p}_{2}$, and ${\bf k}$ lie in the same plane, but, in this case, it is not mandatory that the lepton and the antilepton fly apart along the same straight line, $\vartheta_{1}+\vartheta_{2}\neq\pi$. The deficit in the ordinary law of energy-momentum conservation is compensated in an external field by the isotropic wave vector $k^{\mu}$ (8). Formula (30) represents the three-dimensional form of the conservation law of the covariant projections of the $4$-momenta of all particles on to the direction of the wave vector $(qk)=(pk)+(p^{\prime}k)$, while relation (31) is the conservation law for the ordinary three-dimensional projections of the particle momenta on to the axis that is orthogonal to the vector ${\bf k}$ and which lies in the plane spanned by the $3$-vectors ${\bf p}_{1}$, ${\bf p}_{2}$, and ${\bf k}$. We can now interpret physically the variable $u$ appearing in the integrand on the right-hand side of (2.) and in the argument of the Airy functions in (21). This variable is associated with the angles $\vartheta_{1}$ and $\vartheta_{2}$ of divergence of the lepton-antilepton pair with respect to the $z$ axis in the rest frame of the initial $Z^{0}$-boson: $u=\frac{(pk)}{(qk)}=\frac{\varepsilon_{1}-p_{1}\cos\vartheta_{1}}{M_{Z}}=1-\frac{\varepsilon_{2}-p_{2}\cos\vartheta_{2}}{M_{Z}}$ (32) If the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ is unaffected by an external field, then, by virtue of relations (29), the variable $u$ can be represented in the form $u={1\over 2}\left[1-\left(1-{4m_{\ell}^{2}\over M_{Z}^{2}}\right)^{1/2}\cos\vartheta_{1}\right]={1\over 2}\left[1+\left(1-{4m_{\ell}^{2}\over M_{Z}^{2}}\right)^{1/2}\cos\vartheta_{2}\right].$ (33) The interval of its values admissible from the point of view of decay kinematics in a vacuum is determined by the inequality $u_{1}\leq u\leq u_{2},\quad\mbox{where}\quad u_{1,2}={1\over 2}\left[1\mp\left(1-{4m_{\ell}^{2}\over M_{Z}^{2}}\right)^{1/2}\right]$ (34) The limiting transition $\varkappa\rightarrow 0$ in (2.) reproduces exactly well-known results for the ordinary probability of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ without any field. The kinematically allowed region of values of the variable $u$ (34) corresponds to negative values of the argument of the Airy functions in (21); therefore, the limiting transition to zero values of the external-electromagnetic-field strength is implemented by means of the formal substitution of the Heaviside step $\theta$-function for the Airy function $\Phi_{1}(z)$: $\Phi_{1}(z)\rightarrow\pi\theta(-z)$ (35) In the limit $\varkappa\rightarrow 0$, the differential probability of $Z^{0}$-boson decay with respect to the variable $u$ does not depend on the direction of divergence of leptons and has the form $\frac{dP}{du}(Z^{0}\rightarrow\ell^{-}\ell^{+})=\frac{M_{Z}^{2}}{12\pi q_{0}}\left[g_{V}^{2}+g_{A}^{2}+{2m_{\ell}^{2}\over M_{Z}^{2}}(g_{V}^{2}-2g_{A}^{2})\right]$ (36) As a matter of fact, integration with respect to the invariant variable $u$ now reduces to multiplying expression (36) by the phase space of this variable, $(u_{2}-u_{1})$. As a result, we obtain the well-known formula for the leptonic-decay width of the $Z^{0}$-boson in a vacuum, $\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})=\frac{G_{\rm F}M_{Z}^{3}}{12\pi\sqrt{2}}\sqrt{1-4\delta_{\ell}^{2}}\biggl{(}1-c_{\ell}-\delta_{\ell}^{2}(1+2c_{\ell})\biggr{)},$ (37) where we have introduced the notation $\delta_{\ell}={m_{\ell}\over M_{Z}},\qquad c_{\ell}=4\sin^{2}\theta_{\rm W}-8\sin^{4}\theta_{\rm W}\approx 0,497$ (38) In a crossed electromagnetic field, the kinematics of $Z^{0}$-boson decay changes, first of all, owing to the increase in the phase space of possible states of final leptons $\ell^{\pm}$. In the three-dimensional space of the momenta $(p_{x},p_{y},p_{z})$ of one of the leptons — for example, $\ell^{-}$ — the kinematically allowed region of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a vacuum [see (29)] can be represented as a three-dimensional sphere, each point of this sphere corresponding to one of the final states of this lepton. In an external electromagnetic field, all possible final states of the lepton $\ell^{-}$ already fill some volume in momentum space. At a fixed value of the azimuthal angle $\vartheta_{1}$, the momentum of the lepton $\ell^{-}$ can take any values from the interval $\displaystyle 0\leq p_{1}<p_{\rm max}(\vartheta_{1}),\quad\mbox{where}$ $\displaystyle\ p_{\rm max}(\vartheta)=\frac{M_{Z}}{\sin^{2}\vartheta}\left(\cos\vartheta+\sqrt{1-\delta_{\ell}^{2}\sin^{2}\vartheta}\right).$ (39) The energy $\varepsilon_{1}$ of this particle will be bounded by the inequality $\displaystyle m_{\ell}\leq\varepsilon_{1}<\varepsilon_{\rm max}(\vartheta_{1}),\quad\mbox{where}$ $\displaystyle\varepsilon_{\rm max}(\vartheta)=\frac{M_{Z}}{\sin^{2}\vartheta}\left(1+\cos\vartheta\sqrt{1-\delta_{\ell}^{2}\sin^{2}\vartheta}\right).$ (40) The variables $\varepsilon_{1}$ and $\vartheta_{1}$ determine unambiguously decay kinematics in a crossed field. The energy $\varepsilon_{2}$ and the azimuthal angle $\vartheta_{2}$ of the antilepton $\ell^{+}$ can be expressed in terms of these variables with the aid of the kinematical relations (30) and (31) and the standard relativistic relations $\varepsilon_{1}=\sqrt{p_{1}^{2}+m_{\ell}^{2}},\quad\varepsilon_{2}=\sqrt{p_{2}^{2}+m_{\ell}^{2}}.$ (41) After some simple algebra, we obtain $\displaystyle\varepsilon_{2}=M_{Z}-\varepsilon_{1}+\frac{M_{Z}(\varepsilon_{1}-M_{Z}/2)}{M_{Z}-\varepsilon_{1}+p_{1}\cos\vartheta_{1}}$ (42) $\displaystyle\vartheta_{2}=\arccos\left(\frac{\varepsilon_{1}+\varepsilon_{2}-M_{Z}-p_{1}\cos\vartheta_{1}}{\sqrt{\varepsilon_{2}^{2}-m_{\ell}^{2}}}\right)$ (43) An analysis of the resulting formulas shows that, as a rule, the leptons $\ell^{\pm}$ diverge at arbitrary angles $\vartheta_{1}$ and $\vartheta_{2}$ with respect to the $z$ axis. This is true for almost all of the azimuthal angles, with the exception of the case where the lepton and antilepton move in the direction parallel to the $z$ axis. In the case of $\sin\vartheta_{1}\neq 0$ and $\sin\vartheta_{2}\neq 0$, the lepton and antilepton energies $\varepsilon_{1}$ and $\varepsilon_{2}$, respectively, can be expressed in terms of the emission angles $\vartheta_{1}$ and $\vartheta_{2}$ as $\displaystyle\varepsilon_{1}\approx p_{1}\simeq\frac{M_{Z}\sin\vartheta_{2}}{\sin\vartheta_{1}+\sin\vartheta_{2}-\sin(\vartheta_{1}+\vartheta_{2})}$ (44) $\displaystyle\varepsilon_{2}\approx p_{2}\simeq\frac{M_{Z}\sin\vartheta_{1}}{\sin\vartheta_{1}+\sin\vartheta_{2}-\sin(\vartheta_{1}+\vartheta_{2})}$ (45) Figure 2: Lepton energy $\varepsilon_{1}$ as a function of the antilepton emission angle $\vartheta_{2}$ with respect to the $z$ axis at $\vartheta_{1}=2\pi/3$. These relations were obtained in the relativistic approximation, where $\varepsilon_{1}\gg m_{\ell}$, $\varepsilon_{2}\gg m_{\ell}$ and where the parameter $\delta_{\ell}$ (38) can be disregarded. We can now draw qualitative conclusions on a typical behavior of the energies of the lepton-antilepton pair almost over the entire azimuthal-angle range $0<\vartheta_{1,2}<\pi$. It was indicated above that, at a fixed value of the emission angle ($\vartheta_{1}={\rm const}$) of the lepton $\ell^{-}$, its energy can vary from the rest energy to the maximum value $\varepsilon_{\rm max}(\vartheta_{1})$ (3.). Concurrently, the emission angle $\vartheta_{2}$ of the antilepton $\ell^{+}$ can take an arbitrary value between zero and $\pi$. As for its energy $\varepsilon_{2}$, one can see that, as the energy $\varepsilon_{1}$ increases, $\varepsilon_{2}$ first decreases from the value $\varepsilon_{0}=\frac{M_{Z}}{2}\frac{(1-2\delta_{\ell}+2\delta_{\ell}^{2})}{(1-\delta_{\ell})}\approx\frac{M_{Z}}{2}\left(1-\delta_{\ell}\right)$ (46) to its local minimum $\varepsilon_{2}=\frac{M_{Z}\cos\left(\vartheta_{1}/2\right)}{1+\cos\left(\vartheta_{1}/2\right)},$ (47) whereupon it increases monotonically to infinity as the angle $\vartheta_{2}$ tends to zero, while the lepton energy $\varepsilon_{1}$ tends to its maximum value $\varepsilon_{\rm max}(\vartheta_{1})$. The above relationship between the energies of the lepton-antilepton pair and the directions of lepton and antilepton emission with respect to the $z$ axis is illustrated by the graphs in Figs. 2 and 3. It should be noted that the minimum value of the antilepton energy $\varepsilon_{2}$ (47) is reached at the emission angle of $\vartheta_{2}=\pi-\vartheta_{1}/2$, in which case the energy of the other particle, $\ell^{-}$,is $\varepsilon_{1}=\frac{M_{Z}/2}{1+\cos\left(\vartheta_{1}/2\right)}.$ (48) Figure 3: Antilepton energy $\varepsilon_{2}$ versus the antilepton emission angle $\vartheta_{2}$ with respect to the $z$ axis at a fixed value of the lepton emission angle ($\vartheta_{1}=2\pi/3$). If the lepton escapes in the direction parallel to the $z$ axis, expressions (44) and (45) are inapplicable, so that one must use the exact kinematical relations (30) and (31) or their consequences (42) and (43). In this case, two situations are possible — that in which the leptons fly apart in opposite directions and that in which the leptons move in the same direction. In the first case (at $\vartheta_{1}=0$ and $\vartheta_{2}=\pi$ or at $\vartheta_{1}=\pi$ and $\vartheta_{2}=0$), the energy of the lepton moving along the positive direction of the $z$ axis can take any value from the rest energy $\varepsilon_{\rm min}=m_{\ell}$ to infinity. The energy of the second lepton, which flies in the opposite direction, is virtually constant — it ranges between the initial value $\varepsilon_{0}$ (46) and its maximum value $\varepsilon_{\pi}=\lim_{\vartheta\rightarrow\pi}\varepsilon_{\rm max}(\vartheta)=\frac{M_{Z}}{2}\left(1+\delta_{\ell}^{2}\right).$ (49) which is very close to $\varepsilon_{0}$. The case where both particles $\ell^{+}$ and $\ell^{-}$ produced in $Z^{0}$-boson decay move in the same direction opposite to the positive direction of the $z$ axis ($\vartheta_{1}=\vartheta_{2}=\pi$) is also possible. In this case, their energies $\varepsilon_{1}$ and $\varepsilon_{2}$ can vary from the rest energy to the limiting value $\varepsilon_{0}$ (46). In addition, the following relation can be obtained for this case from (30) in the relativistic approximation (for $\delta_{\ell}\rightarrow 0$): $\varepsilon_{1}+\varepsilon_{2}=\frac{M_{Z}}{2}.$ (50) Thus, one can see that, in relation to the kinematics of the process $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a vacuum [see(29)], the physics of $Z^{0}$-boson decay in an external electromagnetic field is richer in the number of possible final lepton states. From the mathematical point of view, this manifests itself in the fact that all values of the invariant variable $u$ (32) from the domain $0<u<1$ are now admissible. Moreover, it is noteworthy that, in a crossed field, the physical meaning of this variable changes because it now characterizes two independent kinematical quantities simultaneously - for example, $\varepsilon_{1}$ and $\vartheta_{1}$. Disregarding the cases in which leptons fly apart in the direction parallel to the $z$ axis, we can express the variable $u$ in terms of the azimuthal angles $\vartheta_{1}$ and $\vartheta_{2}$ alone by using formulas (44) and (45). The result is $u\simeq\frac{\tan({\vartheta_{1}/2})}{\tan({\vartheta_{1}/2})+\tan({\vartheta_{2}/2})}.$ (51) Figure 4: Differential width with respect to the decay $Z^{0}\rightarrow\ell^{-}\ell^{+}$ as a function of the invariant angular variable $u$ (32) in a weak crossed field (at $\varkappa=0,01$) An analysis of the integrand on the right-hand side of (2.) shows that, in an external field, not all of the directions of divergence of the lepton and antilepton are equiprobable. The graphs in Fig. 4 that represent the dependence of the differential decay width of the $Z^{0}$-boson on the invariant variable $u$ corroborate that, even in a relatively weak external field, there are noticeable deviations from a uniformly distributed probability of the leptonic decay mode in a vacuum [see (36)]. In the case where leptons produced in the process $Z^{0}\rightarrow\ell^{+}\ell^{-}$ fly apart at specific angles $(\vartheta_{1,2}\neq 0,\pi)$ in diametrically opposite directions $(\vartheta_{1}+\vartheta_{2}\approx\pi)$, the invariant variable $u$ takes values of $u\approx 0,5$. In this region, which is kinematically allowed for $Z^{0}$-boson decay in a vacuum, the deviations of the differential decay width $\Gamma^{\prime}_{u}(u,\varkappa)$ from the corresponding static value of $\Gamma^{\prime}_{u}(u,0)\approx 83$ MeV are in fact unobservable. If one of the leptons flies away at a specific angle $\vartheta$ with respect to the $z$ axis and at an energy close to the maximum value $\varepsilon_{\rm max}(\vartheta)$ in (3.) and if the second lepton moves along the direction nearly parallel to this axis (in the positive direction), the variable $u$ takes values in the region around $u\approx 0$ or $u\approx 1$. In this case, one can observe oscillations of the differential decay width of the $Z^{0}$-boson that are characterized by a growing amplitude, which increases and decreases the static vacuum value in (36) by more than 15%. Similar oscillations of probabilities of quantum processes in an external electromagnetic field are characteristic of all particle transformations allowed in a vacuum and were described in detail in the literature [18, 19]. ## 4\. LEPTONIC DECAY WIDTH OF THE $Z^{0}$-BOSON The integral representation of the partial width of the $Z^{0}$-boson with respect to its decay to a pair of charged leptons in an external electromagnetic field can be derived from expression (2.) for the probability by going over to the rest frame ($q_{0}=M_{Z}$) and by making the change of the integration variable $x=u(1-u)$. The result is $\displaystyle\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)$ $\displaystyle=$ $\displaystyle\frac{(g_{V}^{2}+g_{A}^{2})M_{Z}}{6\pi^{2}}\int\limits_{0}^{1/4}\frac{dx}{\sqrt{1-4x}}\biggl{\\{}\biggl{[}1-\delta_{\ell}^{2}(1+3\lambda)\biggr{]}\Phi_{1}(z)-$ (52) $\displaystyle-$ $\displaystyle 2\left(\frac{\varkappa^{2}}{x}\right)^{1/3}\biggl{[}1-2x+\delta_{\ell}^{2}(1+\lambda)\biggr{]}\Phi^{\prime}(z)\biggr{\\}},$ where the argument of the Airy functions is now determined by the expression $z=\frac{\delta_{\ell}^{2}-x}{(\varkappa x)^{2/3}}.$ (53) Let us now consider the asymptotic estimates obtained from this formula at various values of the external-field-strength parameter $\varkappa$ (12). In weak electromagnetic fields such that $\varkappa\ll\delta_{\ell}^{2}\ll 1$, the partial decay width of the $Z^{0}$-boson can be written as the sum of two terms, $\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)=\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})+\Delta\Gamma(\varkappa).$ (54) The first term in this expression is the $Z^{0}$-boson width with respect to the leptonic decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in a vacuum [see expression (37) above]. The experimental value of this quantity is known to a precision of 0.1%: $\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})=83,984\pm 0,086\ \mbox{MeV}$ [2]. The second term characterizes the effect of the external electromagnetic field. In the leading order of the expansion in the small parameters $\varkappa$ and $\delta_{\ell}$, this term can be represented in the form $\displaystyle\Delta\Gamma(\varkappa)=-\frac{G_{\rm F}M_{Z}^{3}}{4\pi\sqrt{6}}\Biggl{[}\ \delta_{\ell}^{2}(1-6c_{\ell})\varkappa\cos\left({1\over 3\varkappa}\right)-$ $\displaystyle-(1-c_{\ell})\varkappa^{2}\sin\left({1\over 3\varkappa}\right)+\frac{8\varkappa^{2}}{\sqrt{3}}(1-c_{\ell})\ \Biggr{]}$ (55) Similar calculations of the probability of $Z^{0}$-boson decay in a magnetic field were performed in [25], where the contribution of the virtual electron- positron loop to the amplitude for elastic $Z^{0}$-boson scattering was considered, its imaginary part being directly related to the $Z^{0}\rightarrow\ell^{+}\ell^{-}$ decay width being studied. It is worth noting, however, that the results of the present calculations [expressions (52) and (4.)] differ somewhat from their counter parts in [25]. For the parameter of the external-electromagnetic-field strength, we will now consider the region specified by the condition $\varkappa\gg\delta_{\ell}^{2}$. We use the fact that the masses of the leptons of all three generations are much smaller than the $Z^{0}$-boson mass and go over to the limit $\delta_{\ell}\rightarrow 0$ in formulas (52) and (53). We can then calculate analytically the integral of the Airy function on the right-hand side of (52) at arbitrary values of the external-field-strength parameter $\varkappa$. The final expression for the leptonic-decay width of the $Z^{0}$-boson is written in terms of the Bessel functions $J_{\nu}(w)$ carrying the noninteger indices $\nu=\pm 1/6$ and $\nu=\pm 5/6$ and depending on the argument $w=(6\varkappa)^{-1}$. We now consider a universal function $R(\varkappa)$ that describes the degree of influence of the external field on the leptonic mode of $Z^{0}$-boson decay: $R(\varkappa)=\frac{\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-}\mid\varkappa)}{\Gamma(Z^{0}\rightarrow\ell^{+}\ell^{-})}.$ (56) Figure 5: Oscillations of the partial width of the $Z$-boson with respect to its decay to a pair of charged leptons in a weak electromagnetic field. In the massless-lepton approximation ($\varkappa\gg\delta_{\ell}^{2}$), this function, which is a relative width of the $Z^{0}$-boson with respect to the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$ in an electromagnetic field, can be represented in the form $R(\varkappa)=\frac{1}{3}\Biggl{[}1+R_{1}(\varkappa)+R_{2}(\varkappa)+R_{3}(\varkappa)\Biggr{]},$ (57) where the following notation has been introduced: $R_{1}(\varkappa)=\frac{\pi(1+3\varkappa^{2})}{12\varkappa}\left[J^{2}_{1/6}\left({1\over 6\varkappa}\right)+J^{2}_{-1/6}\left({1\over 6\varkappa}\right)\right],$ (58) $R_{2}(\varkappa)=\frac{\pi(1-15\varkappa^{2})}{12\varkappa}\left[J^{2}_{5/6}\left({1\over 6\varkappa}\right)+J^{2}_{-5/6}\left({1\over 6\varkappa}\right)\right],$ (59) $\displaystyle R_{3}(\varkappa)$ $\displaystyle=$ $\displaystyle\frac{\pi(2+15\varkappa^{2})}{6}\Biggl{[}\ J_{1/6}\left({1\over 6\varkappa}\right)J_{-5/6}\left({1\over 6\varkappa}\right)-$ (60) $\displaystyle-$ $\displaystyle J_{-1/6}\left({1\over 6\varkappa}\right)J_{5/6}\left({1\over 6\varkappa}\right)\ \Biggr{]}.$ Figure 6: Relative $Z^{0}\rightarrow\ell^{+}\ell^{-}$ decay width $R(\varkappa)$ (56) as a function of the parameter $\varkappa$ at the boundary of the region of oscillations. The graphs representing the dependence of the relative leptonic-decay width (56) normalized to unity at zero external field on the invariant field- strength parameter $\varkappa$ (9) are on display in Figs. 5, 6, 7. In relatively weak fields (such that $\varkappa<0,06$), there arise oscillations of the probability of the decay $Z^{0}\rightarrow\ell^{+}\ell^{-}$, their amplitude increasing quadratically with the parameter $\varkappa$. In this region, one can use the asymptotic expansion for the Bessel functions and approximate formula (57) by the expression Figure 7: Leptonic-decay width of the $Z^{0}$-boson as a function of the external-field-strength parameter $\varkappa$ (12) in the vicinity of the absolute-minimum point. $\displaystyle R(\varkappa)=1-\frac{16}{3}\varkappa^{2}+\frac{3\varkappa^{2}}{\sqrt{3}}\sin\left({1\over 3\varkappa}\right)+\frac{65\varkappa^{3}}{\sqrt{3}}\cos\left({1\over 3\varkappa}\right)+$ $\displaystyle+\frac{880\varkappa^{4}}{3\sqrt{3}}\sin\left({1\over 3\varkappa}\right)-\frac{160}{3}\varkappa^{4}+\ldots$ (61) It is obvious from Figs. 5 and 6 that the maximum deviation of the decay width oscillating in an external field from its vacuum value in (37) does not exceed 2%. A further increase in the parameter of the external-electromagnetic-field strength to values in the region $\varkappa>0,06$ leads to a complete disappearance of oscillations and a monotonic decrease in the relative width $R(\varkappa)$ down to the absolute minimum $R_{\min}=0,843$ in the vicinity of the point $\varkappa_{\rm min}=0,213$ (see Fig. 7). In the region of strong fields (such that $\varkappa>0,213$), the relative width of the $Z^{0}$-boson with respect to its leptonic decay mode grows monotonically with the field strength. In the region $\varkappa\gg 1$, this growth can be approximated by the asymptotic expression. $R(\varkappa)=\frac{15\ \Gamma^{4}(2/3)}{14\pi^{2}}(3\varkappa)^{2/3}+\frac{1}{3}+\frac{3\ \Gamma^{4}(1/3)}{110\pi^{2}\ (3\varkappa)^{2/3}}+\ldots$ (62) ## 5\. HADRONIC MODES OF $Z^{0}$-BOSON DECAY The results obtained in the preceding section admit a straight forward generalization to the case of the decay $Z^{0}\rightarrow\bar{f}f$, where, by $f$, we denote an arbitrarily charged fermion, $q_{f}$ standing for the absolute value of its electric charge. As is well known, the charge is $q_{f}=2/3$ for the up quarks ($u,c,t$) and $q_{f}=1/3$ for the down quarks ($d,s,b$). It is necessary to consider that, for the $Z\bar{f}f$ vertex, the coupling constants $g_{V}$ and $g_{A}$ (1) also depend on the absolute value of the electric charge ($q_{f}$). In the Glashow-Weinberg-Salam model of electroweak interactions, they are determined by the relations $\mid g_{V}(f)\mid=\frac{g(1-4q_{f}\sin^{2}\theta_{\rm W})}{4\cos\theta_{\rm W}},\qquad\mid g_{A}(f)\mid=\frac{g}{4\cos\theta_{\rm W}}.$ (63) Upon making the formal substitution $e\rightarrow q_{f}e$ or $\varkappa\rightarrow q_{f}\varkappa$ in (2.), the differential $Z^{0}\rightarrow\bar{f}f$ decay width as a function of the invariant variables $\varkappa$ (9) and $u$ (32) assumes the form $\displaystyle\frac{d\Gamma}{du}(Z^{0}\rightarrow\bar{f}f)=\frac{G_{\rm F}M_{Z}^{3}}{12\pi^{2}\sqrt{2}}\Biggl{\\{}\left[1-c_{f}-\delta_{f}^{2}(1+2c_{f})\right]\Phi_{1}(z_{f})-$ (64) $\displaystyle-2\left[\frac{q_{f}^{2}\varkappa^{2}}{u(1-u)}\right]^{1/3}\left[(1-c_{f})(1-2u+2u^{2})+\delta_{f}^{2}\right]\Phi^{\prime}(z_{f})\Biggr{\\}}.$ In this expression, use is made of the same notation as in (21) and (38): $\delta_{f}={m_{f}\over M_{Z}},\qquad c_{f}=4q_{f}\sin^{2}\theta_{\rm W}(1-2q_{f}\sin^{2}\theta_{\rm W}),$ (65) $z_{f}=\frac{\delta_{f}^{2}-u(1-u)}{[q_{f}\varkappa u(1-u)]^{2/3}}.$ (66) The limiting transition to an infinitely weak external field ($\varkappa\rightarrow 0$) leads to reproducing the well-known results for the partial widths of the $Z^{0}$-boson with respect to its decay to light quarks; that is, $\Gamma(Z^{0}\rightarrow\bar{u}u)=\frac{G_{\rm F}M_{Z}^{3}}{4\pi\sqrt{2}}\left[1-\frac{8}{3}\sin^{2}\theta_{\rm W}+\frac{32}{9}\sin^{4}\theta_{\rm W}\right].$ (67) $\Gamma(Z^{0}\rightarrow\bar{d}d)=\frac{G_{\rm F}M_{Z}^{3}}{4\pi\sqrt{2}}\left[1-\frac{4}{3}\sin^{2}\theta_{\rm W}+\frac{8}{9}\sin^{4}\theta_{\rm W}\right].$ (68) These relations are obtained from (37) upon the formal substitution $c_{\ell}\rightarrow c_{f}$ and $\delta_{\ell}\rightarrow\delta_{f}$ and the multiplication by the common factor $N_{c}=3$ (number of color quark states). The inclusion of radiative corrections leads to various modifications of expressions (67) and (68), whose explicit form depends on the choice of input parameters in electroweak theory and renormalization scheme for divergent loop diagrams [4, 7]. For example, Sirlin, who was the author of one of the pioneering studies in these realms [26], proposed the parametrization $G_{\rm F}=\frac{\pi\alpha}{\sqrt{2}M_{W}^{2}\sin^{2}\theta_{\rm W}}\frac{1}{(1-\Delta r)},\qquad\sin^{2}\theta_{\rm W}=1-\frac{M_{W}^{2}}{M_{Z}^{2}},$ (69) which has been widely used since then. Here, $\Delta r$ includes all radiative corrections. For the sake of simplicity, we restrict ourselves to the case of massless quarks ($\delta_{f}\ll\varkappa$) and, without going in to details of the calculation of radiative corrections, try to analyze the effect of external electromagnetic fields on the hadronic decay modes of the $Z^{0}$-boson. Without allowance for the external-field effect, the width $\Gamma_{had}=\sum^{5}_{i=1}\Gamma(Z^{0}\rightarrow\bar{q}_{i}q_{i})\simeq 2\Gamma(Z^{0}\rightarrow\bar{u}u)+3\Gamma(Z^{0}\rightarrow\bar{d}d)$ agrees with the experimentally measured value of $\Gamma(Z^{0}\rightarrow hadrons)=1744,4\pm 2,0$ MeV [2] to a high degree of precision. We use this approximation in the region of relatively strong electromagnetic fields ($\varkappa\sim 1$) and estimate the width of the $Z^{0}$-boson with respect to the hadronic decay mode as $\Gamma_{had}(\varkappa)=2\Gamma(Z^{0}\rightarrow\bar{u}u)R\left(\frac{2}{3}\varkappa\right)+3\Gamma(Z^{0}\rightarrow\bar{d}d)R\left(\frac{1}{3}\varkappa\right),$ (70) Figure 8: Hadronic-decay width of the $Z^{0}$-boson in an external electromagnetic field as a function of the parameter $\varkappa$ (12). where the function $R(\varkappa)$ is determined by expressions (56) – (60). Figure 8 shows the results of numerical calculations performed for the total width of the $Z^{0}$-boson with respect to hadronic decay modes in an external field on the basis of expression (70) with allowance for the production of only five quarks ($u,d,s,c,$ and $b$). In this region, the contribution of the heavy $t$ quark is insignificant. This contribution will be considered individually in the next section. One can see from Fig. 8 that, in just the same way as the width with respect to leptonic decay modes (see Fig. 7), the width with respect to the hadronic modes of $Z^{0}$-boson decay in an intense external field behaves nonmonotonically. As the parameter $\varkappa$ increases, the width $\Gamma_{had}(\varkappa)$ first decreases, reaching the absolute minimum of $\Gamma_{had}(\varkappa_{\rm min})=1497$ MeV at the point $\varkappa_{\rm min}=0,501$, where upon it begins growing. In a weak electromagnetic field ($\varkappa\ll 1$), the gradual decrease in the hadronic-decay width $\Gamma_{had}(\varkappa)$ is accompanied by quite complicated oscillations. ## 6\. TOTAL DECAY WIDTH OF THE $Z^{0}$-BOSON AND $t$-QUARK CONTRIBUTION In a vacuum, $Z^{0}$-boson decay to $t$ quarks via the process $Z^{0}\rightarrow\bar{t}t$ is forbidden by the energy-conservation law because the mass of the heavy $t$ quark is rather high, $m_{t}=174,2\pm 3,3$ GeV [2]. The presence of an external electromagnetic field removes this forbiddance, opening new channels of $Z^{0}$-boson decay. In principle, the production of arbitrary charged fermions whose masses satisfy the relation $\delta_{f}=m_{f}/M_{Z}>1/2$ becomes possible. The missing energy for these reactions comes from the external field, and this resembles tunneling through a potential barrier in nonrelativistic quantum mechanics. At small values of the electromagnetic-field strength (such that $\varkappa\ll 1$), the $Z^{0}\rightarrow\bar{f}f$ decay width is exponentially small: Figure 9: Partial width of the $Z^{0}$-boson with respect to its decay to $t$-quarks in an external electromagnetic field (for the description of the curves, see the main body of the text). $\Gamma(Z^{0}\rightarrow\bar{f}f)=\frac{G_{\rm F}M_{Z}^{3}\left(8\delta_{f}^{2}+1-6c_{f}\right)\delta_{f}^{2}q_{f}}{8\pi\sqrt{6}\left(4\delta_{f}^{2}-1\right)\sqrt{8\delta_{f}^{2}+1}}\varkappa\ \exp\left[-\frac{\left(4\delta_{f}^{2}-1\right)^{3/2}}{3q_{f}\varkappa}\right].$ (71) This formula was derived from expression (64) upon integration with respect to the invariant variable $u$ (32) in the approximation $z_{f}\gg 1$ (66). Substituting the $t$-quark charge $q_{f}=2/3$ and performing summation over three color states, we obtain an asymptotic estimate for the partial width of the $Z^{0}$-boson with respect to its decay to $t$-quarks in a relatively weak electromagnetic field; that is, $\displaystyle\Gamma(Z^{0}\rightarrow\bar{t}t)=\frac{G_{\rm F}m_{t}^{2}M_{Z}^{2}\left[8m_{t}^{2}+\left(1-16\sin^{2}\theta_{\rm W}+\frac{64}{3}\sin^{4}\theta_{\rm W}\right)M_{Z}^{2}\right]}{4\pi\sqrt{6}\left(4m_{t}^{2}-M_{Z}^{2}\right)\sqrt{8m_{t}^{2}+M_{Z}^{2}}}\times$ $\displaystyle\varkappa\ \exp\left[-\frac{\left(4m_{t}^{2}-M_{Z}^{2}\right)^{3/2}}{2\varkappa M_{Z}^{3}}\right].$ (72) The results of numerical calculations show that the above approximate formula for $\Gamma(Z^{0}\rightarrow\bar{t}t)$ leads to an error not greater than 4% for all values of the external-field-strength parameter from the region $\varkappa\leq 5$. As the parameter $\varkappa$ grows, the deviation of the results obtained according to the asymptotic formula (6.) from the precise value of the partial width of the $Z^{0}$-boson with respect to the decay $Z^{0}\rightarrow\bar{t}t$ in an external field increases [see Fig.9, where the dotted curve corresponds to the calculation according to formula (6.), while the solid curve represents the results of a numerical integration of expression (64)]. Figure 10: Total decay width of the $Z^{0}$-boson in an external electromagnetic field (for the description of the curves, see the main body of the text). The $t$-quark contribution to the total decay width of the $Z^{0}$-boson remains negligible (less than 1%) up to strength-parameter values of about $\varkappa=6,4$. Figure 10 displays the total decay width of the $Z^{0}$-boson as a function of the parameter $\varkappa$ in the region of strong fields ($\varkappa\leq 10$). The solid curve corresponds to the total decay width of the $Z^{0}$-boson with allowance for the $t$-quark contribution, while the dotted curve represents the same quantity calculated without taking into account the $t$-quark contribution. One can clearly see a trend toward the growth of both the absolute value of the total decay width of the $Z^{0}$-boson and the fraction of the $t$-quark contribution in this decay width. As the external-field strength increases, the process $Z^{0}\rightarrow\bar{t}t$, which is forbidden in a vacuum, becomes dominant in superstrong electromagnetic fields. By way of example, it can be indicated that, in the $Z^{0}$-boson decay width at $\varkappa=100$, the fraction associated with $t$-quarks is as large as 50%. This circumstance clearly demonstrates how an external electromagnetic field can change drastically the standard physics of quantum processes in a vacuum and serve as some kind of a catalyst for a number of new and nontrivial phenomena. ## 7\. CONCLUSIONS The effect of a strong electromagnetic field on the probability of $Z^{0}$-boson decay has been investigated. The present calculations have been performed within the crossed-field model, which makes it possible to trace characteristic variations in the modes of $Z^{0}$-boson decay to known leptons and quarks. We have found that, in the region of relatively strong fields ($\varkappa\sim 1$), all partial decay widths $\Gamma(Z^{0}\rightarrow\bar{f}f)$ decrease in the same manner by about $12$ to $15\%$. This decrease can be described by a universal function $R(\varkappa)$ [see (56)–(60)] depending only on the energy-momentum of the $Z^{0}$-boson and on the external-electromagnetic-field strength. This circumstance becomes quite obvious if we consider that all known fermions observed in $Z^{0}$-boson decays have masses at least an order of magnitude smaller than the $Z^{0}$-boson mass. Therefore, it comes as no surprise that, in the region of strong electromagnetic fields, the massless-fermion approximation works well, also making it possible to calculate the dependence of the total decay width $\Gamma_{Z}(\varkappa)$ of the $Z^{0}$-boson on the external-field-strength parameter $\varkappa$ (9). The results of the present calculations have revealed that, in just the same way as in the case of the $W$-boson [23], the total decay width of the $Z^{0}$-boson in a strong electromagnetic field depends nonmonotonically on the strength parameter $\varkappa$, this dependence featuring the point of a local minimum, $\Gamma_{Z}(\varkappa_{\rm min})=2,164$ GeV, at $\varkappa_{\rm min}=0,445$. The graph in Fig. 11, which represents the behavior of $\Gamma_{Z}(\varkappa)$ in the region $\varkappa\leq 1$, illustrates this observation. Figure 11: Minimal value of the total decay width of the $Z^{0}$-boson in an external field. For the sake of comparison, it can be indicated that the analogous $W$-boson- decay process $W^{-}\rightarrow\ell\bar{\nu}_{\ell}$ in an external field is also characterized by a nonmonotonic dependence of the decay width on the parameter $\varkappa$ with the point of a local minimum at $\varkappa_{\rm W}=0,6112$ [23]. Upon rescaling this value to the $Z^{0}$-boson energy scale, it corresponds to $\varkappa=\varkappa_{\rm W}\left(M_{W}/M_{Z}\right)^{3}=0.419$. Thus, the probabilities for the decays of gauge bosons in an external field, $Z^{0}\rightarrow\bar{f}f$ and $W^{-}\rightarrow\ell\bar{\nu}_{\ell}$, take their minimum values at nearly the same field strength. This agreement is not accidental since the two reactions in question proceed under similar kinematical conditions and since the masses of the $W^{\pm}$ and $Z^{0}$ bosons are almost indistinguishable in order of magnitude. It is also intriguing that, in the region around $\varkappa\sim 1$, the presence or the absence of the initial-gauge-boson electric charge is immaterial because the change in the decay width is due primarily to the increase in the phase space of final particles. In the region of superstrong fields ($\varkappa\geq 10$), a sizable contribution to the total decay width of the $Z^{0}$-boson comes from the $t$-quark-production process $Z^{0}\rightarrow\bar{t}t$, which is forbidden in a vacuum by conservation laws. As the external-field strength increases, this reaction becomes dominant in relation to other modes of $Z^{0}$-boson decay, this being due to a very large mass of the $t$-quark. The external field serves as some kind of a catalyst for super heavy-particle-production processes, which cannot occur under ordinary conditions. In view of the aforesaid, it would be of interest to discuss searches for new (as-yet- undiscovered) hypothetical particles (supersymmetry and so on) among new $Z^{0}$-boson decay modes that arise in an external electromagnetic field. Although external-field strengths necessary for directly observing such reactions are not yet available in experiments (see, for example, [21]), these investigations are of course very interesting from the point of view of fundamental principles of physical theory. ## APPENDIX In the present study, we have employed the special mathematical functions $\Phi(z),\Phi^{\prime}(z)$ and $\Phi_{1}(z)$ generically referred to as Airy functions. The function $\Phi(z)$ is a particular solution to a second-order linear differential equation for specific initial conditions: $\Phi^{\prime\prime}(z)-z\Phi(z)=0,\quad\Phi(0)=\frac{\pi}{3^{2/3}\Gamma(2/3)},\quad\Phi^{\prime}(0)=-\frac{\pi}{3^{1/3}\Gamma(1/3)}.$ (A.1) It has the well-known integral representation $\Phi(z)=\int\limits_{0}^{\infty}\cos\left(zt+{t^{3}\over 3}\right)dt.$ (A.2) Two other functions $\Phi^{\prime}(z)$ and $\Phi_{1}(z)$ can be expressed in terms of this function as $\Phi_{1}(z)=\int\limits_{z}^{\infty}\Phi(t)dt,\qquad\Phi^{\prime}(z)=\frac{d\Phi(z)}{dz}.$ (A.3) At small values of the argument $z$ ($z\ll 1$), the Airy function $\Phi(z)$ can be calculated on the basis of the following expansion in a numerical series: $\Phi(z)={1\over 3^{2/3}}\sum_{n=0}^{\infty}\Gamma\biggl{(}{n+1\over 3}\biggr{)}\sin\biggl{(}{2\pi\over 3}+{2\pi n\over 3}\biggr{)}\frac{3^{n/3}}{n!}\cdot z^{n}.$ (A.4) At large values of the argument ($z\gg 1$), the Airy function can be evaluated by using the asymptotic expressions $\Phi(z)={1\over 2z^{1/4}}\exp\left(-{2\over 3}z^{3/2}\right)\sum_{n=0}^{\infty}(-1)^{n}\frac{\Gamma(3n+1/2)}{(2n)!\ 9^{n}}\cdot\frac{1}{z^{3n/2}},$ (A.5) $\Phi(-z)={1\over z^{1/4}}\sum_{n=0}^{\infty}\sin\left({2\over 3}z^{3/2}+{\pi\over 4}-\frac{\pi n}{2}\right)\frac{\Gamma(3n+1/2)}{(2n)!\ 9^{n}}\cdot\frac{1}{z^{3n/2}}.$ (A.6) The properties of the Airy functions are well known and can be found in mathematical handbooks (see, for example, [27]). ## References * [1] ALEPH Collab. 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Nikishov, Quantum Electrodynamics of Phenomena in Intense Fields, Trudy FIAN SSSR, vol.111 (Nauka, Moscow, 1979) [in Russian]. * [19] I.M. Ternov, V.R. Khalilov, and V.N. Rodionov, Interaction of Charged Particles with Strong Electromagnetic Fields (Izd-vo MGU, Moscow,1982) [in Russian]. * [20] A.V. Borisov, A.S. Vshivtsev, V.Ch. Zhukovsky, and P.A. Eminov, Usp. Fiz. Nauk 167, 241 (1997) [Phys. Usp. 40, 229 (1997)]. * [21] A.V. Kurilin, Nuovo Cimento, 112 A, 977 (1999) [hep-ph/0210194]. * [22] A.V. Kurilin and A.I. Ternov, Yad. Fiz. 63, 1944 (2000) [Phys. At. Nucl. 63, 1855 (2000)]; Pis’ma Zh. Eksp. Teor. Fiz. 63, 305 (1996) [JETP Lett. 63, 311 (1996)]. * [23] A.V. Kurilin, Yad. Fiz. 67, 2116 (2004) [Phys. At. Nucl. 67,2095 (2004)]; [hep-ph/0709.0335]. * [24] V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Quantum Electrodynamics (Nauka, Moscow, 1989; Pergamon, Oxford, 1982). * [25] A.V. Borisov, V.Ch. Zhukovsky, M.Yu. Knizhnikov, and A.N. Prokopenya, Izv. Vyssh. Uchebn. Zaved., Ser. Fiz. 30 (5), 44 (1987). * [26] A. Sirlin, Phys. Rev. D 22, 971 (1980) * [27] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, NewYork, 1964; Nauka, Moscow, 1979) [in Russian].	arxiv-papers	2013-09-11T10:21:06	2024-09-04T02:49:50.703315	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Alexander V. Kurilin", "submitter": "Alexander Kurilin", "url": "https://arxiv.org/abs/1309.2780" }
1309.2864	# Confined colloidal crystals in and out of equilibrium A. Reinmüller Institut für Physik, Johannes-Gutenberg-Universität Mainz, 55128 Mainz, Germany E. C. Oğuz Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany R. Messina Institut de Chimie, Physique et Matériaux (ICPM), Université de Lorraine, 1 Bd. Arago, 57078 Metz - Cedex 3, France H. Löwen Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany H. J. Schöpe Institut für Physik, Johannes- Gutenberg-Universität Mainz, 55128 Mainz, Germany T. Palberg Institut für Physik, Johannes-Gutenberg-Universität Mainz, 55128 Mainz, Germany ###### Abstract Recent studies on confined crystals of charged colloidal particles are reviewed, both in equilibrium and out of equilibrium. We focus in particular on direct comparisons of experiments (light scattering and microscopy) with lattice sum calculations and computer simulations. In equilibrium we address buckling and crystalline multilayering of charged systems in hard and soft slit confinement. We discuss also recent crystalline structures obtained for charged mixtures. Moreover we put forward possibilities to apply external perturbations, in order to drive the system out of equilibrium. These include electrolyte gradients as well as the application of shear and electric fields. ## I Introduction Under strong confinement, as realized by system boundaries or laser-optical external fields, the freezing transition of a colloidal suspension is not only significantly shifted relative to its bulk freezing point Loewen_1994 but the crystalline structure itself can differ much from its stable bulk phase Lowen2012_Advances ; Schoepe2006_Langmuir ; Fontecha_2007 ; Fontecha2008_PRE ; apolinario2006inhomogeneous ; brunner2002phase ; reichhardt2002novel ; fortini2006 ; Qi2013_arXiv ; Rice2009_ChemPhysLett ; Erdal_JPCM . An example of very strong confinement is given in a slit–geometry between two parallel plates which results in a few crystalline layers if the plate distance is small murray1990comparison ; Erdal_2009_EPL . The extreme limit is a quasi- two-dimensional colloidal monolayer. Upon widening the confinement, the latter can buckle into a bilayer pieranski1983thin ; Peeters_PRL_1999 ; Messina_2003 ; Fontecha2005_JPCM ; Schoepe2006_Langmuir ; fortini2006 and subsequent multilayered structures do emerge. Recent research has focused on the structural details of the resulting multilayered crystal. Colloids are excellent model systems to observe these structures on the particle scale using optical microscopy as they can be confined between glass plates. Moreover, various kinds of external fields can be used to bring the system out of equilibrium Lowen2001 ; Villanova-Vidal2009_PhilosMag ; Schoepe2003_JCPM ; Marques-Huesco2008_incoll resulting in novel phenomena not seen in equilibrium or to study the dynamics of the crystallization process in detail. The nature of the confinement can most easily be described by an external one- body potential acting on the colloidal particles, for recent examples see chakrabarti1995reentrant ; bechinger2001phase ; strepp2002phase ; Resnick2003_JColloidInterfaceSci ; Brunner2004_PRL ; apolinario2006inhomogeneous ; brunner2002phase ; reichhardt2002novel ; mikhael2008archimedean ; schmiedeberg2008colloidal ; Schmiedeberg2007_EPJE . One can therefore distinguish between hard and soft confinement: Hard confinement implies a hard external potential, i.e. a potential which is either zero or infinity describing just a steric volume exclusion of the particles. Such an external interaction is athermal, i.e. if scaled with the thermal energy, it is the same at different system temperatures which can reduce the thermodynamic parameters of the total system. Soft confinement, on the other hand, is described by a smooth external potential. An important example are charged glass plates which act as a soft confinement to charged colloids as described by an exponentially screened Coulomb interaction hansen2000 ; Lowen2008_JCPM ; Oguz_2012 . The goal of this minireview is twofold. First, we summarize some recent progress in the experimental determination and theoretical prediction of colloidal multilayers in various kinds of hard and soft confinement under equilibrium conditions. As a result, the stable equilibrium structures reveal a fascinating complexity being very sensitive to both particle interactions and kind of confinement. Most of the structures can be understood by a simple potential energy minimization, i.e. on purely energetic grounds. Second, we focus on nonequilibrium situations which originate from the application of an external field, like gravity, shear flow and an electrolyte gradient. We show that, under appropriate solvent flow conditions, a seed can generate the controlled growth of a colloidal monolayer. Moreover, we discuss how gravity and electric fields can change crystallization. Last but not least, we describe a complex self-organization process of colloidal clustering into a swimming object in an electrolyte gradient. The physical reasons underlying the details of the swimming process still need to be understood and clarified in future studies. The minireview is organized as follows: in chapter 2, we summarize all aspects of confined colloidal crystal in equilibrium. Nonequilibrium effects as induced by different external fields are subsequently discussed in chapter 3 and we conclude in chapter 4. ## II Phase behaviour of confined colloidal crystals in equilibrium ### II.1 Charged colloids in soft confinement Regarding crystallization in slit geometry, hard confinement has been studied for both hard particles and soft particles i.e. particles interacting with a soft pair potential like a Yukawa interaction. More recently, the case of hard particles in soft confinement has also been addressed Curk2012_PRE . Complex cascades of multilayered crystals were predicted and confirmed in colloidal experiments the details of which depend on the interparticle and particle–wall interactions. Detailed comparisons between experiment and theory were performed for hard and soft particles in hard confinement but for soft particles in soft confinement such a direct comparison was done only recently Oguz_2012 , which we describe in more detail in the following. A flexible realization of soft sphere systems is provided by charged colloids suspended in water. The screened Coulomb interaction can conveniently be varied between the theoretical limits of hard spheres (approached at high salt concentrations) and the One Component Plasma (OCP, approached at low salt concentrations). To investigate charged colloidal spheres under deionized conditions in confinement between parallel flat walls, Reinmüller et al. realized a new experimental set–up to be used in combination with a commercial scientific microscope Reinmueller2013_RevSciInst . An experimental set–up for flexible and accurate investigations of such systems under confinement requires a number of minimum properties: It must allow for fast and precise adjustment of the confining geometry; it must facilitate efficient preparation of colloidal suspensions under low salt concentrations at absence of unwanted chemical gradients; it must provide an efficient homogenization mechanism as well as long term stable experimental conditions. The measurement cell of the setup consists of optically flat quartz substrates attached to piezo actuators providing a fast and flexibly adjustable confining geometry. The local wall separations and thus the confining geometry can be quantitatively controlled in situ by interferometric measurements. Proper choice of materials guarantees sufficient chemical inertia against contamination of any kind. Sample preparation and homogenization is performed efficiently using a syringe pump in combination with a conventional closed tubing system including a mixed bed ion exchanger column. Combining optical microscopy, conductivity measurements as well as interferometric measurements, the setup facilitates quantitative control of particle interaction parameters and confining geometry as well as the determination of the structural and dynamic properties of the confined colloids. Two mono-disperse aqueous suspensions of negatively charged polystyrene spheres with diameters of 5.2$\mu$m and 2.6$\mu$m (batch nos PS/Q-F-B1036 and PS-F-B233 by MicroParticles Berlin GmbH, Germany) were used to explore the equilibrium phase diagram of charged colloidal spheres in aqueous suspensions under spatial confinement in parallel plate geometry at low background salt concentrations using the home-made setup described above. In order to suppress gravity, the solvent mass density was matched to that of polystyrene by adding 20 vol% glycerol. During the experiments, the chemical properties of both particle and substrate surfaces were kept constant. The equilibrium phase diagram was determined in terms of the emerging crystal structures depending on the dimensionless parameters the reduced area number density $\eta=n_{A}d^{2}$ and the reduced inverse screening length $\lambda=\kappa d$, where $n_{A}$ denotes the area particle number density, $\kappa$ is the inverse screening length, and $d$ is the width of the confining slit. These parameters could be determined by accurate measurements of the area number density, the cell height, the salt concentration and the particle charge. Varying the reduced area number density $\eta$ and the reduced inverse screening length $\lambda$ in the experiment, the sequence of crystalline structures shown in Figures 1 and 2 was observed. Basically these are hexagonal closed packed crystals (fcc or hcp) oriented in different crystallographic directions with respect to the cell walls. There are no prism structures, transient structures (e.g. hcp like) or exotic structures, like the Belgian waffle iron structure Erdal_2009_EPL , observable. Figure 1: Examples of crystaline structures observed in experiments on charged spheres in soft parallel plate confinement: $1\triangle$ (A), $2\square$ (B), coexistence of $2\square$ and $2\triangle$ (C), $2\textrm{hcp}\perp$ (central large domain) in coexistence with $2\triangle$ (grain at the right margin)(D), $3\triangle$ with different appearances of fcc(111) and hcp(001) faces (E) and Moiré rosettes in coexistence with $2\textrm{hcp}\perp$ and $2\triangle$ (F). (Field of view: $280\times 210\mu\textrm{m}^{2}$.) From Ref. Oguz_2012 . The experimentally observed equilibrium phase behavior of highly charged colloidal crystals in parallel soft confinement was found to be in very good agreement with the theoretical ground state phase diagrams of expponentially screened point-like Coulomb (Yukawa) particles calculated by lattice sum minimizations. The particles were exposed to a soft confining hyperbolic cosine potential as described by the linear screening theory Andelman . The phase diagrams display the same simple multilayer phase sequence with increasing system density. The agreement is documented in Figure 2 which contains a comparison between experiment and theory in the two–dimensional plane spanned by $\eta$ and $\lambda$ for the $5.2\mu\textrm{m}$ size colloids. Figure 2: Comparison of the theoretical prediction (dashed and full lines) for the multilayer stability phase diagram at zero temperature to the experimental data (symbols) taken from the phase diagram for PS $5.2\mu\textrm{m}$ particles in Oguz_2012 . We use $\kappa a=6$ and a particle/wall surface charge ratio of $\nu=0.4$, where $a$ is the particle radius. From Ref. Oguz_2012 . ### II.2 Hard slit confinement One of the simplest situation of confined colloids are sterically-stabilized particles (”hard spheres”) confined between parallel hard walls. In this slit geometry the packing fraction and the reduced plate distance $L/\sigma$ (with $\sigma$ denoting the hard sphere diameter) are the only thermodynamic parameters Schmidt1 ; Schmidt2 . The two-dimensional limit of extreme confinement, $L\to\sigma$, gives a system of hard disks. In this case, the close-packed situation is a hexagonal (or triangular) two-dimensional crystal. For finite packing fractions, the nature of the freezing transition has attracted large attention in the last years Bernard2011 ; Peng10 ; Engel2013_PRE . For slightly larger plate distances, the spheres deviate a bit from their ideal mid-plane positions. In this case, the perturbative correction to the exact partition function can be calculated analytically Franosch2012 . The cascade of close-packing sequences as a function of $L/\sigma$ is nontrivial once the distance exceeds that of two intersecting hexagonal layers, see ivlev2012complex for a review. The phase diagram as determined from free-energy calculations was first investigated in fortini2006 , which reveals the thermodynamic stability of close-packed structures such as prisms. More recently the best packed configuration was found by systematic numerical minimization routines Oguz_2012_PRL . As a result, periodic adaptive prismatic structures which are composed of alternating prisms of spheres are best packed. The internal structure of these prisms adapts to the slit height which results in close packings for a range of plate separations, just above the distance where three intersecting square layers fit exactly between the plates. The adaptive prism phases were also observed in real-space experiments on confined sterically stabilized colloids and in Monte Carlo simulations at finite pressure Oguz_2012_PRL . ### II.3 Cylindrical confinement If particles are confined in cylindrical tubes, there is no sharp freezing transition since phase transitions in general are rounded in one dimension by fluctuations Wilms . However, the ground state at zero temperature can be a periodic crystal. It has been shown, that hard spheres confined to a hollow hard cylinder show complex ordering structures with helical order Pickett2000 . The ground state structures were generalized recently to Yukawa systems in a hard cylindrical pore Oguz_2011 . As a function of screening strength and particle density, the equilibrium phase diagram was found to exhibit a cascade of stable crystals with both helical and non-helical structures. ### II.4 Monolayers of oppositely charged colloids Oppositely charged colloids form stable crystals even at zero pressure due to the mutual attractive Coulomb forces, for a recent review see ivlev2012complex . A model of oppositely charged hard spheres with different diameters is a standard description for the basic interactions. In three dimensions the ground state at zero pressure is textbook knowledge evans1966crystal41to43 . There are three basic equimolar structures whose stability depends on the ratio of the ion radii involving the cesium–chloride, sodium–chloride, and zincblende structures. These (and some more) structures are not only realized for molecular salts but also for mesoscopic colloids leunissen2005 ; hynninen2006 ; hynninen2006cuau . In Refs. Assoud2010_EPL , the same problem was solved for two-dimensional monolayers of oppositely charged hard spheres. Among the stable structures are square, triangular and rhombic crystals as well as “empty” crystals made up of dipoles and chains, which have a vanishing number density. The square structure was already observed for charged granulates Kaufman2009 . As an example, Figure 3 shows the cascade of zero- pressure ground states as a function of the diameter ratio for an interface set–up (interfacial model) where the centers of mass of the two species are all in a common plane. The associated ground state energy per particle is plotted versus the diameter ratio. Figure 3: Stable structures of oppositely charged spheres versus their size asymmetry $R_{B}/R_{A}$ in the interface model, where all sphere centers fall on the same plane: a) side view, b) (scaled) energy per ion ($q$ is the particle charge). The discontinuous transition is indicated by a solid bar. Continuous transitions are denoted by a broken bar. Unit cells of the corresponding stable phases are shown, where the big (small) spheres have a radius $R_{A}$ ($R_{B}$). From Ref. Assoud2010_EPL . ## III Phase behaviour of confined colloidal crystals out of equilibrium ### III.1 Influence of gravity The influence of gravity on the crystal formation in a two layer system of charged colloidal particles in confinement was explored experimentally using commercial polystyrene spheres of 5.2 $\mu$m in diameter in fully deionized water Moire . Here, the colloidal bilayer crystal assembly occurs under influence of both gravity and electrostatics. During a first stage of settling, bilayers form via a line pattern, which quickly gives way to a polycrystalline bilayer structure of AB–stacking of two triangular monolayers. Due to gravity, this bilayer structure is a unidirectional distorted crystal: The lattice constant in vertical direction is noticeable smaller than in lateral direction. In addition to the triangular lattice, two extraordinary transient structures were identified. Each is made from two perfect triangular layers which are rotated by certain angles with respect to each other. The intergrowth of the conventionally stacked two layer crystals results in $1\times 1R\alpha$ super structures. Two specific rotation angles $\alpha=27.8^{\circ}$ and $\alpha=38.2^{\circ}$ dominate in the observed two layer system. These superstructures show fascinating triangular Moiré patterns of rosettes. The morphological function of these structures can be characterized as an “extended grain boundary” known from atomic systems. A similar coherent intergrowth of crystal domains in the bulk are observed e.g. in rare earth boride carbides. Due to their complex structure, the observed crystalline arrangements display extraordinary scattering patterns showing a twelve-fold symmetry. Measurements and model calculations demonstrate the high potential of these extraordinary structures for photonic applications. ### III.2 Crystal formation in parallel confinement under the influence of shearing fields Regarding crystallization in confined geometry, heterogeneous nucleation at the cell wall and the influence of shearing fields are of substantial importance. Colloidal crystals are easily shear molten by gentle mechanical agitation. After cessation of shear, they readily crystallize via nucleation and growth. Applying randomly directed shear before crystallization bcc crystals nucleating at the wall are oriented with their (110)-bcc plane parallel to the wall, but do not show any preferred orientation of the (111)–direction Engelbrecht2011_SoftMatter . The crystal growth in vertical direction was observed to be reaction controlled with a constant growth velocity and as function of meta-stability the growth velocities follow a Wilson-Frenkel-law. The resulting polycrystalline material consist of columnar crystals randomly orientated in the crystallographic direction parallel to the cell wall. Analyzing the heterogeneous nucleation rate densities on the wall as function of metastability, a transition from a nearly time independent nucleation rate density at the phase boundary to a peaked transient nucleation at higher undercooling $\Delta\mu$ was identified. The heterogeneous nucleation barrier height stays first constant and drops down to zero as the metastability increases: In charged sphere model systems, a sharp wetting transition separates a regime at moderate metastability where hemispherical cap nuclei are formed from a strong meta-stability regime, where the crystal phase completely wets the wall. In suspensions molten by unidirectional shear flow this, transition could not be observed. Unidirectional shear leads to shear induced oriented precursor crystals in form of hexagonal layers at the cell wall ((111)–fcc parallel to the wall) and further quasi-epitaxial growth of bcc crystals on this precursor crystal Wette2009_JCPM ; Palberg2012_JCP . The precursor acts as structured substrate reducing the nucleation barrier height to a minimum. The details of this first step are still not well understood. Due to the unidirectional shear, the bcc crystals are uniformly oriented ((110) plane parallel to the wall and (111)–direction in direction of shear flow) leading to a twinned bcc morphology. Like in the first case, vertical growth is reaction controlled with a constant growth velocity. In lateral dimension, the average domain size was observed to increase significantly during vertical growth following the typical power law for coarsening with an exponent of 0.5. Lateral coarsening ceases when the wall crystals from the opposing walls intersect. These investigations and others Messina_2006 show impressively that the heterogeneous nucleation process on the wall, the crystallization process afterwards and the crystalline microstructure is highly modified by shear in confined geometry. Finally, rigid colloidal spheres rotate in shear flow. Flexible neutral polymer chains, on the other hand, tumble He_2010 . If combined with other periodic field there are interesting dynamical modes of polymer propagation He_2011 . The behaviour is expected to be even more complex for charged polymers Allahyarov_2011 . ### III.3 Crystal and microswimmer formation in parallel confinement under the influence of electrostatic and flow fields #### III.3.1 Crystal nucleation The crystallization process in equilibrium two–dimensional colloidal fluids in confinement induced by local electrolyte gradients was studied experimentally using negatively charged polystyrene spheres (diameter $2a=5.2\mu\textrm{m}$) dispersed in water Reinmueller2012 . The suspension was prepared under deionized conditions in confined parallel plate geometry obtaining a fluid monolayer system in thermodynamic equilibrium. After introducing small fragments of ion exchange resin into the confinement cell, a strong interaction between the colloidal spheres and the fragments could be observed. In particular for the case of cation exchange resins (CIEX) a fluid flow leads to particle accumulation at the CIEX-fragment followed by colloidal crystallization. The electrolyte gradient originated by the CIEX-fragment caused an electro-osmotic solvent flow along the anionically charged substrate surface towards the fragment leading to a colloidal particle transport. Once the local particle number density exceeded the salt concentration dependent freezing density, heterogeneous crystallization on the surface of the CIEX- fragment occurred. Although the underlying mechanism generating the flow field is not fully understood, the phenomenology of the crystal formation was observed to be well reproducible. Starting from the thermodynamically stable colloidal fluid multi-domain triangular monolayer crystals were formed with the fragment in the centre, as for experimental snapshots see Figure 4. The domain number is highly correlated with the shape of the fragment acting as seed particle. Analyzing the crystal growth, an exponentially decreasing crystal growth velocity was observed and the final crystallite size is determined by the strength and range of the electrolyte gradients. The same situation was simulated using Brownian dynamics computer simulations in two dimensions and good agreement with the experiment was obtained. Figure 4: Growth of a three–domain monolayer crystal on a seed at $t=0\textrm{s}$ (a), $t=100\textrm{s}$ (b), $t=200\textrm{s}$ (c), $t=300\textrm{s}$ (d), $t=400\textrm{s}$ (e), and $t=500\textrm{s}$ (f); field of view: $360\times 290\mu\textrm{m}^{2}$. From Ref. Reinmueller2012 . Simulation and experiment indicate that there is no nucleation barrier in this special crystallization mechanism and that the number of domains depends neither on thermodynamics nor on nucleation kinetics. The seed geometry determines the microstructure of the formed crystal, see simulation snapshots in Figure 5 for two different seed geometries. In principle these investigations offer the possibility of designing and manufacturing extended thin multidomain colloidal crystal microstructures controlling the positions, shape and size of individual crystals. Figure 5: Simulation snapshots of a three–domain (a) and a mono–domain crystal (b) each at $t=500\tau$ with $\tau$ being a typical Brownian time scale. The green dots in both illustrations indicate the origin of the system. The particle flow into the polygonal (a) and spherical (b) area is prohibited. The red particles are crystalline, while the black ones are not. Field of view: $250\times 250/\kappa^{2}$ where $\kappa$ is the inverse screening length. From Ref. Reinmueller2012 . #### III.3.2 Complex microswimmer formation Repeating the experiment with stronger electro–osmotic solvent flow and larger plate to plate distance ( $\sim 50\mu\textrm{m}$) the formation of self- organized complex microswimmers could be observed Reinmueller2013_Langmuir . The electrolyte gradient inducing a diffusio-osmotic solvent flow towards the CIEX fragment now originates a convection roll at the CIEX fragment. Particles transported to the fragment were lifted up at the CIEX and fall down afterward very close to the fragment. Due to the convection the particles did not crystallize, but still a strong accumulation of particles at the fragment was observed. In some cases the particle concentration within the convection cell was found to be highly asymmetric (rotational symmetry, symmetry axis in the centre of the CIEX) which might be generated by either a shape anisotropy of the CIEX fragment or by a highly asymmetric particle concentration of the colloidal fluid surrounding the CIEX fragment. If the particle concentration within convection cell was highly asymmetric the CIEX fragment and the accumulated particles started to move in the direction of smallest colloid concentration. A possible propulsion mechanism related to the nonuniform colloid concentration (including colloid counterions) might be direct osmotic propulsion. The details of the driving mechanism of this cooperative motion are not yet fully understood. Nevertheless the propulsion was found to be very robust and occurred with varying effectiveness over a large range of experimental parameters. The self propulsion mechanism was self-stabilizing over many minutes and the complex micro swimmers moved with typical velocities of $1\mu\textrm{m}/\textrm{s}-3\mu\textrm{m}/\textrm{s}$. At sufficient speed, further loading from the side became negligible and oncoming particles fall down behind the CIEX fragment after they were lifted up by convective flow. During this stage of constant trail thickness, the complexes cover path lengths of many hundreds of micrometers. ### III.4 Oppositely charged colloidal suspensions in electric fields: laning When oppositely charged colloids are exposed to an external electric field, laning of like-charge particles occurs, for a review see ivlev2012complex ; Sutterlin09 ; Dzubiella02 ; lowen2010instabilities . In three dimensions, by experiment and computer simulation, laning was shown to be continuous for small electric fields and a quantitative comparison between experiment and simulation could be achieved vissers2011 . It is important to note that hydrodynamic interactions between driven particle are of minor importance when charged particle are moving in an electric field Ajdari_Long_EPJE_2001 ; Rex_EPJE_2008 . Therefore they were not found to be crucial for the actual modelling. ## IV Conclusions In conclusion, we have reviewed briefly recent developments in preparing, characterizing and predicting confined colloidal crystals. In equilibrium, a huge variety of equilibrium structures becomes stable depending sensitively on the particle interaction and on the nature of the confinement. These structures may serve as building blocks for filters and sieves as well as for photonic crystals. A coagulated two-dimensional crystal has controlled void space between the particles, which leads to tunable porosity. We have also shown that nonequilibrium effects generated by shear flow, electrolyte gradients and electric fields may generate further structures which are widely different from those stable in equilibrium. These external fields can also be used to steer the crystallization process itself and control the defect structure and distribution of the emerging crystal. One of the most striking consequence was the self-organized formation of a microswimming system which opens the way to possible applications as controlled cargo transport via active colloidal suspensions. Future research activities should focus on microscopic theories for nonequilibrium processes in order to predict nonequilibrium effects of crystallization. Dynamical density functional theory is a promising approach in this respect seed ; Espanol2009 ; Advances . 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1309.2881	# Hidden $T$-Linear Scattering Rate in Ba0.6K0.4Fe2As2 Revealed by Optical Spectroscopy Y. M. Dai Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China LPEM, ESPCI-ParisTech, CNRS, UPMC, 10 rue Vauquelin, F-75231 Paris Cedex 5, France Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA B. Xu Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China LPEM, ESPCI-ParisTech, CNRS, UPMC, 10 rue Vauquelin, F-75231 Paris Cedex 5, France B. Shen H. Xiao Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China H. H. Wen Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China X. G. Qiu Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China C. C. Homes [email protected] Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA R. P. S. M. Lobo [email protected] LPEM, ESPCI-ParisTech, CNRS, UPMC, 10 rue Vauquelin, F-75231 Paris Cedex 5, France (August 27, 2024) ###### Abstract The optical properties of Ba0.6K0.4Fe2As2 have been determined in the normal state for a number of temperatures over a wide frequency range. Two Drude terms, representing two groups of carriers with different scattering rates ($1/\tau$), well describe the real part of the optical conductivity, $\sigma_{1}(\omega)$. A “broad” Drude component results in an incoherent background with a $T$-independent $1/\tau_{b}$, while a “narrow” Drude component reveals a $T$-linear $1/\tau_{n}$ resulting in a resistivity $\rho_{n}\equiv 1/\sigma_{1n}(\omega\rightarrow 0)$ also linear in temperature. An arctan($T$) low-frequency spectral weight is also a strong evidence for a $T$-linear 1/$\tau$. Comparison to other materials with similar behavior suggests that the $T$-linear $1/\tau_{n}$ and $\rho_{n}$ in Ba0.6K0.4Fe2As2 originate from scattering from spin fluctuations and hence that an antiferromagnetic quantum critical point is likely to exist in the superconducting dome. ###### pacs: 72.15.-v, 74.70.-b, 78.30.-j Over the last several decades, it has been observed that the electrical resistivity $\rho$ of some strongly-correlated materials increases linearly with temperature ($T$-linear $\rho$), particularly in the vicinity of an antiferromagnetic quantum critical point (QCP), a striking deviation from Landau’s Fermi-liquid description of metals. This anomalous $T$-linear $\rho$, extensively studied in the high-$T_{c}$ cuprate superconductors Boebinger (2009); Cooper _et al._ (2009); Daou _et al._ (2009); Jin _et al._ (2011), organic Bechgaard salts Doiron-Leyraud _et al._ (2009a, b), as well as heavy- fermion metals Malinowski _et al._ (2005); Tanatar _et al._ (2007); Trovarelli _et al._ (2000); Gegenwart _et al._ (2002), may be intimately related to the emergence of superconductivity Jin _et al._ (2011); Taillefer (2010); Doiron-Leyraud _et al._ (2009a). It is generally believed that in proximity to the antiferromagnetic QCP, spin fluctuations are so strong that the scattering process of quasiparticles is severely modified, inducing non- Fermi-liquid behavior. Such spin fluctuations may be responsible for the pairing of electrons in high-$T_{c}$ superconductors Monthoux _et al._ (2007); Moriya and Ueda (2000); Mazin _et al._ (2008). The presence of the $T$-linear $\rho$ and a QCP in the newly discovered iron- based superconductors is highly desired since superconductivity arises in the vicinity of the spin-density-wave (SDW) instability Kamihara _et al._ (2008); Rotter _et al._ (2008). Up to this point, $T$-linear $\rho$ has been observed by transport, arousing considerable effort to describe it as evidence of possible quantum criticality in iron-pnictides, especially in the “122” family Xu _et al._ (2008); Gooch _et al._ (2009); Shen _et al._ (2011); Maiwald _et al._ (2012); Hashimoto _et al._ (2012); Kasahara _et al._ (2010); Doiron-Leyraud _et al._ (2009a); Shibauchi _et al._ (2013). However, unlike the high-$T_{c}$ cuprates, iron-pnictides fall into the category of multi-band materials Singh and Du (2008); Ding _et al._ (2008). Up to five Fe-3$d$ bands crossing the Fermi level contribute to the Fermi surface, leading to the presence of three hole-like Fermi pockets at $\Gamma$-point and two electron- like pockets at the corners of Brillouin zone. The scattering rate $1/\tau$ and the response of quasiparticles to external electrical field may vary considerably in different Fermi pockets Fang _et al._ (2009); Shen _et al._ (2011); Muschler _et al._ (2009); Rullier-Albenque _et al._ (2010). As a result, the transport properties become extremely complicated in such a system and the question of whether a $T$-linear $\rho$ in iron-pnictides originates from multi-band effects or the presence of a QCP makes it inadequate to investigate the transport properties alone in search of a non-Fermi-liquid behavior and evidence for possible QCP Rullier-Albenque _et al._ (2010, 2009). Although many optical studies on iron-pnictides have been reported Li _et al._ (2008); Charnukha _et al._ (2011a, b), the above issue has never been touched due to the absence of detailed $T$ dependent optical data. In this Letter we address this issue by studying the detailed $T$ dependence of the optical conductivity and low-frequency spectral weight in Ba0.6K0.4Fe2As2. The low-frequency optical conductivity is described by two Drude terms: a broad Drude with a large $1/\tau_{b}$ that is basically $T$-independent alongside a narrow one with a small $T$-linear $1/\tau_{n}$ that reveals a $\rho_{n}\equiv 1/\sigma_{1n}(\omega\rightarrow 0)\propto T$. The low-frequency spectral weight increases with cooling, following an $\arctan(T)$ dependence, which is demonstrated to be a clear signature of $T$-linear $1/\tau$. Comparison with similar behavior found in other materials attributes the $T$-linear $1/\tau_{n}$ and $\rho_{n}$ in Ba0.6K0.4Fe2As2 to spin fluctuation scattering and the presence of a QCP in the superconducting dome. High quality Ba0.6K0.4Fe2As2 single crystals were grown using a self-flux method Shen _et al._ (2011). The inset of Fig. 1 shows the DC resistivity of Ba0.6K0.4Fe2As2 as a function of temperature. Figure 1: (color online) The thick solid lines in the main panel are ab-plane $\sigma_{1}(\omega)$ of Ba0.6K0.4Fe2As2 at different temperatures in the normal state. The smooth lines through the data are fits with Drude-Lorentz model. The inset shows the DC resistivity as a function of temperature from transport measurement (solid curve) and the values derived from the zero frequency extrapolation of $\sigma_{1}(\omega)$ (solid circles). The $\rho(T)$ curve is characterized by a sharp superconducting transition at $T_{c}\simeq 39$ K and a tendency to saturation at room temperature. A visible change in slope occurs at about 175 K. The near-normal incident reflectance, $R(\omega)$, has been measured for light polarized in the a-b plane using FTIR spectrometers and an in situ evaporation technique Homes _et al._ (1993). Data from $\simeq 20-12\,000$ cm-1 were collected at 18 different temperatures from 5 to 300 K on a freshly cleaved surface. The visible and UV range ($10\,000-55\,000$ cm-1) $R(\omega)$ was measured at room temperature with an AvaSpec-2048$\times 14$ fiber optic spectrometer. The real part of the complex optical conductivity, $\sigma_{1}(\omega)$, is determined from $R(\omega)$ via Kramers-Kronig analysis. A Hagen-Rubens form ($R=1-A\sqrt{\omega}$) is used for the low- frequency extrapolation. At high frequencies, $R(\omega)$ is assumed to be constant to 40 eV, above which a free-electron response ($\omega^{-4}$) is used. The main panel of Fig. 1 shows $\sigma_{1}(\omega)$ at 4 selected temperatures in the normal state (thick solid lines); all the spectra exhibit the well- known Drude-like metallic response. In order to quantitatively analyze the optical data, we fit $\sigma_{1}(\omega)$ to the Drude-Lorentz model, $\sigma_{1}(\omega)=\frac{2\pi}{Z_{0}}\\!\left[\sum_{k}\frac{\Omega^{2}_{p,k}}{\tau_{k}(\omega^{2}+\tau_{k}^{-2})}\\!+\\!\sum_{j}\frac{\gamma_{j}\omega^{2}\Omega_{j}^{2}}{(\omega_{j}^{2}-\omega^{2})^{2}+\gamma_{j}^{2}\omega^{2}}\right]$ (1) where $Z_{0}$ is the vacuum impedance. The first term describes a sum of free- carrier Drude responses, each characterized by a plasma frequency $\Omega_{p}^{2}=4\pi ne^{2}/m^{\ast}$, where $n$ is a carrier concentration and $m^{\ast}$ is an effective mass, and a scattering rate $1/\tau$. The second term corresponds to a sum of Lorentz oscillators characterized by a resonance frequency ($\omega_{j}$), a line width ($\gamma_{j}$) and an oscillator strength ($\Omega_{j}$). This Drude-Lorentz model is also used to determine the dc properties of the system Romero _et al._ (1992). As shown in Fig. 2, $\sigma_{1}(\omega)$ at 150 K is described by a broad Drude with a large scattering rate $1/\tau_{b}\approx 936$ cm-1, and a narrow Drude with a small scattering rate $1/\tau_{n}\approx 158$ cm-1 and an overdamped Lorentzian term. Figure 2: (color online) The black solid curve is the measured $\sigma_{1}(\omega)$ of Ba0.6K0.4Fe2As2 at 150 K. The red dashed line through the data is the fit which is decomposed into a broad Drude (blue dotted line), a narrow Drude (pink dotted-dashed line) and a Lorentz (orange short-dashed line) term. The linear superposition of these three components gives very good description to $\sigma_{1}(\omega)$ up to $10\,000$ cm-1 at all the measured temperatures in the normal state. Fits for other temperatures are selectively shown in the main panel of Fig. 1 as smooth thin lines through the corresponding data. The inset of Fig. 1 compares the optical estimate for the DC resistivity $\rho\equiv 1/\sigma_{1}(\omega\rightarrow 0)$ (solid circles) to the transport measurements (solid line). The two-Drude fit indicates the existence of two groups of carriers with different $1/\tau$’s in Ba0.6K0.4Fe2As2, which was first pointed out by Wu _et al._ in various iron-pnictides Wu _et al._ (2010). The disparity of the $1/\tau$’s in different bands is also supported by both theoretical calculations Kemper _et al._ (2011) and measurements using other techniques on similar materials Fang _et al._ (2009); Shen _et al._ (2011); Rullier- Albenque _et al._ (2010); Muschler _et al._ (2009). Tu et al. suggest that it is appropriate to describe the broad Drude term as bound excitations Tu _et al._ (2010), because the mean free path $l=v_{F}\tau$ ($v_{F}$ is the Fermi velocity) associated with the broad Drude is less than the shortest interatomic spacing, violating the Mott-Ioffe-Regel condition Gurvitch (1981). In Ba0.6K0.4Fe2As2, the average Fermi velocities of the electron and hole pockets are estimated to be $v^{e}_{F}\simeq 0.40$ eVÅ and $v^{h}_{F}\simeq 0.36$ eVÅ Ding _et al._ (2011), respectively. Furthermore, it is reported that in iron-pnictides holes have a larger $1/\tau$ than electrons Fang _et al._ (2009); Shen _et al._ (2011); Rullier-Albenque _et al._ (2010); Muschler _et al._ (2009). If we associate the broad Drude component ($1/\tau_{b}\simeq 936$ cm-1) with the hole pockets, a mean free path of $l_{h}\simeq 3$ Å is obtained. This value is close to the lattice parameter $a\simeq 4$ Å of the 122 family compounds and probably too small for coherent transport. Since the broad Drude only produces an incoherent, $T$-independent, background contribution to the total $\sigma_{1}(\omega)$, the nature of the broad Drude component (whether or not bound excitations) does not affect our analysis on the temperature dependence of $\sigma_{1}(\omega)$ and low- frequency spectral weight. Figure 3: (color online) The $T$ dependence of (a) the plasma frequency $\Omega_{p}$, (b) the scattering rate $1/\tau$, (c) the contribution to DC conductivity $\sigma_{1}(\omega\rightarrow 0)$, and (d) the equivalent resistivity $\rho$ for the narrow and broad Drude components, respectively. The dashed lines in panels (b) and (d) are linear fits to $1/\tau_{n}$ and $\rho_{n}$, respectively. The temperature dependence of the two Drude components provides information about the nature of the two different types of carriers in this material. Figure 3 shows the $T$ dependence of the Drude parameters from our fits. The subscripts n and b stand for the narrow and broad Drude terms, respectively. Fig. 3(a) shows the $T$ dependence of the plasma frequencies of the two Drude terms. Upon cooling, $\Omega_{p}$ is roughly a constant for each of the two components, indicating that the band structure and $n/m^{\ast}$ does not change with temperature, in agreement with a previous work Wu _et al._ (2010). Fig. 3(b) portrays the $T$ dependence of the scattering rate of the two Drude components, where $1/\tau_{b}$ is basically $T$-independent while $1/\tau_{n}\propto T$; the black dashed line denotes a linear fit. Fig. 3(c) displays the contribution of the two groups of carries to the DC conductivity. As the broad Drude ($\sigma_{b}$) is $T$-independent, the temperature dependence of the total DC conductivity arises out of the narrow Drude band ($\sigma_{n}$). The $\rho(T)$ curve of Ba0.6K0.4Fe2As2, shown in the inset of Fig. 1, exhibits a tendency to saturation at room temperature, and a change of slope can be seen at about 175 K. This behavior can be explained by the different $T$ dependence of the two Drude bands, which can be considered as a parallel-circuit Wiesmann _et al._ (1977): $\sigma=\sigma_{n}+\sigma_{b}$. A crossover-region, where $\sigma_{n}\simeq\sigma_{b}$, is found in Fig. 3(c) at $\simeq 175$ K. Below this temperature, $\sigma_{n}>\sigma_{b}$, so the total DC conductivity is dominated by $\sigma_{n}$ which exhibits strong temperature dependence. As a result, below 175 K, $\rho(T)$ decreases quickly with decreasing temperature. Above 175 K, $\sigma_{n}<\sigma_{b}$, and the total DC conductivity is dominated by $\sigma_{b}$ which shows no temperature dependence. Hence, above 175 K, the growth of the DC resistivity slows with heating, resulting in the change of slope and the tendency to saturation in $\rho(T)$. Similar conclusions were obtained from investigations of the Hall effect Rullier-Albenque _et al._ (2010, 2009) and theoretical calculations Golubov _et al._ (2011). In the $\omega\rightarrow 0$ limit, the inverse of $\sigma_{1}(0)$ yields the resistivity from the two Drude components, as shown in Fig. 3(d). The resistivity of the broad Drude remains a constant at all measured temperatures, while a $T$-linear $\rho_{n}$ is revealed for the narrow Drude component. This is in accord with transport measurements on hole- doped 122 compounds Shen _et al._ (2011); Maiwald _et al._ (2012); Gooch _et al._ (2009), where $T$-linear $\rho$ was observed at low temperatures in optimally-doped materials. The $T$-linear $\rho$ is only found at low temperatures as this is the region dominated by the narrow Drude component. Further evidence for $T$-linear $1/\tau$ lies in the temperature dependence of the low-frequency spectral weight. The spectral weight is defined as $W_{0}^{\omega_{c}}=\int_{0}^{\omega_{c}}\sigma_{1}(\omega)d\omega,$ (2) where $\omega_{c}$ is a cutoff frequency. In a Drude metal the scattering rate decreases upon cooling, producing a narrowing of the Drude response and an increase of the DC conductivity, resulting in a transfer of spectral weight from high to low frequencies and an increase in $W_{0}^{\omega_{c}}$. To quantitatively analyze the $T$ dependence of the low-frequency spectral weight, we adopt one Drude optical conductivity [see Eq. (1)] into Eq. (2), to obtain the spectral weight as a function of $1/\tau$ $W_{0}^{\omega_{c}}(1/\tau)=\frac{2\pi\Omega_{p}^{2}}{Z_{0}}\arctan\left(\omega_{c}\tau\right).$ (3) In the case of $1/\tau\propto T$, Eq. (3) can be simplified as $W_{0}^{\omega_{c}}(T)=a_{1}\arctan\left(\frac{a_{2}}{T}\right),$ (4) where $a_{1}={2\pi\Omega_{p}^{2}}/Z_{0}$, and $a_{2}\propto\omega_{c}$; both are $T$-independent parameters. Considering the spectral weight arising from the incoherent part and inter-band transition (Lorentz), which are both $T$-independent, we introduce the third $T$-independent parameter $a_{0}$ into Eq. (4). Finally, the low-frequency spectral weight as a function of temperature for $1/\tau\propto T$ is written as $W_{0}^{\omega_{c}}(T)=a_{0}+a_{1}\arctan(\frac{a_{2}}{T}).$ (5) $W_{0}^{\omega_{c}}$ can be easily determined by integrating the measured $\sigma_{1}(\omega)$. The open symbols in Fig. 4 denote $W_{0}^{\omega_{c}}$ with different $\omega_{c}$’s: 150 (squares), 200 (triangles) and 250 cm-1 (diamonds). Figure 4: (color online) Temperature dependence of the low-frequency spectral weight $W_{0}^{\omega_{c}}$ with different $\omega_{c}$’s: 150 (squares), 200 (triangles) and 250 cm-1 (diamonds). The solid lines through the data represent the expected temperature dependence for $1/\tau\propto T$ at each $\omega_{c}$. As expected in a metallic system, for all the $\omega_{c}$’s, $W_{0}^{\omega_{c}}$ increases with decreasing $T$. The solid curves through the data are the least square fits using Eq. (5). The excellent agreement indicates that a $T$-linear $1/\tau$ dominates the low frequency $\sigma_{1}(\omega)$ over a very broad temperature range. We now discuss the possible origin of the $T$-linear $1/\tau_{n}$ and $\rho_{n}$. A $T$-linear $\rho$ is usually expected in a system dominated by electron-phonon scattering in the regime $T\gg\Theta_{D}$, where $\Theta_{D}$ is the Debye temperature; in the regime $T<\Theta_{D}$, the resistivity is approximated by a low-order polynomial that is neither linear nor quadratic in temperature. Transport measurements in the 122 compounds suggest $\Theta_{D}\simeq 250$ K Ni _et al._ (2008); Tu _et al._ (2010). However, the $T$-linear $1/\tau_{n}$ and $\rho_{n}$ spans a very broad temperature range, from $T_{c}$ up to 300 K, which is inconsistent with the electron- phonon dominated scattering process. Further clues on the $T$-linear $1/\tau_{n}$ and $\rho_{n}$ may be revealed by an examination of the phase diagram of the Ba1-xKxFe2As2 system and a comparison with similar behavior found in other materials. The parent compound BaFe2As2 orders in an SDW state below $T_{SDW}\approx 138$ K; $T_{SDW}$ is suppressed by K doping allowing superconductivity to emerge. Here, $1/\tau_{n}\propto T$ is observed at the doping where the SDW order is entirely suppressed, i.e. $T_{SDW}\rightarrow 0$. At this point, spin fluctuations are expected to be very strong, which has been experimentally confirmed by NMR Ning _et al._ (2010); Nakai _et al._ (2010); Li _et al._ (2011). This brings us to models that attribute $T$-linear $\rho$ to spin-fluctuation scattering Moriya and Ueda (2000); Sachdev and Keimer (2011). The $T$-linear $\rho$ (or $1/\tau$) is also found in cuprates such as Nd-doped La2-xSrxCuO4 Daou _et al._ (2009) and electron- doped La2-xCexCuO4 Jin _et al._ (2011), organic superconductors (TMTSF)2PF6 Doiron-Leyraud _et al._ (2009a) and (TMTSF)2ClO4 Doiron-Leyraud _et al._ (2009b), as well as a number of heavy fermions such as CeCoIn5 Malinowski _et al._ (2005); Tanatar _et al._ (2007) and YbRh2Si2 Trovarelli _et al._ (2000); Gegenwart _et al._ (2002). Studies on these materials have shown that $T$-linear $\rho$, arising on the border of antiferromagnetic order, is caused by spin-fluctuation scattering due to the proximity of an antiferromagnetic QCP. Studies on cuprates and Bechgaard salts Taillefer (2010); Jin _et al._ (2011) further show that the strength (or coefficient) of the $T$-linear $\rho$ scales with $T_{c}$ and disappears upon approaching the point where $T_{c}\rightarrow 0$, suggestive of intimate relation between the $T$-linear $\rho$ and superconductivity. Anomalous $T$-linear $\rho$ (or $1/\tau$) and pairing in unconventional superconductors may share a common origin: spin fluctuations. The hidden $T$-linear $1/\tau_{n}$ and $\rho_{n}$ in Ba0.6K0.4Fe2As2 revealed by optical measurements may have the same origin as those found in the cuprates, organic superconductors and heavy fermion metals, because these materials share strikingly similar phase diagrams. Our observations may also imply a possible QCP in the superconducting dome. The existence of the QCP in iron-pnictides is supported by transport properties Gooch _et al._ (2009); Shen _et al._ (2011); Maiwald _et al._ (2012), NMR studies Ning _et al._ (2010); Nakai _et al._ (2010), de Haas-van Alphen effect Walmsley _et al._ (2013), penetration depth measurement Hashimoto _et al._ (2012) and first-principles calculations Xu _et al._ (2008). Spin-fluctuation induced $T$-linear $\rho$ suggests an equivalent $\omega$-linear $1/\tau(\omega)$ Moriya and Ueda (2000), obtainable through the extended Drude model provided that interband contribution is negligible. Low energy interband transitions are important in iron-pnictides, and their contribution has to be subtracted to determine $1/\tau(\omega)$ for mobile carriers Benfatto _et al._ (2011); Charnukha _et al._ (2011b). We calculated $1/\tau(\omega)$ via the extended Drude model with the interband contribution subtracted (supplementary) and found that by taking into account the interband transitions, a large fraction of the frequency dependence in $1/\tau(\omega)$ is eliminated, which is consistent with Charnukha _et al._ ’s analysis on Ba0.68K0.32Fe2As2 Charnukha _et al._ (2011b). There is no confident evidence for the expected $\omega$-linear $1/\tau(\omega)$, since it could be masked by the multiband character of the iron-pnictides. In order to check if the $T$-linear $1/\tau$ is unique in Ba0.6K0.4Fe2As2 or general in iron-pnictides, we applied the same analysis to BaFe2(As0.7P0.3)2 (supplementary). Interestingly, $T$-linear $1/\tau$ is also found for the narrow Drude. This suggests that $T$-linear $1/\tau$ is not unique in Ba0.6K0.4Fe2As2, but most likely, a general behavior in iron-pnictides at the doping where SDW order is completely suppressed, i.e. $T_{SDW}\rightarrow 0$. In summary, the detailed $T$ dependence of the normal state $\sigma_{1}(\omega)$ and low-frequency spectral weight in Ba0.6K0.4Fe2As2 have been examined. Two Drude components with different $1/\tau$’s yield an excellent description of the low-frequency optical response, indicating the existence of two groups of carriers with different quasiparticle lifetimes. The broad Drude component produces an incoherent background conductivity with no temperature dependence, while the narrow Drude component reveals a $T$-linear $1/\tau_{n}$ and $\rho_{n}$. This fact explains the $T$-linear $\rho$ behavior at low temperatures and the tendency to saturation at room temperature observed in transport measurements in optimally hole-doped 122 compounds. An arctan($T$) dependence of the low-frequency spectral weight is also a strong evidence for a $T$-linear $1/\tau$. Comparison with similar behavior found in other materials suggests that the $T$-linear $1/\tau_{n}$ and $\rho_{n}$ in Ba0.6K0.4Fe2As2 may arise out of spin-fluctuation scattering due to the possible existence of an antiferromagnetic QCP in the superconducting dome. We thank Hu Miao, Xiaoxiang Xi, Wei Ku, Cong Ren and Lei Shan for helpful discussion. Work in Beijing was supported by the NSFC (No. 91121004 and No. 11104335) and the MSTC (973 Projects No. 2011CBA00107, No. 2012CB821400, No. 2012CB921302 and No. 2009CB929102). Work at BNL was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DE-AC02-98CH10886. We acknowledge the financial support from the Science and Technology Service of the French Embassy in China. ## References * Boebinger (2009) G. S. Boebinger, Science 323, 590 (2009). * Cooper _et al._ (2009) R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lipscombe, S. M. Hayden, Y. Tanabe, T. 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Boeri, Phys. Rev. B 83, 224514 (2011).	arxiv-papers	2013-09-11T16:33:34	2024-09-04T02:49:50.727722	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Y.M. Dai and B. Xu and B. Shen and H. Xiao and H.H. Wen and X.G. Qiu\n and C.C. Homes and R.P.S.M. Lobo", "submitter": "Yaomin Dai", "url": "https://arxiv.org/abs/1309.2881" }
1309.2900	# Mining for Spatially-Near Communities in Geo-Located Social Networks Joseph Hannigan Dept. of Electrical Engineering and Computer Science U.S. Military Academy West Point, NY [email protected] &Guillermo Hernandez Dept. of Electrical Engineering and Computer Science U.S. Military Academy West Point, NY Guillermo.Hernandez@ usma.edu &Richard M. Medina Dept. of Geography and GeoInformation Science George Mason University Fairfax, VA [email protected] Patrick Roos Dept. of Computer Science University of Maryland College Park, MD [email protected] &Paulo Shakarian Dept. of Electrical Engineering and Computer Science U.S. Military Academy West Point, NY [email protected] ###### Abstract Current approaches to community detection in social networks often ignore the spatial location of the nodes. In this paper, we look to extract spatially- near communities in a social network. We introduce a new metric to measure the quality of a community partition in a geolocated social networks called “spatially-near modularity” a value that increases based on aspects of the network structure but decreases based on the distance between nodes in the communities. We then look to find an optimal partition with respect to this measure - which should be an “ideal” community with respect to both social ties and geographic location. Though an NP-hard problem, we introduce two heuristic algorithms that attempt to maximize this measure and outperform non- geographic community finding by an order of magnitude. Applications to counter-terrorism are also discussed. ## Introduction Community detection in social networks remains an important and active area of research in the study of social network mining (?; ?; ?; ?; ?; ?; ?; ?; ?). However, many real-world social networks also have a geographic context. Social networks are tethered to geographic locations. People and their relationships are tied to places. Even in the information age communications are dependent on access. Though access can seem ubiquitous in many cases, digital interaction cannot yet completely replace face-to-face contact, especially for planned activities of spatiotemporal coincidence and the transfer of tangible objects. Primary considerations for research in many social science disciplines today include characteristics of human activities and interactions in defined spaces. The interactions can be between humans, or between humans and their environments. These characteristics describe aspects of social complexity that are necessary to understand when attempting to model open or closed social systems. In studies of human security there is a new emphasis on implementations of Activity Based Intelligence (ABI) to better understand drivers toward specific actions and interactions, as well as to generate an understanding of the system outside of targeted activities (?). Though the concepts of ABI are new to many, its academic foundations of activity spaces, social interaction, and spatio-temporal research are well established. Attempts to identify sociogeographic based activity spaces, as demonstrated here, are vital to the understanding of human behavior. Multi-spatial or hybrid space (?) studies are much more valuable in this information age than their single space counterparts. Multiple spaces are converging into hybrid spaces as interactions in social systems become more complex. In this paper, we look to develop a framework for deriving communities from social network that is relevant not only with respect to network topology, but also geography. The main geographic concept we use to relate nodes based on space is “nearness.” On a general level, there exists a connection between nearness and similarity. “Near” is a spatial concept, though not necessarily geographically spatial. Social space nearness, or adjacency, typically describes relationships between people or things that interact in some way. Nearness based similarities need not be comprehensive. Single or few similar traits can exist to maintain interaction; however, relatively more similar traits between people can drive further or deeper interaction. Geographers and sociologists have developed concepts that seek to explain the phenomenon of nearness and similarity in their respective disciplines. In geography Tobler’s First Law (TFL) describes this effect in physical space and homophily describes it in social space (?; ?). Geographic and homophilic similarities are inherently connected, as one of the greatest sources of homophily is propinquity. Furthermore, interaction is driven by nearness and similarity. The likelihood for interaction between people increases as distance decreases between them. Community finding at the convergence of geographic and social space nearness will lead to the identification of communities where place and social traits drive interaction. Results of this method may be most meaningful in studies of social systems that are greatly influenced by ethnicity and culture among other geographically based factors. For example, communities identified using geographic and social closeness may apply more to terrorist and criminal networks than globally dispersed business networks. Hence, our intuition is to find communities that are tightly-knit based on network topology, but also spatially “near.” To do this, we create a new measure of partition quality that we term “spatial nearness modularity” that borrows concepts from network modularity (?) and $K$-means clustering (?; ?). Hence, to find a high-quality set of communities with respect to this geography and network connections, it stands to reason to search for an optimal partition with respect to this measure. Unfortunately, we are able to show that doing so is NP-hard based on the results of (?). To address this issue of intractability, we introduce two heuristics and we then experimentally evaluate them, where we find that our approach provides an order-of-magnitude improvement in spatially-near modularity over non- geographic approaches. This is followed by a description of how this technique could apply to counter-terrorism and a discussion of related work. ## Technical Preliminaries We assume the existence of a undirected graph $G=(V,E)$ where set $V$ are vertices and $E$ are edges among them. As the graph in undirected, $(i,j)\in E$ implies $(j,i)\in E$. We shall use $n,m$ to represent the sizes of $V,E$ respectively. Each edge $(i,j)$ will be associated with a positive real weight denoted by $w_{ij}$ (if there is no edge between $i$ and $j$, $w_{ij}=0$). For a given node $i\in V$, we shall use the symbol $\eta_{i}$ to represent the set $\\{j\in V\|\exists(i,j)\in E\\}$ and $k_{i}$ is the size of this set. We shall also assume a distance function $d:V\times V\rightarrow\Re$ that meets the normal distance axioms: $d(i,i)=0$, $d(i,j)=d(j,i)$, and $d(i,j)\leq d(i,j^{\prime})+d(j,j^{\prime})$. For ease of notation, we shall use $d_{ij}$ instead of $d(i,j)$. In this paper we will often use the notation $C=c_{1},\ldots,c_{x}$ to denote a partition of $V$. Hence, $\cup_{c_{i}\in C}c_{i}=V$ and for all $c_{i},c_{j}\in C$ $c_{i}\cap c_{j}=\emptyset$. We define the modularity of a partition ($M(C)$) in accordance with the definition introduced by (?) as follows: NG-Modularity. (?) Given a social network $G=(V,E)$ and partition $C$ the Newman-Girvan (NG) modularity is defined as follows: $\displaystyle M_{NG}(C)=\frac{1}{2m}\sum_{c\in C}\sum_{i,j\in c}w_{ij}-\frac{k_{i}k_{j}}{2m}.$ (1) The modularity of a network partition measures the quality of its partition structure as the density of edges within partitions compared to the density of edges between partitions. The former is ideally very high compared to the latter. Modularity will give a number in $[-1,1]$, a higher value meaning better quality partition. Previous work, such as (?; ?), has focused on finding a partition that optimizes this quantity. However, modularity maximization only considers network topology and does not make any effort to group individuals that are geographically close to each other. An alternative is to find a partition of $K$ clusters of nodes that minimizes the sum-of- squares distance to the center of each cluster. This is known as $K$-means clustering (?; ?). $K$-means clustering algorithms attempt to find a partition of points on a plane into $K$ clusters, such that the following quantity is minimized (here $x_{c}$ is the centroid of the points in cluster $c$): $\displaystyle\sum_{c\in C}\mathbf{agg}_{i\in c}d(i,x_{c})^{2}.$ (2) In the above definition, $\mathbf{agg}$ is some aggregate function. Common aggregates used here are $\max$ and $\sum$. For the purpose of this paper, as modularity is maximized, we wish to minimize some aggregate of the distances to the center of each cluster. Thus, one potential quantity that could be optimized is the following: $\displaystyle\frac{1}{2m}\big{(}\frac{\sum_{c\in C}\sum_{i,j\in c}w_{ij}-\frac{k_{i}k_{j}}{2m}}{1+\sum_{c\in C}\mathbf{agg}_{i\in c}d(i,x_{c})^{2}}\big{)}.$ (3) Note that the additive $1$ in the denominator is to avoid division by zero and to ensure that the result will be within the range $[-1,1]$. The above optimization function has the useful property that we can embed both modularity maximization and $K$-means clustering - the first by placing all nodes in the same location, the second by ignoring edges among any nodes in the network and restricting the number of clusters to be exactly $K$. However, one aspect the above definition misses is that it cannot measure the quality of an individual community. Hence, we introduce an alternative definition below that we term “spatially-near (SN) modularity.” SN-Modularity. Given a social network $G=(V,E)$, partition $C$, and scaling parameter $\sigma\in\Re^{+}$, the spatially-near (SN) modularity is defined as follows: $\displaystyle M_{SN}(C,\sigma)=\frac{1}{2m}\sum_{c\in C}\frac{\sum_{i,j\in c}w_{ij}-\frac{k_{i}k_{j}}{2m}}{1+\mathbf{agg}_{i\in c}\left(\frac{d(i,x_{c})}{\sigma}\right)^{2}}.$ (4) So, for a given community, we can measure its quality with the following: $\displaystyle\frac{1}{2m}\frac{\sum_{i,j\in c}w_{ij}-\frac{k_{i}k_{j}}{2m}}{1+\mathbf{agg}_{i\in c}\left(\frac{d(i,x_{c})}{\sigma}\right)^{2}}.$ (5) We also note that as $\sigma$ increases, distance is de-emphasized. This parameter would be specified based on the relative importance of distance to to network structure as well as the unit of measurement used for distance. Practically, a user could potentially provide this parameter in many different ways. Simple methods would include setting $\sigma$ to $1$, the average distance among all pairs of nodes, or the average distance among all pairs of nodes that have an edge between them. Alternatively, this parameter could also learned from historical data, if such a corpus is available. Another approach is for the user to explore various parameter settings. In this work, we leave advanced methods for determining $\sigma$ to future work and conduct experiments with multiple settings for this parameter. However, we note that for particularly large values of $\sigma$, SN-modularity becomes equivalent to NG-modularity. It is easy to show the following property: $\displaystyle\mathit{lim}_{\sigma\rightarrow\infty}M_{SN}(C,\sigma)=M_{NG}(C).$ (6) However, maximizing $M_{SN}(C,\sigma)$ remains NP-hard. Hence, in this paper we introduce two heuristic algorithms to find a partition $C$ where $M_{SN}(C)$ is near-optimal. Theorem. For a given social network $G=(V,E)$ and scaling factor $\sigma$, identifying a partition $C$ s.t. $M_{SN}(C,\sigma)$ is maximized is NP-hard. Proof. We can embed an instance of finding a partition $C$ that maximizes $M_{NG}(C)$ into the problem from the statement by creating a distance function $d$ where $\forall i,j,d(i,j)=0$ and setting $\sigma$ to an arbitrary value. Hence, any algorithm that maximizes $M_{SN}$ using this construction also maximizes $M_{NG}$. Since finding a partition that maximizes $M_{NG}$ is NP-hard by the results of (?), the statement of the theorem follows. $\blacksquare$ ## Algorithms In the previous section, we found that identifying a spatially-near partition is an NP-hard problem. Hence, in this section, we propose two heuristic approaches to deal with this intractability. We later describe our evaluation of these approaches. In our first heuristic, which we call “Louvain-SN”, we employ the modification of the Louvain algorithm of Blondel et al. (?), only instead of using it to maximize NG-modularity, we use it to maximize SN- modularity (the Louvain algorithm was designed to find a near-optimal parition w.r.t. NG-modularity). Our second algorithm, the SNIC (Spatially Near Iterative Constraining) algorithm, relies on multiple calls to the Louvain-SN algorithm - but each with a limit on the aggregate distance permitted in a community. ### The Louvain-SN Algorithm The original Louvain algorithm of (?) is an iterated, hierarchical process in which two phases are applied repeatedly until maximal modularity is reached: During the first phase, each node $v_{i}\in V$ of the given social network is assigned to a community $c$, creating an initial partition. In (?), the singleton partition was used - which we use in this work as well. Then, for each $v_{i}\in V$, the gains in modularity that would result from moving $v_{i}$ to the community of each of its neighbors $v_{j}\in\eta_{i}$ are calculated, and $v_{i}$ is removed and placed into the community for which the maximum improvement in modularity is achieved (unless no positive gain in modularity is possible). This sub-process is repeated sequentially for each $v_{i}\in V$ until no individual move will result in a gain in modularity, marking the end of the first phase and giving a partition $C$. During the second phase, a new network is built by using each $c_{i}\in C$ as a node in the new network, call these nodes meta-nodes. Weights on the edges between any two meta-nodes in the new network are assigned to be the sum of the weights of the edges between nodes in the two communities corresponding to the meta- nodes. Here, self-loops are created for each meta-node in the new network from the links between nodes of the community corresponding to that meta-node. After this phase is complete, the two phases are reapplied iteratively until there are no more changes. The efficiency of the Louvain algorithm relies on an easy re-calculation of modularity in the first phase of the algorithm. When computing gains in modularity in phase one of the algorithm, removing any node $v_{i}$, the overall increase in modularity if it is placed into community $c$ is proportional to: $\displaystyle k_{i,in}-\sum_{j\in c}\frac{k_{i}k_{j}}{2m}$ (7) In our modification for optimizing SN-modularity, we can retain some of this efficiency by retaining the previous denominator and numerator of Equation 5 (multiplied by $2m$) for each community. By retaining these values along with the value of Equation 7, computing the increase or decrease in modularity for a community can be performed quickly (though this ultimately depends on how the aggregate function $\mathbf{agg}$ and the centers of the communities $x_{c}$ are computed). Additionally, in the creation of the meta-nodes, we use the centers from the previous step as their location. Additionally, we also found that we obtained improvement in performance by allowing a removed node to be moved back to a community containing just itself, as unlike in the maximization of standard modularity, isolating a node in this fashion could potentially increase the overall modularity due to the denominator of Expression 4. ### The SNIC Heuristic Next, we introduce the “Spatially Near, Iterative Constraining” (SNIC) Heuristic. This idea was created as the result from a pilot experiment where we noticed that constraining a node to join only communities where it was geographically “near” to all the members would sometimes improve the resulting quantity of $M_{SN}$. The question is how does one determine where to set this distance constraint. In our experiments we ran our modified Louvain approach iteratively, returning only the maximum distance between two points in the community upon each iteration. This distance is then set as the distance constraint for the next iteration. Once the distance constraint reaches zero (or a maximum number of iterations is reached), the algorithm then returns the partition found which is associated with the greatest value for Expression 4. ## Experimental Results For our experiments, we used information extracted from the Brightkite location-based online social networking sites (?). We built our implementation in Python 2.6 on top of the NetworkX library111http://networkx.github.com/ leveraging code from Thomas Aynaud’s implementation of the Louvain algorithm222http://perso.crans.org/aynaud/communities/. Our implementation took approximately $1000$ lines of code. The experiments were run on a computer equipped with an Intel Core i7 Processor operating at 2.67 GHz (one core utilized) running Microsoft Windows 7 and equipped with 4.0 GB of physical memory. All statistics presented in this section were calculated using SPSS 19. We use our heuristics to find partitions based on Expression 4 where $\mathbf{agg}=\max$. In our first set of tests, we iteratively selected nodes and their neighbors from the Brightkite network dataset provided by the authors of (?) to produce 10 samples of 1000 nodes. To generate the samples, each sample begins with a randomly selected node from the network. The selected node and all of its connected nodes are then included in the next iteration, in which a new random node is chosen. This continues until 1000 nodes are reached for each sample. The minimum edge count for all samples processed is 1729, while the maximum is 2282. The average number of edges is 1929. In our trials, we varied the $\sigma$ parameter with the values $\\{300,500,1000,2000,3000,4000,5000\\}$. For each dataset and each value of $\sigma$, we compare the SN-modularity returned by three approaches: the Louvain algorithm (does not consider geospatial information), the Louvain-SN algorithm (the modified version of the Louvain algorithm for SN-modularity optimization), and the SNIC algorithm (an iterated version of the Louvain-SN that selects the best result based on updating the distance constraint). The SNIC algorithm returned a partition with greater average SN-modularity for each value of $\sigma$ than the partitions returned by the Louvain and Louvain-SN algorithms (see Figure 1). In general, the SNIC algorithm consistently outperformed the Louvain algorithm in terms of SN-modularity - producing a partition of greater SN-modularity on all trials. The Louvain-SN outperformed the standard Louvain in all but $11$ (of $70$) trials, though (as we discuss later in this section) this improvement is likely not statistically significant, unlike the SNIC heuristic.333There were $11$ such trials out of the $70$ trials where the Louvain outperformed the Louvain-SN. Of the cases where there was decreased quality over standard Louvain, the maximum decrease in quality was $26.52\%$ and the average decrease was $15.10\%$. The SNIC algorithm outperformed the Louvain algorithm on all trials. Figure 1: $\sigma$ (in kilometers) vs. (average) SN-modularity for the partitions returned by the Louvain, Louvain-SN, and SNIC algorithms. To determine significant difference in SN-modularity of the three approaches on the Brightkite dataset, analysis of variance (ANOVA) tests were used. Difference in SN-modularity for the three approaches was confirmed with a p-value of 0.006. A Tukey’s Honest Significant Difference (HSD) test was also used to determine pairwise differences between the approaches. No significant differences were found between the Louvain and Louvain-SN algorithms; however, the SNIC algorithm was found to be different than both the Louvain (at p = 0.010) and the Louvain-SN (at p = 0.020). Additionally, the differences for runtimes of the three approaches were found to be different with a p-value of 0.000 (see Figure 2). As with the difference in SN-modularity, runtime differences exist between the Louvain and the SNIC algorithms (at p = 0.000) and between the Louvain-SN and the SNIC algorithms (at p = 0.000). These results are also provided through use of Tukey’s HSD. Differences in SN- modularity and runtimes for the three approaches can be seen in Figures 1 and 2, respectively. Further, we also note that although the SNIC algorithm has significantly greater runtime than the Louvain and Louvain-SN, it still appear to scale linearly with the number of nodes in the network ($R^{2}=0.992$). Hence, it may still be a viable solution for very large networks. We are currently studying the scalability of this algorithm. Figure 2: Network size (by nodes) vs. runtime for the partitions returned by the Louvain, Louvain-SN, and SNIC algorithms. Figure 3 shows the increase in quality of community finding (SN-modularity) over iterations of the SNIC algorithm. Recall that the SNIC algorithm decreases the distance constraint at each iteration. As the geographic constraint decreases, such that community proximity becomes more important, the quality of community (number of connections within vs outside) increases. Here we introduce an axiom - that as the geographic space of interaction for a social network shrinks, it is more likely that those left within the community are more connected. Spatial outliers, which are also social outliers can be conceptualized as weak links (?) and are removed through community proximity limiting iterations. Through 100 iterations, the quality of community increases and in most social networks this value may continue to increase given high enough spatial resolution data. In other words, humans form communities and interact mostly with those they are geographically near, such that the strongest communities will be those shared within small geographic proximities. However, we note that more iterations of the SNIC algorithm will not result in singleton communities, as that is the initial partition considered by the algorithm. Also note that the improvement in SN-modularity as a function of number of iterations of the SNIC heuristic is also likely dependent on the parameter $\sigma$. Figure 3: SN-modularity vs. number of iterations for the partitions returned by the SNIC algorithms for the Britekite network data. As with the example above in Figure 3, Figure 4 shows that a stronger influence of geographic distance on community finding leads to greater quality communities based on the SN-modularity measure. Recall that $\sigma$ is the scaling parameter in SN-modularity. Decreasing the scaling parameter, in turn strengthening the geographic influence on the equation, leads to an increase in the quality of communities identified by the SNIC algorithm. Increasing the $\sigma$ value will result in an asymptotic trend for SN-modularity toward that expected from the non-spatial Louvain algorithm. This trend is shown for the Brightkite network in percent increase in SN-modularity over the Louvain algorithm for (A) the Louvain-SN and (B) the SNIC algorithms. Figure 4: $\sigma$ (in kilometers) vs. percent improvement in SN-modularity for the partition returned by the Louvain-SN (panel A) and SNIC (panel B) algorithms. Not depicted in panel A (Louvain-SN) are results where the Louvain-SN algorithm produced lower-quality results than the standard Louvain (due to the log-scale, see text for further details).3 Note that for the SNIC algorithm (panel B) outperformed the standard Louvain on all trials. The difference in Brightkite communities identified by the Louvain and SNIC algorithms is clear in Figure 5. There is a quantitative difference as suggested by the SN-modularity metric results, but also a very qualitative difference in which the communities identified by the SNIC algorithm are much more spatially constrained. The bottom half of Figure 5 represents the SNIC algorithm results with $\sigma$ = 1. In today’s information age where global networks are common, methods to identify geograhically unconstrained communities, as well as those methods that identify their geographically constrained counterparts are both equally valuable. Implications for strength of ties, activity and operations spaces, and interactions are different when considering geographic network characteristics. Figure 5: Top: Brightkite Communities identified using the Louvain algorithm, Bottom: Communities identified using the SNIC algorithm. Additionally, we also studied the NG-modularity of the partition returned by the SNIC algorithm. We found that although the SNIC algorithm was not designed to maximize NG-modularity, it still provided a positive value - which indicates that there is a greater density of edges within the communities as opposed to between communities (Figure 6). We also found that the solution returned by the NG-modularity of the partition returned by the SNIC algorithm seems to approach the NG-modularity of the solution of the Louvain algorithm as $\sigma$ increases. Although this is not guaranteed theoretically, it should be expected based on the relationship between NG-modularity and SN- modularity shown in Equation 6. Figure 6: NG-modularity of the partition returned by the SNIC algorithm ($10$ iterations) as a function of $\sigma$. ## Applications There are many fitting applications for algorithms that detect sociogeographic communities. In general, any network that requires or benefits from geographic propinquity can serve as a test case for the SNIC algorithm. For example, the diffusion of a disease through a social network requires geographic closeness, or face-to-face interaction, between people. While much of the diffusion may not be social network based, but solely spatial, those that have stronger social ties and in turn interact more in geographic space are more likely to contract or spread a disease. This phenomenon exists at various levels of physical interaction for contagious diseases. This model of diffusion works with the spread of any biologically contagious, material, or even ideological transfer that requires coincidence in space and time. The following example shows the value of the SNIC algorithm for sociogeographic analysis on a transnational terrorist network. Figure 7 illustrates the difference between community finding results using both the Louvain (non-spatial) and SNIC algorithms on a transnational terrorist network dataset. The dataset used for this research is representative of a global Islamist terrorist network from the late 1970s to approximately 2010 (see (?) for a full description and discussion of the dataset). The SNIC algorithm application shown here uses $\sigma$ = 1600. The transnational Islamist terrorist network is a cellular based, decentralized structure and heavily dependent on relative location and proximity (?). Because this is the case, identifying sociogeographic communities requires only a small spatial component additional to the social component. Applications of the SNIC algorithm on other network structures may require more spatial influence to identify sociogeographic communities. For example, the Brightkite application shown in Figure 5 uses $\sigma$ = 1. The terrorist network is much smaller with 358 nodes and 660 edges, and is much more geographically based for operational necessity. As stated previously, research results that identify social closeness vs. those that identify sociogeographic closeness are quantitatively and qualitatively different for many social networks. The top graphic in Figure 7 shows the modularity results using the Louvain algorithm, which highlight the transnational network connections. Many of the Islamist terrorist network cells have foundations or affiliates in Europe and the Middle East. While the strength of social communities can be equal over long distances, especially if network connections were made at some point coincident in space and time, it is beneficial to isolate communities in geographic space for some applications. In this case, the SNIC algorithm successfully identifies operational communities (A) the 9/11 cell planning and preparing for the attack in Southern California and Arizona, (B) a father and son diad working with, specifically financing, al-Qaeda in Canada, (C) a sociogeographic community of al-Qaeda tied members in Montreal, Canada, some of which were plotting to attack Los Angeles International Airport in 1999, (D) two al-Qaeda linked cells in Boston, MA with members in the Boston sleeper cell and plotting a large scale bombing attack in Jordan at multiple sites, (E) the al- Qaeda based cell operating in New York responsible for the first World Trade Center Attack in 1993, and (F) communities of 9/11 hijackers operating in Florida and other eastern US states. The SNIC algorithm can be additionally adjusted to further separate cellular communities based on geography (by varying $\sigma$ and the number of iterations of the algorithm). In systems such as this terrorist network, connected individuals that are close in geographic space, but not as close socially, can be more important to identify when attempting to counter operations. For example, identification of weaker but closer social links, such as those providing materials to a terrorist cell can be used as valuable intelligence to understand and dismantle terrorist operations in local to regional settings. Knowledge of international connections is important for understanding the global terrorist system, and cells in decentralized networks often maintain communications over long distances. However, many of these cells can operate independently, though in most cases they will need proximal resources. These local system interactions can be detected through use of the SNIC algorithm. Figure 7: Top: Terrorist communities identified using the Louvain algorithm, Bottom: Terrorist communities identified using the SNIC algorithm. ## Related Work Modularity maximization for community finding was first introduced in (?). In (?), the Louvain algorithm is introduced, which can scale to very large networks and is shown to provide partitions that nearly maximize modularity. We leverage a modification of the Louvain algorithm in this paper. Finding geographically disperse communities in a social network has also been previously studied (?; ?; ?; ?). Our approach in this paper differs in that we desire to find communities where the nodes are spatially-near and not distant. In addition to the aforementioned approaches, community detection in networks has also been explored in other manners that have potential to be applicable to the geospatial case - though to our knowledge no such application has been presented in the literature. See (?; ?) for examples. There also exist many approaches for community detection in networks not based on modularity maximization. Examples use label propagation (?), random walks (?), or bottom-up voting approaches (?). See (?) for comprehensive surveys. These do not consider spatial interactions - leveraging these approaches in a geospatial context is an important possibility for future work. Geospatial networks have been explored with respect to problems other than community finding such as link-prediction (?) and identifying user location (?). There have also been several empricial studies on social networks with a spatial component such as (?; ?; ?). More domain-specific empirical studies related to this work are also prevalent in the literature. Pertinent to our application are studies on terrorist networks (?) and criminal co-offender networks (?). ## Conclusion In this work, we introduced spatially-near modularity - a measure of the quality of a geographically-near partition in a social network. Though finding an optimal partition with respect to this measure is NP-hard, we were able to obtain quality partitions with two heuristic algorithms that we introduced in this paper and tested on real-world datasets. We have also discussed various ways in which our algorithms can be applied to gain useful knowledge in counter-terrorism applications. Our immediate concern for future work is exploring the scalability of this approach ($10^{6}$ nodes and greater). Additionally, we are also pursuing temporal dynamics of such communities and the differences between the communities formed based on the current state of the nodes (i.e. “work” vs “home”). In our more practical research, we are also working to integrate the generation of geographically-near partitions into our Organizational, Relationship, and Contact Analyzer (ORCA) software (?) that we are currently fielding to several American law-enforcement agencies. Acknowledgements. 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Mining for geographically disperse communities in social networks by leveraging distance modularity. In Proc. 2013 ACM KDD. * [Tobler 1970] Tobler, W. 1970\. A computer movie simulating urban growth in the Detroit region. Economic Geography 46(2):234–240. * [Yang et al. 2009] Yang, T.; Jin, R.; Chi, Y.; and Zhu, S. 2009\. Combining link and content for community detection: a discriminative approach. In Proc. ACM KDD, 927–936. New York, NY, USA: ACM.	arxiv-papers	2013-09-11T17:34:43	2024-09-04T02:49:50.735751	{ "license": "Public Domain", "authors": "Joseph Hannigan, Guillermo Hernandez, Richard M. Medina, Patrcik Roos,\n Paulo Shakarian", "submitter": "Paulo Shakarian", "url": "https://arxiv.org/abs/1309.2900" }
1309.2904	# A System-Theoretic Clean Slate Approach to Provably Secure Ad Hoc Wireless Networking Jonathan Ponniah, Yih-Chun Hu, and P. R. Kumar CSL & ECE, Univ. of Illinois, 1308 West Main St., Urbana, IL 61801. Email: {ponniah1,yihchun}@illinois.edu. Tel: 217-333-4220. Corresponding author: ECE, Texas A&M University, 3259 TAMU, College Station, TX 77843-3259. Email: [email protected]. Tel: 979-862-3376. This paper is partially based on work supported by NSF under Contract Nos. CNS-1302182, CCF-0939370 and CNS-1232602, AFOSR under Contract No. FA-9550-13-1-0008, and USARO under Contract No. W911NF-08-1-0238. ###### Abstract Traditionally, wireless network protocols have been designed for performance. Subsequently, as attacks have been identified, patches have been developed. This has resulted in an “arms race” development process of discovering vulnerabilities and then patching them. The fundamental difficulty with this approach is that other vulnerabilities may still exist. No provable security or performance guarantees can ever be provided. We develop a system-theoretic approach to security that provides a complete protocol suite with provable guarantees, as well as proof of min-max optimality with respect to any given utility function of source-destination rates. Our approach is based on a model capturing the essential features of an ad-hoc wireless network that has been infiltrated with hostile nodes. We consider any collection of nodes, some good and some bad, possessing specified capabilities vis-a-vis cryptography, wireless communication and clocks. The good nodes do not know the bad nodes. The bad nodes can collaborate perfectly, and are capable of any disruptive acts ranging from simply jamming to non- cooperation with the protocols in any manner they please. The protocol suite caters to the complete life-cycle, all the way from birth of nodes, through all phases of ad hoc network formation, leading to an optimized network carrying data reliably. It provably achieves the min-max of the utility function, where the max is over all protocol suites published and followed by the good nodes, while the min is over all Byzantine behaviors of the bad nodes. Under the protocol suite, the bad nodes do not benefit from any actions other than jamming or cooperating. This approach supersedes much previous work that deals with several types of attacks including wormhole, rushing, partial deafness, routing loops, routing black holes, routing gray holes, and network partition attacks. ###### Index Terms: Ad hoc wireless networks, security. ## I Introduction Our focus is on the problem of security of ad-hoc, multi-hop, wireless networks. The wireless nodes in these types of networks need to determine when to transmit packets and at what power levels, discover routes from sources to destinations, and ensure overall end-to-end reliability, all without any centralized controller guiding the process. This requires a suite consisting of multiple protocols. Several candidate have been proposed. Medium access control protocols include IEEE 802.11 [10] and MACAW [2], power control protocols include COMPOW [12] and PCMA [15], routing protocols include DSDV [17], AODV [16], DSR [11], and OLSR [5], and transport protocols include TCP [20] and variations for ad hoc networks [13, 6, 3, 21]. All the above protocols are designed on the assumption that all nodes are “good,” and will conform to the protocol. Some nodes can however be malicious, deliberately intent on disrupting the network, a vulnerability especially acute since the very purpose of ad hoc networks is to allow any node to join a network. For wireless networks used in safety-critical applications, e.g., vehicular networks, vulnerabilities can be dangerous. Moreover, many wireless networking protocols have been based on wireline protocols, with possible susceptibilities to novel over the air attacks. The assumption of benignness, implicit or explicit, has been the traditional starting point of protocol development. Systems have been first designed to provide high performance. Subsequently, as vulnerabilities have been discovered, they have been patched on a case by case basis. For example, the “wormhole” attack was discovered in [7], where an attacker sets up a false link between two nodes. It is countered by a fix using temporal and geographical packet leashes [7, 19]. The “rushing” attack against DSR was discovered in [8], in which attackers manipulate the network topology. This is countered by a fix using network discovery chains. The “partial deafness” attack against 802.11 was discovered in [4], in which an attacker artificially reduces its link quality to draw more network resources. It is countered by a fix using queue regulation at the access point. Other attacks against DSR are the routing loop attack in which an attacker generates forged routing packets causing data packets to cycle endlessly; the routing black hole attack in which an attacker simply drops all packets it receives; and the network partition attack in which an attacker injects forged routing packets to prevent one set of nodes from reaching another. These attacks are all countered in the Ariadne protocol [9] by the joint use of routing chains, encryption, and packet leashes. Some protocols such as Watchdog and Pathrater [14] try to pre-empt attacks by maintaining a blacklist that tracks malicious behavior, but this backfires if an attacker maligns a good node, causing other good nodes to add that node to their blacklists. These attacks are not targeted at violating privacy of communications between nodes, which can be avoided simply by encryption. Rather, they are generally Denial of Service attacks (DoS), which usually take advantage of algorithms that assume the participating users are good or cooperative. The basic problem with this arms race approach of hardening algorithms initially designed for good performance is that one never knows what other vulnerabilities or attacks exist. Thus no guarantees can be provided about the security of the protocols at any stage of the arms race process. Our goal in this paper is to propose an alternate clean slate system-theoretic approach to security that provides provable performance guarantees. We pursue a model-based approach, comprising a physical model of node capabilities, clocks, cryptography, and wireless communication. It is an initial attempt to holistically model the entire dynamics of an ad-hoc wireless network that has been infiltrated with hostile nodes. Our goal is to design a protocol suite for the complete life-cycle of the wireless system, all the way from the very birth of the nodes, and continuing through all phases of the network formation process, to a long-term operation where the network is carrying data reliably from sources to their destinations. The good nodes don’t know who the bad nodes are, and are required to follow the published protocol suite. Throughput all phases, the bad nodes can perfectly collaborate and incessantly indulge in any disruptive behavior to make the network formation and operation dysfunctional. They could just “jam,” or engage in more intricate behavior such as not relay a packet, advertise a wrong hop count, advertise a wrong logical topology, cause packet collisions, disrupt attempts at cooperative scheduling, drop an ACK, refuse to acknowledge a neighbor’s handshake, or behave inconsistently. We design a protocol suite that is provably secure against all such attacks by the malicious nodes. Not only that, it guarantees min-max optimal performance. The performance is described by a given utility function, which the good nodes wish to maximize by publishing a complete protocol suite and conforming to it. The bad nodes on the other hand aim to minimize this utility by indulging in all manner of “Byzantine” behavior described above not conforming to the protocol. This leads to a zero-sum game. Since the good nodes first announce the protocol, the best value of the utility function that the good nodes can hope to attain is its max-min, where the maximization is over all protocol suites, and the minimization is over all Byzantine behaviors of the bad nodes. We will prove that the protocol suite designed attains this max-min to within any $\epsilon>0$. Moreover, we establish three even stronger results. First, this game actually has a saddle point, i.e., the protocol suite attains the min-max (to within any $\epsilon>0$). (Generally, min-max results in a higher utility than max-min, since the bad nodes have to first disclose their tactics). Second, the bad nodes can do no better than just jamming or conforming to the published protocol suite on each “concurrent transmission vector,” a generalization of the notion of an “independent set” of nodes that can simultaneously transmit. They do not benefit from more elaborate Byzantine antics. Third, the protocol optimally exploits any non-hostile behavior of the bad nodes. If they behave suboptimally, i.e., are not as hostile as they could be, then it will take advantage. This is a desirable feature since while one wants to design protocols that are guaranteeably secure in the worst case, one would want them to exploit any benignness in the environment. Some important qualifications need to be noted. First, the results are valid only for the postulated model of the network. Future research may identify technological capabilities outside the model that can attack the protocol suite. Such discoveries will, one hopes, lead to the development of more general models and procotols provably secure in them. The research enterprise will thereby be elevated to a higher level; instead of reacting to each proposed protocol one reacts to each proposed model, with provable guarantees provided at each step. Section VII provides some such directions for model generalization. Second, though not merely asymptotic, the optimality is over a large time period, and the overhead of transient phases of the protocol may be high. However, there is much scope for optimizing protocol overhead while preserving security. Third, how should one view the proposed protocol suite? The answer is layered. At a minimum, it can be regarded as a constructive existence proof that one can indeed provide optimal performance while guaranteeing security, with the identified model class only serving as an exemplar of conditions under which this can be done. To a more receptive reader, the designed protocol suite is suggestive of of how one can do so. The architectural decomposition into several phases could perhaps be kept in mind by future protocol designers. At any rate, one hopes that this approach will trigger several critical reactions among a skeptical readership, and lead to follow up work that designs protocols with guaranteed security and performance for more general model classes. Section II describes the model, Section III the main results, Section IV an outline of the approach, Section V the protocol suite, and Section VI proves feasibility and optimality. ## II The Model The model of an ad-hoc wireless network infiltrated by hostile nodes can be organized into four categories: the nodal model (N), communication model (CO), clock behavior (CL), and cryptographic capabilities (CR). Nodal model: (N1) There are $n$ nodes, some good and some bad. Let $G$ denote the set of good nodes, and its complement $B$ the set of bad nodes. (N2) The good nodes do not know who the bad nodes are a priori. (N3) The bad nodes are able to fully coordinate their actions, and are fully aware of their collective states (equivalent to unlimited bandwidth between them). (N4) The good nodes are all initially powered off, and they all turn on within $U_{0}$ time units of the first good node that turns on. Communication model: (CO1) Each node $i$ can choose from among a finite set of transmission/reception modes $M_{i}$ at each time. Each mode corresponds, if transmitting, to a joint choice of power level, modulation scheme and encoding scheme for each other intended receiver node, or to just listening and not transmitting, or even to “jamming,” which simply consists of using its power output to emit noise. (CO2) The good nodes are half-duplex, i.e., cannot transmit and receive simultaneously. (CO3) We call $c=(c_{1},c_{2},\ldots,c_{n})$ denoting the mode choices of all the nodes made at a certain time, as a “concurrent transmission vector” (CTV). (It is more general than an independent set that is sometimes used to model wireless networks). We will denote by $c_{G}=(c_{i}:i\in G)$ and $c_{B}=(c_{i}:i\in B)$ the vectors of choices of modes made by the good and bad nodes respectively, with each $c_{i}\in{\cal{M}}_{i}$, and let ${\cal{C}}_{G}$ and ${\cal{C}}_{B}$ denote the sets of all such choices. We will denote by ${\cal{C}}:={\cal{C}}_{G}\times{\cal{C}}_{B}$, the set of all CTVs. (CO4) Each $c$ results in a “link-rate vector” $r(c)$ of dimension $n(n-1)$. Its $ij$-th component, $r_{ij}(c)$, is the data rate at which bits can be sent from node $i$ to node $j$ at that time. Due to the shared nature of the wireless medium, the rate depends on the transmission mode choices made by all the other nodes, as well as the geographic locations of the nodes, the propagation path loss, the ambient noise, and all other physical characteristics affecting data rate. A component $r_{ij}(c)$ may be zero, for example if the SINR at $j$ is below a threshold value for decoding, or if node $i$ is not transmitting to node $j$. (CO5) If a certain rate vector is achievable then lower rates are also achievable. To state this, let $\Lambda:=\\{r_{ij}(c):i\neq j,c\in{\cal{C}}\\}$ denote the finite set of all possible rates than can be achieved. We suppose that for every $c$, and $r^{\prime}\leq r(c)$ (understood component wise) with all elements in $\Lambda$, there is a choice $c^{\prime}\in{\cal{C}}$ such that $r(c^{\prime})=r^{\prime}$. This assumption is not strictly necessary, but it helps to simplify the statement that bad nodes can claim to receive only at low rates. (CO6) In the case of a bad node $j$, the rate $r_{ij}(c)$ may be the result of some other bad node being able to decode the packet from $i$ at that rate, and then passing on that packet to $j$, since bad nodes can collaborate perfectly. In the case of a bad node $i$, the rate $r_{ij}(c)$ may be the result of some other bad node being able to transmit the packet successfully to $j$ at that rate, pretending to be $i$. Meanwhile, in either case, the bad node may be jamming. Thus a bad node can both jam and appear to be cooperating, whether transmitting or receiving, at the same time. (CO7) The bad nodes can claim to have received transmissions from each other at any of the rates in the finite set $\Lambda$, as they please. To state this, for $c=(c_{G},c_{B})$, we will partition the resulting link-rate vector as $r(c)=(r_{GG}(c),r_{GB}(c),r_{BG}(c),r_{BB}(c))$, where $r_{BG}$ denotes the link-rates from the bad nodes to the good nodes, etc. We suppose that for every $c=(c_{G},c_{B})$ and every $r^{\prime}$ with all elements in $\Lambda$, there is a $c^{\prime}_{B}\in{\cal{C}}_{B}$ such that $r(c_{G},c^{\prime}_{B})=(r_{GG}(c),r_{GB}(c),r_{BG}(c),r^{\prime})$. (CO8) The good nodes know $\Lambda$, and an upper bound on the cardinalities of the ${\cal{M}}_{i}$’s, but do not know the values of the vectors $r(c)$ for any $c\in{\cal{C}}$. (CO9) The assumption that the link-rate vector $r(c)$ does not change with time implicitly assumes that nodes are not mobile to any significant extent. We comment further about this assumption in Section VII, Clock model: (CL1) Each good node $i$ has a local continuous-time clock that it initializes to zero when it turns on. Its time $\tau^{i}(t)$ is affine with respect to some reference time $t\geq 0$, i.e., $\tau^{i}(t)=a_{i}t+b_{i}$ where $a_{i}$ and $b_{i}$ are called the skew and offset respectively. Wlog, the time $t$ above and in (N4) is taken equal to the clock time of the first good node to turn on. (CL3) Denoting the relative skew and offset between nodes $i$ and $j$ by $a_{ij}:=\frac{a_{i}}{a_{j}}$ and $b_{ij}:=b_{i}-a_{ij}b_{j}$, node $i$’s time with respect to node $j$’s time $s$ is $\tau^{i}_{j}(s)=a_{ij}s+b_{ij}$. We assume $0<a_{ij}\leq a_{max}$. As a corollary of (N4,CL1,CL3), $\|b_{ij}\|\leq a_{max}U_{0}$, since $\tau^{i}(U_{0})\geq 0$. (CL4) The good nodes do not know their skew or offset a priori. (CL5) Finally, due to its digital processor, a good node $i$ can only observe a quantized version of its continuous-time local clock $\tau^{i}(t)$. Cryptographic capabilities: (CR1) Each node is assigned a public key and a private key; information encrypted by a private key can only be decrypted with the corresponding public key. The private key is never revealed by a good node to any other node. Possession of a public key does not enable an attacker to forge, alter, or tamper with an encrypted packet generated with the corresponding private key. The good nodes encrypt all their transmissions. (CR2) Each node possesses the public key of a central authority. (CR3) Each node possesses an identity certificate, signed by the central authority, containing node $i$’s public key and ID number. The certificate binds node $i$’s public key to its identity. (CR4) Each node possesses a list of all the other $n$ node IDs. ## III The Main Results Each time that the good nodes make a certain choice $c_{G}$, the bad nodes could respond with some choice drawn only from a certain subset ${\cal{C}}_{B,c_{G}}\subseteq{\cal{C}}_{B}$. In this way they could ensure that only the subset ${\cal{E}}:=\\{(c_{G},c_{B}):c_{G}\in{\cal{C}}_{G},c_{b}\in{\cal{C}}_{B,c_{G}}\\}$ is ever employed by the network. If so, we will say that ${\cal{E}}$ is _enabled_ , while its complement ${\cal{D}}:={\cal{C}}\setminus\\{(c_{G},c_{B}):c_{G}\in{\cal{C}}_{G},c_{b}\in{\cal{C}}_{B,c_{G}}\\}$ is _disabled_ by the bad nodes. We will denote by $\Delta$ the set of all such sets ${\cal{D}}$ that they have the capability to disable. For any set ${\cal{E}}$ of enabled CTVs, let ${{R(\cal{E})}}:=$ ConvexHull$(\\{r(c):c\in{\cal{E}}\\})$ _be the set of link rate-vectors supported by_ ${\cal{E}}$, i.e., generated by time sharing over ${\cal{E}}$. Let ${\cal{G}}({{\cal E}})$ be a directed graph over the nodes, where there is an edge $ij$ if and only if $r_{ij}(c)>0$ for some $c\in{{R}}({{\cal E}})$. We assume that the good nodes can communicate in a multi-hop fashion with each other over bidirectional links at some minimal positive rate, regardless of what the bad nodes do: Connectedness Assumption (C): Let ${\cal{G}}^{}:={\cal{G}}({{\cal C}}\setminus{{\cal D}}^{})$ be the graph resulting from the maximum set ${\cal{D}}^{}\in\Delta$ that the bad nodes can disable. We will assume that the good nodes are connected in the subgraph of ${\cal{G}}^{}$ that consists only of edges $ij$ for which both $ij$ as well as $ji$ are edges in ${\cal{G}}^{}$. Denoting by $P_{ij}$ the set of all paths from $i$ to $j$, the _multi-hop capacity region_ of $n(n-1)$-dimensional end-to-end source-destination throughput vectors is defined in the standard way as ${{C}}({\cal{E}}):=\\{x:$ For some vector $y\geq 0$ with $0\leq\sum_{p:\ell\in p}y_{p}\leq r_{\ell}$ for some $r\in{{R(\cal{E})}}$, $x_{ij}=\sum_{p\in P_{ij}}y_{p}$ for all $1\leq i,j\leq n,j\neq i\\}$. We employ a utility function defined over the throughputs of any subset of source-destination pairs of interest: Utility function assumption (U): For any subset $S\subseteq\\{1,2,\ldots,n\\}$ and any throughput vector $x$, let $U(x,S)$ depend only on $x_{ij}$ for $i,j\in S$. For every $S$, $U(x,S)$ is continuous and monotone increasing in the components of $x$. We now consider the game where the good nodes wish to maximize it for the nodes perceived to be good, while the bad nodes wish to minimize it over all their Byzantine behaviors. To obtain an upper bound on utility, suppose that the bad nodes disable only the CTVs in ${\cal{D}}$ and reveal this choice to the good nodes. Let ${\cal{E}}:={\cal{C}}\setminus{\cal{D}}$. If ${\cal{G}}({\cal{E}})$ has several strongly connected components, then, by the connectedness assumption (C), the good nodes are all in the same component, denoted by $F({\cal{E}})$, and thus know that the nodes outside $F({\cal{E}})$ are bad. They will therefore only consider the utility accrued as $U(x,F({\cal{E}}))$, and maximize it over all $x\in{\cal{C}}({\cal{E}})$. Hence an upper bound on achievable utility is $\displaystyle\min_{\begin{subarray}{c}{{\cal{D}}\in\Delta}\end{subarray}}\hskip 5.69054pt\displaystyle\max_{\begin{subarray}{c}x\in{\cal{C}}({\cal{C}}\setminus{\cal{D}})\end{subarray}}U(x,F({\cal{C}}\setminus{\cal{D}}))$. Our main result, elaborated on in Theorem VI.2, is: ###### Theorem III.1 Consider a network that satisfies (N), (CO), (CL), (CR), (C) and (U). Given an arbitrary $\epsilon$, where $0<\epsilon<1$, the protocol described in Section V ensures that all the good nodes obtain a common estimate of the component that they are all members of, and achieves the utility $\displaystyle(1-\epsilon)\displaystyle\min_{\begin{subarray}{c}{{\cal{D}}\in\Delta}\end{subarray}}\hskip 5.69054pt\displaystyle\max_{\begin{subarray}{c}x\in{\cal{C}}({\cal{C}}\setminus{\cal{D}})\end{subarray}}U(x,F({\cal{C}}\setminus{\cal{D}})).$ (1) Some important consequences are the following. Normally, one would expect that since the good nodes have to first declare their protocol and follow it, they can only attain “max-min,” which is generally smaller than min-max. Since the latter can be attained (arbitrarily closely), it shows firstly that the bad nodes are unable to benefit from having a priori knowledge of the protocol. Second, since all that the bad nodes can benefit from is deciding which sets to disable, they are effectively limited to jamming and/or cooperating in each CTV. Other more Byzantine behaviors are not any more effective. The example below shows why a bad node may prefer to “conform” rather than jam for some utility functions. ###### Example III.1 Consider the network of Figure 1. Nodes 1 and 2 are good and in close proximity, while node 3 is bad and located far away. Consider the “fairness- based ”utility function $U(x):=\min\\{x_{12},x_{32}\\}$. If node 3 jams, then the connected component becomes $\\{1,2\\}$, and the good nodes proceed to maximize only $x_{12}$, which node 3 can only slightly impinge because it is so far away from node 2. However, if node 3 cooperates, then the connected component is $\\{1,2,3\\}$, and the optimal solution for this “fair” utility function is to make $x_{32}=x_{12}$. However, link 32 being weak, it requires much more airtime than link 12, thus considerably reducing $x_{12}$. Figure 1: Example III.1. ## IV The Outline of the Approach The heart of the approach is to investigate different CTVs, exploiting the fact that the operation of the network consists of invoking which such set to use at any given instant. If a good node fails to receive a scheduled packet transmitted during a CTV set, then that good node alerts the rest of the network during a verification phase, and the offending CTV set is never used again. After each such pruning the network then re-optimizes its utility over the remaining CTVs. The decreasing sequence of remaining sets of CTVs necessarily converges to an operational collection of CTVs, over which the utility is optimized by time sharing. Since the set of disabled CTVs is determinable by the network, as we show, it is the same as if it were revealed to the good nodes a priori, which allows achievement of min-max. It also shows why more complex Byzantine behaviors than jamming or cooperating are not any more effective for the bad nodes. There are however several problems that lie along the way to realizing this scheme. First, all of the above presumes that all the nodes are good, and, second, also that the nodes know the network topology and other parameters, both of which are false. This leads to the challenge: How to determine the network, while under attack from bad nodes when one does not know the network? We present a complete protocol suite that proceeds through several phases to achieve this end result. After their birth, the nodes need to first discover who their neighbors are. This requires a two-way handshake, which presents one problem already. Two good nodes that are neighbors can successfully send packets to each other if there are no primary (half-duplex) or secondary (collision) conflicts. To achieve this we employ an Orthogonal MAC Code [18]. Next, the two nodes need to update their clock parameters. After this, the nodes propagate their neighborhood information so that everyone learns about the network topology. This also poses some challenges when there are intermediary bad nodes. This is addressed by a version of the Byzantine General’s algorithm of [1], by capitalizing on connectedness assumption (C). Next, even though all the good nodes converge to a common network view, that view may be internally inconsistent, especially with respect to clocks. To resolve this we employ a certain consistency check algorithm. Next, the nodes proceed to determine an optimal schedule for time sharing over the set of CTVs that have performed consistently from the very beginning, and execute it. However, a bad node that has cooperated hitherto may not cooperate at this point. Hence the results of this operational phase need to be verified, the dysfunctional CTV pruned, the schedule re-optimized, and the procedure iterated. The reader may wonder: Why do we even need a notion of “time”? First, without it, we cannot even speak of throughput or thus of utility. Second, we use local clocks to schedule transmissions and coordinate activity (as is quite common, e.g., time-outs in MAC and transport protocols). On the other hand, dependence on distributed synchronized clocks for coordinated activity opens yet another avenue for bad nodes to sabotage the protocol – interfering with the clock synchronization algorithm. Therefore, topics like scheduling, clock synchronization, utility maximization, and security, are deeply interwoven. Therefore one needs a holistic approach that addresses all these issues at every stage of the operating lifetime, and guarantees overall security and min-max optimality. This is the raison d’$\rm{\hat{e}}$tre for this paper. ## V The Phases of the Protocol Suite The protocol suite consists of six phases: Neighbor Discovery, Network Discovery, Consistency Check, Scheduling, Data Transfer, and Verification. Proofs are deferred to Section VI. We first note the necessity for a key ingredient. Even two good nodes that are neighbors as in assumption (C) are only guaranteed to be able to successfully send packets to each other provided one is transmitting, the other is listening (since good nodes are half-duplex), and the remaining good nodes are all silent. The Orthogonal MAC Code (OMC) of [18] ensures the simultaneity of all these events, even though the clocks of different nodes have different skews and offsets. For each pair of nodes $i,j$, it defines certain zero-one valued functions of local time at each node, such that if $i$ transmits a packet of duration $W$ to $j$ at that time, then the packet is successfully received, and the delay involved in waiting for such an eventuality is never more than a certain $T_{MAC}(W)$. ### V-A The Neighbor Discovery Phase In this phase, each node $i$ will determine the identity and relative clock parameters of nodes in its neighborhood ${\cal N}_{i}$, and include this data in a mutually authenticated link certificate. In the first two steps, each node $i$ attempts a handshake with a neighbor node $j$ by broadcasting a probe packet $PRB_{ij}$ and waiting for an acknowledgement $ACK_{ji}$. The probe packet contains an identity certificate signed by a central authority. Given ${\cal N}_{i}:=\\{1,\ldots,n\\}$\$i$, an initial candidate for the set of bidirectional neighbors of $i$ (as in (C)), to indicate that node $i$ transmits $PRB_{ij}$ to each node $j\in{\cal N}_{i}$ via the OMC, and receives $PRB_{jj}$ from each node $j\in{\cal N}_{i}$, we use $\text{TxRxMAC}(PRB_{i\rightarrow{\cal N}_{i}}$,$PRB_{{\cal N}_{i}\rightarrow i})$. If a probe packet is not received from some node $j$, then $j$ is pruned from ${\cal N}_{i}$. Next, node $i$ transmits an acknowledgment $ACK_{ij}$ to node $j$ containing a signed confirmation of the received probe packet $PRB_{j}$. Node $i$ also listens for an acknowledgment $ACK_{ji}$ from node $j$. Node $i$ further removes from ${\cal N}_{i}$ any nodes that failed to return acknowledgements. Then node $i$ transmits to each node $j\in{\cal N}_{i}$ a pair of timing packets $TIM^{(1)}_{i,j}$ and $TIM^{(2)}_{j,i}$ that contain the send-times $s^{(1)}_{ij}$ and $s^{(2)}_{ij}$ respectively as recorded by its local clock $\tau^{j}(t)$. Node $i$ also receives a corresponding pair of timing packets $TIM^{(1)}_{i,j}$ and $TIM^{(2)}_{j,i}$ from node $j$, and records the corresponding receive-times $r^{(1)}_{ji}$ and $r^{(2)}_{ji}$ respectively, as measured by the local clock $\tau^{i}(t)$. Any node that fails to deliver timing packets to node $i$ is further removed from ${\cal N}_{i}$. The timing packets are used to estimate the relative skew $a_{ji}$ by $\hat{a}_{ji}:=\frac{r^{(2)}_{ji}-r^{(1)}_{ji}}{s^{(2)}_{ji}-s^{(1)}_{ji}}$. The relative skew is used at the end of the Network Discovery Phase, to estimate a reference clock with respect to the local continuous-time clock. In the last two steps, node $i$ creates a link certificate $LNK^{(1)}_{ij}$ containing the computed relative clock skew with respect to node $j$, and transmits this link to node $j$ using the OMC. Node $i$ also listens for a similar link certificate $LNK^{(2)}_{ji}$ from node $j$. Finally, node $i$ verifies and signs the received link certificate, and transmits the authenticated version $LNK^{(2)}_{ij}$ back to node $j$. Node $i$ listens for a similar authenticated link certificate $LNK^{(2)}_{ji}$ from $j$. Any nodes that fail to return link certificates are removed from the set ${\cal N}_{i}$. This set now represents the nodes in the neighborhood of node $i$ with whom node $i$ has established mutually authenticated link certificates. The Neighbor Discovery Phase’s pseudocode is shown in Algorithm 1. One problem is that the algorithm must be completed in a partially coordinated manner even though the nodes are asynchronous; the completion of any stage in the Exponential Information Gathering (EIG) algorithm (see below) depends on the successful completion of the previous stages by all other good nodes. Consequently, we assign increasingly larger intervals $S_{k}:=[t_{k},t_{k+1}),k=1,\ldots 6,$ to each successive protocol stage; see Section VI. Algorithm 1 The Neighbor Discovery Phase procedure NeighborDiscovery ${\cal N}_{i}:=\\{1,\ldots,n\\}\setminus i$ while $t\in S_{1}$ do TxRxMAC($PRB_{i\rightarrow{\cal N}_{i}}$,$PRB_{{\cal N}_{i}\rightarrow i}$) Update(${\cal N}_{i}$) end while while $t\in S_{2}$ do TxRxMAC($ACK_{i\rightarrow{\cal N}_{i}}$,$ACK_{{\cal N}_{i}\rightarrow i}$) end while while $t\in S_{3}$ do TxRxMAC($TIM^{(1)}_{i\rightarrow{\cal N}_{i}}$,$TIM^{(1)}_{{\cal N}_{i}\rightarrow i}$) Update(${\cal N}_{i}$) end while while $t\in S_{4}$ do TxRxMAC($TIM^{(2)}_{i\rightarrow{\cal N}_{i}}$,$TIM^{(2)}_{{\cal N}_{i}\rightarrow i}$) Update(${\cal N}_{i}$) end while while $t\in S_{5}$ do TxRxMAC($LNK^{(1)}_{i\rightarrow{\cal N}_{i}}$,$LNK^{(1)}_{{\cal N}_{i}\rightarrow i}$) Update(${\cal N}_{i}$) end while while $t\in S_{6}$ do TxRxMAC($LNK^{(2)}_{i\rightarrow{\cal N}_{i}}$,$LNK^{(2)}_{{\cal N}_{i}\rightarrow i}$) Update(${\cal N}_{i}$) end while end procedure ### V-B The Network Discovery Phase The purpose of this Phase is to allow the good nodes to obtain a _common_ view of the network topology and _consistent_ estimates of all clock parameters. To accomplish this, the good nodes must disseminate their lists of neighbors to all nodes, so that all can decide on the same topology view. However good nodes do not know a priori which nodes are bad, and so bad nodes can selectively drop lists or introduce false lists to prevent consensus. We resolve this by using a version of the Byzantine General’s algorithm of [1], requiring an EIG tree data structure. Let $T_{i}$ denote node $i$’s EIG tree, which by construction has depth $n$. The root of $T_{i}$ is labelled with node $i$’s neighborhood, i.e., the nodes in ${\cal N}_{i}$ and the corresponding collection of link certificates. First node $i$ transmits to every node $j\in{\cal N}_{j}$ in its neighborhood, the list of nodes in ${\cal N}_{i}$ and corresponding link certificates, while receiving similar lists from each node in ${\cal N}_{j}$. Node $i$ updates its EIG tree with the newly received lists from its neighbors, by assigning each received list to a unique child vertex of the root of $T_{i}$. Node $i$ then transmits the set of level 1 vertices of $T_{i}$ to every node in its neighborhood, receiving a set of level 1 vertcies from each neighbor in turn. The EIG tree $T_{i}$ is updated again. This process continues through all $n$ levels of the EIG tree. The notation $T^{(k)}_{i}$ in Algorithm 2 indicates the $k$-level vertices of the EIG tree $T_{i}$. The notation $\text{TxRxMAC}(T^{(k)}_{i\rightarrow{\cal N}_{i}},T^{(k)}_{{\cal N}_{i}\rightarrow i})$ indicates that, using the OMC, node $i$ transmits $T^{(k)}_{i}$ to each node $j\in{\cal N}_{i}$, and receives $T^{(k)}_{j}$ from each node $j\in{\cal N}_{i}$. We use $\text{UPDATE}(T_{i})$, to update the EIG tree $T_{i}$ after the arrival of new information, and the procedure $\text{DECIDE}(T_{i})$ to infer the network topology based on the EIG tree. The $n$-stage EIG algorithm guarantees that if the subgraph of good nodes is connected, then each good node will decide on the same topological view. Algorithm 2 The EIG Byzantine General’s Algorithm procedure EIGByzMAC(${\cal N}_{i}$) $T^{(0)}_{i}:={\cal N}_{i}$ for $k=1,\ldots n$ do while $t\in S_{6+k}$ do TxRxMAC($T^{(k)}_{i\rightarrow{\cal N}_{i}},T^{(k)}_{{\cal N}_{i}\rightarrow i}$) Update($T_{i}$) end while end for Decide($T_{i}$) end procedure ### V-C The Consistency Check Phase Unfortunately, a fundamental difficulty is that malicious nodes along a path $1,\ldots,n$ may have generated false time stamps in the Neighbor Discovery Phase, and thus corrupted the measured relative skews between adjacent nodes. There may be several connecting paths infiltrated by bad nodes that thereby generate different values for the relative skew. It is impossible to determine the correct path from the relative skews alone. Every pair of such inconsistent paths corresponds to an inconsistent cycle in which the skew product is not equal to one. We use an algorithm called Consistency Check to identify the path that generated the correct relative skew. Consistency Check works by circling a timing packet around every cycle in which the skew product differs from one by more than $\epsilon_{a}$, a desired maximum skew error. At the conclusion of the test, at least one link with a malicious endpoint will be removed from the cycle, eliminating a connecting path. During the test, each node in such a cycle is obliged to append a receive time-stamp and a send time-stamp generated by the local clock before forwarding the packet to the next node. These time-stamps must satisfy a delay bound condition; the send time and receive time cannot differ by more than 1 clock count. A node fails the consistency check otherwise, or if its time stamps do not agree with its declared relative skew. The key idea is that if the test starts after a sufficiently large amount of time has elapsed, the clock estimates based on faulty relative skews will have diverged so extensively from the actual clocks that at least one malicious node in the cycle will find it impossible to generate time-stamps that are consistent with its declared relative clock skew and satisfy the delay bound condition (all proofs are in Section VI):. ###### Theorem V.1 Let $T_{j}$ be the start-time of the Consistency Check for the $jth$ inconsistent cycle, consisting of nodes $i_{1},\ldots,i_{m}$. At least one malicious node in cycle $j$ will violate a consistency check condition, if $T_{j}>\frac{\hat{a}_{i_{m},i^{}}(m+1)K+\epsilon_{b}}{\epsilon_{a}}$ where $i^{}$ is the node with the smallest skew product $\hat{a}_{i^{},i_{1}}$. Algorithm 3 depicts Consistency Check. Given a cycle, $j$, $k$ and $m$ denote nodes that follow and precede node $i$ respectively in the cycle. If node $i$ is the leader of the cycle, i..e., the node with smallest ID, then node $i$ initiates the timing packet that traverses the cycle and transmits it to node $k$. Otherwise, node $i$ waits for the timing packet to arrive from node $m$ before forwarding it to node $k$. Algorithm 3 Consistency Check Algorithm at Node $i$ procedure ConsistencyCheck $START:=\frac{(n+1)(a_{max})^{n+1}+(n+1)(a_{max})^{n+1}U_{0}}{\epsilon_{a}}$ for each cycle $C_{j}$ do $k=$Next($C_{j}$) $m=$Prev($C_{j}$) if $i$=Leader($C_{j}$) and $t\geq START$ then Transmit($TIM_{i\rightarrow k}$) else if $i\in C_{j}$ then Receive($TIM_{m\rightarrow i}$) Transmit($TIM_{i\rightarrow k}$) end if end for end procedure After all inconsistent cycles have been tested, each node $i$ disseminates the set of all timing packets ${\cal T}_{i}$ it received to other nodes. The EIG algorithm is used to ensure a common view of the timing packets generated. Each node removes from the topology any link whose endpoints generate time- stamps inconsistent with its declared relative skew or violated the delay bound. The complete Phase is shown in Algorithm 4. Algorithm 4 The Network Discovery Phase at Node $i$ procedure NetworkDiscovery EIGByzMAC(${\cal N}_{i}$) ConsistencyCheck EIGByzMAC(${\cal T}_{i}$) end procedure At the conclusion of Network Discovery Phase node $i$ shares a common view of the network topology with all other good nodes. As a result, the network can designate the node with smallest ID as the _reference clock_. Furthermore, each node $i$ has an estimate of the reference clock $\tau^{r}_{i}(t)$ with respect to its local clock $t$ using the formula $\hat{\tau}^{r}_{i}(t):=\hat{a}_{ri}t$, where estimated $\hat{a}_{ri}$ and actual relative skews $a_{ri}$ differ by at most $\epsilon_{a}$. ### V-D The Scheduling Phase In the Scheduling Phase the good nodes in the network obtain a common schedule governing the transmission and reception of data packets. A “schedule” is simply a sequence of CTVs, each with specified start and end times. Each node $i$ divides the Data Transfer Phase into time-slots, and assigns a CTV to each time-slot so that the resulting throughput vector is utility optimal. All the good nodes independently arrive at the same schedule since they independently optimize the same utility function over the same ${\cal C}$ (ties broken lexicographically). Since the good nodes must conform to a common schedule, each node $i$ generates a local estimate of the reference clock $\hat{\tau}^{r}_{i}(t)$ with respect to its local clock $t$, as described in the Network Discovery Phase. However, this estimate may not be perfectly accurate; some of the nodes on a path along which relative skew is estimated may be malicious and can introduce an error of at most $\epsilon_{a}$ into the computed relative skew. To address this, the time-slots are separated by a dead-time of size $D$, where given any pair of nodes $(i,j)$, $D$ is chosen to satisfy $\|\hat{\tau}^{r}_{i}(t)-\hat{\tau}^{r}_{j}(\tau^{j}_{i}(t))\|\leq D$. Finally, $n^{2}(n-1)$ time-slots are enough to guarantee that every pair of nodes can communicate once in either direction, via multihop routing, during Data Transfer Phase. The algorithm $\text{UtilityMaximization}({\cal C})$ for the Scheduling Phase is depicted in Algorithm 5. At the end of Scheduling Phase, node $i$ shares a common utility maximizing schedule with other good nodes. Algorithm 5 The Scheduling Phase at Node $i$ procedure Scheduling UtilityMaximization(${\cal C}$) end procedure ### V-E The Data Transfer Phase In this Phase the nodes exchange data packets using the generated schedule. It is divided into time-slots, with each assigned a CTV, a rate vector, and set of packets for each transmitter in the set. To prevent collisions resulting from two nodes assigning themselves to different time slots due to timing error, node $i$ begins transmission $D$ time-units after the start of the time-slot. The transmitted packet is then guaranteed to arrive at the receiver in the same time slot, for appropriate choice of $D$ and time-slot size $B_{slot}$. Algorithm 7 defines this phase, with $m_{k}$ denoting a message to be transmitted or received by node $i$ in the $k_{th}$ slot, $T_{start}$ the start time of the phase measured by the local estimate of the reference clock $\hat{\tau}^{r}_{i}(t))$, $S_{k}=[t_{k},t_{k+1}),k=1,\ldots,N$ the time-slots of the phase with $N=n^{2}(n-1)$, $t_{1}=T_{start}$, and $t_{k+1}:=t_{k}+B_{slot}+2D$, and $TX(k)$ and $RX(k)$ the CTV, and receiving nodes during slot $k$. Algorithm 6 The Data Transfer Phase at Node $i$ procedure DataTransfer($T_{start}$) for k=1,…,N do if $t\in S_{k}$ and $t\geq t_{k}+D$ and $i\in TX(k)$ then Transmit($m_{k}$) else if $t\in S_{k}$ and $i\in RX(k)$ then Receive($m_{k}$) end if end for end procedure ### V-F The Verification Phase However, malicious nodes may not cooperate in the Data Transfer Phase. So whenever a scheduled packet fails to arrive at node $j$, it adds the offending CTV and associated packet number to a list, and disseminates the list in the Verification Phase using the EIG Byzantine General’s algorithm. These CTVs are then permanently further pruned from the collection of feasible CTVs. With $L_{k}$ denoting the list that failed during the $k$th iteration of the Data Transfer Phase, the set ${\cal C}_{k}$ of feasible CTVs during the $k$th iteration of the Scheduling Phase is updated to ${\cal C}_{k+1}={\cal C}_{k}\setminus L_{k}$ in Algorithm 7. All communication can be scheduled into slots separated by a dead-time of $2D$. Within each of the $n$ stages of the EIG Byzantine General’s algorithm, there are $n(n-1)$ pairs of nodes that may communicate, and at most $n$ nodes on the connecting path. Therefore, the total number of time slots required is $n^{3}(n-1)$. At the conclusion of the phase, the good nodes again share a common view of the set of feasible CTVs for the next iteration of the Scheduling Phase. Algorithm 7 The Verification Phase at Node $i$ procedure Verification EIGByz($L_{k}$) Update(${\cal C}_{k+1}$) end procedure ### V-G The Steady State The network cycles through Scheduling, Data Transfer, and Verification Phases for $n_{iter}$ iterations. Eventually, by finiteness, it converges to a set of CTVs, and a utility-maximizing schedule over it. The overall protocol is in Algorithm 8. Algorithm 8 The Complete Protocol NeighborDiscovery NetworkDiscovery for $k=1,\ldots,n_{iter}$ do Scheduling(${\cal C}_{k}$) DataTransfer(t) Verification end for ## VI Feasibility of Protocol and Optimality Proof For the distributed wireless nodes to exchange data over the network, they must not only have the same topological view, in order to independently arrive at a common schedule, but they must also have a consistent view of a reference clock so that any activity will conform to this common schedule. For this, we consider the consistency check algorithm of Section V-C. Consider a chain network $1,\ldots,n$, where the endpoints, nodes $1$ and $n$ are good, and the intermediate nodes $2,\ldots,n-1$ are bad. Note that this network can also be reduced to a cycle of size $n-1$ by making both endpoints the same node. We assume that the two good endpoints do not know if any of the intermediate nodes are bad. Now suppose that each pair of adjacent nodes $(i,i-1)$ for $i=2,\ldots,n$ has declared a set of relative skews and offsets $\\{\hat{a}_{i,i-1},\hat{b}_{i,i-1}\\}$, and that each node in the chain knows this set. The two good nodes wish to determine whether the declared skews are accurate, i.e., whether $a_{n,1}=\prod^{n}_{i=2}\hat{a}_{i,i-1}$. Unfortunately, the good nodes have no way of directly measuring $a_{n,1}$. The estimate of $a_{n,1}$ is obtained from the skew product itself, which is the very quantity that needs to be verified. So, instead, the good nodes carry out the consistency check described earlier. After waiting a sufficiently long time, node $1$ initiates a timing packet that traverses the chain from left to right. Each node in the chain is obligated to forward the packet after appending receive and time-stamps that satisfy the skew consistency and delay bound conditions. In order to defeat this test, the bad nodes, having collectively declared a false set of relative skews and offsets, must support two sets of clocks for each node $i\in\\{2,\ldots,n\\}$: a “left” clock $\tau^{i,l}(t)$ to generate receive time-stamps, and a “right” clock $\tau^{i,r}(t)$ to generate send-time stamps. Unlike the clocks of the good nodes, the left and right clocks of the bad nodes need not be affine with respect to the global reference clock. In fact, the bad nodes are free to jointly select any set of clocks $\\{\tau^{i,l}(t),\tau^{i,r}(t),\forall i=2,\ldots,n-1\\}$ that are arbitrary functions of $t$, a much larger set than the affine clocks being emulated. However, we will show that if node $1$ waits sufficiently long enough, there is no set of clocks $\\{\tau^{i,l}(t),\tau^{i,r}(t),i=2,\ldots,n-1\\}$ that can generate time-stamps which satisfy both conditions of the consistency check. Let $r_{i,i-1}$ and $s_{i,i+1}$ denote the receive and send time-stamps generated by a bad node $i$ with respect to the left and right clocks $\tau^{i,l}(t)$ and $\tau^{i,r}(t)$ respectively. Let $t_{i,l}$ and $t_{i,r}$ denote the time with respect to the global reference clock at which the receive and send time-stamps are generated at node $i$. We have $r_{i-1,i}:=\tau^{i,l}(t_{i,l})$ and $s_{i,i+1}:=\tau^{i,r}(t_{i,r})$. Let $t_{1}$ and $t_{n}$ denote the time with respect to the global reference clock at which the timing packet was transmitted by node $1$ and received by node $n$ respectively. We have $s_{1,2}:=\tau^{1}(t_{1}),_{n-1,n}:=\tau^{n}(t_{n})$. To simplify notation we will define left and right clocks at the endpoints so that $t_{1,r}:=t_{1},t_{n,l}:=t_{n}$ and $\tau^{1,r}(t_{1,r}):=\tau^{1}(t_{1}),\tau^{n,l}(t_{n,l}):=\tau^{n}(t_{n})$. In order to prove that both conditions of the consistency check cannot be satisifed by any set of clocks $\\{\tau^{i,l}(t),\tau^{i,r}(t),i=2,\ldots,n-1\\}$, we will assume that the first condition is satisfied, and show that second must fail. Therefore, the clocks must satisfy: $\displaystyle\tau^{i,l}(t_{i,l})$ $\displaystyle=a_{i,i-1}\tau^{i-1,r}(t_{i-1,r})+b_{i,i-1}\mbox{ for }i\leq 2\leq n.$ (2) In addition, by virtue of causality, we also have: $\displaystyle\tau^{i,l}(t_{i,l})$ $\displaystyle\leq\tau^{i,r}(t_{i,r}).$ (3) We prove that delay bound condition must be violated if node $1$ waits for a sufficiently large period of time before before initiating the timing packet, i.e., if $\tau^{1}(t_{1})$ is sufficiently large, then for some $i$, we have $\tau^{i,r}(t_{i,r})-\tau^{i,l}(t_{i,l})>K$. More precisely, we show $\sum^{n-1}_{i=2}\left(\tau^{i,r}(t_{i,r})-\tau^{i,l}(t_{i,l})\right)>nK$, which implies that some node has violated delay bound condition. The sum $\sum^{n-1}_{i=2}\left(\tau^{i,r}(t_{i,r})-\tau^{i,l}(t_{i,l})\right)$ cannot be directly evaluated because the left and right clocks $\\{\tau^{i,l}(t),\tau^{i,r}(t)\\}$ are arbitrary functions of $t$. However, we have the following equality by repeated addition and subtraction $\tau^{n,l}(t_{n,l})=\tau^{1,r}(t_{1,r})+\sum^{n}_{i=2}\left(\tau^{i,l}(t_{i,l})-\tau^{i-1,r}(t_{i-1,r})\right)=\sum^{n-1}_{i=2}\left(\tau^{i,l}(t_{i,l})-\tau^{i-1,r}(t_{i-1,r})\right)=\tau^{1,r}(t_{1,r})+S_{1}+S_{2}$, where $S_{1}:=\sum^{n}_{i=2}\left(\tau^{i,l}(t_{i,l})-\tau^{i-1,r}(t_{i-1,r})\right),S_{2}:=\sum^{n-1}_{i=2}\left(\tau^{i,l}(t_{i,l})-\tau^{i-1,r}(t_{i-1,r})\right)$. The value $S_{2}$ is the sum of the forwarding delays. We will use (2) and (3) to obtain an upper bound on $S_{1}$. Inserting this upper bound and using the fact that $\tau^{n,l}(t)$ and $\tau^{1,r}(t)$ are both affine functions of $t$, will allow us to obtain a lower bound on $S_{2}$. The proof will then follow easily. We now obtain an upper bound on $S_{1}$ when the forward skew product $\prod^{j}_{i=2}\hat{a}_{i,i-1}\geq 1$ for all $j\geq 2$. ###### Lemma VI.1 Suppose $\prod^{j}_{i=2}a_{i,i-1}\geq 1$ for $2\leq i\leq n$. Then $\sum^{n}_{i=2}(\tau^{i,l}(t_{i,l})-\tau^{i-1,r}(t_{i-1,r}))\leq\left(\frac{\hat{a}_{n,1}-1}{\hat{a}_{n,1}}\right)\tau^{n,l}(t_{n,l})\sum^{n}_{i=2}\frac{\hat{b}_{i,i-1}}{\hat{a}_{i,1}}$. ###### Proof: We have by definition $\tau^{n+1,l}(t_{n,l}):=\hat{a}_{n+1,n}\tau^{n,r}(t_{n,r})+\hat{b}_{n+1,n}$. For $n=2$, we have $\tau^{2,l}(t_{2,l})-\tau^{1,r}(t_{1,r})=\left(\frac{a_{2,1}-1}{a_{2,1}}\right)\tau^{2,l}(t_{2,l})+\frac{b_{2,1}}{a_{2,1}}$. Now assume the lemma is true for $n$. We will show that it also holds for $n+1$: $\sum^{n+1}_{i=2}(\tau^{i,l}(t_{i,l})-\tau^{i-1,r}(t_{i-1,r}))=\sum^{n}_{i=2}(\tau^{i,l}(t_{i,l})-\tau^{i-1,r}(t_{i-1,r}))+\tau^{n+1,l}(t_{n+1,l})-\tau^{n,r}(t_{n,r})\leq\left(\frac{\hat{a}_{n,1}-1}{\hat{a}_{n,1}}\right)\tau^{n,l}(t_{n,l})+\sum^{n}_{i=2}\frac{\hat{b}_{i,i-1}}{\hat{a}_{i,1}}+\tau^{n+1,l}(t_{n+1,l})-\tau^{n,r}(t_{n,r})\leq\left(\frac{\hat{a}_{n,1}-1}{\hat{a}_{n,1}}\right)\tau^{n,r}(t_{n,r})+\sum^{n}_{i=2}\frac{\hat{b}_{i,i-1}}{\hat{a}_{i,1}}+\tau^{n+1,l}(t_{n+1,l})-\tau^{n,r}(t_{n,r})=\left(\frac{\hat{a}_{n,1}-1}{\hat{a}_{n,1}}\right)\left(\frac{\tau^{n+1,l}(t_{n+1,l})-\hat{b}_{n+1,n}}{\hat{a}_{n+1,n}}\right)+\sum^{n}_{i=2}\frac{\hat{b}_{i,i-1}}{\hat{a}_{i,1}}+\left(\frac{\hat{a}_{n+1,n}-1}{\hat{a}_{n+1,n}}\right)\hat{\tau}^{n+1,l}(t_{n+1,l})+\frac{\hat{b}_{n+1,n}}{\hat{a}_{n+1,n}}=\left(\frac{\hat{a}_{n+1,1}-1}{\hat{a}_{n+1,1}}\right)\tau^{n+1,l}(t_{n+1,l})+\sum^{n+1}_{i=2}\frac{\hat{b}_{i,i-1}}{\hat{a}_{i,1}}$, which follow from the induction hypothesis above in the Lemma statement, and the fact that $\tau^{n,r}(t_{n,r})\geq\tau^{n,l}(t_{n,l})$ and $a_{i,1}\geq 1$ for all $2\leq i\leq n+1$ (that is, the coefficient $\hat{a}_{i,1}-1$ is negative). ∎ We next bound $S_{1}$ in the special case when the reverse skew product $\prod^{j}_{i=1}\hat{a}_{n-(i-1),n-i}\leq 1$ for all $j\geq 1$. ###### Lemma VI.2 Suppose $\prod^{j}_{i=1}a_{n-(i-1),n-i}\leq 1$ for $2\leq j\leq n-1$. Then $\sum^{j}_{i=1}(\tau^{n-(i-1),l}(t_{n-(i-1),l})-\tau^{n-i,r}(t_{n-i,r}))$ $\leq\left(\hat{a}_{n,n-j}-1\right)\tau^{n-j,r}(t_{n-j,r})+\hat{b}_{n,n-1}+\sum^{n-1}_{i=n-j+1}\hat{a}_{n,i}\hat{b}_{i,i-1}$. ###### Proof: We have by definition $\tau^{n-(k-1),l}(t_{n-(k-1),l}):=\hat{a}_{n-(k-1),n-k}\tau^{n-k,r}(t_{n-k,r})+\hat{b}_{n-(k-1),n-k}$. For $j=1$, $\tau^{n,l}(t_{n,l})-\tau^{n-1,r}(t_{n-1,r})=(a_{n,n-1}-1)\tau^{n-1,r}(t_{n-1,r})$. Now assume the Lemma holds for $j$. We will show that it must hold for $j+1$: $\sum^{j+1}_{k=1}(\tau^{n-(k-1),l}(t_{n-(k-1),l})-\tau^{n-k,r}(t_{n-k,r}))=\sum^{j}_{k=1}(\tau^{n-(k-1),l}(t_{n-(k-1),l})-\tau^{n-k,r}(t_{n-k,r}))+\tau^{n-j,l}(t_{n-j,l})-\tau^{n-(j+1),r}(t_{n-(j+1),r})\leq(\hat{a}_{n,n-j}-1)\tau^{n-j,r}(t_{n-j,r})+\hat{b}_{n,n-1}+\sum^{n-1}_{k=n-j+1}\hat{a}_{n,k}\hat{b}_{k,k-1}+\tau^{n-j,l}(t_{n-j,l})-\tau^{n-(j+1),r}(t_{n-(j+1),r})\leq(\hat{a}_{n,n-j}-1)\tau^{n-j,l}(t_{n-j,l})+\hat{b}_{n,n-1}+\sum^{n-1}_{k=n-j+1}\hat{a}_{n,k}\hat{b}_{k,k-1}+\tau^{n-j,l}(t_{n-j,l})-\tau^{n-(j+1),r}(t_{n-(j+1),r})\leq(\hat{a}_{n,n-(j+1)}-1)\tau^{n-(j+1),r}(t_{n-(j+1),r})+\hat{b}_{n,n-1}+\sum^{n-1}_{k=n-j}\hat{a}_{n,k}\hat{b}_{k,k-1}$. The above follow from induction hypothesis in Lemma VI.2, since $\tau^{i,l}(t_{i,l})\leq\tau^{i,r}(t_{i,r})$ and $\hat{a}_{n,n-j}\leq 1$ for $1\leq j\leq n-1$ (that is, the coefficient $\hat{a}_{n,n-j}-1$ is negative), and from substitution into $\tau^{n-j,l}(t_{n-j,l})$ and simplification. ∎ We will combine both special cases in Lemma VI.1 and Lemma VI.2 to obtain an upper bound on $S_{1}$. First we define $i^{}$ as the node with the smallest skew product $\hat{a}_{i^{},1}$ in the chain network, that is less than one. That is, $\hat{a}_{i^{},1}=\displaystyle\min_{k}\hat{a}_{k,1}$ and $\hat{a}_{i^{},1}\leq 1$. If no such node exists, set $i^{}=1$. Now we consider an arbitrary set of skews $\\{\hat{a}_{i,i-1},i=2,\ldots,n\\}$. Next we show that if $i^{}\geq 2$ then the forward skew product starting from $i^{}$ is greater than 1, and the reverse skew product starting from $i^{}-1$ is always less than one. ###### Lemma VI.3 If $i^{}\geq 2$ then $\hat{a}_{j,i^{}}\geq 1$ for $i^{}+1\leq j\leq n$ and $\hat{a}_{i^{},i^{}-k+1}\leq 1$ for $1\leq k\leq i^{}$. Otherwise, $\hat{a}_{j,1}\geq 1$ for $2\leq j\leq n$. ###### Proof: Consider $i^{}\geq 2$, and suppose the first part of the assertion is false. I.e., for some $j^{\prime}$, $\hat{a}_{j^{\prime}i^{}}<1$. It follows that $\hat{a}_{j^{\prime}1}=\hat{a}_{j^{\prime}i^{}}\hat{a}_{i^{}1}\leq\hat{a}_{i^{}1}$. But then $j^{\prime}$ is a node with a smaller skew product $\hat{a}_{j1}$ than node $i^{}$, which contradicts the definition of $i^{}$. Now suppose that the second part of the assertion is false. I.e., for some $j^{\prime}$ we have $\hat{a}_{i^{}j^{\prime}}>1$. It follows that $\hat{a}_{i^{}1}=\hat{a}_{i^{}j^{\prime}}\hat{a}_{j^{\prime}1}\geq\hat{a}_{j^{\prime}1}$. But then $j^{\prime}$ is a node with a smaller skew product than node $i^{}$, which again contradicts the definition of $i^{}$. Now consider the case when $i^{}=1$. Then by definition of $i^{}$ it follows that $\hat{a}_{j1}\geq 1$ for all $2\leq j\leq n$. ∎ We now obtain an upper bound on $S_{1}$ for arbitrary skews. ###### Lemma VI.4 Suppose $i^{}\geq 2$. We have the following inequality: $\sum^{n}_{j=2}\tau^{j,l}(t_{j,l})-\tau^{j-1,r}(t_{j-1,r})\leq(\hat{a}_{i^{},1}-1)\tau^{1,r}(t_{1,r})$ $+\left(\frac{\hat{a}_{n,i^{}}-1}{\hat{a}_{n,i^{}}}\right)\tau^{n,l}(t_{n,l})+\frac{\hat{b}_{n,1}}{\hat{a}_{n,i^{}}}$. ###### Proof: $\sum^{n}_{j=2}\tau^{j,l}(t_{j,l})-\tau^{j-1,r}(t_{j-1,r})$ $\sum^{i^{}}_{j=2}\tau^{j,l}(t_{j,l})-\tau^{j-1,r}(t_{j-1,r})+\sum^{n}_{j=i^{}+1}\tau^{j,l}(t_{j,l})-\tau^{j-1,r}(t_{j-1,r})$ $=(\hat{a}_{i^{},1}-1)\tau^{1,r}(t_{1,r})+\hat{b}_{i^{},i^{}-1}$ $+\sum^{i^{}}_{j=2}\hat{a}_{i^{},j}\hat{b}_{j,j-1}+\left(\frac{\hat{a}_{n,i^{}}-1}{\hat{a}_{n,i^{}}}\right)\tau^{n,l}(t_{n,l})+\sum^{n}_{i=i^{}+1}\frac{\hat{b}_{i,i-1}}{\hat{a}_{i,i^{}}}$ $=(\hat{a}_{i^{},1}-1)\tau^{1,r}(t_{1,r})+\left(\frac{\hat{a}_{n,i^{}}-1}{\hat{a}_{n,i^{}}}\right)\tau^{n,l}(t_{n,l})$ $+\sum^{n}_{j=2}\frac{\hat{a}_{n,j}\hat{b}_{j,j-1}}{\hat{a}_{n,i^{}}}$ $=(\hat{a}_{i^{},1}-1)\tau^{1,r}(t_{1,r})+\left(\frac{\hat{a}_{n,i^{}}-1}{\hat{a}_{n,i^{}}}\right)\tau^{n,l}(t_{n,l})+\frac{\hat{b}_{n,1}}{\hat{a}_{n,i^{}}}.$which follow by applying Lemma VI.2 and Lemma VI.1, by multiplying the terms in each summation by $\frac{\hat{a}_{n,i^{}}}{\hat{a}_{n,i^{}}}$ and simplifying, and from the definitions of $\hat{b}_{ij}$ and $\hat{d}^{(i)}_{ji}$. ∎ Now that we have an upper bound on $S_{1}$, we can obtain a lower bound on $S_{2}$, the sum of the forwarding delays. ###### Lemma VI.5 The sum of forwarding delays in the chain network satisfies: $\sum^{n-1}_{j=2}\left(\tau^{j,l}(t_{j,l})-\tau^{j,r}(t_{j,r})\right)\geq\frac{(a_{n,1}-\hat{a}_{n,1})}{\hat{a}_{n,i^{}}}\tau^{1,r}(t_{1,r})+\frac{(b_{n,1}-\hat{b}_{n,1})}{\hat{a}_{n,i^{}}}$. ###### Proof: $\sum^{n-1}_{j=2}\left(\tau^{j,l}(t_{j,l})-\tau^{j,r}(t_{j,r})\right)$ $=\tau^{n,l}(t_{n,l})-\tau^{n,r}(t_{n,r})-\sum^{n}_{j=2}\tau^{j,l}(t_{j,l})-\tau^{j-1,r}(t_{j-1,r})$ $\geq\tau^{n,l}(t_{n,l})-\tau^{n,r}(t_{n,r})-(\hat{a}_{i^{},1}-1)\tau^{1,r}(t_{1,r})$ $-\left(\frac{\hat{a}_{n,i^{}}-1}{\hat{a}_{n,i^{}}}\right)\tau^{n,l}(t_{n,l})-\frac{\hat{b}_{n,1}}{\hat{a}_{n,i^{}}}$ $=\frac{\tau^{n,l}(t_{n,l})}{\hat{a}_{n,i^{}}}-\hat{a}_{i^{},1}\tau^{1,r}(t_{1,r})-\frac{\hat{b}_{n,1}}{\hat{a}_{n,i^{}}}$ $\geq\frac{\tau^{n,l}(t_{1,r})}{\hat{a}_{n,i^{}}}-\hat{a}_{i^{},1}\tau^{1,r}(t_{1,r})-\frac{\hat{b}_{n,1}}{\hat{a}_{n,i^{}}}$ $=\frac{a_{n,1}\tau^{1,r}(t_{1,r})+b_{n,1}}{\hat{a}_{n,i^{}}}-\hat{a}_{i^{},1}\tau^{1,r}(t_{1,r})-\frac{\hat{b}_{n,1}}{\hat{a}_{n,i^{}}}$ $=\frac{(a_{n,1}-\hat{a}_{n,1})}{\hat{a}_{n,i^{}}}\tau^{1,r}(t_{1,r})+\frac{(b_{n,1}-\hat{b}_{n,1})}{\hat{a}_{n,i^{}}}$, which follow by noting from repeated addition and subtraction that $\tau^{n,l}(t_{n,l})=\tau^{1,r}(t_{1,r})+\sum^{n}_{j=2}\left(\tau^{j,l}(t_{j,l})-\tau^{j-1,r}(t_{j-1,r})\right)+\sum^{n-1}_{j=2}\left(\tau^{j,l}(t_{j,l})-\tau^{j,r}(t_{j,r})\right)$, by applying Lemma VI.4, because $t_{n,l}\geq t_{1,r}$ since node $n$ could not have received the timing packet before node $1$ transmitted it, and since node $n$’s clock is relatively affine with respect to node $1$’s clock. ∎ We now complete the proof of consistency check for a chain network. We show that if the start time of the consistency check is sufficiently large, and the left and right clocks $\\{\tau^{i,l}(t_{i,l}),\tau^{i,r}(t_{i,r})\\}$ satisfy the parameter consistency condition, then at least one node will violate delay bound condition. Hence there are no left and right clocks that can pass both conditions of consistency check if start time is large. ###### Proof: We assume node $1$ is a good node. Now $\frac{(a_{n,1}-\hat{a}_{n,1})}{\hat{a}_{n,i^{}}}\tau^{1,r}(t_{1,r})+\frac{(b_{n,1}-\hat{b}_{n,1})}{\hat{a}_{n,i^{}}}>nK$. But by Lemma VI.5 the LHS of this inequality is the lower bound of the sum of the delays in the chain $\sum^{n}_{j=2}\left(\tau^{j,l}(t_{j,l})-\tau^{j,r}(t_{j,r})\right)$. By substitution, $\sum^{n}_{j=2}\left(\tau^{j,l}(t_{j,l})-\tau^{j,r}(t_{j,r})\right)>nK$. It follows that for some malicious node $j\in\\{2,\ldots,n\\}$, $\tau^{j,l}(t_{j,l})-\tau^{j,r}(t_{j,r})>K$ which violates the delay bound condition. ∎ Now we can show that neighbor and network discovery phases together allow the good nodes to form a rudimentary network, where the good nodes have the same topological view and consistent estimates of a reference clock. The first obstacle is that the protocol is composed of stages that must be completed sequentially by all the nodes in the network, even prior to clock synchronization. Suppose that $[t_{k},t_{k+1})$ is the interval allocated to the $kth$ stage. Any messages transmitted between adjacent good nodes must arrive in the same interval they were transmitted. Since send-times are measured with respect to the source clock, and receive-times with respect to the destination clock, the intervals must be chosen large enough to compensate for the maximum clock divergence caused by skew $a_{ij}\leq a_{max}$ and offset $b_{ij}\leq a_{max}U_{0}$. ###### Lemma VI.6 There exists a sequence of adjacent time-intervals $[t_{k},t_{k+1})$ and corresponding schedule that guarantees any message of size $W$ transmitted (via OMC) by node $i$ in the interval $[t_{k},t_{k+1})$ (as measured by $i$’s clock) will be received by node $j$ in the same interval as measured by node $j$’s clock. ###### Proof: Set $t_{k+1}:=(a_{max})^{2}t_{k}+2(a_{max})^{3}U_{0}\\\ +(a_{max})^{3}T_{MAC}(W)$. Suppose a message from node $i$ to node $j$ during $[t_{k},t_{k+1})$ is transmitted (via the OMC) at $t_{s}:=a_{max}t_{k}+(a_{max})^{2}U_{0}$ with respect to node $i$’s clock. By substitution and simplification it follows that $\tau^{j}_{i}(t_{s})\geq t_{k}$ and $\tau^{j}_{i}(t_{s}+T_{MAC}(W))<t_{k+1}$. Hence $\tau^{j}_{i}([t_{s},t_{s}+T_{MAC}(W)))\subset[t_{k},t_{k+1})$, and so $j$ receives this message during the same interval with respect to $j$’s clock. ∎ ###### Theorem VI.1 After Network Discovery, the good nodes have a common view of the topology and consistent estimates (to within $\epsilon_{a}$) of the skew of the reference clock. ###### Proof: From Lemma VI.6 all good nodes will proceed through each stage of Neighbor and Network Discovery Phases together, and therefore establish link certificates with their good neighbors. Since they form a connected component, the good nodes obtain a common view of their link certificates using the EIGByzMAC algorithm and the schedule in Lemma VI.6. The good nodes can therefore infer the network topology and the relative skews of all adjacent nodes based upon the collection of link certificates. Using Consistency Check, the good nodes can eliminate paths along which bad nodes have provided false skew data. The good nodes can disseminate this information to each other using the EIGByzMAC algorithm and Lemma VI.6 and thus obtain consistent estimates of the reference clock to within $\epsilon_{a}$. ∎ ###### Lemma VI.7 The sequence of adjacent intervals $[t_{j},t_{j+1})$, $j=0,\ldots,k$ is contained in $[t_{0},c_{1}t_{0}+c_{2}W)$ where constants $c_{1}$ and $c_{2}$ depend on $a_{max}$, $k$, $U_{0}$, and $n$. ###### Proof: For the OMC $T_{MAC}(W)\leq cW$, where $c$ depends on $a_{max}$, and $n$. The result for $k=1$ follows from definition of $t_{k}$, and substitution of $cW$ into $T_{MAC}(W)$, and for general $k$ by induction and definition of $t_{k}$. ∎ ###### Lemma VI.8 The time to complete Neighbor and Network Discovery Phases $T_{nei}+T_{net}$ is less than $c_{1}\log T_{life}+\frac{c_{2}}{\epsilon_{a}}$ where $c_{1},c_{2}$ depend only on $n,a_{max},U_{0}$. ###### Proof: From Algorithms 1, 2, 3 and 4 there are at most $6+n+n\|C\|+n$ protocol stages in the Neighbor and Network Discovery Phases. Hence the time required is at most $c_{1}t_{0}+c_{2}W$, where $W$ is the size of a message to be transmitted, and $c_{1},c_{2}$ are constants depending on the number of protocol stages $a_{max},U_{0},n$. The maximum size of a message is proportional to the timing packet size $\log T_{life}$. To account for the effect of the minimum start-time $T_{s}$ for the consistency check, we can assume the worst case that the $T_{s}$ comes into effect during the first protocol stage (instead of later in the Network Discovery Phase). From Theorem V.1 the consistency check start-time is at most $\frac{c}{\epsilon_{a}}$, where $c$ depends on $U_{0},a_{max},n$. Substitution into $t_{0}$ proves the lemma. ∎ ###### Lemma VI.9 The time required for the Data Transfer Phase is at most $c_{3}B+c_{4}D$ where $B$ is the time spent transmitting data packets, $D$ is the size of the dead- time separating time slots, and $c_{3},c_{4}$ depend on $n$ alone. ###### Proof: The total number of time-slots for data transfer between all source- destination pairs is $n^{2}(n-1)$, each supporting data transfer of size $B_{s}$ and a dead-time $D$. ∎ ###### Lemma VI.10 The time required for the Verification Phase is at most $c_{5}D$ where $c_{5}$ depends on $n$ alone. ###### Proof: In each stage of the EIG Byzantine General’s algorithm, there are at most $n!$ vertex values that must be transmitted with each node in the neighborhood. The value of a vertex is a list of CTVs. There are at most $2^{n}$ CTVs and at most $n$ nodes in a CTV. Therefore the size of any message to be transmitted by a node during EIG algorithm is at most $cD$, where $c$ is a constant dependent on $n$. Since there are $n(n-1)$ possible source-destination pairs, there are at most $n(n-1)$ time slots in each stage, separated at the beginning and end by a dead-time $D$. Therefore the duration of each stage is at most $cD+n(n-1)2D$. There are at most $n$ stages. ∎ We can now prove the main theorem of this paper. ###### Theorem VI.2 The protocol ensures that the network proceeds from startup to a functioning network carrying data. There exists a selection of parameters $n_{iter}$, $D$, $B$, $\epsilon_{a}$ and $T_{life}$ that achieves min-max utility over the enabled set, to within a factor $\epsilon$, where the min is over all policies of the bad nodes that can only adopt two actions in each CTV: conform to the protocol and/or jam. The achieved utility is $\epsilon$-optimal. ###### Proof: We begin by choosing parameters so that the protocol overhead, which includes Neighbor Discovery, Network Discovery, Verification, all dead-times, and iterations converging to the final rate vector, is an arbitrarily small fraction of the total operating lifetime. With $\hat{\tau}^{r}_{i}(t):=\hat{a}_{ri}t$ the estimate of reference clock $r$ with respect to the local clock at node $i$, the maximum difference in nodal estimates is bounded as $\|\hat{\tau}^{r}_{i}(\tau^{i}(t))-\hat{\tau}^{r}_{k}(\tau^{k}_{i}(\tau^{i}(t)))\|\leq 2(a_{max})^{2}\epsilon_{a}T_{life}+(a_{max})^{2}U_{0}$. With $k_{r}$ be the number of rate vectors in the rate region, we can choose $n_{iter}$, $D$, $B$, $\epsilon_{a}$ and $T_{life}$ to satisfy: $\frac{n_{iter}}{n_{iter}+2^{n}k_{r}}\geq 1-\epsilon_{l}$, $\frac{B}{c_{1}\log T_{life}+\frac{c_{2}}{\epsilon_{a}}+B+c_{3}D+c_{4}D}\geq 1-\epsilon_{d}$, $n_{iter}((c_{1}\log T_{life}+\frac{c_{2}}{\epsilon_{a}}+B+c_{3}D+c_{4}D)\leq T_{life}$, $2(a_{max})^{2}\epsilon_{a}T_{life}+(a_{max})^{2}U_{0}\leq D$. These ensure that the rate loss due to failed CTVs is arbitrarily small, the time spent transmitting data is an arbitrarily large fraction of the duration of that iteration, the operating lifetime is large enough to support $n_{iter}$ protocol iterations, and the dead-time $D$ is large enough to tolerate the maximum divergence in clock estimates caused by skew error $\epsilon_{a}$. Let $\\{{\cal{D}}(t)\\}$ be the decreasing sequence of sets of disabled CTVs, with limit $\bar{\cal{D}}$ attained at some finite time $T$. Suppose $x$ achieves the maximum utility for $\bar{\cal{D}}$ over the nodes in the same component as the good nodes. No protocol can do better when $\bar{\cal{D}}$ is disabled. The proposed protocol attains $x(1-\epsilon_{d})(1-\epsilon_{l})$. ∎ ## VII Concluding Remarks We have presented a complete suite of protocols that enables a collection of good nodes interspersed with bad nodes to form a functioning network from start-up, operating at a utility-optimal rate vector, regardless of what the bad nodes conspire to do, under a certain system model. Further, the attackers cannot decrease the utility any more than they could by just conforming to the protocol or jamming on each CTV. This paper is only an initial attempt to obtain a theoretical foundation for a much needed holistic all-layer approach to secure wireless networking, and there are several open issues. An important potential generalization is to allow probabilistic communication. 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Wang and Y. Zhang. Improving TCP performance over mobile ad-hoc networks with out-of-order detection and response. In Proceedings of the third ACM international symposium on Mobile ad hoc networking and computing, pages 217–225. ACM Press, 2002.	arxiv-papers	2013-09-11T17:47:57	2024-09-04T02:49:50.745450	{ "license": "Public Domain", "authors": "Jonathan Ponniah, Yih-Chun Hu and P. R. Kumar", "submitter": "P. R. Kumar", "url": "https://arxiv.org/abs/1309.2904" }
1309.2963	∎ 11institutetext: P. Shakarian 22institutetext: Network Science Center and Dept. Electrical Engineering and Computer Science U.S. Military Academy West Point, NY 10996 Tel.: 845-938-5576 22email: [email protected] 33institutetext: S. Eyre 44institutetext: Network Science Center and Dept. Electrical Engineering and Computer Science U.S. Military Academy West Point, NY 10996 Tel.: 845-938-5576 44email: [email protected] 55institutetext: Damon Paulo Network Science Center and Dept. Electrical Engineering and Computer Science U.S. Military Academy West Point, NY 10996 Tel.: 845-938-5576 55email: [email protected] # A Scalable Heuristic for Viral Marketing Under the Tipping Model Paulo Shakarian Sean Eyre Damon Paulo ###### Abstract In a “tipping” model, each node in a social network, representing an individual, adopts a property or behavior if a certain number of his incoming neighbors currently exhibit the same. In viral marketing, a key problem is to select an initial ”seed” set from the network such that the entire network adopts any behavior given to the seed. Here we introduce a method for quickly finding seed sets that scales to very large networks. Our approach finds a set of nodes that guarantees spreading to the entire network under the tipping model. After experimentally evaluating 31 real-world networks, we found that our approach often finds seed sets that are several orders of magnitude smaller than the population size and outperform nodal centrality measures in most cases. In addition, our approach scales well - on a Friendster social network consisting of 5.6 million nodes and 28 million edges we found a seed set in under 3.6 hours. Our experiments also indicate that our algorithm provides small seed sets even if high-degree nodes are removed. Lastly, we find that highly clustered local neighborhoods, together with dense network- wide community structures, suppress a trend’s ability to spread under the tipping model. ###### Keywords: social networks viral marketing tipping model ## 1 Introduction A much studied model in network science, tippingGran78 ; Schelling78 ; jy05 (a.k.a. deterministic linear thresholdkleinberg ) is often associated with “seed” or “target” set selection, chen09siam (a.k.a. the maximum influence problem). In this problem, we have a social network in the form of a directed graph and thresholds for each individual. Based on this data, the desired output is the smallest possible set of individuals (seed set) such that, if initially activated, the entire population will become activated (adopting the new property). This problem is NP-Complete kleinberg ; Dreyer09 so approximation algorithms must be used. Though some such algorithms have been proposed, leskovec07 ; chen09siam ; benzwi09 ; chen10 none seem to scale to very large data sets. Here, inspired by shell decomposition, ShaiCarmi07032007 ; InfluentialSpreaders_2010 ; baxter11 we present a method guaranteed to find a set of nodes that causes the entire population to activate - but is not necessarily of minimal size. We then evaluate the algorithm on $31$ large, real-world, social networks and show that it often finds very small seed sets (often several orders of magnitude smaller than the population size). We also show that the size of a seed set is related to Louvain modularity and average clustering coefficient. Therefore, we find that dense community structure combined with tight-knit local neighborhoods inhibit the spreading of activation under the tipping model. We also found that our algorithm outperforms the classic centrality measures and is robust against the removal of high-degree nodes. The rest of the paper is organized as follows. In Section 2, we provide formal definitions of the tipping model. This is followed by the presentation of our new algorithm in Section 3. We then describe our experimental results in Section 4. Finally, we provide an overview of related work in Section 5. ## 2 Technical Preliminaries Throughout this paper we assume the existence of a social network, $G=(V,E)$, where $V$ is a set of vertices and $E$ is a set of directed edges. We will use the notation $n$ and $m$ for the cardinality of $V$ and $E$ respectively. For a given node $v_{i}\in V$, the set of incoming neighbors is $\eta^{in}_{i}$, and the set of outgoing neighbors is $\eta^{out}_{i}$. The cardinalities of these sets (and hence the in- and out-degrees of node $v_{i}$) are $d^{in}_{i},d^{out}_{i}$ respectively. We now define a threshold function that for each node returns the fraction of incoming neighbors that must be activated for it to become activate as well. ###### Definition 1 (Threshold Function) We define the threshold function as mapping from V to $(0,1]$. Formally: $\theta:V\rightarrow(0,1]$. For the number of neighbors that must be active, we will use the shorthand $k_{i}$. Hence, for each $v_{i}$, $k_{i}=\lceil\theta(v_{i})\cdot d^{in}_{i}\rceil$. We now define an activation function that, given an initial set of active nodes, returns a set of active nodes after one time step. ###### Definition 2 (Activation Function) Given a threshold function, $\theta$, an activation function $A_{\theta}$ maps subsets of V to subsets of V, where for some $V^{\prime}\subseteq V$, $A_{\theta}(V^{\prime})=V^{\prime}\cup\\{v_{i}\in V\ s.t.\ \|\eta^{in}_{i}\cap V^{\prime}\|\geq k_{i}\\}$ (1) We now define multiple applications of the activation function. ###### Definition 3 (Multiple Applications of the Activation Function) Given a natural number $i>0$, set $V^{\prime}\subseteq V$, and threshold function, $\theta$, we define the multiple applications of the activation function, ${A^{i}_{\theta}}(V^{\prime})$, as follows: $A^{i}_{\theta}(V^{\prime})=\begin{cases}A_{\theta}(V^{\prime})&\text{if $i=1$}\\\ A_{\theta}(A^{i-1}_{\theta}(V^{\prime}))&\text{otherwise}\end{cases}$ (2) Clearly, when $A^{i}_{\theta}(V^{\prime})=A^{i-1}_{\theta}(V^{\prime})$ the process has converged. Further, this always converges in no more than $n$ steps (as, prior to converging, a process must, in each step, activate at least one new node). Based on this idea, we define the function $\Gamma$ which returns the set of all nodes activated upon the convergence of the activation function. ###### Definition 4 ($\Gamma$ Function) Let j be the least value such that $A^{j}_{\theta}(V^{\prime})=A^{j-1}_{\theta}(V^{\prime})$. We define the function $\Gamma_{\theta}:2^{V}\rightarrow 2^{V}$ as follows. $\mathbf{\Gamma_{\theta}}(V^{\prime})=A^{j}_{\theta}(V^{\prime})$ (3) We now have all the pieces to introduce our problem - finding the minimal number of nodes that are initially active to ensure that the entire set $V$ becomes active. ###### Definition 5 (The MIN-SEED Problem) The MIN-SEED Problem is defined as follows: given a threshold function, $\theta$, return $V^{\prime}\subseteq V\ s.t.\ \Gamma_{\theta}(V^{\prime})=V$, and there does not exist $V^{\prime\prime}\subseteq V$ where $\|V^{\prime\prime}\|<\|V^{\prime}\|$ and $\Gamma_{\theta}(V^{\prime\prime})=V$. The following theorem is from the literature kleinberg ; Dreyer09 and tells us that the MIN-SEED problem is NP-complete. ###### Theorem 2.1 (Complexity of MIN-SEED kleinberg ; Dreyer09 ) MIN-SEED in NP-Complete. ## 3 Algorithms In this section, we introduce an integer program that solved the MIN-SEED problem exactly and our new decomposition-based heuristic. ### 3.1 Exact Approach Below we present SEED-IP, an integer program that if solved exactly, guarantees an exact solution to MIN-SEED (see Proposition 1). Though, in general, solving an integer program is also NP-hard, suggesting that an exact solution will likely take exponential time, good approximation techniques such as branch-and-bound exist and mature tools such as QSopt and CPLEX can readily take and approximate solutions to integer programs. ###### Definition 6 (SEED-IP) $\displaystyle\min\sum_{i}x_{i,1},\,\,\,\,\,\,\,\,\,\,\mathit{w.r.t.}$ (4) $\displaystyle\forall i,t\in\\{1,\ldots,n\\},$ $\displaystyle x_{i,t}\in\\{0,1\\}$ (5) $\displaystyle\forall i,$ $\displaystyle x_{i,n}=1$ (6) $\displaystyle\forall i,\forall t>0,$ $\displaystyle\,\,\,\,\,\,\,\,\,\,x_{i,t}\leq x_{i,t-1}+\frac{1}{d^{in}_{i}\theta(v_{i})}\sum_{v_{j}\in\eta^{in}_{i}}x_{j,t-1}$ (7) ###### Proposition 1 If $V^{\prime}$ is a solution to MIN-SEED, then setting $\forall v_{i}\in V^{\prime},x_{i,1}=1$ and $\forall v_{i}\notin V^{\prime},x_{i,1}=0$ is a solution to SEED-IP. If the vector $[x_{i,t}]$ is a solution to SEED-IP, then $\\{v_{i}\|x_{i,1}=1\\}$ is a solution to MIN-SEED. ###### Proof Claim 1: If $V^{\prime}$ is a solution to MIN-SEED, then setting $\forall v_{i}\in V^{\prime},x_{i,1}=1$ and $\forall v_{i}\notin V^{\prime},x_{i,1}=0$ is a solution to SEED-IP. Let $[x_{i,t}]$ be a vector for SEED-IP created as per claim 1. Suppose, by way of contradiction (BWOC), there exists vector $[x^{\prime}_{i,t}]$ s.t. $\sum_{i}x^{\prime}_{i,1}<\sum_{i}x_{i,1}$. However, consider the set of nodes $V^{\prime\prime}=\\{v_{i}\|x^{\prime}_{i,1}=1\\}$. By Constraint 7 of SEED-IP, we know that, for $t>1$, that if $x^{\prime}_{i,t}=1$, we have $v_{i}\in A^{t}_{\theta}(V^{\prime\prime})$. Hence, by Constaint 6 $V^{\prime\prime}$ is a solution to MIN-SEED. This means that $\|V^{\prime\prime}\|<\|V^{\prime}\|$ as $\sum_{i}x^{\prime}_{i,1}<\sum_{i}x_{i,1}$, which is a contradiction. Claim 2: If the vector $[x_{i,t}]$ is a solution to SEED-IP, then $\\{v_{i}\|x_{i,1}=1\\}$ is a solution to MIN-SEED. Suppose, BWOC, there exists set $V^{\prime\prime}$ that is a solution to MIN- SEED s.t. $\|V^{\prime\prime}\|<\|\\{v_{i}\|x_{i,1}=1\\}\|$. Consider the vector $[x^{\prime}_{i,t}]$ where $\forall i,$ $x^{\prime}_{i,0}=1$ iff $v_{i}\in V^{\prime\prime}$. By Constraint 7 of SEED-IP, we know that, for $t>1$, that if $v_{i}\in A^{t}_{\theta}(V^{\prime\prime})$, we have $x^{\prime}_{i,t}=1$. Hence, as $A^{t}_{\theta}(V^{\prime\prime})=V$, know that $[x^{\prime}_{i,t}]$ satisfies Constraint 6. Hence, as $\|V^{\prime\prime}\|<\|\\{v_{i}\|x_{i,1}=1\\}\|$, we know $\sum_{i}x^{\prime}_{i,1}<\sum_{i}x_{i,1}$, which is a contradiction. Proof of theorem: Follows directly form claims 1-2. However, despite the availability of approximate solvers, SEED-IP requires a quadratic number of variables and constraints (Proposition 2), which likely will prevent this approach from scaling to very large datasets. As a result, in the next section we introduce our heuristic approach. ###### Proposition 2 SEED-IP requires $n^{2}$ variables and $2n^{2}$ constraints. ### 3.2 Heuristic To deal with the intractability of the MIN-SEED problem, we design an algorithm that finds a non-trivial subset of nodes that causes the entire graph to activate, but we do not guarantee that the resulting set will be of minimal size. The algorithm is based on the idea of shell decomposition often cited in physics literature Seidman83 ; ShaiCarmi07032007 ; InfluentialSpreaders_2010 ; baxter11 but modified to ensure that the resulting set will lead to all nodes being activated. The algorithm, TIP_DECOMP is presented in this section. Algorithm 1 TIP_DECOMP 0: Threshold function, $\theta$ and directed social network $G=(V,E)$ 0: $V^{\prime}$ 1: For each vertex $v_{i}$, compute $k_{i}$. 2: For each vertex $v_{i},\ dist_{i}=d_{i}^{in}-k_{i}$. 3: FLAG = TRUE. 4: while FLAG do 5: Let $v_{i}$ be the element of $v$ where $dist_{i}$ is minimal. 6: if $dist_{i}=\infty$ then 7: FLAG = FALSE. 8: else 9: Remove $v_{i}$ from $G$ and for each $v_{j}$ in $\eta_{i}^{out}$, if $dist_{j}>0$, set $dist_{j}=dist_{j}-1$. Otherwise set $dist_{j}=\infty$. 10: end if 11: end while 12: return All nodes left in $G$. Intuitively, the algorithm proceeds as follows (Figure 1). Given network $G=(V,E)$ where each node $v_{i}$ has threshold $k_{i}=\lceil\theta(v_{i})\cdot d^{in}_{i}\rceil$, at each iteration, pick the node for which $d^{in}_{i}-k_{i}$ is the least but positive (or $0$) and remove it. Once there are no nodes for which $d^{in}_{i}-k_{i}$ is positive (or $0$), the algorithm outputs the remaining nodes in the network. Figure 1: Example of our algorithm for a simple network depicted in box A. We use a threshold value set to $50\%$ of the node degree. Next to each node label (lower-case letter) is the value for $d^{in}_{i}-k_{i}$ (where $k_{i}=\lceil\frac{d^{in}_{i}}{2}\rceil$). In the first four iterations, nodes e, f, h, and i are removed resulting in the network in box B. This is followed by the removal of node j resulting in the network in box C. In the next two iterations, nodes a and b are removed (boxes D-E respectively). Finally, node c is removed (box F). The nodes of the final network, consisting of d and g, have negetive values for $d_{i}-\theta_{i}$ and become the output of the algorithm. Now, we prove that the resulting set of nodes is guaranteed to cause all nodes in the graph to activate under the tipping model. This proof follows from the fact that any node removed is activated by the remaining nodes in the network. ###### Theorem 3.1 If all nodes in $V^{\prime}\ \subseteq\ V$ returned by TIP_DECOMP are initially active, then every node in $V$ will eventually be activated, too. ###### Proof Let $w$ be the total number of nodes removed by TIP_DECOMP, where $v_{1}$ is the last node removed and $v_{w}$ is the first node removed. We prove the theorem by induction on $w$ as follows. We use $P(w)$ to denote the inductive hypothesis which states that all nodes from $v_{1}$ to $v_{w}$ are active. In the base case, $P(1)$ trivially holds as we are guaranteed that from set $V^{\prime}$ there are at least $k_{1}$ edges to $v_{1}$ (or it would not be removed). For the inductive step, assuming $P(w)$ is true, when $v_{w+1}$ was removed from the graph $dist_{w+1}\geq 0$ which means that $d_{w+1}^{in}\geq k_{w+1}$. All nodes in $\eta^{in}_{w+1}$ at the time when $v_{w+1}$ was removed are now active, so $v_{w+1}$ will now be activated - which completes the proof. We also note that by using the appropriate data structure (we used a binomial heap in our implementation), for a network of $n$ nodes and $m$ edges, this algorithm can run in time $O(m\log n)$. ###### Proposition 3 The complexity of TIP_DECOMP is $O(m\cdot log(n))$. ###### Proof If we use a binomial heap as described in cormen , we can create a heap where we store each node and assign it a key value of $dist_{i}$ for each node $v_{i}$. The creation of a heap takes constant time and inserting the $n$ vertices will take $O(nlog(n))$ time. We can also maintain a list data structure as well. In the course of the while loop, all nodes will either be removed (as per the algorithm), decreased in key-value no more than $d^{in}_{i}$ or increased to infinity (which we can implement as being removed and added to the list). Hence, the number of decrease key or removal operations is bounded by $n+\sum_{i}d^{in}_{i}$. As $\sum_{i}d^{in}_{i}=m$ (where $m$ is the number of edges). As $O(m\cdot log(n))$, the statement follows. ## 4 Results In this section we describe the results of our experimental evaluation. We describe the datasets we used for the experiments in Section 4.1. We evaluate the run-time of TIP_DECOMP in Section 4.1.5. In Section 4.1.6, we evaluate the size of the seed-set returned by the algorithm and we compare this to the seed size returned by known centrality measures in Section 4.2. The speed of the activiation process initiated with seed sets discovered by our algorithm is described in Section 4.3. We then study how the removal of high-degree nodes and community structure affect the results of the algorithm in Sections 4.4 and 4.4.1 respectively. The algorithm TIP_DECOMP was written using Python 2.6.6 in 200 lines of code that leveraged the NetworkX library available from http://networkx.lanl.gov/. The code used a binomial heap library written by Björn B. Brandenburg available from http://www.cs.unc.edu/$\sim$bbb/. The experiments were run on a computer equipped with an Intel X5677 Xeon Processor operating at 3.46 GHz with a 12 MB Cache running Red Hat Enterprise Linux version 6.1 and equipped with 70 GB of physical memory. All statistics presented in this section were calculated using R 2.13.1. ### 4.1 Datasets In total, we examined $36$ networks: nine academic collaboration networks, three e-mail networks, and $24$ networks extracted from social-media sites. The sites included included general-purpose social-media (similar to Facebook or MySpace) as well as special-purpose sites (i.e. focused on sharing of blogs, photos, or video). All datasets used in this paper were obtained from one of four sources: the ASU Social Computing Data Repository, Zafarani+Liu:2009 the Stanford Network Analysis Project, snap the University of Michigan, umich and Universitat Rovira i Virgili.uvi $31$ of the networks considered were symmetric – i.e. if a directed edge from vertex $v$ to $v^{\prime}$ exists, there is also an edge from vertex $v^{\prime}$ to $v$. Tables 1 (A-C) show some of the pertinent qualities of the symmetric networks. The networks are categorized by the results for the MIN-SEED experiments (explained later in this section). Additionally, we also looked at several non-symmetric (directed) networks and placed them in their own category. In what follows, we provide their real- world context. #### 4.1.1 Category A * • BlogCatalog is a social blog directory that allows users to share blogs with friends. Zafarani+Liu:2009 The first two samples of this site, BlogCatalog1 and 2, were taken in Jul. 2009 and June 2010 respectively. The third sample, BlogCatalog3 was uploaded to ASU’s Social Computing Data Repository in Aug. 2010. * • Buzznet is a social media network designed for sharing photographs, journals, and videos. Zafarani+Liu:2009 It was extracted in Nov. 2010. * • Douban is a Chinese social medial website designed to provide user reviews and recommendations. Zafarani+Liu:2009 It was extracted in Dec. 2010. * • Flickr is a social media website that allows users to share photographs. Zafarani+Liu:2009 It was uploaded to ASU’s Social Computing Data Repository in Aug. 2010. * • Flixster is a social media website that allows users to share reviews and other information about cinema. Zafarani+Liu:2009 It was extracted in Dec. 2010. * • FourSquare is a location-based social media site. Zafarani+Liu:2009 It was extracted in Dec. 2010. * • Frienster is a general-purpose social-networking site. Zafarani+Liu:2009 It was extracted in Nov. 2010. * • Last.Fm is a music-centered social media site. Zafarani+Liu:2009 It was extracted in Dec. 2010. * • LiveJournal is a site designed to allow users to share their blogs. Zafarani+Liu:2009 It was extracted in Jul. 2010. * • Livemocha is touted as the “world’s largest language community.” Zafarani+Liu:2009 It was extracted in Dec. 2010. * • WikiTalk is a network of individuals who set and received messages while editing WikiPedia pages. snap It was extracted in Jan. 2008. #### 4.1.2 Category B * • Delicious is a social bookmarking site, designed to allow users to share web bookmarks with their friends. Zafarani+Liu:2009 It was extracted in Dec. 2010. * • Digg is a social news website that allows users to share stories with friends. Zafarani+Liu:2009 It was extracted in Dec. 2010. * • EU E-Mail is an e-mail network extracted from a large European Union research institution. snap It is based on e-mail traffic from Oct. 2003 to May 2005. * • Hyves is a popular general-purpose Dutch social networking site. Zafarani+Liu:2009 It was extracted in Dec. 2010. * • Yelp is a social networking site that allows users to share product reviews. Zafarani+Liu:2009 It was extracted in Nov. 2010. #### 4.1.3 Category C * • CA-AstroPh is a an academic collaboration network for Astro Physics from Jan. 1993 - Apr. 2003. snap * • CA-CondMat is an academic collaboration network for Condense Matter Physics. Samples from 1999 (CondMat99), 2003 (CondMat03), and 2005 (CondMat05) were obtained from the University of Michigan. umich A second sample from 2003 (CondMat03a) was obtained from Stanford University. snap * • CA-GrQc is a an academic collaboration network for General Relativity and Quantum Cosmology from Jan. 1993 - Apr. 2003. snap * • CA-HepPh is a an academic collaboration network for High Energy Physics - Phenomenology from Jan. 1993 - Apr. 2003. snap * • CA-HepTh is a an academic collaboration network for High Energy Physics - Theory from Jan. 1993 - Apr. 2003. snap * • CA-NetSci is a an academic collaboration network for Network Science from May 2006. * • Enron E-Mail is an e-mail network from the Enron corporation made public by the Federal Energy Regulatory Commission during its investigation. snap * • URV E-Mail is an e-mail network based on communications of members of the University Rovira i Virgili (Tarragona). uvi It was extracted in 2003. * • YouTube is a video-sharing website that allows users to establish friendship links. Zafarani+Liu:2009 The first sample (YouTube1) was extracted in Dec. 2008. The second sample (YouTube2) was uploaded to ASU’s Social Computing Data Repository in Aug. 2010. #### 4.1.4 Non-Symmetric Networks * • Epinions is a consumer review website that allows members to establish directed trust relationships. snap * • WikiVote is a sample of Wikipedia users voting beahavior (who votes for whom). snap * • Slashdot formerly had a feature called “Slashdot Zoo” that allowed users to tag each other as friend or foe. We looked at three samples based on friendship relationships: one sample from 2008 (Slashdot1) and two from 2009 (Slashdot2-Slashdot3). snap Table 1: Information on the networks in Categories A, B, and C. Table 2: Information on non-symmetric networks. #### 4.1.5 Runtime First, we examined the runtime of the algorithm (see Figure 2 and Table 3). Our experiments aligned well with our time complexity result (Proposition 3). For example, a network extracted from the Dutch social-media site Hyves consisting of $1.4$ million nodes and $5.5$ million directed edges was processed by our algorithm in at most $12.2$ minutes. The often-cited LiveJournal dataset consisting of $2.2$ million nodes and $25.6$ million directed edges was processed in no more than $66$ minutes - a short time to approximate an NP-hard combinatorial problem on a large-sized input. Figure 2: $m\ln n$ vs. runtime in seconds (log scale, $m$ is number of edges, $n$ is number of nodes). The relationship is linear with $R^{2}=0.9015$, $p=2.2\cdot 10^{-16}$. Table 3: Runtime data on the datasets used in the experiments. #### 4.1.6 Seed Size For each network, we performed $10$ “integer” trials. In these trials, we set $\theta(v_{i})=\min(d^{in}_{i},k)$ where $k$ was kept constant among all vertices for each trial and set at an integer in the interval $[1,10]$. We evaluated the ability of a network to promote spreading under the tipping model based on the size of the set of nodes returned by our algorithm (as a percentage of total nodes). For purposes of discussion, we have grouped our networks into three categories based on results (Figure 3 and Table 4). We have also included results for symmetric networks (Figure 4 and Table 5). In general, online social networks had the smallest seed sets - $13$ networks of this type had an average seed set size less than $2\%$ of the population (these networks were all in Category A). We also noticed, that for most networks, there was a linear realtion between threshold value and seed size. Figure 3: Threshold value (assigned as an integer in the interval $[1,10]$) vs. size of initial seed set as returned by our algorithm in our three identified categories of networks (categories A-C are depicted in panels A-C respectively). Average seed sizes were under $2\%$ for Categorty A, $2-10\%$ for Category B and over $10\%$ for Category C. The relationship, in general, was linear for categories A and B and lograthimic for C. CA-NetSci had the largest Louvain Modularity and clustering coefficient of all the networks. This likely explains why that particular network seems to inhibit spreading. Figure 4: Threshold value assigned as both an integer in the range $[1,10]$ (panel 1) as well as a fraction of node degree (panel 2) for the non-symmetric networks. Category A can be thought of as social networks highly susceptible to influence - as a very small fraction of initially activated individuals can lead to activation of the entire population. All were extracted from social media websites. For some of the lower threshold levels, the size of the set of seed nodes was particularly small. For a threshold of three, $11$ of the Category A networks produced seeds smaller than $0.5\%$ of the total populationa. For a threshold of four, nine networks met this criteria. Networks in Category B are susceptible to influence with a relatively small set of initial nodes - but not to the extent of those in Category A. They had an average initial seed size greater than $2\%$ but less than $10\%$. Members in this group included two general purpose social media networks, two specialty social media networks, and an e-mail network. Non-symmetric networks generally perofrmed somewhat poorer than Category B networks (though in general, not as poorly as those in Category C). The initial seed sizes for the non-symmmetric networks ranged from $3\%$ to $29\%$. Category C consisted of networks that seemed to hamper diffusion in the tipping model, having an average initial seed size greater than $10\%$. This category included all of the academic collaboration networks, two of the email networks, and two networks derived from friendship links on YouTube. We also studied the effects on spreading when the threshold values were assigned as a specific fraction of each node’s in-degree jy05 ; wattsDodds07 , which results in heterogeneous $\theta_{i}$’s across the network. We performed $12$ trials for each network. Thresholds for each trial were based on the product of in-degree and a fraction in the interval $[0.05,0.60]$ (multiples of $0.05$). The results (Figure 5 and Table 4; for non-symmertic networks see Figure 4 and Table 5) were analogous to our integer tests. We also compared the averages over these trials with $M$ and $C$ and obtained similar results as with the other trials (Figure 14 B). Figure 5: Threshold value (assigned as a fraction of node in-degree as a multiple of $0.05$ in the interval $[0.05,0.60]$) vs. size of initial seed set as returned by our algorithm in our three identified categories of networks (categories A-C are depicted in panels A-C respectively, categories are the same as in Figure 1). Average seed sizes were under $5\%$ for Categorty A, $1-7\%$ for Category B and over $3\%$ for Category C. In general, the relationship between threshold and initial seed size for networks in all categories was exponential. ### 4.2 Comparison with Centrality Measures We compared our results with six popular centrality measures: degree, betweenness, closeness, shell number, eigenvector, and PageRank. Here, we define degree centrality is simply the number of outgoing adjacent nodes. 111Note that in the symmetric networks we examined, our empirical results hold for the number of incoming adjacent edges as well as the total number of adjacent edges. The intuition behind high betweenness centrality nodes is that they function as “bottlenecks” as many paths in the network pass through them. Hence, betweenness is a medial centrality measure. Let $\sigma_{st}$ be the number of shortest paths between nodes $s$ and $t$ and $\sigma_{st}(v)$ be the number of shortest paths between $s$ and $t$ containing node $v$. In freeman77 , betweenness centrality for node $v$ is defined as $\sum_{s\neq v\neq t}\frac{\sigma_{st}(v)}{\sigma_{st}}$. In most implementations, including the ones used in this paper, the algorithm of brandes01 is used to calculate betweenness centrality. Another common measure from the literature that we examined is closeness freeman79cent . Given node $i$, its closeness $C_{c}(i)$ is the inverse of the average shortest path length from node $i$ to all other nodes in the graph. Intuitively, closeness measures how “close” it is to all other nodes in a network. Formally, if we define the shortest path between nodes $i$ to $j$ as function $d_{G}(i,j)$, we can express the average path length from $i$ to all other nodes as $L_{i}=\frac{\sum_{j\in V\setminus i}d_{G}(i,j)}{\|V\|-1}.$ (8) Hence, the closeness of a node can be formally written as $C_{c}(i)=\frac{1}{L_{i}}=\frac{\|V\|-1}{\sum_{j\in V\setminus i}d_{G}(i,j)}.$ (9) The idea of shell number is based on a core to which a node lies in. A $c$-core of a network is the subgraph in which every node is connected to the rest of the network by at least $c$ edges. A node is assigned a shell number based on the maximal core that contains it. This value can be derived exactly using shell decomposition Seidman83 . The eigenvector centrality bona72 of a node is assigned based on the associated entry in the eigenvector of the adjacency matrix corresponding to the largest real eigenvalue. The PageRank Page98 for each node based on the PageRank of its neighbors. An initial value for rank is considered for each node and the relationship is then computed iteratively until convergence is reached. Intuitively, PageRank can be thought of as the importance of a node based on the importance of its neighbors. We note that shell number, eigenvector, and PageRank are often associated with diffusion processes. A more complete discussion of centrality measures can be found in wasserman1994social . We evaluated the performance of centrality measures in finding a seed set by iteratively selecting the most central nodes with respect to a given measure until the $\Gamma_{\theta}$ of that set returns the set of all nodes. Due to the computational overhead of calculating these centrality measures and the repeated re-evaluation of $\Gamma_{\theta}$, we limited this comparison to only BlogCatalog3, CA-HepTh, CA-NetSci, URV E-Mail, and Douban (no betweeness calcualted for Douban). As with the experiments in the previous section, we studied threshold settings based on an integer in the interval $[1,10]$ (see Figure 6) and as a fraction of incoming neighbors in the interval $[0.05,0.60]$ (multiples of $0.05$, see Figure 8). In general, selecting highly-central nodes is an inefficient method for finding small seed sets. In all but the lowest threshold settings, the use of centrality measures for the integer-threshold trials proved to significantly underperformed when the method presented in this paper - often returning seed-sets several orders of magnitude larger and in many cases including the majority of nodes in the network. Even for the centrality measures outperformed our method in these trials, the reduction in seed set size was minimal (the greatest reduction in seed set size experienced in a centrality-measure test over the algorithm of this paper was $0.09\%$, while often producing seed sets orders of magnitude greater than our method). This held even for the centrality measures associated with diffusion (shell number, eigenvector, and PageRank). Figure 6: The use of degree, betweenness and closeness to find seed-sets on select networks when the threshold is set to an integer in the interval $[1,10]$. For these trials, centrality measure returned significantly larger (several orders of magnitude) larger seed sets than our approach. Figure 7: The use of shell number, Eigenvector, and PageRank to find seed-sets on select networks when the threshold is set to an integer in the interval $[1,10]$. Our tests using fractional-based thresholds tell a slightly different story. While our method still generally outperformed the centrality measures for the fractional tests, there were a few cases where the centrality measures fared better. With BlogCatalog3 all of the centrality measures outperformed our algorithm in the fraction-based experiments. For that dataset, centrality- based algorithm consistently outperformed our method finding seed sets with less members (by $3.13-3.29\%$ of the population, on average). With URV-Email, many trials that utilized a lower threshold setting outperformed our method, but never finding a feed set with smaller by more than $8\%$ of the total population. However, in the larger threshold settings, our method consistently found smaller seeds. For a given centrality measure for this dataset, centrality measures on average provided poorer results than our algorithm ranged - returning seed sets which were, on average $10.22-67.14\%$ (by overall population) larger than that returned by our algorithm. Perhaps the most interesting result among the centrality measures were the PageRank fraction-based tests on CA-NetSci, which is associated with the largest seed sets. PageRank found seed sets that were, on average $14.45\%$ smaller (by population) than our approach. Additionally, though centrality measures outperformed TIP_DECOMP for BlogCatalog3, this does not appear to hold for all social networks as the seed sets returned using centrality measures for the Douban approaches at least an order of magnitude increase over our method for nearly every fractional threshold setting for all centrality measures. Hence, we conclude that for fraction-based thresholds, using centrality measures to find seed sets provides inconsistent results, and when it fails, it tends to provide a large portion of the network. A possibility for a practical algorithm that could combine both methods would be to first run TIP_DECOMP, returning some set $V^{\prime}$. Then, create set $V^{\prime\prime}$ by selecting the most central nodes until either $\|V^{\prime}\|=\|V^{\prime\prime}\|$ or $\Gamma_{\theta}(V^{\prime\prime})=V$ (whichever ensures the lower cardinality for $V^{\prime\prime}$. If $\|V^{\prime}\|=\|V^{\prime\prime}\|$, return $V^{\prime}$, otherwise return $V^{\prime\prime}$. For such an approach, we would likely recommend using degree centrality due to its ease of computation and performance in our experiments. However, we note that highly-central nodes often may not be realistic targets for a viral-marketing campaign. For instance, it may be cost-prohibitive to create a seed set consisting of major celebrities in order to spread the use of a product. As such is a practical concern, we look at the performance of TIP_DECOMP when high-degree nodes are removed in the next section. Figure 8: The use of degree, betweenness and closeness to find seed-sets on select networks when the threshold is set to an fraction in the interval $[0.05,0.60]$ (multiples of $0.05$). Figure 9: The use of shell number, Eigenvector, and PageRank to find seed-sets on select networks when the threshold is set to an fraction in the interval $[0.05,0.60]$ (multiples of $0.05$). ### 4.3 The Speed of the Activation Process and Sets of “Critical Mass” An important aspect to consider in viral marketing is the speed of the activation process. We illustrate this speed for several networks under a threshold of $2$ as well as a majority threshold (half of each nodes neighbors) in Figure 10. Interestingly, we found that the size of the initial seed set was not indicative of the speed of spreading. For instance, in BlogCatalog3, a Category A network (for which our algorithm found a very small seed set) the activation process proceeded quickly when compared to the others examined. However, this was also true for CA-NetSci, a Category C network (large seed set). Conversely, the activation process in the Douban and CA- HepTh networks (also Category A and C, respectively) proceeded more slowly than the rest. Another interesting feature we learned in exploring the speed of the activation process was that in all of our experiments there was a single time step where the number of activated nodes increased significantly more than the other time periods - sometimes by several orders of magnitude (see Figure 11). We can think of such a set of activated nodes as when the population reaches a “critical mass” which results in mass adoption in the next interval. In many cases, such a critical mass is reached early - normally in the first few time- steps. Finding a subset of the population of “critical mass” may be an important problem in its own right. Though the critical mass point will often be larger than the seed set found by an algorithm in this paper, we can be assured that in one time step of the model, the number of individuals reached (with a certain number of signals from their neighbors) is substantially larger than the investment. In practice, this could lead to quicker spreading of information in an advertising campaign, for example. Further, our experiments indicate that order-of-magnitude critical mass sets exist in several networks. We are currently conducting further research on this topic. Figure 10: An examination of several of speed of activation initiated from the seed set using a threshold of two (panel 1) and a majority threshold (panel 2). Figure 11: Greatest Percent increase experienced in a single time step (the effect of reaching “critical mass”) for integer-based and percentage- based thresholds (panel 1 and 2 respectively). ### 4.4 Effect of Removing High-Degree Nodes In the last section we noted that high-degree nodes may not always be targetable in a viral marketing campaign (i.e. it may be cost prohibitive to include them in a seed set). In this section, we explore the affect of removing high-degree nodes on the size of the seed-set returned by TIP_DECOMP. This type of node removal has also recently been studied in a different context in nodeRemRef . In these trials, we studied all $31$ networks and looked at two specific threshold settings: an integer threshold of $2$ (Figure 12) and a fractional threshold of $0.5$ (Figure 13). We then studied the effect of removing up to $50\%$ of the nodes in order from greatest to least degree. With an integer threshold of $2$, networks in category A still retained a seed-size (as returned by TIP_DECOMP) two orders of magnitude smaller than the population size up to the removal of $10\%$ of the top degree nodes, and for many networks this was maintained to $50\%$. Networks in category B retained seed sets an order of magnitude smaller than the population for up to $50\%$ of the nodes removed. For most networks in category C, the seed size remained about a quarter of the population size up to $15\%$ of the top degree nodes being removed. Figure 12: Size of the seed set returned by TIP_DECOMP (as a fraction of the popualtion) as a function of the percent of the highest degree nodes removed from the network with an integer theshold of $2$ for networks in categories A-C. With a fractional threshold of $0.5$, we noted that many networks in category A actually had larger seed sets (as returned by TIP_DECOMP) when more high degree nodes are removed. Further, networks in categories A-B retained seed sets of at least an order of magnitude smaller than the population in these tests while most networks in category C retained sizes of about a quarter of the population. Figure 13: Size of the seed set returned by TIP_DECOMP (as a fraction of the popualtion) as a function of the percent of the highest degree nodes removed from the network with an fractional theshold of $0.5$ for networks in categories A-C. #### 4.4.1 Seed Size as a Function of Community Structure In this section, we view the results of our heuristic algorithm as a measurement of how well a given network promotes spreading. Here, we use this measurement to gain insight into which structural aspects make a network more likely to be “tipped.” We compared our results with two network-wide measures characterizing community structure. First, clustering coefficient ($C$) is defined for a node as the fraction of neighbor pairs that share an edge - making a triangle. For the undirected case, we define this concept formally below. ###### Definition 7 (Clustering Coefficient) Let $r$ be the number of edges between nodes with which $v_{i}$ has an edge and $d_{i}$ be the degree of $v_{i}$. The clustering coefficient, $C_{i}=\dfrac{2r}{d_{i}(d_{i}-1)}$. Intuitively, a node with high $C_{i}$ tends to have more pairs of friends that are also mutual friends. We use the average clustering coefficient as a network-wide measure of this local property. Second, we consider modularity ($M$) defined by Newman and Girvan. newman04 . For a partition of a network, $M$ is a real number in $[-1,1]$ that measures the density of edges within partitions compared to the density of edges between partitions. We present a formal definition for an undirected network below. ###### Definition 8 (Modularity newman04 ) Given partition $C=\\{c_{1},\ldots,c_{q}\\}$, modularity, $M=\dfrac{1}{2m}\sum_{c\in C}\sum_{i,j\in c}w_{ij}-P_{ij}$ where $m$ is the number of undirected edges; $w_{ij}=1$ if there is an edge between nodes $i$ and $j$ and $w_{ij}=0$ otherwise; $P_{ij}=\frac{k_{i}k_{j}}{2m}$ The modularity of an optimal network partition can be used to measure the quality of its community structure. Though modularity-maximization is NP-hard, the approximation algorithm of Blondel et al. blondel08 (a.k.a. the “Louvain algorithm”) has been shown to produce near-optimal partitions.222Louvain modularity was computed using the implementation available from CRANS at http://perso.crans.org/aynaud/communities/. We call the modularity associated with this algorithm the “Louvain modularity.” Unlike the $C$, which describes local properties, $M$ is descriptive of the community level. For the $31$ networks we considered, $M$ and $C$ appear uncorrelated ($R^{2}=0.0538$, $p=0.2092$). We plotted the initial seed set size ($S$) (from our algorithm - averaged over the $10$ threshold settings) as a function of $M$ and $C$ (Figure 14a) and uncovered a correlation (planar fit, $R^{2}=0.8666$, $p=5.666\cdot 10^{-13}$, see Figure 14 A). The majority of networks in Category C (less susceptible to spreading) were characterized by relatively large $M$ and $C$ (Category C includes the top nine networks w.r.t. $C$ and top five w.r.t. $M$). Hence, networks with dense, segregated, and close-knit communities (large $M$ and $C$) suppress spreading. Likewise, those with low $M$ and $C$ tended to promote spreading. Also, we note that there were networks that promoted spreading with dense and segregated communities, yet were less clustered (i.e. Category A networks Friendster and LiveJournal both have $M\geq 0.65$ and $C\leq 0.13$). Further, some networks with a moderately large clustering coefficient were also in Category A (two networks extracted from BlogCatalog had $C\geq 0.46$) but had a relatively less dense community structure (for those two networks $M\leq 0.33$). Figure 14: (A) Louvain modularity ($M$) and average clustering coefficient ($C$) vs. the average seed size ($S$). The planar fit depicted is $S=43.374\cdot M+33.794\cdot C-24.940$ with $R^{2}=0.8666$, $p=5.666\cdot 10^{-13}$. (B) Same plot at (A) except the averages are over the 12 percentage-based threshold values. The planar fit depicted is $S=18.105\cdot M+17.257\cdot C-10.388$ with $R^{2}=0.816$, $p=5.117\cdot 10^{-11}$. Table 4: Regression analysis and network-wide measures for the networks in Categories A, B, and C. Table 5: Regression analysis and network-wide measures for the non-symmetric networks. ## 5 Related Work Tipping models first became popular by the works of Gran78 and Schelling78 where it was presented primarily in a social context. Since then, several variants have been introduced in the literature including the non- deterministic version of kleinberg (described later in this section) and a generalized version of jy05 . In this paper we focused on the deterministic version. In wattsDodds07 , the authors look at deterministic tipping where each node is activated upon a percentage of neighbors being activated. Dryer and Roberts Dreyer09 introduce the MIN-SEED problem, study its complexity, and describe several of its properties w.r.t. certain special cases of graphs/networks. The hardness of approximation for this problem is described in chen09siam . The work of benzwi09 presents an algorithm for target-set selection whose complexity is determined by the tree-width of the graph - though it provides no experiments or evidence that the algorithm can scale for large datasets. The recent work of reichman12 proves a non-trivial upper bound on the smallest seed set. Our algorithm is based on the idea of shell-decomposition that currently is prevalent in physics literature. In this process, which was introduced in Seidman83 , vertices (and their adjacent edges) are iteratively pruned from the network until a network “core” is produced. In the most common case, for some value $k$, nodes whose degree is less than $k$ are pruned (in order of degree) until no more nodes can be removed. This process was used to model the Internet in ShaiCarmi07032007 and find key spreaders under the SIR epidemic model in InfluentialSpreaders_2010 . More recently, a “heterogeneous” version of decomposition was introduced in baxter11 \- in which each node is pruned according to a certain parameter - and the process is studied in that work based on a probability distribution of nodes with certain values for this parameter. ### 5.1 Notes on Non-Deterministic Tipping We also note that an alternate version of the model where the thresholds are assigned randomly has inspired approximation schemes for the corresponding version of the seed set problem.kleinberg ; leskovec07 ; chen10 Work in this area focused on finding a seed set of a certain size that maximizes the expected number of adopters. The main finding by Kempe et al., the classic work for this model, was to prove that the expected number of adopters was submodular - which allowed for a greedy approximation scheme. In this algorithm, at each iteration, the node which allows for the greatest increase in the expected number of adopters is selected. The approximation guarantee obtained (less than $0.63$ of optimal) is contingent upon an approximation guarantee for determining the expected number of adopters - which was later proved to be $\\#P$-hard. chen10 Recently, some progress has been made toward finding a guarantee Borgs12 . Further, the simulation operation is often expensive - causing the overall time complexity to be $O(x\cdot n^{2})$ where $x$ is the number of runs per simulation and $n$ is the number of nodes (typically, $x>n$). In order to avoid simulation, various heuristics have been proposed, but these typically rely on the computation of geodesics - an $O(n^{3})$ operation - which is also more expensive than our approach. Additionally, the approximation argument for the non-deterministic case does not directly apply to the original (deterministic) model presented in this paper. A simple counter-example shows that sub-modularity does not hold here. Sub-modularity (diminishing returns) is the property leveraged by Kempe et al. in their approximation result. ### 5.2 Note on an Upper Bound of the Initial Seed Set Very recently, we were made aware of research by Daniel Reichman that proves an upper bound on the minimal size of a seed set for the special case of undirected networks with homogeneous threshold values. reichman12 The proof is constructive and yields an algorithm that mirrors our approach (although Reicshman’s algorithm applies only to that special case). We note that our work and the work of Reichman were developed independently. We also note that Reichman performs no experimental evaluation of the algorithm. Given undirected network $G$ where each node $v_{i}$ has degree $d_{i}$ and the threshold value for all nodes is $k$, Reichman proves that the size of the minimal seed set can be bounded by $\sum_{i}\min\\{1,\frac{k}{d_{i}+1}\\}$. For our integer tests, we compared our results to Reichman’s bound. Our seed sets were considerably smaller - often by an order of magnitude or more. See Figure 15 for details. ## 6 Conclusion As recent empirical work on tipping indicates that it can occur in real social networks,centola10 ; zhang11 our results are encouraging for viral marketers. Even if we assume relatively large threshold values, small initial seed sizes can often be found using our fast algorithm - even for large datasets. For example, with the FourSquare online social network, under majority threshold ($50\%$ of incoming neighbors previously adopted), a viral marketeer could expect a $297$-fold return on investment. As results of this type seem to hold for many online social networks, our algorithm seems to hold promise for those wishing to “go viral.” An important open question to address in future work is if a similar decomposition-based approach is viable for finding seed sets under other diffusion models, such as the independent cascade model kleinberg and evolutionary graph theory lieberman_evolutionary_2005 as well as probabilistic variants of the tipping model and diffusion processes on multi- modal networks snops-iclp . Exploring other models can lead to the development of decomposition algorithms in domains where social behavior is more dynamic such as cell-phone networks dyag13 ; otherPho . ###### Acknowledgements. We would like to thank Gaylen Wong (USMA) for his technical support. Additionally, we would like to thank (in no particular order) Albert-László Barabási (NEU), Sameet Sreenivasan (RPI), Boleslaw Szymanski (RPI), Patrick Roos (UMD), John James (USMA), and Chris Arney (USMA) for their discussions relating to this work. Finally, we would also like to thank Megan Kearl, Javier Ivan Parra, and Reza Zafarani of ASU for their help with some of the datasets. The authors are supported under by the Army Research Office (project 2GDATXR042) and the Office of the Secretary of Defense (project F1AF262025G001). The opinions in this paper are those of the authors and do not necessarily reflect the opinions of the funders, the U.S. Military Academy, or the U.S. Army. Figure 15: Integer threshold values vs. the seed size divided by Reichman’s upper bound reichman12 the three categories of networks (categories A-C are depicted in panels A-C respectively). 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In: Proceedings of International Conference on Game Theory for Networks (GameNets) (2011)	arxiv-papers	2013-09-11T20:40:11	2024-09-04T02:49:50.758588	{ "license": "Public Domain", "authors": "Paulo Shakarian, Sean Eyre, Damon Paulo", "submitter": "Paulo Shakarian", "url": "https://arxiv.org/abs/1309.2963" }
1309.2966	# Sustainable Software Ecosystems for Open Science: 15 Years of Practice and Experience at Kitware Marcus D. Hanwell Kitware, Inc., 28 Corporate Drive, Clifton Park, NY 12065 USA. http://www.kitware.com/ Amitha Perera Kitware, Inc., 28 Corporate Drive, Clifton Park, NY 12065 USA. http://www.kitware.com/ Wes Turner Kitware, Inc., 28 Corporate Drive, Clifton Park, NY 12065 USA. http://www.kitware.com/ Patrick O’Leary Kitware, Inc., 28 Corporate Drive, Clifton Park, NY 12065 USA. http://www.kitware.com/ Katie Osterdahl Kitware, Inc., 28 Corporate Drive, Clifton Park, NY 12065 USA. http://www.kitware.com/ Bill Hoffman Kitware, Inc., 28 Corporate Drive, Clifton Park, NY 12065 USA. http://www.kitware.com/ Will Schroeder Kitware, Inc., 28 Corporate Drive, Clifton Park, NY 12065 USA. http://www.kitware.com/ ## Introduction Mathematics is the core language of science, and for centuries it was necessary to show the mathematical underpinnings of new research as part of scientific explorations. This lingua franca provided an essential level of understandability and precision; providing for unambiguous communication and rigorous verification of scientific claims beyond the inaccuracies of spoken languages. However, the last few decades have seen an erosion in this paradigm. The increased reliance of science on complex computational codes and large data makes the description of all but the most basic research error prone, impenetrable, and unverifiable. This issue is not restricted to just one field of science, but is endemic throughout the broader scientific community and the consequences of opaque processes and lack of reproducibility are not trivial. Cases of irreproducible studies and clinical trials have been making headlines, from Bayer Health Care stopping nearly two-thirds of its target-validation projects because of inconsistencies with the initially published claims, to global economic policy being based on a single fundamentally flawed study by Harvard economists. These costly mistakes can be remedied much earlier, and before key decisions are made, simply by returning transparency and precision to the process of publication and review. The question we must address is how best to reinstate a common language and what that language should be. We believe that the only practical choice is to require that disclosures of scientific research based on complex codes and data use the very same complex codes and data as the common language of publication. This means that as new studies and new scientific explorations are undertaken, the data, methods, and software used by the researchers to arrive at their conclusions must be made available and accessible to other researchers and the general populace. If this goal is to be realized then the standard of software engineering in science must be improved, and sustainable software ecosystems with meaningful credit must be realized. This is not simply limited to teaching scientists to write code; if sustainable software projects are to be established in science then issues such as testing, licensing, and collaboration must be addressed. Sufficient engineering discipline is required to realize robust foundations that can be extended through the use of code review, regression testing and proper citation and attribution of software used in research. ## Vision Kitware has been developing open-source solutions and laying the building blocks for open science for over a decade. We contribute to a wide and varied range of open-source projects in the scientific domain across many disciplines including supercomputing, scientific data visualization, large data management and informatics, computational fluid dynamics, medical imaging, chemistry and computer vision. Kitware’s specific expertise in this area, including business practices, collaboration patterns, and growth strategies, are discussed. Based on this experience, we believe practical frameworks for science revolve around well-defined and open application programming interfaces (APIs) between key steps in the analysis workflow from raw data to final figures in published manuscripts. Although not directly related to the scientific goals, it is important to consider licensing in the context of scientific communities. It is not enough to simply make code and data available for free if we wish to see communities flourish—shared ownership and access must be provided under permissive licenses wherever possible. These should encourage reuse, derivatives, and allow those products to be shared. Permissive licenses such as MIT, BSD, and Apache 2 for software; and CC0 or CC-BY for data should be strongly encouraged over copyleft, non-commercial and no derivative clauses that can hamper further reuse of code or data. When a wealth of data and code is available under permissive licenses that encourage reuse it is much easier to grow sustainable software ecosystems—suddenly the method a student developed in their research can be incorporated into a larger framework even after a student has moved on to the next challenge. This relies upon the code being published and made available under permissive licenses allowing for reuse in other contexts. The same path can be considered for data, where individual findings can be combined into larger collections and analyses performed on aggregate data to make new discoveries outside the scope of the original research in many cases. ## Practice and Experience Kitware started as an experiment in open source education beginning when three General Electric research employees wanted to author a book on visualization using a new programming language called C++. The book included code, and in order to encourage reuse and improve the general standard in the field they were granted authorization by GE to retain copyright and publish the code with the book. This would later become the Visualization Toolkit (VTK). In 1998 two of the authors left GE and with several others founded a small company in upstate New York named Kitware. Fifteen years later the adventure continues, with over 100 employees and sustained year-on-year growth. ### Baseline Implementations As noted, VTK began life as code to support a book. The goal was to improve the state-of-the-art in the visualization community by putting tested and verified implementations of important visualization and analysis algorithms into a software library that was permissively licensed and available for use by a broad and open community. The intent was to provide these implementations as a concrete instantiation of complex algorithms so that researchers around the world did not have to do this from the papers of the time, with varying degrees of success and different levels of verification. This same goal lies at the heart of the Insight Toolkit in medical imaging for image segmentation and registration. Over the years things such as the Insight Journal were added so that new algorithms could be proposed in formal publications that included code, data, and baselines demonstrating the algorithm. If accepted, the algorithm would be merged into the main codebase and made available to the wider community. ### Business Model The majority of well-known software companies use a licensing and intellectual property model in order to derive revenue from their development activity. In the sciences this is often bolstered by funding from major agencies as well. Kitware generates no intellectual property, and has moved away from the licensing model to a large extent—focusing our business activity on a services model. This is an extremely effective strategy when working in high- performance computing, and addresses one of the major flaws in many open- source projects where support is difficult to find. Services make up a larger share of the industry sector, with companies such as IBM and Red Hat deriving most of their income from services rather than licensing. This model enables us to work with academics, national laboratories, and industry as partners often with joint funding streams to develop new features and make headway in different avenues of research and development. Through the use of permissive licensing models, rigorous software processes and agile development methodologies we have been able to make significant progress and received recognition for major projects such as VTK and ParaView. ### Developing Communities One of our core focus areas at Kitware is software process and developing robust communities around software projects that can grow beyond individual contracts. This process is shown at a high level in Figure 1 where developers commit code, it is automatically tested, dashboards gather testing information and disseminate that information. This process is powered by CMake, which began as a build system for ITK when existing build systems proved inadequate, with the simple goal of building C++ code on all major platforms. This was later augmented with CTest that would run and report the results of automated testing to Dart, and later CDash. Packaging was addressed using CPack, enabling some degree of abstraction when creating binary installers which are now routinely generated each night so that non-developers can install the latest build of a software project to see if a bug has been fixed. Figure 1: Overview of the Kitware software process. Kitware has built up software workflows around version control systems such as RCS, CVS, Subversion and more recently Git (among other distributed version control systems). These processes are designed to minimize the barrier to entry while maintaining the high quality of the software using automated repository hooks, continuous integration, and advanced testing capabilities such as image-based regression images to verify that visualization algorithms produce consistent and correct output on a large number of platforms beyond the capability of any individual contributor. Any community needs to establish a strong process to remain viable over the long term. This requires a mixture of technical and social resources to facilitate productive engagement and growth. Scientific projects often have limited resources, with a high turnover of developers/engineers as they move through their academic careers. This means that it is essential to establish procedures to help new community members get up to speed, and to retain group knowledge as people move on to new positions. Some will maintain long-term ties through different positions, as is normal in more mainstream open-source projects, others will not and it is important to ensure the parts of the project they developed do not atrophy. ### Open Software Process Kitware engineers and researchers have experimented with a number of techniques to improve induction of new community members and employees into our company and software communities, to support open-source development, and the growth of our projects. Most importantly, these experiments are more than just an academic exercise. All of the techniques are applied as part of our ongoing, commercial, software business. The techniques that offer pragmatic benefits to our business and open-source communities survive. Figure 2 shows a more detailed view of the software process used by the Visualization Toolkit (VTK). Figure 2: VTK’s software process incorporating online code review, automated testing of topic branches, and nightly testing with distributed mirrors. Recently we started using Gerrit to conduct online code reviews, and added automated triggers so that proposed changes are built on all major operating systems. This simple modification allows us to highlight potential platform- specific issues before engineers review code, saving scarce engineering time. Our software projects are developed using test-driven development methodologies, and require the submission of tests for new code or features ensuring that these features are automatically tested and verified before they are merged. We have a strong commitment to testing. We run nightly dashboards to provide wider coverage, ensuring that any problems are normally highlighted within 24 hours of merging them into the repository and allowing developers to review and remedy regressions while the changes are still fresh in their mind. All of these processes are learning opportunities, enabling developers to quickly discover what will and won’t work with different environments as new features are developed. These advantages were only made possible in recent years by migrating to distributed version control systems that enable development outside of the main repository and for changes to be pushed to staging locations before being merged into the main development branches. We have worked hard to close the gap between developers of the frameworks and applications and their users by automatically generating binary installers. This enables them to download the latest nightly binary of any given project to check if their bug still exists, or work more closely with developers on exercising new features before the final release is made. Underlying all of this are multiple communication mediums including mailing lists, wikis, and bug trackers to offer a comprehensive set of community tools. Having a commercial company backing up these projects also offers the ability to provide books, webinars, and on-site courses. ### Mature Communities Kitware hosts or plays a leading role in the maintenance and engineering on a number of projects, and over the years has learned to be adaptable. The oldest Kitware-led community is VTK, which also has one of the longest running version histories of any project whose history is publicly available. This is a community that grew up organically around the VTK book, and many pieces of the code were experimented with and added as the community grew. According to Ohloh the project has had nearly 55,000 commits made by over 200 contributors and includes more than 3.5 million lines of code. The first recorded VTK commit occurred nearly 20 years ago and development remains active. Even when looking at the last 12 months the project has seen over 3,000 commits from 66 contributors under a permissive BSD license. Figure 3 shows a graphical overview of the project’s activity from 1994 to present. Figure 3: Statistics generated by Ohloh showin lines of code, and commit rates for VTK. Even so, these numbers understate the total significance. For instance, they do not count the number of commercial contracts where Kitware supported the toolkit by providing engineers to extend VTK, or create/augment focused applications that leverage VTK, or research conducted using VTK. The development is also funded by several ongoing contracts with established groups, and funding streams from various federal agencies such as the DOE, NIH, DOD, DARPA, and NSF. Due to VTK’s importance to researchers in the medical area, it was granted a rare R01 maintenance grant from the NIH. Under this contract a consortium of companies and universities led by Kitware is working to update and overhaul aspects of the VTK rendering subsystems and to make other improvements over the next four years. VTK was able to garner a large number of support letters from the community of users and developers making use of VTK in their research, applications and products. After the success of VTK, the ITK project was, at least in part, founded to emulate the success of VTK in the medical segmentation and registration field. ITK receives ongoing funding from the National Library of Medicine, initially to provide algorithms that could operate on and make sense of the data from the visible human project. It is a little younger, with its first commit in 2000, and is interesting as it was one of the first major projects that began after Kitware’s founding and was designed from the ground up as an open-source project. It has seen nearly 45,000 commits in that time from over 200 contributors with over 1.5 million lines of code. Over the last year more than 1300 commits were made by 67 contributors under a permissive Apache 2 license. The ITK project also has an associated non-profit organization that is used to coordinate community efforts and manage some of the funding and development. The history of some of these older projects that have seen one or two decades of development effort should be contrasted with younger projects at different stages of development. Experimentation is encouraged, and not all of the projects that are started will achieve the critical mass required to become successful open source projects, but our established processes seek to make experimentation easy and the permissive open-source licensing means that even “failed” projects will often live on in other forms if the code developed for the project is useful. A brief summary of some of these projects include Arbor/Avatol—an NSF funded project where Kitware is an engineering subcontractor in a larger project led by the University of Idaho. The goals of this project are to provide tools for the analysis and exploration of the Tree of Life. The XDATA project is a new project funded by DARPA with a large consortium (led by Kitware as the prime contractor) tasked with creating open-source technology to address big data analytics. The Open Chemistry project was funded by a DOD SBIR awarded to Kitware, with the aim of creating a suite of desktop applications for computational chemists. In addition to creating, growing and shepherding our own communities Kitware has more recently begun helping other organizations to build successful communities. This includes the VA with the OSEHRA project, NA-MIC, and caBIG, all of whom use Kitware’s expertise to assist them in growing vibrant and successful communities around their projects. ### Concluding Remarks Sustainable software ecosystems are difficult to build, and require concerted effort, community norms and collaborations. In science it is especially important to establish communities in which faculty, staff, students and open- source professionals work together and treat software as a first-class product of scientific investigation—just as mathematics is treated in the physical sciences. Kitware has a rich history of establishing collaborative projects in the science, engineering and medical research fields, and continues to work on improving that model as new technologies and approaches become available. This approach closely follows and is enhanced by the movement towards practicing open, reproducible research in the sciences where data, source code, methodology and approach are all available so that complex experiments can be independently reproduced and verified. ### License This document is released under the Creative Commons Attribution 3.0 license (CC-BY), see http://creativecommons.org/licenses/by/3.0/.	arxiv-papers	2013-09-11T20:52:56	2024-09-04T02:49:50.769299	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Marcus D. Hanwell, Amitha Perera, Wes Turner, Patrick O'Leary, Katie\n Osterdahl, Bill Hoffman and Will Schroeder", "submitter": "Marcus Hanwell", "url": "https://arxiv.org/abs/1309.2966" }
1309.2975	These are not the k-mers you are looking for: efficient online k-mer counting using a probabilistic data structure Qingpeng Zhang1, Jason Pell1, Rosangela Canino-Koning1, Adina Chuang Howe2,3, C. Titus Brown1,2∗ 1 Department of Computer Science and Engineering, Michigan State University, East Lansing, MI, USA 2 Department of Microbiology and Molecular Genetics, Michigan State University, East Lansing, MI, USA 3 Department of Plant, Soil, and Microbial Sciences, Michigan State University, East Lansing, MI, USA $\ast$ E-mail: [email protected] ## Abstract K-mer abundance analysis is widely used for many purposes in nucleotide sequence analysis, including data preprocessing for de novo assembly, repeat detection, and sequencing coverage estimation. We present the khmer software package for fast and memory efficient online counting of k-mers in sequencing data sets. Unlike previous methods based on data structures such as hash tables, suffix arrays, and trie structures, khmer relies entirely on a simple probabilistic data structure, a Count-Min Sketch. The Count-Min Sketch permits online updating and retrieval of k-mer counts in memory which is necessary to support online k-mer analysis algorithms. On sparse data sets this data structure is considerably more memory efficient than any exact data structure. In exchange, the use of a Count-Min Sketch introduces a systematic overcount for k-mers; moreover, only the counts, and not the k-mers, are stored. Here we analyze the speed, the memory usage, and the miscount rate of khmer for generating k-mer frequency distributions and retrieving k-mer counts for individual k-mers. We also compare the performance of khmer to several other k-mer counting packages, including Tallymer, Jellyfish, BFCounter, DSK, KMC, Turtle and KAnalyze. Finally, we examine the effectiveness of profiling sequencing error, k-mer abundance trimming, and digital normalization of reads in the context of high khmer false positive rates. khmer is implemented in C++ wrapped in a Python interface, offers a tested and robust API, and is freely available under the BSD license at github.com/ged-lab/khmer. ## Introduction The goal of k-mer counting is to determine the number of occurrences for each fixed-length word of length k in a DNA data set [1]. Efficient k-mer counting plays an important role in many bioinformatics approaches, including data preprocessing for de novo assembly, repeat detection, and sequencing coverage estimation [2]. Short-read shotgun sequencing data is both relatively sparse in k-mers and contains many erroneous k-mers. For typical values of k such as 32 these data sets are sparse, as only a small fraction of the total possible number of k-mers ($4^{32}$) are actually present in any genome or read data sets derived from the genome. The high error rate (e.g. Illumina has a 0.1-1% per-base error rate [3]) generates many unique k-mers. As the total number of generated reads increases, the total number of errors grows with it linearly. This leads to data sets where the erroneous k-mers vastly outnumber the true k-mers [4]. Tracking and counting the resulting large number of k-mers, most of which are erroneous, has become an unavoidable and challenging task in sequence analysis [5]. A variety of k-mer counting approaches, and standalone software packages implementing them, have emerged in recent years; this includes Tallymer, Jellyfish, BFCounter, DSK, KMC, Turtle and KAnalyze [2, 1, 6, 7, 8, 9, 10]. These approaches and implementations each offer different algorithmic trade- offs and enable a non-overlapping set of functionality. Tallymer uses a suffix tree to store k-mer counts in memory and on disk [2]. Jellyfish stores k-mer counts in in-memory hash tables, and makes use of disk storage to scale to larger data sets [1]. BFCounter uses a Bloom filter as a pre-filter to avoid counting unique k-mers, and is the first published probabilistic approach to k-mer counting [6]. DSK adopts an approach to k-mer counting that enables time- and memory-efficient k-mer counting with an explicit trade-off between disk and memory usage [7]. KMC and KAnalyze rely primarily on fast and inexpensive disk access to count k-mers in low memory [8, 10]. Turtle provides several different containers that offer different false positive and false negative tradeoffs when counting k-mers [9]. Our motivation for exploring efficient k-mer counting comes from our work with metagenomic data, where we routinely encounter data sets that contain $300\times 10^{9}$ bases of DNA and over 50 billion distinct k-mers [11]. To efficiently filter, partition, and assemble these data, we need to store counts for each of these k-mers in main memory, and query and update them in realtime — a set of functionality not readily offered by current packages. Moreover, we wish to enable the use of cloud and desktop computers, which may have poor I/O performance or limited memory. These needs have dictated our exploration of efficient in-memory k-mer counting techniques. Below, we describe an implementation of a simple probabilistic data structure for k-mer counting. This data structure is based on a Count-Min Sketch [12], a generalized probabilistic data structure for storing the frequency distributions of distinct elements. Our implementation extends an earlier implementation of a Bloom filter [13], which has been previously used in bioinformatics applications, such as sequence matching [14], k-mer counting [6], and de Bruijn graph storage and traversal [15, 16]. Many other variations of Bloom filters have been proposed [17], including counting Bloom filters [18], multistage filters [19], and spectral Bloom filters [20], which are related to the Count-Min Sketch and our khmer implementation. Probabilistic approaches can be particularly memory efficient for certain problems, with memory usage significantly lower than any exact data structure [15]. However, their use introduces set membership or counting false positives, which have effects that must be analyzed in the context of specific problems. Moreover, unlike existing techniques, the Count-Min Sketch stores only counts; k-mers must be retrieved from the original data set. In exchange, the low memory footprint enabled by this probabilistic approach enables online updating and retrieval of k-mer counts entirely in memory, which in turn supports streaming applications such as digital normalization [21]. We use the Amazon cloud to compare time, memory, and disk usage of our k-mer counting implementation with that of other k-mer counting software packages, for two problems. First, we generate a k-mer abundance distribution for large data sets; and second, we query many individual k-mer counts at random from a previously constructed k-mer count database. We show that khmer is competitive in speed, memory, and disk usage for these problems. We also analyze the effects of counting error on calculations of the k-mer count in sequencing data sets, and in particular on metagenomic data sets. Finally, we discuss khmer’s miscount performance in the context of two specific applications: low- abundance k-mer trimming of reads, and digital normalization. The khmer software [22] is implemented in C++ in a Python wrapper, enabling flexible use and reuse by users with a wide range of computational expertise. The software package is freely available for academic and commercial use and redistribution under the BSD license at github.com/ged-lab/khmer/. khmer comes with substantial documentation and many tutorials, and contains extensive unit tests. Moreover, we have built several applications on top of khmer, including memory-efficient de Bruijn graph partitioning [15] and lossy compression of short-read data sets for assembly [21]. ## Results ### Implementing a Count-Min Sketch for k-mers The two basic operations supported by khmer are increment_count(kmer) and c = get_count(kmer). Both operate on the data structure in memory, such that neither incrementing a count nor retrieving a count involves disk access. The implementation details are similar to those of the Bloom filter in [15], but with the use of 8 bit counters instead of 1 bit counters. Briefly, Z hash tables are allocated, each with a different size of approximately H bytes ($H_{1},H_{2},...,H_{Z}$); the sum of these hash table sizes must fit within available main memory. To increment the count for a particular k-mer, a single hash is computed for the k-mer, and the modulus of that hash with each hash table’s size H gives the location for each hash table; the associated count in each hash table is then incremented by 1. We use different sizes for each hash table so as to vary the hash function. Even if two k-mers have the same modulus in one hash table (a collision), they are unlikely to collide in the other hash tables. To retrieve the count for a k-mer, the same hash is computed and the minimum count across all hash tables is computed. While different in implementation detail from the standard Bloom filter, which uses a single hash table with many hash functions, the performance details are identical [15]. One particularly important feature of the Count-Min Sketch is that the counting error is one-sided [12]. Because counts are only incremented, collisions result in inflated miscounts; if there is no collision for a particular k-mer, the count is correct. An additional benefit of the Count-Min Sketch is that it is extremely easy to implement correctly, needing only about 3 dozen lines of C++ code for a simple threadsafe implementation. (We have described how khmer scales with multiple threads in [23].) To determine the expected false positive rate — the average frequency with which a given k-mer count will be incorrect when retrieved — we can look at the hash table load. Suppose N distinct k-mers have been counted using Z hash tables, each with size H. The probability that no collisions happened in a specific entry in one hash table is $(1-1/H)^{N}$, or approximately $e^{-N/H}$. The individual collision rate in one hash table is then $\approx 1-e^{-N/H}$. The total collision rate, which is the probability that a collision occurred in each entry where a k-mer maps across all Z hash tables, is $\approx(1-e^{-N/H})^{Z}$, which is also the expected false positive rate. While the false positive rate can easily be calculated from the hash table load, the average miscount — the degree to which the measured count differs from the true count — depends on the k-mer frequency distribution, which must be determined empirically. We analyze the effects of this below. ### Choosing number and size of hash tables used for k-mer counting The false positive rate depends on the number of distinct k-mers $N$, the number of hash tables $Z$, and the size of the hash tables $H$: $f\approx(1-e^{-N/H})^{Z}$, with an associated memory usage of $M=HZ$. We face two common scenarios: one in which we have a fixed number of k-mers $N$ and fixed memory $M$ and we want to calculate the optimal number of hash tables $Z$; and one in which we have a desired maximum false positive rate $f$ and a fixed number of k-mers $N$, and we want to calculate the minimum memory usage required to achieve $f$. For fixed memory $M$ and number of distinct k-mers $N$, the optimal number of hash tables can be found by minimizing $f$; taking the derivative, $df/dZ$, with $f\approx\exp(Z\log(1-e^{-ZN/M}))$ and solving for 0, we find that $f$ is minimized when $Z=\log(2)(M/N)$ (see [24] for details). Given a desired false positive rate $f$ and a fixed number of k-mers $N$, the optimal memory usage can be calculated as follows. First, the optimal number of hash tables is determined by the expected false positive rate alone: $Z=\log_{0.5}f$. Using this $Z$, the minimum average hash table size $H$ necessary to achieve $f$ can be calculated as $H=(\log_{0.6185}(f)\times N)/Z$ (see [24] for details). A remaining problem is that the number of distinct k-mers $N$ is typically not known. However, memory- and time-efficient algorithms for calculating $N$ do exist and we plan to implement this in khmer in the future [25]. ### khmer efficiently calculates k-mer abundance histograms We measured time and memory required to calculate k-mer abundance histograms in five soil metagenomic read data sets using khmer, Tallymer, Jellyfish, DSK, KMC, Turtle, and KAnalyze (Table 1; Figures 1 and 2). We chose to benchmark abundance histograms because this functionality is common to all the software packages, and is a common analysis approach for determining assembly parameters [26]. We applied each package to increasingly large subsets of a 50m read soil metagenome data set [11]. For the BFCounter, KMC, Turtle and KAnalyze packages, which do not generate k-mer abundance distribution directly, we output the frequency of each k-mer to a file but do no further analysis. Figure 1 shows that the time usage of the khmer approach is comparable to DSK and BFCounter, and, as expected, increases linearly with data set size. Tallymer is the slowest of the four tools in this testing, while KMC, Turtle, and Jellyfish are the fastest. From Figure 2, we see that the memory usage of Jellyfish, Tallymer, BFCounter, and Turtle increases linearly with data set size. Tallymer uses more memory than Jellyfish generally, while BFCounter and Turtle have considerably lower memory usage. DSK, KMC, and KAnalyze use constant memory across the data sets, but at the cost of more limited functionality (discussed below). The memory usage of khmer also increases linearly with data set size as long as we hold the false positive rate constant. However, the memory usage of khmer varies substantially with the desired false positive rate: we can decrease the memory usage by increasing the false positive rate as shown in Figure 2. We also see that with a low false positive of 1%, the memory usage is competitive with Tallymer and Jellyfish; with a higher 5% false positive rate, the memory usage is lower than all but the disk-based DSK; with an false positive rate as high as 20%, the memory usage is further lower, close to DSK, KAnalyze, and KMC. We also measured disk usage during counting. Figure 3 shows that the disk usage also increases linearly with the number of k-mers in the data set. For a high-diversity metagenomic data set of 5 GB, the disk usage of both Jellyfish and Tallymer is around 30 GB. khmer counts k-mers entirely in working memory and does not rely on any on-disk storage to store or retrieve k-mer counts, although for practicality the hash tables can be saved for later reuse; the uncompressed disk usage for khmer in Figure 3 is the same as its memory. At the expense of more time, khmer supports saving and loading gzip-compressed hash tables, which are competitive in size to DSK’s on-disk database (Figure 3, dashed line). ### khmer accesses k-mer counts efficiently We measured the time it took to access 9.7m 22-mers across five different data sets after the initial databases had been built (Figure 4). Note that Tallymer, Jellyfish, and khmer all support random access to k-mer counts, while BFCounter, DSK, KMC, Turtle and KAnalyze do not. Here, khmer performed well, dramatically outperforming Jellyfish and Tallymer. In all three cases, system time dominated the overall time required to retrieve k-mers, suggesting that the primary reason for the increase in retrieval time was due to the increased size of the database on the disk (data not shown). In particular, khmer is independent of the size of the database in retrieval time once the hash tables are loaded into memory. ### The measured counting error is low on short-read data Due to the use of Count-Min Sketch and its lack of collision tracking, khmer will report some incorrect counts for k-mers; these counts are always higher than the true counts, up to the bound of 255 (a limit imposed by our use of 8-bit counters). The frequency with which incorrect counts are reported can be estimated from the hash table load. However, the expected miscount — the difference between the true k-mer frequency and the reported k-mer frequency — cannot be calculated without knowing the distribution of k-mer abundances; in general, the average miscount will be small if the data is left-skewed. As noted by Melsted and Pritchard, a large number of k-mers in short-read data are low-abundance, leading to precisely the skew that would yield low miscounts [6]. Here we use both real and simulated data sets to evaluate the counting performance in practice. Figure 5 shows the relationship between average miscount and counting false positive rate for five different test data sets with similar numbers of distinct k-mers: one metagenome data set; a simulated set of random k-mers; a simulated set of reads, chosen with 3x coverage and 1% error; a simulated set of reads (3x) with no error; and a set of E. coli reads (Table 2). Even when the counting false positive rate is as high as 0.9 — where 90% of k-mers have an incorrect count — the average miscount is still below 4. We separately analyzed the average percentage miscount between true and false k-mers; e.g. an miscount of 4 for a k-mer whose true count is 1 would be 400%. Figure 6 shows the relationship between average miscount and counting false positive rate for the same five data sets as in Figure 5. For a false positive rate of 0.1 (10% of k-mer counts are incorrect), the average percentage miscount is less than 10% for all five data sets; this will of course generally be true, because the average miscount is bounded by the product of the false positive rate with k-mer abundance. We see here that for a fixed false positive rate, the simulated reads without error have the highest average miscount, and the randomly generated k-mers have the lowest average miscount. This is because these two abundance distributions have the least and most left-skew, respectively: the simulated reads without error have no abundance-1 k-mers, while the randomly generated k-mers are entirely low abundance. ### Sequencing error profiles can be measured with k-mer abundance profiles One specific use for khmer is detecting random sequencing errors by looking at the k-mer abundance distribution within reads [27]. This approach, known also as “k-mer spectral analysis”, was first proposed in by [28] and further developed in [29]. The essential idea is that low-abundance k-mers contained in a high-coverage data set typically represent random sequencing errors. A variety of read trimming and error correcting tools use k-mer counting to reduce the error content of the read data set, independent of quality scores or reference genomes [30]. This is an application where the counting error of the Count-Min Sketch approach used by khmer may be particularly tolerable: it will never falsely call a high-abundance k-mer as low-abundance because khmer never underestimates counts. In Figure 7, we use khmer to examine the sequencing error pattern of a 5m-read subset of an Illumina reads data set from single-colony sequencing of E. coli [31]. The high rate of occurrence of unique k-mers close to the 3’ end of reads is due to the increased sequencing error rate at the 3’ end of reads. ### khmer can be applied iteratively to read trimming We next evaluated the effect of false-positive induced miscounts on read trimming, in which reads are truncated at the first low-abundance k-mer. Because the Count-Min Sketch never undercounts k-mers, reads will never be erroneously trimmed at truly high-abundance k-mers; however, reads may not be trimmed correctly when miscounts inflate the count of low-abundance k-mers. In cases where many errors remain, read trimming can potentially be applied multiple times, with each round reducing the total number of k-mers and hence resulting in lower false positive rates for the same memory usage. We performed six iterations of unique k-mer trimming on 5 million Illumina reads from sequencing of E. coli, with memory usage less than 30 MB. For each iteration we measured empirical false positive rate compared with number of bases trimmed as well as the total number of k-mers (Table 3). In the first round, the estimated false positive rate was 80.0%, and 13.5% of the total bases were removed by trimming reads at low-abundance k-mers; the second iteration had a false positive rate of 37.7%, and removed only 1.5% additional data; and by the fourth iteration the false positive rate was down to 23.2% with 0.0% of the data removed. The elimination of so many unique k-mers (column 5) in the first pass was unexpected: the high false positive rate should have resulted in fewer k-mers being identified as unique, were the erroneous k-mers independent of each other. Upon examination, we realized that in Illumina data erroneous k-mers typically come from substitution errors that yield runs of up to k erroneous k-mers in a row [30]. When trimming reads with high false positive rates, these runs are typically trimmed after the first few unique k-mers, leaving unique k-mers at the 3’ end. Because of this we hypothesized that high-FP rate trimming would result in the retention of many unique k-mers at the 3’ end of the read, and this was confirmed upon measurement (Table 3, column 6, pass 1 vs pass 2). In comparison to quality-based trimming software such as seqtk and FASTX, trimming at unique k-mers performed very well: in this data set, all unique k-mers represent errors, and even with an initial false positive rate of 80%, khmer outperformed all but the most stringent seqtk run (Table 3). With a lower false positive rate or multiple passes, khmer eliminates more erroneous k-mers than seqtk or FASTX. The tradeoff here is in memory usage: for larger data sets, seqtk and FASTX will consume the same amount of memory as on smaller data sets, while khmer’s memory usage will need to grow with the data set size. ### Using khmer for digital normalization, a streaming algorithm Digital normalization is a lossy compression algorithm that discards short reads based on saturating coverage of a de Bruijn graph [21]. While several non-streaming implementations exist, including Trinity’s in silico normalization [32, 33], digital normalization can be efficiently implemented as a streaming algorithm. In the streaming implementation, if a read is not kept, it is not loaded into the Count-Min Sketch structure, and the false positive rate does not increase. For high coverage data sets, the digital normalization algorithm is sublinear in memory because it does not collect the majority of k-mers in those data sets [21]. This has the advantage of enabling low-memory preprocessing of both high-coverage genomic data sets, as well as mRNAseq or metagenomic data sets with high-coverage components [21, 11]. While digital normalization is already implemented inside khmer, previous work did not explore the lower bound on memory usage for effective digital normalization. In particular, the effects of high false positive rates have not been examined in any prior work. We applied digital normalization to the E. coli data set used above, and chose seven different Count-Min Sketch sizes to yield seven different false positive rates 4. The data set was normalized to a k-mer coverage of 20 and the resulting data were evaluated for retention of true and erroneous k-mers, as in [21] (Table 4). The results show that digital normalization retains the same set of underlying “true” k-mers until the highest false positive rate of 100% (Table 4, column 5), while discarding only about 2% additional reads (Table 4, column 6). To evaluate the effect of digital normalization with high false positive rates on actual genome assembly, we next performed normalization to a coverage of 20 with the same range of false positive rates as above. We then assembled this data with Velvet [34] and compared the resulting assemblies to the known E. coli MG1655 genome using QUAST (Table 5). To our surprise, we found that even after executing digital normalization with a false positive rate of 83.2%, a nearly complete assembly was generated. No progressive increase in misassemblies (measured against the real genome with QUAST) was seen across the different false positive rates (data not shown). This suggests that below 83.2% FP rate, the false positive rate of digital normalization has little to no effect on assembly quality with Velvet. (Note that the Velvet assembler itself used considerably more memory than digital normalization.) While these results are specific to Velvet and the coverage parameters used in digital normalization, they do suggest that no significant information loss occurs due to false positive rates below 80%. Further evaluation of assembly quality in response to different normalization parameters and assemblers is beyond the scope of of this paper. ## Discussion ### khmer enables fast, memory-efficient online counting khmer enables memory- and time-efficient online counting (Figures 1, 2, and 4). This is particularly important for the streaming approaches to data analysis needed as data set sizes increase. Because query and updating of k-mer counts can be done directly as data is being loaded, with no need for disk access or an indexing step, khmer can also perform well in situations with poor disk I/O performance. (Note that BFCounter also supports online k-mer counting [6].) ### khmer is a generally useful k-mer counting approach In addition to online counting, khmer offers a general range of useful performance tradeoffs for disk I/O, time and memory. From the performance comparison between khmer and other k-mer counting packages in calculating k-mer abundance distributions, khmer is comparable with existing packages. In time, khmer performs competitively with DSK and BFCounter (Figure 1); khmer also provides a way to systematically trade memory for miscounts across a wide range of parameters (Figure 2). khmer’s uncompressed disk storage is competitive with Jellyfish, and, in situations where disk space is at a premium, khmer can take advantage of gzip compression to provide storage similar to that of DSK (Figure 3, purple line with boxes). KMC, DSK, and KAnalyze perform especially well in memory usage for calculating the abundance distribution of k-mers. However, in exchange for this efficiency, retrieving specific k-mer counts at random is likely to be quite slow, as DSK is optimized for iterating across partition sets of k-mers rather than randomly accessing k-mer counts. For retrieving the counts of individual k-mers, khmer is significantly faster than both Tallymer and Jellyfish. This is not surprising, since this was a primary motivation for the development of khmer. ### khmer memory usage is fixed and low The memory usage of the basic Count-Min Sketch approach is fixed: khmer’s memory usage does not increase as data is loaded. While this means that khmer will never crash due to memory limitations, and all operations can be performed in main memory without recourse to disk storage, the false positive rate may grow too high. Therefore the memory size must be chosen in light of the false positive rate and miscount acceptable for a given application. In practice, we recommend choosing the maximum available memory, because the false positive rate decreases with increasing memory and there are no negative effects to minimizing the false positive rate. For any given data set, the size and number of hash tables will determine the accuracy of k-mer counting with khmer. Thus, the user can control the memory usage based on the desired level of accuracy (Figure 2). The time usage for the first step of k-mer counting, consuming the reads, depends on the total amount of data, since we must traverse every k-mer in every read. The second step, k-mer retrieval, is algorithmically constant for fixed k; however, for practicality, the hash tables are usually saved to and loaded from disk, meaning that k-mer retrieval time depends directly on the size of the database being queried. The memory usage of khmer is particularly low for sparse data sets, especially since only main memory is used and no disk space is necessary beyond that required for the read data sets. This is no surprise: the information theoretic comparison in [15] shows that, for sparse sequencing data sets, Bloom filters require considerably less memory than any possible exact information storage for a wide range of false positive rates and data set sparseness. In our implementation we use 1 byte to store the count of each k-mer in the data structure. Thus the maximum count for a k-mer will be 255. In cases where tracking bigger counts is required, khmer also provides an option to use an STL map data structure to store counts above 255, with the trade-off of significantly higher memory usage. In the future, we may extend khmer to counters of arbitrary bit sizes. ### False positive rates in k-mer counting are low and predictable The Count-Min Sketch is a probabilistic data structure with a one-sided error that results in random overestimates of k-mer frequency, but does not generate underestimates. In the Count-Min Sketch, the total memory usage is fixed; the memory usage, the hash functions, and the total number of distinct objects counted all influence the accuracy of the count. While the probability of an inaccurate count can easily be estimated based on the hash table load, the miscount size is dependent on details of the frequency distribution of k-mers [12]. More specifically, in the analysis of the Count-Min Sketch, the difference between the incorrect count and actual count is related to the total number of k-mers in a data set and the size of each hash table [12]. Further study has shown that the behavior of Count-Min Sketch depends on specific characteristics of the data set under consideration, especially left-skewness [35, 36]. These probabilistic properties suit short reads from next generation sequencing data sets: the miscounts are low because of the highly left-skewed abundance distribution of k-mers in these data sets. Figures 5 and 6 demonstrate these properties well. We see more correct counting for error-prone reads from a genome than for error-free reads from a genome, with a normal distribution of k-mer abundance. Thus, this counting approach is especially suitable for high diversity data sets, such as metagenomic data, in which a larger proportion of k-mers are low abundance or unique due to sequencing errors. ### Real-world applications for khmer For many applications, an approximate k-mer count is sufficient. For example, when eliminating reads with low abundance k-mers, we can tolerate a certain number of low-frequency k-mers remaining in the resulting data set falsely. If RAM-limited we can do the filtering iteratively so that at each step we are making more effective use of the available memory. In practice, we have found that a false positive rate of between 1% and 10% offers acceptable miscount performance for a wide range of tasks, including error profiling, digital normalization and low-abundance read-trimming. Somewhat surprisingly, false positive rates of up to 80% can still be used for both read trimming and digital normalization in memory-limited circumstances, although multiple passes across the data may be needed. For many applications, the fact that khmer does not break an imposed memory bound is extremely useful, since for many data sets — especially metagenomic data sets — high memory demands constrain analysis [11, 37]. Moreover, because the false positive rate is straightforward to measure, the user can be warned that the results should be invalidated when too little memory is used. When combined with the graceful degradation of performance for both error trimming and digital normalization, khmer readily enables analysis of extremely large and diverse data sets [38]. In an experiment to assemble the reads of a soil metagenomic sample collected from Iowa prairie, the number of reads to assemble drops from 3.3 million to 2.2 million and the size of the data set drops from 245GB to 145GB accordingly after digital normalization [11]. 240GB memory was used in the process. This also shows that khmer works well to analyze large, real-world metagenomic data sets. ### Conclusion K-mer counting is widely used in bioinformatics, and as sequencing data set sizes increase, graceful degradation of data structures in the face of large amounts of data has become important. This is especially true when the theoretical and practical effects of the degradation can be predicted (see e.g. [6, 15, 9]). This is a key property of the Count-Min Sketch approach, and its implementation in khmer. The khmer software implementation offers good performance, a robust and well- tested Python API, and a number of useful and well-documented scripts. While Jellyfish, DSK, KMC, and Turtle also offer good performance, khmer is competitive, and, because it provides a Python API for online counting, is flexible. In memory-limited situations with poor I/O performance, khmer is particularly useful, because it will not break an imposed memory bound and does not require disk access to store or retrieve k-mer counts. However, in exchange for this memory guarantee, counting becomes increasingly incorrect as less memory is used or as the data set size grows large; in many situations this may be an acceptable tradeoff. ### Future considerations Applying khmer to extremely large data sets with many distinct k-mers requires a large amount of memory: approximately 446 GB of memory is required to achieve an false positive rate of 1% for $50\times 10^{9}$ k-mers. It is possible to reduce the required memory by dividing k-mer space into multiple partitions and counting k-mers separately for each partition. Partitioning k-mer space into $M$ partitions results in a linear decrease in the number of k-mers under consideration, thus reducing the occupancy by a constant factor $M$ and correspondingly reducing the collision rate. Partitioning k-mer space is a generalization of the systematic prefix filtering approach, where one might first count all k-mers starting with AA, then AC, then AG, AT, CA, etc., which is equivalent to partitioning k-mer space into 16 equal-sized partitions. These partitions can be calculated independently, either across multiple machines or iteratively on a single machine, and the results stored for later comparison or analysis. This is similar to the approach taken by DSK [7], and could easily be implemented in khmer. Further optimization of khmer on single machines, e.g. for multi-core architectures, is unlikely to achieve significantly greater speed. Past a certain point k-mer counting is fundamentally I/O bound [23]. Perhaps the most interesting future direction for probabilistic k-mer counting is that taken by Turtle [9], in which several data structures are provided, each with different tradeoffs, but with a common API. We hope to pursue this direction in the future by integrating such approaches into khmer. ## Methods ### Code and data set availability The version of khmer used to generate the results below is available at http://github.com/ged-lab/khmer.git, tag ’2013-khmer-counting’. Scripts specific to this paper are available in the paper repository at https://github.com/ged-lab/2013-khmer-counting. The IPython[39] notebook file and data analysis to generate the figures are also available in that github repository. Complete instructions to reproduce all of the results in this paper are available in the khmer-counting repository; see README.rst. ### Sequence Data One human gut metagenome reads data set (MH0001) from the MetaHIT (Metagenomics of the Human Intestinal Tract) project [40] was used. It contains approximately 59 million reads, each 44bp long; it was trimmed to remove low quality sequences. Five soil metagenomics reads data sets with different size were taken from the GPGC project for benchmark purpose (see Table 1). These reads are from soil in Iowa region and they are filtered to make sure there are less than 30% Ns in the read and each read is longer than 30 bp. The exact data sets used for the paper are available on Amazon S3 and the instructions to acquire these data sets are available in the paper repository on github.com. We also generated four short-read data sets to assess the false positive rate and miscount distribution. One is a subset of a real metagenomics data set from the MH0001 data set, above. The second consists of randomly generated reads. The third and fourth contain reads simulated from a random, 1 Mbp long genome. The third has a substitution error rate of 3%, and the fourth contains no errors. The four data sets were chosen to contain identical numbers of distinct 22-mers. The scripts necessary to regenerate these data are available in the paper repository on github.com. ### Count-Min Sketch implementation We implemented the Count-Min Sketch data structure, a simple probabilistic data structure for counting distinct elements [12]. Our implementation uses $Z$ independent hash tables, each containing a prime number of counters $H_{i}$. The hashing function for each hash table is fixed, and reversibly converts each DNA k-mer (for $k\leq 32$) into a 64-bit number to which the modulus of the hash table size is applied. This provides $Z$ distinct hash functions. To increment the count associated with a k-mer, the counter associated with the hashed k-mer in each of the $N$ hash tables is incremented. To retrieve the count associated with a k-mer, the minimum count across all $N$ hash tables is chosen. In this scheme, collisions are explicitly not handled, so the count associated with a k-mer may not be accurate. Because collisions only falsely increment counts, however, the retrieved count for any given k-mer is guaranteed to be no less than the correct count. Thus the counting error is one-sided. ### Hash function and khmer implementation The current khmer hash function works only for $k\leq 32$ and converts DNA strings exactly into 64-bit numbers. However, any hash function would work. For example, a cyclic hash would enable khmer to count k-mers larger in size than 32; this would not change the scaling behavior of khmer at all, and is a planned extension. By default khmer counts k-mers in DNA, i.e. strandedness is disregarded by having the hash function choose the lower numerical value for the exact hash of both a k-mer and its reverse complement. This behavior is configurable via a compile-time option. ### Comparing with other k-mer counting programs We generated k-mer abundance distribution from five soil metagenomic reads data sets of different sizes using khmer, Tallymer, Jellyfish, DSK, BFCounter, KMC, Turtle and KAnalyze. If the software does not include function to generate k-mer abundance distribution directly, we output the frequency of each k-mer in an output file. We fixed k at 22 unless otherwise noted. #### khmer: For khmer, we set hash table sizes to fix the false positive rate at either 1%, 5% or 20%, and used 8 threads in loading the data. We did the khmer random-access k-mer counting benchmark with a custom-written Python script khmer-count-kmers which loaded the database file and then used the Python API to query each k-mer individually. #### Tallymer: Tallymer is from the genometools package version 1.3.4. For the suffixerator subroutine we used: -dna -pl -tis -suf -lcp. We use the mkindex subroutine to generate k-mer abundance distribution, we used: -mersize 22. The Tallymer random access k-mer counting benchmark was done using the ’tallymer search’ routine against both strands; see the script tallymer- search.sh. #### Jellyfish: The Jellyfish version used was 1.1.10 and the multithreading option is set to 8 threads. Jellyfish uses a hash table to store the k-mers and the size of the hash table can be modified by the user. When the specified hash table fills up, Jellyfish writes it to the hard disk and initializes a new hash table. Here we use a similar strategy as in [6] and chose the minimum size of the hash tables for Jellyfish so that all k-mers were stored in memory. We ran Jellyfish with the options as below: jellyfish count -m 22 -c 2 -C for k=22. The Jellyfish random access k-mer counting benchmark was performed using the ’query’ routine and querying against both strands; see the script jelly- search.sh. #### DSK: We ran DSK with default parameters with -histo option to generate k-mer abundance distribution. The DSK version used was 1.5031. #### BFCounter: The BFcounter version used was 1.0 and the multithreading option is set to 8 threads. We ran BFCounter count subroutine with the options as below: BFCounter count -k 22 -t 8 -c 100000. -n option representing the estimated number of k-mers is adjusted to the different test data sets. This subroutine produces the actual count of k-mers in input files. We ran BFCounter dump subroutine with the options as below: BFCounter dump -k 22. This subroutine can write k-mer occurrences into a tab-separated text file. #### KMC: The KMC version used was 0.3. We ran both kmc and kmc_dump subroutines with default parameters. #### Turtle: The Turtle version used was 0.3. 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All programs executed in time approximately linear with the number of input reads. Figure 2: Memory usage of k-mer counting tools when calculating k-mer abundance histograms, with maximum resident program size (y axis, in GB) plotted against the total number of distinct k-mers in the data set (x axis, billions of k-mers). Figure 3: Disk storage usage of different k-mer counting tools to calculate k-mer abundance histograms in GB (y axis), plotted against the number of distinct k-mers in the data set (x axis). ∗Note that khmer does not use the disk during counting or retrieval, although its hash tables can be saved for reuse. Figure 4: Time for several k-mer counting tools to retrieve the counts of 9.7m randomly chosen k-mers (y axis), plotted against the number of distinct k-mers in the data set being queried (x axis). BFCounter, DSK, Turtle, KAnalyze, and KMC do not support this functionality. Figure 5: Relation between average miscount — amount by which the count for k-mers is incorrect — on the y axis, plotted against false positive rate (x axis), for five data sets. The five data sets were chosen to have the same total number of distinct k-mers: one metagenome data set; a set of randomly generated k-mers; a set of reads, chosen with 3x coverage and 1% error, from a randomly generated genome; a simulated set of error-free reads (3x) chosen from a randomly generated genome and a set of E. coli reads. Figure 6: Relation between percent miscount — amount by which the count for k-mers is incorrect relative to its true count — on the y axis, plotted against false positive rate (x axis), for five data sets. The five data sets are the same as in Figure 5. Figure 7: Number of unique k-mers (y axis) by starting position within read (x axis) in an untrimmed E. coli 100-bp Illumina shotgun data set, for k=17 and k=32. The increasing numbers of unique k-mers are a sign of the increasing sequencing error towards the 3’ end of reads. Note that there are only 69 starting positions for 32-mers in a 100 base read. ## Tables Table 1: Benchmark soil metagenome data sets for k-mer counting performance, taken from [11]. Data set \| size of file (GB) \| number of reads \| number of distinct k-mers \| total number of k-mers ---\|---\|---\|---\|--- subset 1 \| 1.90 \| 9,744,399 \| 561,178,082 \| 630,207,985 subset 2 \| 2.17 \| 19,488,798 \| 1,060,354,144 \| 1,259,079,821 subset 3 \| 3.14 \| 29,233,197 \| 1,445,923,389 \| 1,771,614,378 subset 4 \| 4.05 \| 38,977,596 \| 1,770,589,216 \| 2,227,756,662 entire data set \| 5.00 \| 48,721,995 \| 2,121,474,237 \| 2,743,130,683 Table 2: Data sets used for analyzing miscounts. Data set \| Size of data set file \| Number of total k-mers \| Number of distinct k-mers ---\|---\|---\|--- Real metagenomics reads \| 7.01M \| 2,917,200 \| 1,944,996 Totally random reads with randomly generated k-mers \| 3.53M \| 2,250,006 \| 1,973,059 Simulated reads from simulated genome with error \| 5.92M \| 3,757,479 \| 2,133,592 Simulated reads from simulated genome without error \| 9.07M \| 5,714,973 \| 1,989,644 Real E. coli reads \| 4.85M \| 4,004,911 \| 2,079,302 Table 3: Iterative low-memory k-mer trimming. The results of trimming reads at unique (erroneous) k-mers from a 5m read E. coli data set (1.4 GB) in under 30 MB of RAM. After each iteration, we measured the total number of distinct k-mers in the data set, the total number of unique (and likely erroneous) k-mers remaining, and the number of unique k-mers present at the 3’ end of reads. \| FP rate \| bases trimmed \| distinct k-mers \| unique k-mers \| unique k-mers at 3’ end ---\|---\|---\|---\|---\|--- untrimmed \| - \| - \| 41.6m \| 34.1m \| 30.4% khmer iteration 1 \| 80.0% \| 13.5% \| 13.3m \| 6.5m \| 29.8% khmer iteration 2 \| 40.2% \| 1.7% \| 7.6m \| 909.9k \| 12.3% khmer iteration 3 \| 25.4% \| 0.3% \| 6.8m \| 168.1k \| 3.1% khmer iteration 4 \| 23.2% \| 0.1% \| 6.7m \| 35.8k \| 0.7% khmer iteration 5 \| 22.8% \| 0.0% \| 6.6m \| 7.9k \| 0.2% khmer iteration 6 \| 22.7% \| 0.0% \| 6.6m \| 1.9k \| 0.0% filter by FASTX \| - \| 9.1% \| 26.6m \| 20.3m \| 26.3% filter by seqtk(default) \| - \| 8.9% \| 17.7m \| 12.1m \| 12.3% filter by seqtk(-q 0.01) \| - \| 15.4% \| 9.9m \| 5.1m \| 5.2% filter by seqtk(-b 3 -e 5) \| - \| 8.0% \| 34.5m \| 27.7m \| 25.3% Table 4: Low-memory digital normalization. The results of digitally normalizing a 5m read E. coli data set (1.4 GB) to C=20 with k=20 under several memory usage/false positive rates. The false positive rate (column 1) is empirically determined. We measured reads remaining, number of “true” k-mers missing from the data at each step, and the number of total k-mers remaining. Note: at high false positive rates, reads are erroneously removed due to inflation of k-mer counts. memory \| FP rate \| retained reads \| retained reads % \| true k-mers missing \| total k-mers ---\|---\|---\|---\|---\|--- before diginorm \| - \| 5,000,000 \| 100.0% \| 170 \| 41.6m 2400 MB \| 0.0% \| 1,656,518 \| 33.0% \| 172 \| 28.1m 240 MB \| 2.8% \| 1,655,988 \| 33.0% \| 172 \| 28.1m 120 MB \| 18.0% \| 1,652,273 \| 33.0% \| 172 \| 28.1m 60 MB \| 59.1% \| 1,633,182 \| 32.0% \| 172 \| 27.9m 40 MB \| 83.2% \| 1,602,437 \| 32.0% \| 172 \| 27.6m 20 MB \| 98.8% \| 1,460,936 \| 29.0% \| 172 \| 25.7m 10 MB \| 100.0% \| 1,076,958 \| 21.0% \| 185 \| 20.9m Table 5: E. coli genome assembly after low-memory digital normalization. A comparison of assembling reads digitally normalized with low memory/high false positive rates. The reads were digitally normalized to C=20 (see [21] for more information) and were assembled using Velvet. We measured total length of assembly, as well as percent of true MG1655 genome covered by the assembly using QUAST. memory \| FP rate \| N contigs \| total length(bases) \| % of true genome covered ---\|---\|---\|---\|--- before diginorm \| - \| 106 \| 4,546,051 \| 97.84% 2400 MB \| 0.0% \| 617 \| 4,549,235 \| 98.05% 240 MB \| 2.8% \| 87 \| 4,549,253 \| 98.04% 120 MB \| 18.0% \| 86 \| 4,549,335 \| 98.04% 60 MB \| 59.1% \| 90 \| 4,548,619 \| 98.03% 40 MB \| 83.2% \| 89 \| 4,550,599 \| 98.11% 20 MB \| 98.8% \| 85 \| 4,550,014 \| 98.04% 10 MB \| 100.0% \| 97 \| 4,545,871 \| 97.97%	arxiv-papers	2013-09-11T21:08:59	2024-09-04T02:49:50.776380	{ "license": "Public Domain", "authors": "Qingpeng Zhang, Jason Pell, Rosangela Canino-Koning, Adina Chuang\n Howe, C. Titus Brown", "submitter": "Qingpeng Zhang", "url": "https://arxiv.org/abs/1309.2975" }
1309.2980	# High Fidelity Detection of the Orbital Angular Momentum of Light by Time Mapping Paul Bierdz, Minho Kwon, Connor Roncaioli, Hui Deng Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA [email protected] ###### Abstract We demonstrate high-fidelity detection of the orbital angular momentum (OAM) of light using a compact and practical OAM spectrometer that maps the OAM spectrum to time. The spectrometer consists of a single optical delay loop to achieve timing mapping, a vortex phase plate that iteratively decreases the OAM value, and a single mode fibre to distinguish zero from non-zero OAM states. Light with arbitrarily OAM compositions can be measured. For light with OAM up to $4\hbar$, we measured an average crosstalk of -21.3 dB, which is mainly limited by the purity of the input states and optical alignment. ## 1 Introduction The orbital angular momentum (OAM) of light has been exploited for a wide range of modern applications. With a discrete, yet unbounded Hilbert space [1], OAM has been used to increase the capacity of free-space communications [2, 3], generate high order entanglement [4, 5] and increase security of non- classical communications [6]. The unique topological properties of OAM states have been utilized to create optical tweezers that can apply varying degrees of torque [7, 8], high-sensitivity vortex coronagraphs for extrasolar planet detection[9] and revealing topological properties of objects[10]. A recent paper suggests that light emitted nearby rotating black holes is twisted such that the angular momentum of the black hole can be inferred from the OAM spectrum [11]. For these applications, it is often crucial to discern different OAM states with high fidelity. Several methods have been demonstrated for measuring the OAM spectrum including those using fork diffraction gratings coupled with single mode fibres (SMF) [4], cascading Mach-Zehnder interferometers [12] and transformation optics [13]. However, all these methods map different OAM states to spatially separated modes; as a result, complexity and size of the experimental setups increase with the highest measurable OAM state. Fidelity of the detection is often limited by the complexity of the experimental setup and optical diffraction. There are other techniques that do not spatially separate the modes. Courtial et. al. proposed a scheme based on OAM-dependent rotational Doppler shift [14]; yet experimental implementation has not been possible for optical wavelengths. Recently, we proposed a scheme based on OAM to time mapping [15]. The scheme utilizes the concept of quantum counterfactual measurement and quantum Zeno effect to allow non-destructive sorting of the OAM spectrum with unit efficiency, if optical loss or misalignment is neglected. Experimental implementation with lossy optics, however, will significantly reduce the efficiency for high-order OAM state. Here we demonstrate a simplified, and practical, OAM-to-time mapping scheme, which achieved a record high extinction ratio among five OAM states at an operation speed of 80 MHz. Since OAM is mapped to time, the same apparatus can be used to measure an arbitrary number of unique OAM states. We note that a similar scheme was also adopted in a recent experiment on time-division multiplexing of OAM, although the fidelity and repetition rate were orders of magnitude lower than reported here [16]. We also note that, without employing a quantum Zeno investigator, time-division schemes are not suitable for improving channel capacity in communication. ## 2 Principle of the OAM spectrometer As shown in figure 1a, our OAM spectrometer consists of a single optical loop to perform time mapping, a vortex phase plate (VPP) of topological charge 1 as an OAM ladder operator, and an SMF as a filter for the fundamental Gaussian mode with zero OAM. Consider an incident pulse consisting of a fraction $\beta_{\ell}$ of OAM components with OAM value $\ell$, where $\ell$ is an integer and $\sum_{\ell=-\infty}^{\infty}\beta_{\ell}=1$. The optical loop converts the pulse into a sequence of pulses equally separated by the round trip propagation time $T$ in the loop. The loop needs an even number of reflections to maintain the same sign of OAM. Per loop, the VPP decreases the OAM value of each OAM component by $1$. Hence, after $N$ loops, the fractional $\beta_{\ell}$ of the original pulse will have OAM value $\ell-N$. Only the component with zero OAM value in each pulse can pass through the SMF to be detected. Thus, the OAM component $\beta_{\ell}$ with OAM value $\ell$ in the original pulse will exit the spectrometer at a time $t=lT$ (figure 1b). Figure 1: (a) Schematic of the OAM spectrometer, consisting of an optical loop that converts an input pulse into a sequence of pulses equally spaced in time, a VPP that decreases the OAM value by 1 per pass, and an SMF to filter out states with non-zero OAM. (b) Showing how the OAM value changes in time for an example input state with $\ell_{0}=3$. ### 2.1 Energy distribution The distribution of the input-pulse energy among the sequence of pulses can be pre-calibrated and controlled by the beam splitter with polarization optics. We control the distribution with a polarizing beam-splitter (PBS) and half- wave plate (labelled as R in figure 1a) in the loop. The incident light is first set to be linearly polarized at an angle $\phi_{0}$ with respect to the vertical. The PBS splits off the fraction of $\sin^{2}\\!\phi_{0}$ into the first time window and sends the rest, now vertically polarized, into the loop. The wave plate R, rotated at an angle $\theta/2$ with respect to the vertical, rotates the vertically polarized light by an angle $\theta$. The corresponding energy distribution is described below: $\displaystyle P(l)=\cases{\sin^{2}\\!\phi_{0}&\mbox{if $\ell=0$,}\\\ \cos^{2}\\!\phi_{0}\cos^{2}\\!\theta&\mbox{if $\ell=1$,}\\\ \cos^{2}\\!\phi_{0}\sin^{4}\\!\theta\cos^{2(\ell-2)}\\!\theta&\mbox{if $\ell\geq 2$.}}$ (1) Here $\ell$ represents the $\ell$th output pulse. Alternatively, the PBS and wave plate can be replaced by a non-polarizing beam-splitter with a chosen splitting ratio for polarization-insensitive measurements. This would allow for information to be encoded in the polarization degree of freedom. ## 3 Experimental implementation We implemented an OAM spectrometer as illustrated in figure 1(a). To test its performance, we used OAM eigenstates as input pulses and detected the output pulses using a Hamamatsu streak camera. The input pulse was generated by diffracting a pulsed Gaussian laser beam off fork-diffraction patterns [17] on a Holoeye PLUTO LCoS spatial light modulator (SLM) with a period of about 10 lines per millimeter and pixel pitch of 8$\mu$m. The initial laser beam was from a Tsunami Ti-Sapphire laser centered at 730 nm, with a pulse width of 100 fs and repetition rate of 80 MHz, and collimated with a 10x objective lens from an SMF ensuring an initial $M^{2}$ close to 1 and a spot size within the frame of the SLM. We first verify the input states by measuring its far-field intensity and phase distributions, as shown in figure 2. The intensity distributions were measured with a charge-coupled device. The higher-order OAM beams show a larger spatial size as expected. Inhomogeneity in the radial intensity distribution reflects the lack of purity of the OAM value. We measured the phase distribution by interfering the OAM beam with a reference Gaussian beam from the original laser. This produces a forked interference pattern. From this interference pattern we extract the phase front, as plotted in figure 2(f-j), by performing a Fourier transform, selecting the first diffracted mode, centring and then applying an inverse Fourier transform. The phase should twist about by $2\pi\ell$ for an $\ell^{\mbox{th}}$-order OAM beam. Deviation from this results implies impure initial OAM states from optical aberrations or phase ripples from modulating the liquid crystals. To calibrate the time-mapping and energy distribution of the OAM spectrometer, we used the zero-OAM Gaussian state as the input and replaced the VPP by a flat glass plate of the same thickness. The output from the SMF was detected by a Hamamatsu streak camera with a time resolution of 0.02 ns. As shown in figure 1(b), the input pulse was mapped into a sequence of pulses separated by $T=1.03$ ns. The timing and dynamic range of the streak camera allowed us to measure the first five output pulses, or, the first five OAM states, at the laser repetition rate of 80 MHz. The experiment can also be redesigned to allow the measurement of a larger number of pulses by using different laser repetition rates, loop sizes or measurement devices. Figure 2: (a-e) The intensity measured with a camera of the different initial OAM beams from the SLM with $\ell=0-4$. (f-j) The phase fronts of the input OAM beams. They are calculated from the interference patterns between the OAM beams and a reference Gaussian beam, as explained in the text. The energy distribution among the pulses was calculated by integrating the intensity over each output pulse. The results are shown in figure 3(a) (the $\ell_{0}=0$ curve). We performed the same measurement with the input state $\ell_{0}=1$. The ratio of the two energy normalization curves at each $\ell T$ gives an estimate of the misalignment of the SMF-coupling (figure 3(b)) for the $\ell^{\mbox{th}}$ output pulse, as will be discussed in detail in Section 4.2 later. The uniformity of ratio among different $\ell$ confirms that the loop was well aligned. Finally, we benchmark the performance of the OAM spectrometer by comparing the correct versus incorrect detection of an input OAM eigenstate. Figure 4(a-e) show examples of the output pulse sequences for input states $\ell_{0}=0-4$. The integrated intensity under each peak versus the input state $\ell_{0}$ and the output time bin $\ell$ is shown in figure 4(f), where each row has been renormalized to the peak of the correct detection. We define the crosstalk to be the ratios between the left and right adjacent incorrect detections versus the correct detection, i.e., $\frac{P(\ell_{0}=\ell,N=\ell\pm 1)}{P(\ell_{0}=\ell,N=\ell)}$ where $P(\ell_{0},N)$ refers to the intensity measured at time $NT$ with input OAM state $\ell_{0}$. For $\ell_{0}=0\mbox{ to }4$ we measured crosstalk values: $-12.3$, $-18.8$, $-20.0$, $-21.1$, $-21.6$, $-23.2$, $-24.8$ and $-28.2$ dB. The geometric mean of the crosstalk is $7.47\times 10^{-3}$ or $-21.3$ dB. Figure 3: (a) Unnormalized power through the fibre versus the number of passes through the loop without a VPP and different initial OAM values ($\ell_{0}=0,1$) set by the SLM. (b) Computed misalignment of SMF. Figure 4: (a-e) Streak camera images for OAM eigenstates ($\ell=0,2,4$) normalized based on (1). (f) Tabulated results of the OAM spectrometer up to $\ell=4$. ## 4 Analysis of the spectrometer performance Below we simulate the performance of the spectrometer and analyze the main sources of error. We model the laser pulses as a superposition of Laguerre- Gaussian (LG) modes, which are a complete orthonormal set of solutions to the paraxial wave equation. Each optical element operating on the pulse maps each LG mode into a different superposition of other LG modes. In particular, we focus on SMF, VPP and SLM. The remaining optics: mirrors, beam splitters, waveplates, etc. are insensitive to the transverse mode of the beam and thus preserve the LG mode. We then simulate the propagation of an input pulse through the spectrometer, and the coefficients of superposition obtained after the SMF corresponds to the measurement results. ### 4.1 Laguerre-Gaussian modes The electric fields of LG modes can be described in cylindrical coordinates $(\rho,\phi,z)$ as [18]: $\displaystyle u_{\ell,p}(\rho,\phi,z)=\langle\rho,\phi,z\|\ell,p\rangle=\frac{1}{w(z)}\sqrt{\frac{2}{\pi}\frac{p!}{(p+\|\ell\|)!}}\left(\frac{\sqrt{2}\rho}{w(z)}\right)^{\|\ell\|}\exp\left({-\frac{\rho^{2}}{w^{2}(z)}}\right)L_{p}^{\|\ell\|}\left(\frac{2\rho^{2}}{w^{2}(z)}\right)\rme^{\rmi\ell\phi}$ $\displaystyle\exp\left(\frac{\rmi k_{0}\rho^{2}z}{2\left(z^{2}+z_{\rm R}^{2}\right)}\right)\exp\left({-\rmi(2p+\|\ell\|+1)\tan^{-1}\left(\frac{z}{z_{\rm R}}\right)}\right)$ (2) Here $\ell\hbar$ ($\ell\in\mathbb{Z}$) is the OAM per photon, $p\geq 0$ labels the radial modes, $w(z)$ is the beam waist, $z_{\rm R}$ is Rayleigh range, and $k_{0}$ is the wave number of the fundamental Gaussian mode. ### 4.2 Single mode fibre Ideally, an SMF selects only the fundamental Gaussian mode ($\ell=0,p=0$) while all the other orthogonal spatial modes do not propagate through. In reality, higher order LG modes may still couple through the fibre due to finite apertures of optics and the fibre, imperfections of the fibre, mismatched beam waists between free-space and fibre-modes, and transverse misalignment between the propagation axes of the free-space and fibre-modes. In our experiments, the last effect, the transverse misalignment, is by far the dominant, while the other effects were too small to be measured. Hence we restrict our discussion to the transverse misalignment only. For a misalignment of $\Delta$ (in units of beam waist) between the two optical axes, the coupling efficiency of the LG beam $(\ell,p)$ through the SMF is given by (A.2): $\|{}_{\Delta}\langle 0,0\|\ell,p\rangle\|^{2}=\frac{1}{p!(p+\|\ell\|)!}\left(\frac{\Delta^{2}}{2}\right)^{2p+\|\ell\|}\rme^{-\Delta^{2}}$ (3) Experimentally, the above misalignment of the SMF can be estimated using the data for calibrating the energy distribution, as shown in figure 3(a). Nonzero output intensity from an input pulse of $\ell_{0}=1$ modes implies a misaligned (or imperfect) SMF. The amount of misalignment $\Delta$ can be calculated from the ratio of the two output intensities corresponding to the two input modes $\ell_{0}=0$ versus $\ell_{0}=1$ via: $\frac{P(\ell_{0}=1)}{P(\ell_{0}=0)}=\frac{\|{}_{\Delta}\langle 0,0\|M_{\mbox{VPP1}}\|0,0\rangle\|^{2}}{\|{}_{\Delta}\langle 0,0\|0,0\rangle\|^{2}}$ (4) $\|{}_{\Delta}\langle 0,0\|M_{\mbox{VPP1}}\|0,0\rangle\|^{2}=\frac{\pi}{8}\Delta^{2}\rme^{-\frac{3}{2}\Delta^{2}}\left(I_{0}\left(\frac{\Delta^{2}}{4}\right)+I_{1}\left(\frac{\Delta^{2}}{4}\right)\right)^{2}$ (5) The left hand side of (4) is measured experimentally. $M_{\mbox{VPP1}}$ represents the operation by the SLM to create the input beam with $\ell_{0}=1$ and in a superposition of different $p$-states [19]. Equation (5) is obtained using (3) and (6) (See A.3 for details). We plotted the derived misalignments in figure 3(b). The misalignment is less than 10.1% for all loops and converges to 3.7% for higher loop numbers, indicating only a minor accumulative loop misalignment. ### 4.3 VPP/SLM The VPP and SLM are used to change the OAM state. The SLM, when applying the forked phase pattern, is mathematically equivalent to VPP for small diffraction angles [20, 21]. For a VPP of topological charge $\beta$, its operation on the laser beam can be described by a four-dimensional tensor $M_{VPP\beta}(z)$. We solve for the tensor elements $m_{\ell_{1},p_{1};\ell_{2},p_{2};\beta}(z)$ analytically (A.1) in the LG-mode basis: $\displaystyle m_{\ell_{1},p_{1};\ell_{2},p_{2};\beta}(z)=\left[\sqrt{\frac{p_{1}!p_{2}!}{(p_{1}+\|\ell_{1}\|)!(p_{2}+\|\ell_{2}\|)!}}\right.\sum_{k=0}^{p_{1}}\sum_{m=0}^{p_{2}}(-1)^{k+m}\frac{(p_{1}+\|\ell_{1}\|)!}{(p_{1}-k)!(\|\ell_{1}\|+k)!k!}$ $\displaystyle\frac{(p_{2}+\|\ell_{2}\|)!}{(p_{2}-m)!(\|\ell_{2}\|+m)!m!}\left.\Gamma\left(\frac{\|\ell_{1}\|+\|\ell_{2}\|}{2}+k+m+1\right)\right]$ $\displaystyle\left[\exp\left(\rmi(2(p_{1}-p_{2})+\|\ell_{1}\|-\|\ell_{2}\|)\tan^{-1}\left(\frac{z}{z_{\rm R}}\right)\right)\right]$ $\displaystyle\left[\frac{\exp[2\pi\rmi(\ell_{2}+\beta-\ell_{1})]-1}{2\pi\rmi(\ell_{2}+\beta-\ell_{1})}\right]$ (6) This equation consists of a product of three terms denoted by square brackets. The first term with the double sum comes from the amplitude overlap between the different LG modes. The second term, consisting of an exponential, describes the effects of Gouy phases. The third term comes from OAM conservation and can describe the error in the topological charge of the VPP. We discuss the effect of each term below. #### 4.3.1 Gouy phase effects Different LG modes have different Gouy phases which also vary differently as the beam propagates; therefore, whenever there is a superposition of modes, there can be interference effects between the modes that varies under free space propagation [22] and can lead to the loss of efficiency. However, the effect does not produce crosstalk between different $\ell$-states, as OAM remain unchanged. In our experiment, the effect of the Gouy phase accounts for less than $0.05\%$ loss in efficiency. This upper bound of $0.05\%$ was calculated based on distances between the optics in the experimental setup and using (6). In general, the effect is negligible when the propagation distance is much smaller than the Rayleigh range ($z\ll z_{\rm R}$). When the size of the OAM spectrometer becomes comparable to the Rayleigh range, the use of 4-f systems between all phase elements would eliminate any Gouy phase effects. #### 4.3.2 Topological charge error If the topological charge of the VPP is an integer, then the OAM of the beam is changed by that amount. If the laser’s wavelength is different from the nominal wavelength of the VPP, or if the beam comes at a skew angle to the VPP, then the VPP will appear thicker or thinner. As a result, the OAM of the beam will be changed by a fractional value instead, or equivalently, become a superposition of LG modes of many OAM values. This leads to a loss in both the efficiency and fidelity of the OAM spectrometer. Such topological charge error can be modelled by non-integer $\beta$ in (6). In our experiment, both the laser wavelength and the angle of incidence are tightly controlled. Even if we consider a rather generous 0.5% error in the topological charge, it results in less than 0.05% loss of efficiency and less than -34dB crosstalk. Therefore the error in the topological charge is negligible. #### 4.3.3 Lateral misalignment Although topological charge error will reduce the fidelity, it is, by far, not the leading cause. If the optical axes of the beam and VPP are displaced relative to each other, the VPP produces a superposition of not just $p$, but $\ell$ states [23] and thus reduces the fidelity. Such lateral misalignment of the VPP affects all three terms in (6). To model it, we first express the VPP tensor $M_{\beta}=\rme^{\rmi\beta\phi}$ in a coordinate displaced from the common optical axis of the spectrometer. We then numerically evaluate $m_{\ell_{1},p_{1};\ell_{2},p_{2};\beta}(z)$ in a subspace of ${\ell,p}=[-7,7]\times[0,10]$. This chosen subspace yields less than 2% error on theoretical crosstalk. To compare with the experimental result, we calculated the average crosstalk versus VPP misalignment, assuming a transverse misalignment of the SMF of 3.7% to 10.1% as obtained in 4.2. The results are shown in 5(a), where the x-direction corresponds to the direction of SMF misalignment. The measured crosstalk of $-21.3$ dB (3) corresponds to a VPP misalignment of 4.0% to 6.1% of the beam waist $w_{0}$. A larger (smaller) VPP misalignment is allowed when its direction is aligned with (perpendicular to) that of the SMF misalignment. ### 4.4 Limiting factors of fidelity In short, we show that misalignments of 3.7% to 10.1% at the SMF and 4.0% to 6.1% at the VPP are the leading causes for the loss of fidelity in our experiment, with an average crosstalk of -21.3 dB. Gouy phase effects do not affect fidelity and topological charge error contributes less than -34 dB to crosstalk. The fidelity of the spectrometer can be further increased by improving the optical alignment at the VPP and SMF. We show in figure 5(b) the average crosstalk as a function of the VPP misalignment alone, neglecting SMF misalignment, and as a function of the SMF misalignment alone, neglecting VPP misalignment. It is more sensitive to VPP misalignment than to SMF misalignment, but decreases superlinearly with the reduction of either misalignment. With a VPP (SMF) misalignment of $<1\%$, the crosstalk is reduced to $<-34.4$ dB ($<-43.4$ dB). When including both SMF and VPP misalignments, the crosstalk varies depending on the relative angle between the directions of the two misalignments. The crosstalk is the greatest when the misalignments are in perpendicular directions from each other. We show this upper bound crosstalk in 5(c). When the misalignments are in opposite directions, we generally see a reduction in crosstalk, sometimes up to 20 dB. Curiously, if the SMF is misaligned, crosstalk can be _decreased_ by introducing some VPP misalignment, and vice versa. This is because with multiple elements misaligned, they can compensate each other. However, zero crosstalk is reached only when both misalignments are exactly zero. Figure 5: Calculated the geometric mean of crosstalk (in dB) as a function of the lateral misalignment of the SMF and VPP (normalized by $w_{0}$). (a) Crosstalk (in dB) as a function of the lateral misalignment of the VPP. SMF misalignment (measured in 3(c)) is taken along the x direction for convenience, since only the relative angle between VPP and SMF misalignment is important. The measured crosstalk of $-20.9$ to $-21.6$ dB corresponds to the values between the two black lines. (b) The crosstalk versus VPP misalignment with no SMF misalignment (green solid line), and the crosstalk versus SMF misalignment with no VPP misalignment (blue dashed line). (c) The upper bound of crosstalk (in dB) as a function of the SMF and VPP misalignments. The upper bound corresponds to when the directions of the SMF and VPP misalignments are perpendicular to each other. The crosstalk can be significantly reduced if the directions of the SMF and VPP lateral displacements are anti-parallel to each other. As misalignment is reduced, other subtle effects will need to be taken under consideration. Such as imperfections in the surface roughness or other aberrations in the optics, which could further reduce efficiency and fidelity. Eventually, the fidelity will be limited by overlap in time between adjacent pulses. We did not include these effects in our model. ## 5 Conclusions In summary, we present a practical OAM spectrometer that can map OAM to time up to arbitrarily large OAM values. We have shown an average nearest neighbor crosstalk of -21.3dB among 5 OAM states, limited mainly by optical misalignment. The high fidelity of the demonstrated OAM spectrometer may enable accurate measurements of topological properties of objects such as in spiral imaging [10] and the angular momentum of black holes, which is encoded into the OAM spectrum of light from the accretion disc due to strong gravitational effects predicted by general relativity [11]. We also demonstrated speeds (80 MHz) orders of magnitude faster than the switching times of SLMs. Miniaturization of optics could allow for GHz detection rate. ## Acknowledgments We would like to thank Professor Miles Padgett for his helpful advice. And also thank Professor Theo Lasser and Matthias Geissbühler for their _Morgenstemning_ colormap [24]. ## Appendix A In this appendix section we seek to derive the various actions and overlaps between the LG modes and optics. We first derive the action of the VPP in the LG basis. Then we derive the overlap of any LG mode with a misaligned SMF. Lastly, we derive the overlap of OAM states created by a VPP with a misaligned SMF. This is done by explicit integration and definitions of various special functions. We use the inner product between Laguerre-Gaussian modes defined in 2 in cylindrical coordinates: $\displaystyle\langle\ell_{1},p_{1}\|\ell_{2},p_{2}\rangle=\int_{0}^{\infty}\rho\rmd\rho\int_{0}^{2\pi}\rmd\phi\langle\ell_{1},p_{1}\|\rho,\phi,z\rangle\langle\rho,\phi,z\|\ell_{2},p_{2}\rangle$ $\displaystyle\times\int_{0}^{\infty}\rho\rmd\rho\int_{0}^{2\pi}\rmd\phi u_{\ell_{1},p_{1}}^{}(\rho,\phi,z)u_{\ell_{2},p_{2}}(\rho,\phi,z)$ (7) ### A.1 Derivation of VPP tensor We derive the four dimensional tensor of the action of a VPP in the LG basis as seen in (6). We start by explicitly writing down the overlap integral: $\langle\ell_{1},p_{1}\|M_{\mbox{VPP}\beta}\|\ell_{2},p_{2}\rangle=\int_{0}^{\infty}\rho\rmd\rho\int_{0}^{2\pi}\rmd\phi LG^{}_{\ell_{1},p_{1}}(\rho,\phi)\rme^{+\rmi\beta\phi}LG_{\ell_{2},p_{2}}(\rho,\phi)$ (8) The LG functions are given by (2). Due to separability of variables, we will solve the $\phi$-integral first: $\int_{0}^{2\pi}\rmd\phi\rme^{-\rmi\ell_{1}\phi}\rme^{+\rmi\beta\phi}\rme^{+\rmi\ell_{2}\phi}=\frac{\exp\left(2\pi\rmi\left(\ell_{2}+\beta-\ell_{1}\right)\right)-1}{\rmi\left(\ell_{2}+\beta-\ell_{1}\right)}$ (9) In the limit where $\beta$ is an integer, the $\phi$-integral yields the OAM conserving solution of $2\pi\delta_{\ell_{1}+\beta-\ell_{2},0}$, where $\delta_{a,b}$ is the Kronecker delta. The remaining $\rho$-integral is found by expanding the rest of the LG modes and the generalized Laguerre polynomials given by (10). This yields a finite polynomial of degree $\|\ell_{1}\|+\|\ell_{2}\|+2p_{1}+2p_{2}+1$ (the +1 is from the Jacobian) multiplied by the Gaussian function $\exp\left(\frac{-\rho^{2}}{2w_{0}^{2}}\right)$. This integral can be easily solved by using one of the definitions of the Gamma function as shown in (11) to yield the result in the text (6). $L_{p}^{k}(x)=\sum_{m=0}^{p}(-1)^{m}\frac{(p+k)!}{(p-m)!(k+m)!m!}x^{m}$ (10) $\int_{0}^{\infty}\rmd xx^{k}\rme^{-x^{2}}=\frac{1}{2}\Gamma\left(\frac{1+k}{2}\right)$ (11) ### A.2 Overlap between fundamental Gaussian and Laguerre-Gaussian We first derive the overlap between a misaligned fundamental Gaussian mode and generic higher order LG modes. This is (3) in the text. We start with the fundamental Gaussian mode defined in cylindrical coordinates ($\rho,\phi,z$) with an origin shifted by $w_{0}\Delta$ along the $\phi=0$ direction. The propagation direction is along the z-axis. ${}_{\Delta}\langle 0,0\|\rho,\phi,z\rangle=\frac{1}{w_{0}}\sqrt{\frac{2}{\pi}}\exp\left({\frac{-\left(\rho^{2}+2\rho w_{0}\Delta\cos\phi+w_{0}^{2}\Delta^{2}\right)}{w_{0}^{2}}}\right)$ (12) If the origin were shifted along a different $\phi$ angle, the only change would be the overall phase which does not affect the intensity. These calculations are performed at the beam waist since the beam should be focused on the fibre to have any substantial coupling. With a change of variables $x=\frac{\sqrt{2}\rho}{w_{0}}$ the complete overlap integral is now: $\displaystyle{}_{\Delta}\langle 0,0\|\ell,p\rangle=\int_{0}^{\infty}\int_{0}^{2\pi}x\rmd x\rmd\phi\frac{1}{\pi}\sqrt{\frac{p!}{(p+\|\ell\|)!}}x^{\|\ell\|}\exp\left(\frac{-x^{2}}{2}\right)L_{p}^{\|\ell\|}\left(x^{2}\right)\rme^{\rmi\ell\phi}$ $\displaystyle\exp\left(-\left(\frac{x^{2}}{2}+\sqrt{2}x\Delta\cos\phi+\Delta^{2}\right)\right)$ (13) The $\phi$-integral is related to the well known Bessel integral (14), so (A.2) becomes (15). $J_{n}(z)=\frac{1}{2\pi\rmi^{n}}\int_{0}^{2\pi}\rmd\phi\rme^{\rmi z\cos\phi}\rme^{\rmi n\phi}$ (14) ${}_{\Delta}\langle 0,0\|\ell,p\rangle=\sqrt{\frac{p!}{(p+\|\ell\|)!}}\int_{0}^{\infty}x\rmd xx^{\|\ell\|}\rme^{-\left(x^{2}+\Delta^{2}\right)}L_{p}^{\|\ell\|}\left(x^{2}\right)2\rmi^{-\|\ell\|}(-1)^{\|\ell\|}J_{\|\ell\|}\left(\rmi\sqrt{2}\Delta x\right)$ (15) The Bessel function can be converted into an infinite sum of generalized Laguerre polynomials as given in (16) below with $\sqrt{w}=\rmi\Delta/\sqrt{2}$. Using (16) and the orthogonality relationship between generalized Laguerre polynomials (17). Equation (15) magnitude squared surprisingly reduces to the rather simple expression of (3) in the text. $\displaystyle\sum_{n=0}^{\infty}\frac{L_{n}^{k}(x)}{\Gamma(n+k+1)}w^{n}=e^{w}(xw)^{\frac{-k}{2}}J_{k}\left(2\sqrt{xw}\right)$ (16) $\displaystyle\int_{0}^{\infty}\rmd x\rme^{-x}L_{n}^{k}(x)L_{m}^{k}(x)=\frac{(n+k)!}{n!}\delta_{m,n}$ (17) ### A.3 Overlap between fundamental Gaussian and vortex phase plate mode We create states with OAM by passing a Gaussian beam through a VPP. This state can be expressed as a sum of Laguerre Gaussian modes as given in (18), which is a special case of the more general equation solved in section A.1. In the previous section, we calculated the overlap of a shifted fundamental Gaussian mode and any Laguerre Gaussian mode. Therefore, by combining these two calculations, we can derive the overlap of a shifted Gaussian mode and an OAM mode created by a VPP (19), here $z=\frac{\Delta^{2}}{2}$. $\displaystyle M_{\mbox{VPPN}}\|0,0\rangle=\sum_{p=0}^{\infty}m_{0,0;N,p;N}\|N,p\rangle$ (18) $\displaystyle\mbox{where }m_{0,0;N,p;N}=\frac{N}{2}\sqrt{\frac{1}{(p+N)!}}\Gamma\left(p+\frac{N}{2}\right)$ $\displaystyle{}_{\Delta}\langle 0,0\|M_{\mbox{VPPN}}\|0,0\rangle=\frac{e^{-z}z^{\frac{N}{2}}(-1)^{N}N}{2}\sum_{p=0}^{\infty}\frac{(-z)^{p}\Gamma\left(p+\frac{N}{2}\right)}{(p+N)!p!}$ (19) Equation (19) can be written in the form of a regularized confluent hypergeometric function (20) which can be evaluated using (21)111http://functions.wolfram.com/07.21.03.0011.01 to yield the result in (22), where $I_{\alpha}(x)$ is the modified Bessel function of the first kind. The expression can be further simplified by noting $I_{\alpha}(-z)=\rmi^{2\alpha}I_{\alpha}(z)$. In the case of $N=1$ and taking the magnitude squared, this reduces to the form in the text (5). ${}_{1}\tilde{F}_{1}(a;b;z)=\sum_{k=0}^{\infty}\frac{\Gamma(a+k)}{\Gamma(b+k)\Gamma(a)}\frac{z^{k}}{k!}$ (20) ${}_{1}\tilde{F}_{1}\left(\frac{N}{2};N+1;z\right)=\frac{e^{\frac{z}{2}}\sqrt{\pi}}{2\Gamma\left(\frac{N}{2}+1\right)}z^{\frac{1}{2}-\frac{N}{2}}\left(I_{\frac{N}{2}-\frac{1}{2}}\left(\frac{z}{2}\right)-I_{\frac{N}{2}+\frac{1}{2}}\left(\frac{z}{2}\right)\right)$ (21) $\frac{\rme^{\frac{-3\Delta^{2}}{4}}\Delta^{2}\sqrt{\pi}}{2\sqrt{2}}\left(I_{\frac{N}{2}-\frac{1}{2}}\left(\frac{\Delta^{2}}{4}\right)+I_{\frac{N}{2}+\frac{1}{2}}\left(\frac{\Delta^{2}}{4}\right)\right)$ (22) ## References ## References * [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Physical Review A, 45(11):8185, June 1992. 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URL: http://www.opticsexpress.org/abstract.cfm?URI=oe-21-8-9862, doi:10.1364/OE.21.009862.	arxiv-papers	2013-09-11T21:56:31	2024-09-04T02:49:50.787967	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Paul Bierdz, Minho Kwon, Connor Roncaioli, Hui Deng", "submitter": "Paul Bierdz", "url": "https://arxiv.org/abs/1309.2980" }
1309.2988	EPJ Web of Conferences INPC 2013 11institutetext: Mississippi State University, MS 39762-5167, USA # Feasibility of the Spin-Light Polarimetry Technique for Longitudinally Polarized Electron Beams Prajwal Mohanmurthy 11 [email protected] Dipangkar Dutta 11 ###### Abstract A novel polarimeter based on the asymmetry in the spacial distribution of synchrotron radiation will make for a fine addition to the existing Møller and Compton polarimeters. The spin light polarimeter consists of a set of wiggler magnet along the beam that generate synchrotron radiation. The spacial distribution of synchrotron radiation will be measured by ionization chambers. The up-down (below and above the wiggle) spacial asymmetry in the transverse plain is used to quantify the polarization of the beam. As a part of the design process, effects of a realistic wiggler magnetic field and an extended beam size were studied. The perturbation introduced by these effects was found to be negligible. Lastly, a full fledged GEANT-4 simulation was built to study the response of the ionization chamber. ## 1 Introduction A 1993 proposal from Karabekov and Rossmanith explored the possibility of measuring the electron beam polarization using the synchrotron radiation produced by a magnet karabekov93 . In this paper we examine the feasibility of a “spin-light” polarimetery technique for measuring longitudinal polarization of multi-GeV electron beams while building on the 1993 proposal. The simulated wiggler magnetic field was implemented in a full-fledged Geant4 simulation of the polarimeter. A polarimeter based on spin-light would provide for a polarization measurement independent of both Compton and Møller polarimeters. A relative spin-light polarimeter could also be used in association with either a Compton or a Møller polarimeter. Highly precise, multiple independent polarimeters are a must if the ambitious goal of $\sim$ 0.5 % uncertainity in polarimetry is to be achieved at an Electron Ion Collider (EIC) in order to meet the experimental demands. ## 2 Spin-Light Charecteristics The spin-dependent SR distribution as given by Sokolov et. al. [4] , ignoring higher order effects, is of particular interest as it expresses the distribution in terms of physical parameters such as spin of the electron - ’$j$’ and the vertical angle - $\psi$ (in the electron’s frame of reference) [5] . The verticle angle is important as it determines the geometry of the apparatus besides other design parameters such as position of collimators. $\displaystyle N_{\gamma}(long)$ $\displaystyle=$ $\displaystyle\frac{9n_{e}}{16\pi^{3}}\frac{e^{2}}{cm_{e}R^{2}}\gamma^{4}\int_{0}^{\infty}\frac{y^{2}dy}{(1+\xi y)^{4}}\oint d\Omega(1+\alpha^{2})^{2}\times$ (1) $\displaystyle\left[K^{2}_{2/3}(z)+\frac{\alpha^{2}}{1+\alpha^{2}}K^{2}_{1/3}(z)+j\xi y\frac{\alpha}{\sqrt{1+\alpha^{2}}}K_{1/3}(z)K_{2/3}(z)\right]$ where $\xi=\frac{3B}{2B_{c}}\gamma$, $B_{c}$ being the magnetic field under the influenze of which the entire kinetic energy of the electron is expelled as one SR photon, $y=\frac{\omega_{o}}{\omega_{c}}$, $K_{n}(x)$ are modified Bessel functions, $n_{e}$ is the number of electrons and, $z=\frac{\omega}{2\omega_{C}}(1+\alpha^{2})^{3/2}$, and $\alpha=\gamma\psi$. For an electron that is polarized, the power below (i.e. $-\frac{\pi}{2}\leq\psi\leq 0$) and above (i.e. $0\leq\psi\leq\frac{\pi}{2}$) are spin dependent. More importantly the difference between the power radiated above and power radiated below, called Spin-Light with an assymetry which can be defined as $A=\frac{\Delta N_{\gamma}}{N_{\gamma}}$, is directly spin dependent and this opens up the possibility of a direct measurement technique. Figure 1: Schematic diagram of a differential spin-light polarimeter ## 3 Conceptual Design The wiggler magnet is at the heart of the setup where the SR photons are produced and the ionization chambers can be used to characterize the SR in order to measure the asymmetry. It is important to note the presence of collimators on the faces of the wiggler magnets in order to prevent intermixing of the SR light fans. Collimation creates four SR spots, with each ionization chamber receiving two collimated SR spots. Two major variables in this setup are the electron beam energy and the wiggler pole strength. In Figures 2(A) & (B), Spin-Light spectra and the asymmetry are plotted for various wiggler pole strengths with a $11GeV$ beam and in Figures 2(C) & (D), the same are plotted for various beam energies with a $4T$ wiggler field. A numerical integration code is used to generate the SR spectra and asymmetry using Eq.(1). Figure 2: (Left- Right): A. Plot of spin light spectra for various pole strengths from $2T-5T$; B. Plot of asymmetry vs. photon energy for various pole strengths.; C. Plot of spin light spectra for various beam energies ranging from $4GeV-12GeV$.; D. Plot of Asymmetry vs. the photon energy for various beam energies. ### 3.1 Effects of realistic dipole magnetic field with fringes A field map of the wiggler magnets can be generated by solving Maxwell’s equations with appropriate boundary conditions. In LANL Poisson SuperFish [18] , the magnet contours can be easily defined. The field map of the magnet can then be plotted. Here, the field map at the edge where the electron beam enters the magnet is presented in Figure 3(A). Note that the beam pipe is going at the center below the magnet pole. In Figure 3(A), the physical taper of the cores can be noticed, since it is at the edge of the magnet face. The singularities seen in Figure 3(A) are the areas where the current cuts the plane. Also, it is important to note that the entire ‘C’ magnet is not visible in the field-map, only one half of the ‘C’ magnet is shown in the field map. The field map obtained can be used in the numerical integration code, in place of a constant pole strength, to plot the SR spectra and the asymmetry which are presented in Figures 4(A) & (B). Even though there is a reduction in the total power output of SR light by introducing a realistic dipole field, the asymmetry has not changed. This implies that the changes introduces by the realistic dipoles are negligible. Figure 3: (Anticlockwise from Top-Left): A. Field map of the dipole face at the edge of the dipole.; B. Plot of both the $x$ and $y$ components of the magnetic field on the transverse plane at the the edge of the dipole (Beam pipe is centered around $15$cm mark along the ’x’ axis).;C. Integrated power spectra of SR Light at the IC due to Gaussian beam. (The difference between the profile has been enlarged for clarity); D. Integrated power spectra of SR Light at the IC due to a point beam. ### 3.2 Effects of Extended Beam Size Figure 4: (Left- Right): A. Plot showing the SR - Light and Spin - Light power spectra with a realistic wiggler magnetic field (Power spectra for uniform magnetic field have also been presented as .; B. Plot of the assymetry with a realistic wiggler magnetic field. The effect of having an extended beam size of about $100\mu m$ was studied by essentially superimposing the SR Spectra generated by each differential element of the beams weighted with Gaussian distribution in order to make the extended beam a perfect Gaussian beam. The cumulative spectra for a point beam was obtained by setting the weighting factor to one. The cumulative spectra for a Gaussian beam when plotted has approximately the same structure as the spectra for the point - cross section beam. This is so because the size of the beam ($R_{beam}=100\mu m$) is small compared to the size of the collimated SR - Light spot which is about $1mm$ big. For the beam with a point cross section, the SR - profile is rather ’box’ like at the ionization chamber. When an extended beam, that is of Gaussian profile, is introduced, the SR - profile gets a taper which is Gaussian in nature too as illustrated in Figures 3(C) & (D). ## 4 Spin Light in Geant4 and SR-Spectra A Geant4 simulation with a geometry as in Figure 1 using EM Extra process list was constructed. Even though the integrated energy spectrum is validated in Geant4, it does not contain the angular dependence of SR light. We implement the angular dependence of SR light at the stacking action level. SR photons are killed with a probability equal to asymmetry which can be calculated from standard Geant4 track parameters. This creates an up-down asymmetry in the SR cones which is vital for this simulation of a Spin-Light polarimeter. As seen in Figure 5, the simulated SR-Spectra closely matches the physics SR-Spectra within $1\%$. Spin light component is obtained by subtracting the remaining tracks with positive momentum (corresponding to $0\leq\psi\leq\frac{\pi}{2}$) from tracks with negative momentum (corresponding to $-\frac{\pi}{2}\leq\psi\leq 0$). Figure 5 also shows non-SR events in green which are significantly small in number compared to the corresponding spin- light events in a particular photon energy bin. ## 5 Conclusion Figure 5: Geant4 SR & SL spectra as compared with physics SR & SL spectra. The figure of merit for such a polarimeter increases with electron beam energy and the strength of magnetic field used. On the other hand, the SR profile becomes more compact with increase in electron beam energy. This makes it difficult for an IC to charecterize the SR profile given that the SR load increases as fourth power of beam energy. A 3 pole wiggler with a field strength of $4T$ and a pole length of $10cm$ would be adequate for such a polarimeter. Locating a reasonable piece of beam-line real estate is however very challenging. Given that the eRHIC design of EIC involves using a Gatalin gun [25] at a very high rate, the recovery time of the spin-light IC will need to be extremely small if every bunch of the electron beam is to be measured for polarization, a goal which is nearly impossible with this design. A Spin-light polarimeter is apt for measuring averaged polarization of a number of beam bunches. A survey of all experiments beings proposed and their corresponding polarimetry requirements both in terms of precision of polarimetry required and the rate of measurement will go a long way in helping pin down the instrument specifications. ## References * (1) I. P. Karabekov, R. Rossmanith, Proc. of the 1993 PAC, Washington, v. 1, p. 457 (1993); I. P. Karabekov and S. I. Karabekian, Proceedings of 5th European Particle Accelerator Conference (EPAC 96), Sitges, Spain, 10-14 Jun 1996, pp 1743-1745 (1996); A. V. Airapetian, R. O. Avakian, I. P. Karabekov, E. L. Saldin, and M. V. Yurkov, Proc. of the SPIN-96, Amsterdam, The Netherlands, Vol.1, p762 (1996). * (2) A. A. Sokolov, N. P. Klepikov and I. M. Ternov, JETF 23, 632 (1952). * (3) A. A. Sokolov and I. .M. Ternov, Radiation from Relativistic Electrons, A.I.P. Translation Series, New York (1986) ; I. M. Ternov, Physics - Uspekhi 38, 409 (1995). * (4) K. Sato, J. of Synchrotron Rad., 8, 378 (2001). * (5) B. Norum, CEBAF Technical note, TN-0019 (1985). * (6) M. Sands, SLAC Technical note, SLAC-121 (1970). * (7) Poisson SuperFish 2D EM Solver, laacg1.lanl.gov/laacg/services/sfu_04_04_03.phtml, 2007. * (8) V. N. Litvinenko, Gatling Gun: High Average Polarized Current Injector for eRHIC, EIC BNL Whitepapers (2012).	arxiv-papers	2013-09-11T22:17:08	2024-09-04T02:49:50.795819	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Prajwal Mohanmurthy and Dipangkar Dutta", "submitter": "Prajwal Mohanmurthy", "url": "https://arxiv.org/abs/1309.2988" }
1309.3065	UT–13–34 Probing Bino Contribution to Muon $g-2$ Motoi Endo, Koichi Hamaguchi, Teppei Kitahara, and Takahiro Yoshinaga**Research Fellow of the Japan Society for the Promotion of Science Department of Physics, University of Tokyo, Tokyo 113-0033, Japan We study SUSY models in which Bino contributions solve the muon $g-2$ anomaly. The contributions are enhanced by large left-right mixing of the smuons. However, it is constrained by the vacuum stability condition of the slepton–Higgs potential. Therefore, there are upper bounds on masses of sleptons and Bino. When the slepton soft masses are universal, the upper bound on the smuon mass becomes $330~{}(460)\,{\rm GeV}$ in order to solve the $g-2$ anomaly at the $1\sigma~{}(2\sigma)$ level. It is within the reach of LHC and ILC. If the stau is heavier than the smuon, the bound can be as large as $1.4~{}(1.9)\,{\rm TeV}$. Such non-universal slepton mass spectrum generically predicts too large LFV/CPV. We show that the models are expected to be probed by LHC/ILC and LFV/CPV complementarily in future. ## 1 Introduction The anomalous magnetic moment of the muon, $a_{\mu}=(g-2)_{\mu}/2$, (muon $g-2$) has been measured very precisely [1]. It is compared with the Standard Model (SM) prediction, and the latest result is $\Delta a_{\mu}\equiv a_{\mu}({\rm exp})-a_{\mu}({\rm SM})=(26.1\pm 8.0)\times 10^{-10},$ (1) where Ref. [2] is referred to for contributions of the hadronic vacuum polarization, and the hadronic light-by-light contribution is from Ref. [3]. Similarly, Ref. [4] provides $\Delta a_{\mu}=(28.6\pm 8.0)\times 10^{-10}$. Therefore, there is discrepancy at more than $3\sigma$ confidence levels. It is noticed that the difference is as large as SM electroweak contributions, $a_{\mu}({\rm EW})=(15.36\pm 0.1)\times 10^{-10}$ [5, 6]. If this is a signature of physics beyond the SM, and if the new physics exists in the TeV scale, new physics contributions to the muon $g-2$ are necessarily enhanced by some mechanisms, because they are naively estimated as $a_{\mu}({\rm NP})\sim(\alpha_{\rm NP}/4\pi)\times(m_{\mu}^{2}/m_{\rm NP}^{2})$, which is required to be comparable to $a_{\mu}({\rm EW})$. The supersymmetry (SUSY) is a good candidate for such new physics models. The model can provide sizable contributions to the muon $g-2$ [7, 8, 9], which are enhanced by $\tan\beta=\langle H_{u}\rangle/\langle H_{d}\rangle$. The muon $g-2$ anomaly is solved if the superparticles (muonic sleptons, neutralinos and/or charginos) have a mass around $\mathcal{O}(100)\,{\rm GeV}$ for $\tan\beta=\mathcal{O}(10)$. They are light enough to be produced at the LHC experiments. Recently, LHC phenomenology of the superparticles that are relevant for the muon $g-2$ has been studied in Ref. [10]. It has been shown that they can be discovered in the near future in most of the parameter regions, especially in regions where the SUSY contributions to the muon $g-2$ are dominated by chargino–sneutrino diagrams. However, the searches rely on the assumption that the Wino is light. This assumption is not necessary to explain the muon $g-2$ discrepancy, when the SUSY contribution is mainly from Bino–smuon diagrams. If the Wino is heavy, collider searches differ significantly from those in Ref. [10]. In this paper, we study searches for the models in which only the Bino and the left- and right-handed smuons are light, while the other superparticles including the Wino are decoupled from the LHC sensitivity. The Bino–smuon contribution is enhanced when the left-right mixing of the smuon is large. If the left-right mixing were allowed to be arbitrarily large, the superparticles could be extremely heavy while keeping the contribution to the muon $g-2$, thereby escaping any collider searches. However, too large left-right mixing spoils stability of the electroweak vacuum. Hence, the superparticle masses are bounded from above. We will show that, when staus and smuons have comparable masses to each others, upper bounds on the slepton masses are within the reach of LHC/ILC. If the staus are much heavier than smuons, the mass bounds are relaxed, while lepton flavor violations (LFV) and CP violations (CPV) generically become too large. Therefore, we will show that almost all the parameter regions can be tested in future complementarily by LHC/ILC and LFV/CPV, if the muon $g-2$ anomaly is solved by the Bino and the left- and right-handed smuons. The rest of the paper is organized as follows. The mass spectrum is provided in Sec. 2. The SUSY contributions to the muon $g-2$ and the vacuum stability conditions are explained in Sec. 3. In Sec. 4, experimental searches are studied. Sec. 5 is devoted to the conclusion. ## 2 Mass spectrum We assume that the SUSY contributions to the muon $g-2$ are dominated by the Bino–smuon contribution. The Bino and the left- and right-handed smuons contribute to the diagram. It is enhanced by the left-right mixing of the smuon, which is determined by the muon Yukawa coupling constant, the Higgsino mass parameter, $\mu$, and $\tan\beta$. Details will be discussed in the next section. Since too large $\tan\beta$ spoils perturbativity of the down-type Yukawa interactions, $\mu$ is favored to be large to explain the muon $g-2$ discrepancy (1). Thus, we focus on large Higgsino mass regions. The Winos are also supposed to be decoupled. The collider searches depend on the slepton mass spectrum. The left- and right-handed selectron masses are assumed to be degenerate with those of the smuons, respectively. The following conclusion does not depend on this assumption.111 If the selectron masses are non-universal in the first two generations, the LFV bound is severe. See the discussions in Sec. 4.2. On the other hand, the vacuum stability condition and the LFV/CPV bounds depend on the stau masses (see Sec. 3.2 and 4.2). All colored superparticles are set to be very heavy. In fact, none of them have been discovered at LHC. The Higgs boson mass of $126\,{\rm GeV}$ favors the scalar top masses to be $\mathop{\mathcal{O}}(10-100)\,{\rm TeV}$, if the trilinear coupling of the top squark is not large. Similarly, the heavy Higgs bosons of the two Higgs doublets are assumed to be heavy. In this paper, all of them are considered to be decoupled. Consequently, we consider the low-energy effective theory, in which only the following superparticles are light, $\tilde{B},~{}\tilde{\ell}_{L},~{}\tilde{\ell}_{R}.$ (2) Here, $\tilde{\ell}$ denotes the selectron and the smuon (and the stau, depending on the mass spectrum). In addition, the Higgsinos can contribute to the effective Lagrangian, if they are light due to the vacuum stability condition. The model parameters are $M_{1},m_{\tilde{\ell}_{L}}^{2},m_{\tilde{\ell}_{R}}^{2},m_{\tilde{\ell}_{LR}}^{2}.$ (3) Here, $M_{1}$ is the Bino mass. Since the Higgsinos (and the Wino) are heavy, the lightest neutralino is almost composed of the Bino. On the other hand, $m_{\tilde{\ell}_{L}}^{2}$ and $m_{\tilde{\ell}_{R}}^{2}$ are soft SUSY- breaking masses of the left- and right-handed sleptons, respectively. $m_{\tilde{\ell}_{LR}}^{2}$ is off-diagonal components of the slepton mass matrices. This controls the observables and constraints that are studied in this paper. Before proceeding to the phenomenology, let us comment on the left-right mixing of the smuon. It includes the scalar trilinear coupling of the muon, $A_{\mu}$, as well as $\mu$ and $\tan\beta$. The Bino–smuon contribution could be enhanced by $A_{\mu}$ with $\mu\tan\beta$ kept small. However, this requires $A_{\mu}$ to be extraordinary large. If the trilinear coupling is universal among the matter scalar fermions, this implies extremely large trilinear coupling for the stop sector, resulting in either too large Higgs boson mass or rapid decays of our electroweak vacuum into charge/color breaking vacua. In this paper, $A_{\mu}$ is set to be zero for simplicity, and the left-right mixing of the smuon is determined by $\mu$ and $\tan\beta$. ## 3 Muon $g-2$ and Vacuum Stability ### 3.1 Muon $g-2$ Figure 1: The Bino–smuon contribution to the magnetic dipole operators. The Bino–smuon contribution to the muon $g-2$ is represented by Fig. 1, where the lepton $\ell$ is the muon. In the mass insertion approximation, it is estimated as (cf. [9]) $\displaystyle a_{\mu}({\rm SUSY})$ $\displaystyle=-(1+\delta^{\rm 2loop})\frac{\alpha_{Y}}{4\pi}\frac{m_{\mu}M_{1}m_{\tilde{\mu}_{LR}}^{2}}{m^{2}_{\tilde{\mu}_{L}}m^{2}_{\tilde{\mu}_{R}}}\,f_{N}\left(\frac{m^{2}_{\tilde{\mu}_{L}}}{M^{2}_{1}},\frac{m^{2}_{\tilde{\mu}_{R}}}{M^{2}_{1}}\right)$ $\displaystyle=\frac{1+\delta^{\rm 2loop}}{1+\Delta_{\mu}}\frac{\alpha_{Y}}{4\pi}\frac{m_{\mu}^{2}M_{1}\mu}{m^{2}_{\tilde{\mu}_{L}}m^{2}_{\tilde{\mu}_{R}}}\tan\beta\cdot f_{N}\left(\frac{m^{2}_{\tilde{\mu}_{L}}}{M^{2}_{1}},\frac{m^{2}_{\tilde{\mu}_{R}}}{M^{2}_{1}}\right)$ $\displaystyle\simeq 1.5\times 10^{-9}~{}\frac{1+\delta^{\rm 2loop}}{1+\Delta_{\mu}}\left(\frac{\tan\beta}{10}\right)\left(\frac{(100\,{\rm GeV})^{2}}{m_{\tilde{\mu}_{L}}^{2}m_{\tilde{\mu}_{R}}^{2}/M_{1}\mu}\right)\left(\frac{f_{N}}{1/6}\right),$ (4) at the leading order of $\tan\beta$. The loop function $f_{N}(x,y)$ is defined as $f_{N}(x,y)=xy\left[\frac{-3+x+y+xy}{(x-1)^{2}(y-1)^{2}}+\frac{2x\log x}{(x-y)(x-1)^{3}}-\frac{2y\log y}{(x-y)(y-1)^{3}}\right],$ (5) which satisfies $f_{N}(1,1)=1/6$ and $0\leq f_{N}(1,1)\leq 1$. It is noticed that the contribution is proportional to the left-right mixing, $\displaystyle m_{\tilde{\ell}_{LR}}^{2}=-Y_{\ell}\,v_{u}\mu=-\frac{m_{\ell}}{1+\Delta_{\ell}}\mu\tan\beta,$ (6) where $v_{u}$ is the vacuum expectation value (VEV) of the up-type Higgs field. Note that the heavy Higgs bosons are decoupled. In the last equation, $\Delta_{\ell}$ is a correction to the lepton Yukawa coupling constant [11]. It appears when the lepton Yukawa coupling is matched to the physical lepton mass, $m_{\ell}$, or the Yukawa coupling in SM. In the low-energy effective theory, it becomes $\displaystyle\Delta_{\ell}=\frac{\alpha_{Y}}{4\pi}M_{1}\mu\tan\beta\cdot I(M_{1}^{2},m_{\tilde{\ell}_{L}}^{2},m_{\tilde{\ell}_{R}}^{2}),$ (7) where Higgsino diagrams are discarded, because they are suppressed in large Higgsino mass regions. Also, the terms that are not enhanced by $\tan\beta$ are omitted. The loop function is defined as $\displaystyle I(a,b,c)=-\frac{ab\ln a/b+bc\ln b/c+ca\ln c/a}{(a-b)(b-c)(c-a)},$ (8) which satisfies $f(a,a,a)=1/2a$. In particular, when $\mu\tan\beta$ is very large, $\Delta_{\mu}$ becomes as large as or larger than $\mathop{\mathcal{O}}(0.1-1)$. The correction $\delta^{\rm 2loop}$ denotes leading contributions of two loop diagrams. It is estimated as $\displaystyle 1+\delta^{\rm 2loop}=\left(1-\frac{4\alpha}{\pi}\ln\frac{m_{\rm soft}}{m_{\mu}}\right)\left[1+\frac{1}{4\pi}\left(2\alpha_{Y}\Delta b+\frac{9}{4}\alpha_{2}\right)\ln\frac{M_{\rm soft}}{m_{\rm soft}}\right].$ (9) In the right-hand side, the first bracket is QED corrections to the muon $g-2$ [12], or renormalization group contributions to the effective operator of the magnetic dipole operator from the (smuon or Bino) soft mass scale, $m_{\rm soft}$, to the muon mass scale. In the numerical analysis, $m_{\rm soft}$ is chosen to be the smuon mass. This correction is $\sim 10\%$. Non-logarithmic terms evaluated in Ref. [12] are found to be very small in the parameter regions of this paper. The second bracket in Eq.(9) is corrections to the Bino couplings with the smuons. When SUSY is exact, the gaugino coupling is equal to the gauge coupling constant. This equality is violated after heavy superparticles are decoupled at a scale, $M_{\rm soft}$. The Bino–muon–smuon interactions are $\displaystyle\mathcal{L}_{\text{int}}=-\frac{1}{\sqrt{2}}\tilde{g}_{L}\,\overline{\tilde{B}}\mu_{L}\,\tilde{\mu}^{}_{L}+\sqrt{2}\tilde{g}_{R}\,\overline{\tilde{B}}\mu_{R}\,\tilde{\mu}_{R}^{}+{\rm h.c.},$ (10) where the coefficients are $\displaystyle\tilde{g}_{L}$ $\displaystyle=g_{Y}+\delta\tilde{g}_{L}\simeq g_{Y}\left[1+\frac{1}{4\pi}\left(\alpha_{Y}\Delta b+\frac{9}{4}\alpha_{2}\right)\ln\frac{M_{\rm soft}}{m_{\rm soft}}\right],$ (11) $\displaystyle\tilde{g}_{R}$ $\displaystyle=g_{Y}+\delta\tilde{g}_{R}\simeq g_{Y}\left[1+\frac{\alpha_{Y}}{4\pi}\Delta b\,\ln\frac{M_{\rm soft}}{m_{\rm soft}}\right].$ (12) Here, the terms of $\alpha_{Y}$ are corrections to the Bino self-energy, which are SUSY analog of the oblique corrections [13, 14, 15]. The U(1)Y gauge coupling constant, $g_{Y}$, is evaluated at $m_{\rm soft}$, and the corrections are represented by the difference between the beta functions of the U(1)Y gauge and Bino couplings. In our setup, the SM particles and the sleptons contribute to the beta functions. Thus, the coefficient becomes $\Delta b=41/6-n_{\rm slepton}$, where $n_{\rm slepton}$ is number of the generations of light sleptons; for instance, $n_{\rm slepton}=3$ if all the sleptons are light, and $n_{\rm slepton}=2$ when the staus are decoupled. On the other hand, the term of $\alpha_{2}$ is non-oblique corrections after the Wino decoupled (cf. Ref. [16]). The weak boson couples only to the left-handed (s)leptons at the Bino–muon–smuon vertices. Since the Bino–smuon contribution to the muon $g-2$ includes both $\tilde{g}_{L}$ and $\tilde{g}_{R}$, the loop correction (9) is obtained. It yields a correction of $5-10\%$ for $M_{\rm soft}=10-100\,{\rm TeV}$ with $m_{\rm soft}\sim 100\,{\rm GeV}$. In the analysis, only the logarithmic contributions are included. Non-logarithmic terms are expected to be suppressed. Figure 2: The SUSY contributions to the muon $g-2$ and the vacuum stability bounds are shown. In the orange (yellow) regions, the muon $g-2$ discrepancy (1) is explained at the $1\sigma$ $(2\sigma)$ level. The black solid lines represent the upper bound on $\mu$ from the vacuum stability bound of the stau-Higgs potential. The stau masses are set to be $m_{\tilde{\tau}}/m_{\tilde{\mu}}=1,2,5,10$ and 20 from bottom to top. Above the black dashed line, $\Delta_{\mu}$ becomes larger than unity. The parameters are $M_{1}=m_{\tilde{\mu}_{L}}=m_{\tilde{\mu}_{R}}$, $\tan\beta=40$ and $M_{\rm soft}=30\,{\rm TeV}$. In the right panel, a part of the parameter region of the left panel is magnified. In Eq. (9), the corrections that can be as large as or larger than $\mathop{\mathcal{O}}(10)\%$ are included. Other two loop contributions are unknown or expected to be small. In Refs. [17, 18], SUSY corrections to the SM one-loop diagrams are evaluated. When the superparticles except those of (2) are decoupled, the contributions in Refs. [17, 18] are negligibly small. Other corrections that have not yet been estimated include electroweak and SUSY two- loop contributions to SUSY one-loop diagrams. They might provide $\delta a_{\mu}\sim 10^{-10}$, according to Ref. [19]. In addition, non-logarithmic corrections to $\delta^{\rm 2loop}$ could be a few percents of the SUSY one- loop contributions, similarly to the discussions in Ref. [19]. Apart from the Bino–smuon diagrams, there are other one-loop contributions to the muon $g-2$. The Bino–Higgsino–smuon contribution can be $\lesssim\mathop{\mathcal{O}}(10^{-10})$ when the Higgsinos are light due to the vacuum stability bound (Sec. 3.2). It is included in the numerical analysis for completeness.222 This contribution can dominate the SUSY contributions to the muon $g-2$, when $\mu$ is small while decoupling the Wino. Since they are enhanced only by $\tan\beta$, superparticles are required to be light to explain (1). They are detectable in colliders. In particular, the Higgsino production can be significant. On the other hand, the chargino–muon sneutrino contributions are less than $\mathop{\mathcal{O}}(10^{-11})$ for $M_{2}>10\,{\rm TeV}$, i.e., negligible. In Fig. 2, contours of the SUSY contributions to the muon $g-2$ are shown. The horizontal and vertical axises are the lightest smuon mass, $m_{\tilde{\mu}_{1}}$, and $\mu$, respectively. The parameters are set as $M_{1}=m_{\tilde{\mu}_{L}}=m_{\tilde{\mu}_{R}}$, $\tan\beta=40$ and $M_{\rm soft}=30\,{\rm TeV}$. In the orange (yellow) regions, the SUSY contributions explain the muon $g-2$ discrepancy (1) at the $1\sigma$ $(2\sigma)$ level. It is found that they are enhanced by large $\mu$, and the smuon masses can be $1\,{\rm TeV}$ for $\mu=\mathop{\mathcal{O}}(10-100)\,{\rm TeV}$. This is contrasted to the chargino–muon sneutrino contributions to the muon $g-2$, where $\mu$ is favored to be small [10]. On the other hand, detailed dependences on the superparticle mass spectrum are determined by the loop function (5) and the vacuum stability condition. They will be discussed in the next subsection. ### 3.2 Vacuum stability As shown in Sec. 3.1, the Bino–smuon contribution to the muon $g-2$ is enhanced by a large left-right mixing of the smuon. However, too large mixing spoils the stability of the electroweak vacuum. The trilinear coupling of the sleptons and the SM-like Higgs boson is given by $\displaystyle V$ $\displaystyle\simeq\frac{1}{\sqrt{2}v}m_{\tilde{\ell}_{LR}}^{2}\tilde{\ell}_{L}^{}\tilde{\ell}_{R}h^{0}+{\rm h.c.}$ $\displaystyle=-\frac{m_{\ell}}{\sqrt{2}v(1+\Delta_{\ell})}\mu\tan\beta\cdot\tilde{\ell}_{L}^{}\tilde{\ell}_{R}h^{0}+{\rm h.c.},$ (13) where $v\simeq 174\,{\rm GeV}$ is the Higgs VEV. As the trilinear coupling increases, disastrous charge-breaking minima in the scalar potential become deeper, and our electroweak vacuum could decay to them. By requiring that the lifetime of the electroweak vacuum should be longer than the age of the Universe, $m_{\tilde{\ell}_{LR}}^{2}$ is constrained. The vacuum stability conditions have been studied. The fitting formula of the stability condition is obtained as $\displaystyle\left\|m_{\tilde{\ell}_{LR}}^{2}\right\|$ $\displaystyle\leq\eta_{\ell}\bigg{[}1.01\times 10^{2}\,{\rm GeV}\sqrt{m_{\tilde{\ell}_{L}}m_{\tilde{\ell}_{R}}}+1.01\times 10^{2}\,{\rm GeV}(m_{\tilde{\ell}_{L}}+1.03m_{\tilde{\ell}_{R}})$ $\displaystyle-2.27\times$ $\displaystyle 10^{4}\,{\rm GeV}^{2}+\frac{2.97\times 10^{6}\,{\rm GeV}^{3}}{m_{\tilde{\ell}_{L}}+m_{\tilde{\ell}_{R}}}-1.14\times 10^{8}\,{\rm GeV}^{4}\left(\frac{1}{m^{2}_{\tilde{\ell}_{L}}}+\frac{0.983}{m^{2}_{\tilde{\ell}_{R}}}\right)\bigg{]}.$ (14) This is consistent with the result of Ref. [20] in a case of the stau–Higgs potential. The vacuum decay rate is evaluated by the bounce method [21]. The public package CosmoTransitions 1.0.2 [23] is used for the numerical evaluation. The scalar potential of the left- and right-handed sleptons and the SM-like Higgs boson is analyzed at the zero temperature.333 Thermal corrections can change the transition rate. They could be significant when the staus are light, for instance, for $m_{\tilde{\tau}_{1}}\lesssim 200\,{\rm GeV}$ [22]. The Higgs potential is set to reproduce the mass of $126\,{\rm GeV}$. Only the renormalizable terms are taken into account. Higher- dimensional terms depend on the superparticles that decouple at $M_{\rm soft}$. Since $M_{\rm soft}$ is very large, their contributions to (14) are considered to be small. The fitting formula reproduces results of CosmoTransitions at better than the 1% level. The fitting formula (14) is universal for all the slepton flavors up to corrections, $\eta_{\ell}\sim 1$. Given the soft scalar masses, the bound is independent of the flavor except through Yukawa interactions in the quartic terms of the scalar potential. Since the quartic terms are dominated by gauge interactions, $\eta_{\ell}$ changes little around unity. It also depends on $\tan\beta$ when the Yukawa coupling is large. In practice, $\eta_{\tau}=1$ is set for the stau with $\tan\beta/(1+\Delta_{\tau})=70$. Numerical result of $\eta_{\tau}$ for the stau with various $\tan\beta$ is found in Fig. 2 of Ref. [20]. For instance, it is $\eta_{\tau}\simeq 0.94$ for $\tan\beta/(1+\Delta_{\tau})=40$. On the other hand, we obtain $\eta_{\ell}\simeq 0.88$ for the smuons and the selectrons, which is independent of the flavor and $\tan\beta$ because of small Yukawa couplings. The most severe constraint on $\mu\tan\beta$ is obtained from the stability condition of the stau-Higgs potential. This is because the left-right mixing is proportional to the Yukawa coupling. In Fig. 2, the upper limits are shown for $\tan\beta=40$. Combined with the muon $g-2$, the smuon masses are bounded from above for given stau masses. For $m_{\tilde{e}}=m_{\tilde{\mu}}=m_{\tilde{\tau}}$, the lightest smuon is limited to be $m_{\tilde{\mu}_{1}}\lesssim 300~{}(420)\,{\rm GeV}$ at the $1\sigma$ ($2\sigma$) level of the muon $g-2$. Figure 3: The SUSY contributions to the muon $g-2$ as a function of the lightest smuon mass, $m_{\tilde{\mu}_{1}}$, and the lightest neutralino mass, $m_{\tilde{\chi}^{0}_{1}}$. In the orange (yellow) regions, the muon $g-2$ discrepancy (1) is explained at the $1\sigma$ $(2\sigma)$ level. The left- right mixing is maximized under the vacuum stability condition. The parameters are $m_{\tilde{\ell}_{L}}=m_{\tilde{\ell}_{R}}$, $\tan\beta=40$ and $M_{\rm soft}=10\,{\rm TeV}$. The stau soft masses are equal to those of the selectrons and smuons. The region below the green line is excluded by LHC. The SUSY contributions to the muon $g-2$ depend on superparticle mass spectra mainly through the loop function (5). In Fig. 3, contours of the muon $g-2$ are displayed for various smuon and neutralino masses. In the orange (yellow) regions, the muon $g-2$ discrepancy is explained at the $1\sigma$ ($2\sigma$) level. The left-right mixing is maximized with satisfying the vacuum stability condition. Here, the stau soft mass is supposed to be degenerate with that of the smuon. It is found that the lightest smuon mass can be $330~{}(460)\,{\rm GeV}$ at the $1\sigma$ ($2\sigma$) level of the muon $g-2$, when the lightest neutralino mass, $m_{\tilde{\chi}^{0}_{1}}$, is almost a half of $m_{\tilde{\mu}_{1}}$. Note that $m_{\tilde{\chi}^{0}_{1}}$ is almost equal to $M_{1}$. On the other hand, $m_{\tilde{\chi}^{0}_{1}}\simeq 560\,{\rm GeV}$ is realized for $m_{\tilde{\mu}_{1}}\simeq 250\,{\rm GeV}$ at the $2\sigma$ level of the muon $g-2$. For a fixed value of the lightest smuon mass, the SUSY contributions to the muon $g-2$ is maximized when $m_{\tilde{\mu}_{L}}$ is equal to $m_{\tilde{\mu}_{R}}$. Eq. (4) depends on the left- and right-handed smuon masses as $f_{N}(x,y)/xy$ with $x=m_{\tilde{\mu}_{L}}^{2}/M_{1}^{2}$ and $y=m_{\tilde{\mu}_{R}}^{2}/M_{1}^{2}$. When $xy$ is fixed, it is maximized for $x=y$ and rapidly decreases for $x\neq y$. On the other hand, the vacuum stability condition (14) is relaxed for $m_{\tilde{\ell}_{L}}\neq m_{\tilde{\ell}_{R}}$. Since the former dependence is stronger than the latter, the SUSY contributions to the muon $g-2$ become largest when $m_{\tilde{\mu}_{L}}$ is equal to $m_{\tilde{\mu}_{R}}$ for given $M_{1}$ and $\tan\beta$. In other words, the orange/yellow regions in Fig. 3, i.e., the parameter regions favored by the muon $g-2$, shrink toward smaller superparticle masses for $m_{\tilde{\mu}_{L}}\neq m_{\tilde{\mu}_{R}}$. Figure 4: The left-right mixing of the smuon (blue solid) and the stau (red solid) as a function of $\mu$. The blue (red) dashed line represents the vacuum meta-stability bound with $\eta=0.88~{}(0.94)$. Above the black dashed line, sleptons become tachyonic. Below the blue dotted line, $\Delta_{\mu}<1$ is satisfied. The parameters are $m_{\tilde{\ell}_{L}}=m_{\tilde{\ell}_{R}}=2M_{1}=500\,{\rm GeV}$, $\tan\beta=40$ and $M_{\rm soft}=30\,{\rm TeV}$. The vacuum stability bound from the stau-Higgs potential is relaxed for heavier staus. When they are decoupled, the constraint almost disappears, and larger $\mu$ is allowed. As $\mu$ increases, the left-right mixing (6) approaches to a maximal value. It is determined by the soft scalar masses as $\displaystyle m_{\tilde{\ell}_{LR}}^{2}=-\frac{m_{\ell}}{1+\Delta_{\ell}}\mu\tan\beta~{}\to~{}-\frac{4\pi}{\alpha_{Y}}\frac{m_{\ell}}{M_{1}}\frac{1}{I(M_{1}^{2},m_{\tilde{\ell}_{L}}^{2},m_{\tilde{\ell}_{R}}^{2})}~{}~{}~{}(\text{for}~{}\mu\to\infty).$ (15) Note that the correction $\Delta_{\ell}$ is much larger than unity in this limit. In Fig. 4, the left-right mixings of the smuon and the stau, $\|m_{\tilde{\ell}_{LR}}^{2}\|$, are plotted as a function of $\mu$, by the blue and red solid lines, respectively. Since $m_{\tilde{\ell}_{LR}}^{2}$ is proportional to the Yukawa coupling, $\|m_{\tilde{\tau}_{LR}}^{2}\|$ is larger than $\|m_{\tilde{\mu}_{LR}}^{2}\|$. When the staus are light, $\mu$ is bounded from above by the stability condition of the stau–Higgs potential (red dashed line). For heavier staus, the red dashed line (as well as the stau tachyonic bound or the black dashed line) is lifted up, and larger $\mu$ is allowed. Even if the staus are decoupled, the left-right mixing is limited by the vacuum stability condition of the smuon-Higgs potential. When $M_{1}$ is smaller than the smuon mass with $m_{\tilde{\mu}_{L}}=m_{\tilde{\mu}_{R}}$, the vacuum constraint (14) is more severe than Eq. (15). It can be estimated that the lightest smuon can be as large as $1.4~{}(1.9)\,{\rm TeV}$ at the $1\sigma~{}(2\sigma)$ level of the muon $g-2$.444 In this limit, the situation is essentially the same as the soft Yukawa coupling models [24, 25]. In this paper, the vacuum stability condition is analyzed seriously and is found to restrict the SUSY contributions to the muon $g-2$, resulting in tighter bound on the smuon mass than Ref. [25]. Since it is very weak, the bound does not appear in the parameter range of Fig. 2. It is noticed that $\Delta_{\ell}$ is larger than unity when $\mu$ is very large. Then, the radiative correction exceeds the tree level contribution to the lepton mass. For $\Delta_{\ell}\gg 1$, the Yukawa coupling constant is suppressed by $\Delta_{\ell}$, as observed by Eq. (6). This may be disfavored by a naturalness argument [26]. In Fig. 2, $\Delta_{\mu}$ becomes larger than unity above the black dashed line, and above the blue dotted line in Fig. 4. ## 4 Searches Mass spectrum \| Smuon \| Vacuum \| LHC/ILC \| LFV/EDM ---\|---\|---\|---\|--- $m_{\tilde{e}}=m_{\tilde{\mu}}=m_{\tilde{\tau}}$ (Sec. 4.1) \| $<330/460\,{\rm GeV}$ \| $\tilde{\tau}$ \| ✓ \| $m_{\tilde{e}}=m_{\tilde{\mu}}<m_{\tilde{\tau}}$ (Sec. 4.2) \| $<1.4/1.9\,{\rm TeV}$ \| $\tilde{\tau}$ or $\tilde{\mu}$ \| \| ✓ Table 1: Summary of searches. The lightest smuon mass is restricted to explain the muon $g-2$ discrepancy at the $1\sigma/2\sigma$ level. The left-right mixing is limited by the vacuum stability condition of the stau– or smuon–Higgs potential, depending on the stau masses. Models of the universal mass spectrum can be tested by LHC/ILC, while those of the non-universal spectrum predict large LFV/EDM. In the previous section, it has been shown that when the muon $g-2$ anomaly (1) is solved by the Bino–smuon contribution (4), soft masses of the Bino and the smuon are bounded from above by the vacuum stability condition (14). The result is summarized in Tab. 1. In this section, we study experimental status and future prospects to search for such SUSY models. The mass bounds depend on the slepton mass spectrum. When the stau is degenerate with the smuon, a tight constraint is imposed by the stau stability condition. In Sec. 4.1, it will be discussed that the limit is strong enough for the superparticles to be detectable directly in colliders. On the other hand, if the staus are heavier or decoupled, the stability bound is relaxed. The superparticle masses can exceed the collider sensitivities. In Sec. 4.2, it will be argued that such hierarchical mass spectrum can be probed by LFV and CPV. ### 4.1 Universal slepton mass In this section, we study collider searches for the SUSY models with the universal slepton mass spectrum. Here, the left- (right-) handed selectron, smuon and stau have a common soft SUSY-breaking mass, $\displaystyle m_{\tilde{e}_{L}}=m_{\tilde{\mu}_{L}}=m_{\tilde{\tau}_{L}},~{}~{}~{}m_{\tilde{e}_{R}}=m_{\tilde{\mu}_{R}}=m_{\tilde{\tau}_{R}}.$ (16) Then, the vacuum stability condition from the stau–Higgs potential restricts slepton masses up to $330-460\,{\rm GeV}$ to solve the muon $g-2$ anomaly. This is within the reach of the LHC or ILC sensitivity. In fact, some of the parameter regions have already been excluded by LHC, as shown later. Collider signatures depend on the lightest superparticle (LSP). In the case of universal soft slepton masses, either the lightest neutralino or the lightest stau is LSP among the MSSM particles.555 The lightest stau is lighter than sneutrinos when $\mu$ is large. In the latter case, the stau is likely to be long-lived.666 The stau could decay in detectors, for instance, through R-parity violations. The SUSY signatures depend on decay channels. Such cases are not discussed here. The (meta-) stable staus leave charged tracks in detectors. Such signatures have been studied by ATLAS [27] and CMS [28]. The CMS constraint on the cross section of the stau direct production provides the 95% CL exclusion limit, $m_{\tilde{\tau}_{1}}>339\,{\rm GeV}$. If this is imposed in addition to the vacuum stability condition, all the parameter regions of the long-lived stau are excluded in Fig. 3. Figure 5: Same as Fig. 3, but with a condition that the neutralino is LSP, $m_{\tilde{\tau}_{1}}>m_{\tilde{\chi}^{0}_{1}}$. In Fig. 5, we show the SUSY contributions to the muon $g-2$ when the left- right mixing is maximized under the two conditions; (i) the vacuum stability and (ii) the neutralino LSP, $m_{\tilde{\tau}_{1}}>m_{\tilde{\chi}^{0}_{1}}$. Then, SUSY signature is (opposite-sign same-flavor) di-lepton with large missing transverse energy. Selectrons and smuons are produced by collisions. They subsequently decay into the lightest neutralino and a partner lepton. Recently, this signature was studied by ATLAS [29] and CMS [30]. In particular, the 95% CL exclusion limit is obtained for $m_{\tilde{\mu}_{L}}=m_{\tilde{\mu}_{R}}$ by ATLAS. The result is shown by the green solid line in Fig. 5 (and in Fig. 3). Here, the region below the line is excluded. In detail, the left-right mixing is negligible in the ATLAS analysis. However, the left- and right-handed smuons maximally mix with each other in Fig. 5. The total cross section of the smuon productions decreases by 10% compared to the ATLAS setup. Since it is sufficiently small, the exclusion limit is considered to be almost the same as the ATLAS result. As a result, it is found that a large fraction of the $1\sigma$ parameter region of the muon $g-2$ is already excluded.777 In the ATLAS analysis, $m_{\tilde{e}}=m_{\tilde{\mu}}$ is assumed. The selectron production provides almost the same constraint as the smuon [29]. Thus, if selectrons are decoupled, the mass bound becomes weaker. However, LFV/CPV constraints are severe, as will be mentioned in Sec. 4.2. The sensitivity will be improved by the upgrade of the energy and the luminosity. In Fig. 6, the total cross section of the smuon productions are presented for $\sqrt{s}=8\,{\rm TeV}$ and $14\,{\rm TeV}$. The cross section is estimated at the leading order. For instance, it becomes $1\,{\rm fb}$ for the lightest smuon mass of $330\,{\rm GeV}$ at $\sqrt{s}=8\,{\rm TeV}$, which corresponds to the current LHC bound. The same cross section is obtained for $450\,{\rm GeV}$ at $14\,{\rm TeV}$. Studies for the future sensitivity are required. Figure 6: The total cross section of the smuon productions at LHC with $\sqrt{s}=8\,{\rm TeV}$ (dashed) and $14\,{\rm TeV}$ (solid). The parameters satisfy $m_{\tilde{\ell}_{L}}=m_{\tilde{\ell}_{R}}=2M_{1}$. The left-right mixing is maximized under the vacuum stability condition with $\tan\beta=40$. When the neutralino is LSP, the stau productions have also been studied in LEP and LHC. If their mass difference is larger than $15\,{\rm GeV}$, the stau mass is constrained to be larger than $81.9\,{\rm GeV}$ by LEP [31]. LHC is still ineffective to search for di-tau events from the direct stau productions [32]. These limits are sufficiently weak. Fig. 5 does not change even if they are imposed. \| $m_{\tilde{\ell}}$ \| $M_{1}$ \| $\mu$ \| $m_{\tilde{e}_{1}}$ \| $m_{\tilde{e}_{2}}$ \| $m_{\tilde{\mu}_{1}}$ \| $m_{\tilde{\mu}_{2}}$ \| $m_{\tilde{\tau}_{1}}$ \| $m_{\tilde{\tau}_{2}}$ \| $m_{\tilde{\chi}^{0}_{1}}$ \| $\Delta a_{\mu}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- A \| 300 \| 200 \| 756 \| 303 \| 304 \| 298 \| 309 \| 199 \| 380 \| 199 \| 16.1 $\rm{A^{\prime}}$ \| 300 \| 200 \| 699 \| 303 \| 304 \| 299 \| 308 \| 209 \| 375 \| 199 \| 14.6 B \| 470 \| 250 \| 1680 \| 472 \| 472 \| 465 \| 479 \| 329 \| 581 \| 250 \| 10.2 C \| 340 \| 160 \| 1138 \| 343 \| 343 \| 336 \| 350 \| 199 \| 442 \| 160 \| 18.0 Table 2: Model parameters and mass spectra at several model points in Fig. 5. The masses are in units of GeV, and the muon $g-2$ is scaled by $10^{-10}$. Here, $m_{\tilde{\ell}}$ denotes $m_{\tilde{\ell}_{L}}=m_{\tilde{\ell}_{R}}$, and $\tan\beta=40$ is set. In Tab. 2, superparticle mass spectra are listed for several points in Fig. 5. The lightest neutralino mass is close to the Bino mass, since $\mu$ is very large. Given $m_{\tilde{\ell}}\equiv m_{\tilde{\ell}_{L}}=m_{\tilde{\ell}_{R}}$, the slepton masses are hierarchical except for those of the selectrons due to a large left-right mixing. The lightest stau mass is closest to the neutralino mass among the sleptons. In Fig. 5, the point A (A’) is around the upper side of the contour of the muon $g-2$. In the vicinity of this side, $m_{\tilde{\tau}_{1}}$ has the closest value to $m_{\tilde{\chi}^{0}_{1}}$ in the allowed range. If $m_{\tilde{\tau}_{1}}>m_{\tilde{\chi}^{0}_{1}}$ is imposed, it satisfies $m_{\tilde{\tau}_{1}}=m_{\tilde{\chi}^{0}_{1}}$ to maximize the SUSY contributions to the muon $g-2$ around the upper side of the contour. Above it, the regions are excluded by the long-lived stau search, or the muon $g-2$ becomes too small. On the other hand, if the mass difference between the stau and the neutralino is assumed to be larger than $\delta M$, they satisfy $m_{\tilde{\tau}_{1}}=m_{\tilde{\chi}^{0}_{1}}+\delta M$ in the vicinity of the upper side. At the point A’, $\delta M=10\,{\rm GeV}$ is imposed, and $m_{\tilde{\tau}_{1}}$ is larger than $m_{\tilde{\chi}^{0}_{1}}$ by $10\,{\rm GeV}$. In contrast, the points B and C are away from this region. B (C) is close to the maximal end point of the lightest smuon mass which explains the muon $g-2$ at the $2\sigma$ ($1\sigma$) level. Here, the left-right mixing is determined by the vacuum stability condition of the stau. Figure 7: The production cross section of the lightest smuon at ILC with $\sqrt{s}=1\,{\rm TeV}$. The beams are assumed to be polarized at 80% $(e^{-})$ and 30% $(e^{+})$. The left-right mixing is maximized under the vacuum stability condition with $\tan\beta=40$. The $1\sigma$ ($2\sigma$) regions of the muon $g-2$ are also shown by the orange (yellow) bands. The parameters are $M_{1}=200\,{\rm GeV}$ and $M_{\rm soft}=10\,{\rm TeV}$. Linear colliders of $e^{+}e^{-}$ (ILC) are very useful (see e.g., Ref. [33, 34]). They can provide rich informations of the models. Moreover, they are superior to LHC when the slepton masses are close to that of the lightest neutralino [35]. In the orange/yellow regions of Fig. 5, selectrons, smuons and staus can be produced at the linear colliders. In Fig. 7, production cross sections of the lightest smuon are shown for the left- and right-handed smuon masses. Here, it is assumed that the collision energy is $\sqrt{s}=1\,{\rm TeV}$ at ILC, and the beams are polarized at 80% for the electron and 30% for the positron. The left-right mixing is maximized under the vacuum stability condition. The other parameters are $M_{1}=200\,{\rm GeV}$ and $\tan\beta=40$. It is found that the cross section is larger than 20 (1)$\,{\rm fb^{-1}}$ for the $1\sigma$ ($2\sigma$) parameter region of the muon $g-2$. The estimation is based on Ref. [36].888 The cross section is calculated at the leading order in Fig. 7. It can be enhanced by $\sim 10\%$ near the mass threshold [36]. It is expected that the smuons can be discovered in almost all the parameter region that are kinematically allowed (cf. [34]). It is possible to measure masses of the smuon and the neutralino from event distributions and the cross section [37, 13]. Also, the chirality structure of the smuons could be determined by the beam polarization. We assume $(P_{e^{-}},P_{e^{+}})=(+0.8,-0.3)$ in the left panel of Fig. 7, and $(-0.8,+0.3)$ in the right panel. The cross section of a chiral smuon is sensitive to the polarization, because the productions proceed by the s-channel $\gamma/Z$ exchanges. On the other hand, when $m_{\tilde{\mu}_{L}}=m_{\tilde{\mu}_{R}}$, the cross section is insensitive to the beam polarization. Since the left-right mixing is large, the left- and right-handed smuons are maximally mixed with each other. Both of them can be produced, and the mass difference is expected to be measured, for instance, by measuring threshold productions. At the model points in Tab. 2, the difference is $\gtrsim 10\,{\rm GeV}$, which is much larger than the uncertainty of the mass measurement at ILC, $\mathop{\mathcal{O}}(10-100)\,{\rm MeV}$ [33]. It is emphasized that the smuon productions are clean and direct signatures of the SUSY contributions to the muon $g-2$. Figure 8: The production cross section of $\tilde{e}^{+}_{R}\tilde{e}^{-}_{L}$ and $\tilde{e}^{+}_{L}\tilde{e}^{-}_{R}$ at ILC with $\sqrt{s}=1\,{\rm TeV}$. The parameters are same as Fig. 7, but $M_{1}$ is varied with $m_{\tilde{e}_{L}}=m_{\tilde{e}_{R}}\equiv m_{\tilde{e}}$. The corrections to the Bino coupling due to decoupling of heavy superparticles are included (discarded) in the black solid (dashed) lines. The selectron productions proceed not only by the s-channel $\gamma/Z$ exchanges, but also by the t-channel Bino exchange. The latter contribution can enhance the cross section and provide individual information in addition to mass measurements of the selectron and the neutralino. The productions, $e^{+}_{L}e^{-}_{L}\to\tilde{e}^{+}_{R}\tilde{e}^{-}_{L}$ and $e^{+}_{R}e^{-}_{R}\to\tilde{e}^{+}_{L}\tilde{e}^{-}_{R}$, proceed by the t-channel. In Fig. 8, their total cross sections are shown by the black solid lines. Here, the parameters are $m_{\tilde{e}_{L}}=m_{\tilde{e}_{R}}$, and the beam polarization is 80% for the electron and 30% for the positron. It is found that the cross sections are $\mathop{\mathcal{O}}(10)\,{\rm fb}$ in the muon $g-2$ parameter regions. They are larger than those of the smuon in Fig. 7. Importantly, this channel is useful to measure the Bino–electron–selectron couplings [13, 14]. They are deviated from the U(1)Y gauge coupling constant by decoupling heavy superparticles, as discussed in Sec. 3.1. In Fig. 8, the cross sections are estimated with (without) $\delta\tilde{g}_{L}$ and $\delta\tilde{g}_{R}$ (see Eqs. (11) and (12)). The results are shown by the solid (dashed) lines. It is found that the corrections enhance the cross section by $8-10\%$ for $M_{\rm susy}=10\,{\rm TeV}$. This is measurable at ILC [13, 14]. Cross sections of $e^{+}_{L}e^{-}_{R}\to\tilde{e}^{+}_{R}\tilde{e}^{-}_{R}$ and $e^{+}_{R}e^{-}_{L}\to\tilde{e}^{+}_{L}\tilde{e}^{-}_{L}$ also include the t-channel contribution of the Bino (see e.g., Ref. [36]). In particular, the former cross section is enhanced well compared to those solely by the s-channel $\gamma/Z$ exchanges. It can be $\gtrsim 100\,{\rm fb}$ when $M_{1}$ is relatively small in the muon $g-2$ parameter regions. However, in the latter process, the t-channel contribution interferes destructively with the s-channel contribution. On the other hand, the cross section of $e^{+}_{L}e^{-}_{R}\to\tilde{e}^{+}_{R}\tilde{e}^{-}_{R}$ differs by $5-6\%$ between the cases with and without $\delta\tilde{g}_{L}$ and $\delta\tilde{g}_{R}$. This is smaller than the above channels. The stau productions proceed similarly to the smuon at ILC. The stau mass parameters can be determined [38, 13, 39]. The produced staus decay into the lightest (Bino-like) neutralino and a tau lepton. The tau polarization can be measured from the energy spectra of tau hadronic decays [38]. If productions of both the heavy and light staus are kinematically allowed, all the components of the stau mass matrix can be measured. In particular, the left- right mixing is determined.999 Strictly, it is difficult to determine $\mu$ and $A_{\tau}$ separately by the stau productions, because the lightest neutralino is almost composed of the Bino [13, 39]. Let us discuss the dark matter in the present model. The lightest neutralino is a candidate of the dark matter. The neutralino relic abundance becomes consistent with the measured cold dark matter abundance [40, 41] by the stau co-annihilation. The mass difference between the stau and the neutralino is required to be $5-10\,{\rm GeV}$. This corresponds to the model points in the vicinity of the upper side of the muon $g-2$ contours in Fig. 5, including the points A or A’. This region has not been excluded by the studies of the di- lepton signature at LHC [29, 30]. Masses of the selectrons and the smuons are too close to that of the neutralino for LHC. ILC is superior to study the region [33, 34]. Selectrons and smuons as well as staus can be produced, as discussed above. Finally, let us mention the branching ratios of the Higgs boson decays. When the staus are light, and their left-right mixing is large, the branching ratio of the di-photon channel can be enhanced sizably [42]. We have estimated it in the muon $g-2$ parameter regions of Fig. 5. In the $1\sigma$ parameter region, the ratio increases by $10-40\%$ compared to the SM prediction, while it is enhanced by $5-10\%$ in the $2\sigma$ region. The latter is comparable to the sensitivity of the high-luminosity LHC with $\int\mathcal{L}=3000\,{\rm fb^{-1}}$ [43, 44]. On the other hand, the ratio of the Higgs boson decaying to the di-muon is almost unchanged. Even when the radiative corrections to the Higgs boson coupling with the muon is very large, i.e., $\Delta_{\mu}\gg 1$, the ratio does not change, because the tree level (Yukawa) coupling decreases by $\Delta_{\mu}$, and the sum of the tree and radiative contributions is not varied. ### 4.2 Non-universal slepton mass Figure 9: The lower bound on the stau mass to satisfy the vacuum stability bound of the stau–Higgs potential. In the orange (yellow) region, the SUSY contribution to the muon $g-2$ can explain the discrepancy (1) at the $1\sigma$ $(2\sigma)$ level. The parameters are $m_{\tilde{\mu}_{L}}=m_{\tilde{\mu}_{R}}=M_{1}$, $m_{\tilde{\tau}_{L}}=m_{\tilde{\tau}_{R}}$, $\tan\beta=40$ and $M_{\rm soft}=30\,{\rm TeV}$. In this section, we discuss the case without slepton mass universality. The slepton soft SUSY-breaking masses satisfy $\displaystyle m_{\tilde{e}_{L}}=m_{\tilde{\mu}_{L}}<m_{\tilde{\tau}_{L}},~{}~{}~{}m_{\tilde{e}_{R}}=m_{\tilde{\mu}_{R}}<m_{\tilde{\tau}_{R}}.$ (17) As discussed in Sec. 3.2, the vacuum stability bound of the stau–Higgs potential is relaxed by heavy staus, and the smuon masses are allowed to be larger in order to solve the muon $g-2$ anomaly. In other words, the stau masses are required to be large relative to the smuons, when the smuons are heavy. In Fig. 9, the lower bound on the stau mass is shown. In the orange (yellow) regions, the muon $g-2$ anomaly is solved at the $1\sigma$ ($2\sigma$) level, while the vacuum stability constraint is avoided. The ratios of the stau and smuon masses are defined as $\displaystyle R_{L}\equiv\frac{m_{\tilde{\tau}_{L}}}{m_{\tilde{\mu}_{L}}},~{}~{}~{}R_{R}\equiv\frac{m_{\tilde{\tau}_{R}}}{m_{\tilde{\mu}_{R}}}.$ (18) The parameters are $M_{1}=m_{\tilde{\mu}_{L}}=m_{\tilde{\mu}_{R}}$, $m_{\tilde{\tau}_{L}}=m_{\tilde{\tau}_{R}}$, $\tan\beta=40$ and $M_{\rm soft}=30\,{\rm TeV}$. The vertical axis is $R\equiv R_{L}=R_{R}$. If the staus are decoupled, the left-right mixing is bounded either by Eq. (15) or the vacuum stability bound of the smuon–Higgs potential, as discussed in Sec. 3.2. The smuon masses can be $1.4\,{\rm TeV}$ $(1.9\,{\rm TeV})$ for the $1\sigma$ $(2\sigma)$ level of the muon $g-2$. Even though it becomes difficult to produce such sleptons at LHC or ILC, non-universal slepton masses generically cause problems of too large lepton flavor violations (LFV) and CP violations (CPV). LFV and CPV are sensitive to off-diagonal (generation mixing) components of the slepton mass matrices. They are suppressed if the slepton mass matrices are universal among the flavors, such as in the gauge mediated SUSY-breaking models. However, even when the mass matrices are diagonal in the model basis, sizable FCNC and CPV are generically induced as long as the diagonal components are not equal to each others. In fact, a lot of models have been proposed to explain the SM Yukawa couplings, and many of them predict non- diagonal Yukawa matrices in the model basis. Even if the sfermion mass matrices are diagonal in this basis, off-diagonal components are generated in the fermion mass eigenstate basis, which are obtained by rotating the mass matrices of the model basis with unitary matrices. In other words, the super GIM mechanism does not work generically unless the sfermion mass matrices are universal. When the muon $g-2$ anomaly is solved by the SUSY contributions, LFV and lepton electric dipole moments (EDM) are generically sizable. In this section, the following setup is considered. In the model basis, the slepton mass matrices are diagonal among the flavors, $\displaystyle(m_{\tilde{\ell}_{L}}^{2})_{ij}$ $\displaystyle=\text{diag}(m_{\tilde{e}_{L}}^{2},m_{\tilde{\mu}_{L}}^{2},m_{\tilde{\tau}_{L}}^{2}),$ (19) $\displaystyle(m_{\tilde{\ell}_{R}}^{2})_{ij}$ $\displaystyle=\text{diag}(m_{\tilde{e}_{R}}^{2},m_{\tilde{\mu}_{R}}^{2},m_{\tilde{\tau}_{R}}^{2}).$ (20) Moreover, $m_{\tilde{e}_{L}}=m_{\tilde{\mu}_{L}}$ and $m_{\tilde{e}_{R}}=m_{\tilde{\mu}_{R}}$ are imposed. Otherwise, LFV becomes too large, as shown later. The Yukawa matrices are generally non-diagonal in the model basis. The mass eigenstate basis of the charged leptons is obtained by the left- and right-handed unitary matrices, $U_{L},U_{R}$, as $\displaystyle U_{R}\,M_{\ell}\,U_{L}^{\dagger}\equiv U_{R}\,Y_{\ell}\,v_{d}\,U_{L}^{\dagger}={\rm diag}\left(\frac{m_{e}}{1+\Delta_{e}},~{}\frac{m_{\mu}}{1+\Delta_{\mu}},~{}\frac{m_{\tau}}{1+\Delta_{\tau}}\right),$ (21) where $v_{d}$ is the down-type Higgs VEV, and $\Delta_{\ell}$ is given by Eq. (7). The unitary matrices are generally represented as $U_{L,R}=\exp\left[\begin{pmatrix}0&(\delta_{L,R})_{12}&(\delta_{L,R})_{13}\\\ -(\delta_{L,R})^{}_{12}&0&(\delta_{L,R})_{23}\\\ -(\delta_{L,R})^{}_{13}&-(\delta_{L,R})^{}_{23}&0\\\ \end{pmatrix}\right].$ (22) Non-vanishing mixings, $(\delta_{L,R})_{ij}$, induce LFV and EDM, as long as the slepton masses are non-universal.101010 Similar setup has been studied in Ref. [45], where squarks of the first two generations are light. Note that quark FCNC and CPV are suppressed by heavy colored superparticles in this paper. The magnetic dipole contributions to LFV are represented by the following effective Lagrangian, $\displaystyle\mathcal{L}_{\rm eff}=e\frac{m_{\ell_{j}}}{2}\bar{\ell}_{i}\sigma_{\mu\nu}\left(A^{L}_{ij}P_{L}+A^{R}_{ij}P_{R}\right)\ell_{j}F^{\mu\nu}+{\rm h.c.},$ (23) where $i,j$ are flavor indices. Contributions to the other higher dimensional operators are subdominant in this paper. The Wilson coefficients, $A^{L}_{ij}$ and $A^{R}_{ij}$, are dominated by the Bino–slepton contributions, similarly to the muon $g-2$. In the mass insertion approximation, they are estimated as (cf. [46]) $\displaystyle A^{L}_{ij}$ $\displaystyle=(1+\delta^{\rm 2loop})\frac{\alpha_{Y}}{8\pi}\frac{M_{1}\mu\tan\beta}{m_{\ell_{j}}}\sum_{a,b=1,2,3}\Big{[}U_{R}\Big{]}_{ib}\Big{[}M_{\ell}\Big{]}_{ba}\Big{[}U_{L}^{\dagger}\Big{]}_{aj}\,F_{a,b},$ (24) $\displaystyle A^{R}_{ij}$ $\displaystyle=(1+\delta^{\rm 2loop})\frac{\alpha_{Y}}{8\pi}\frac{M_{1}\mu\tan\beta}{m_{\ell_{j}}}\sum_{a,b=1,2,3}\Big{[}U_{L}\Big{]}_{ia}\Big{[}M_{\ell}^{\dagger}\Big{]}_{ab}\Big{[}U_{R}^{\dagger}\Big{]}_{bj}\,F_{a,b}.$ (25) The two loop factor $\delta^{\rm 2loop}$ is found in Eq. (9). The loop function $F_{a,b}$ is defined as $\displaystyle F_{a,b}=\frac{1}{m_{\tilde{\ell}_{La}}^{2}m_{\tilde{\ell}_{Rb}}^{2}}f_{N}\left(\frac{m_{\tilde{\ell}_{La}}^{2}}{\|M_{1}\|^{2}},\frac{m_{\tilde{\ell}_{Rb}}^{2}}{\|M_{1}\|^{2}}\right).$ (26) The muon FCNC’s are the most sensitive to the non-universal slepton mass matrices and the non-diagonal Yukawa matrix. The decay rate of $\mu\to e\gamma$ is represented as $\displaystyle\Gamma(\mu\to e\gamma)=\frac{\alpha}{4}m_{\mu}^{5}\left(\left\|A^{L}_{12}\right\|^{2}+\left\|A^{R}_{12}\right\|^{2}\right),$ (27) where the unitarity matrices in $A^{R}_{12}$ are expanded as $\displaystyle\sum_{a,b=1,2,3}\Big{[}U_{L}\Big{]}_{1a}\Big{[}M_{\ell}^{\dagger}\Big{]}_{ab}\Big{[}U_{R}^{\dagger}\Big{]}_{b2}\,F_{a,b}$ $\displaystyle=-\frac{m_{\mu}}{1+\Delta_{\mu}}(\delta_{L})_{12}\left(F_{1,2}-F_{2,2}\right)$ (28) $\displaystyle~{}~{}~{}+\frac{m_{\tau}}{1+\Delta_{\tau}}(\delta_{L})_{13}(\delta_{R})_{23}^{}\left(F_{1,2}-F_{1,3}-F_{3,2}+F_{3,3}\right),$ at the leading order of $(\delta_{L})_{ij}$, $(\delta_{R})_{ij}$ and $m_{\mu}/m_{\tau}$. Here and hereafter, $m_{e}=0$ is set, for simplicity. Similarly, $A^{L}_{12}$ is obtained by replacing $L\leftrightarrow R$. In the last term, $F_{1,2}$ is dominant when the staus are heavy, i.e., $F_{1,2}\gg F_{1,3},F_{3,2},F_{3,3}$ for $m_{\tilde{e}_{L}},m_{\tilde{\mu}_{L}}\ll m_{\tilde{\tau}_{L}}$ and $m_{\tilde{e}_{R}},m_{\tilde{\mu}_{R}}\ll m_{\tilde{\tau}_{R}}$. On the other hand, the right-hand side vanishes when the slepton masses are universal, as expected from the super GIM mechanism. In particular, when the sleptons are degenerate in the first two generations, the first term becomes zero due to $F_{1,2}=F_{2,2}$. Otherwise, the muon LFV is induced at the order of $(\delta_{L})_{12}$. The above rate is compared with the SUSY contribution to the muon $g-2$. In the non-universal slepton mass spectrum, it is represented by $A^{L}_{ij}$ and $A^{R}_{ij}$ as $\displaystyle a_{\mu}({\rm SUSY})$ $\displaystyle=m_{\mu}^{2}\,{\rm Re}\left[A^{L}_{22}+A^{R}_{22}\right]$ $\displaystyle=(1+\delta^{\rm 2loop})\frac{\alpha_{Y}}{4\pi}m_{\mu}M_{1}\mu\tan\beta\left[\frac{m_{\mu}}{1+\Delta_{\mu}}F_{2,2}+\kappa\right].$ (29) This is same as Eq. (4) up to a correction $\kappa$, which is represented as $\displaystyle\kappa=\frac{m_{\tau}}{1+\Delta_{\tau}}{\rm Re}\left[(\delta_{L})_{23}(\delta_{R})_{23}^{}\right]\left(F_{2,2}-F_{2,3}-F_{3,2}+F_{3,3}\right)+\cdots.$ (30) Here, the omitted terms are suppressed by $(\delta_{L})_{ij}$, $(\delta_{R})_{ij}$ or $m_{\mu}/m_{\tau}$. If the slepton mass matrices are universal, $\kappa$ vanishes. It is noticed that Eqs. (27) and (29) are tightly correlated to each other. For the mass spectrum (17) with $R_{L},R_{R}\gg 1$, the ratio is $\displaystyle\frac{{\rm Br}(\mu\to e\gamma)}{a_{\mu}({\rm SUSY})^{2}}\simeq\frac{1}{\Gamma_{\rm tot}}\frac{\alpha m_{\mu}}{16}\,\big{\|}\delta_{13}\delta_{23}\big{\|}^{2}\left(\frac{m_{\tau}}{m_{\mu}}\frac{1+\Delta_{\mu}}{1+\Delta_{\tau}}\right)^{2},$ (31) where $\Gamma_{\rm tot}$ is the total decay rate of muon, and the mixing is defined as $\displaystyle\big{\|}\delta_{13}\delta_{23}\big{\|}^{2}\equiv\big{\|}(\delta_{R})_{13}(\delta_{L})_{23}\big{\|}^{2}+\big{\|}(\delta_{L})_{13}(\delta_{R})_{23}\big{\|}^{2}.$ (32) It is independent of superparticle mass spectra except through $\Delta_{\mu}$ and $\Delta_{\tau}$. Thus, when the muon $g-2$ discrepancy (1) is explained by the SUSY contributions, $\mu\to e\gamma$ is induced by the non-universal slepton mass sizably. It is important that the decay is not suppressed by heavy slepton masses, for instance, $m_{\tilde{\mu}_{1}}=1.4\,{\rm TeV}$ or $1.9\,{\rm TeV}$ in Tab. 1, for given SUSY contributions to the muon $g-2$. Figure 10: Contours of ${\rm Br}(\mu\to e\gamma)$ (left) and $d_{e}$ (right) for $a_{\mu}({\rm SUSY})=10^{-9}$ with $M_{\rm soft}=30\,{\rm TeV}$. In the left panel, the upper black solid line is the current bound by MEG, ${\rm Br}(\mu\to e\gamma)<5.7\cdot 10^{-13}$ at 90% CL [47]. The lower one is the sensitivity of the MEG upgrade, ${\rm Br}(\mu\to e\gamma)=6\cdot 10^{-14}$ [48]. The blue dashed line is the Mu3e sensitivity, ${\rm Br}(\mu\to e\bar{e}e)=10^{-16}$ [49]. The $\mu-e$ conversion is expected to probe down to $R_{\mu e}=3\cdot 10^{-17}$ (upper red dotted) by COMET/Mu2e [50, 51] and $2\cdot 10^{-19}$ (lower red dotted) by PRISM/PRIME [52]. In the right panel, the black solid line is the current bound by using YbF, $\|d_{e}\|<1.05\cdot 10^{-27}\,e{\rm cm}$ at 90% CL [53]. The sensitivity is planned to be improved: $\|d_{e}\|=10^{-29}\,e{\rm cm}$ by Fr or ThO [54, 55, 56], and $10^{-30}\,e{\rm cm}$ by YbF or WN [57, 58]. Here, $m_{\tilde{\ell}_{L}}=m_{\tilde{\ell}_{R}}$ is assumed. In the left panel of Fig. 10, contours of ${\rm Br}(\mu\to e\gamma)$ are shown. Here, $m_{\tilde{\ell}_{L}}=m_{\tilde{\ell}_{R}}$ is assumed. The SUSY contributions to the muon $g-2$ is fixed to be $a_{\mu}({\rm SUSY})=1\times 10^{-9}$ with $M_{\rm soft}=30\,{\rm TeV}$. The small corrections are taken at $M_{1}=m_{\tilde{\mu}}=400\,{\rm GeV}$ and $\tan\beta=40$ as a reference, though the result is almost independent of them. In the figure, the contours correspond to the current limit and future sensitivities of experiments, * • the current limit of the MEG experiment, ${\rm Br}(\mu\to e\gamma)<5.7\times 10^{-13}$ at 90% CL [47] (upper black solid line in the figure). * • the sensitivity of the MEG upgrade, ${\rm Br}(\mu\to e\gamma)=6\times 10^{-14}$ [48] (lower black solid line). * • the sensitivity of the Mu3e experiment, ${\rm Br}(\mu\to e\bar{e}e)=10^{-16}$ at Phase II [49] (blue dashed line). * • the sensitivity of the COMET experiment, $R_{\mu e}=3\times 10^{-17}$ at Phase II [50] (upper red dotted line). The Mu2e experiment has a similar sensitivity [51]. * • the proposal of the PRISM/PRIME project, $R_{\mu e}=2\times 10^{-19}$ [52] (lower red dotted line). Note that $\mu\to e\bar{e}e$ and $\mu-e$ conversion experiments have better sensitivity in future than those of $\mu\to e\gamma$. In particular, the latter experiment has low (accidental) backgrounds. On the other hand, the current constraint and future sensitivities of the tau LFV’s are weaker than those of the muon, though they are also induced by the non-universal slepton masses with finite $(\delta_{L})_{23}$ and $(\delta_{R})_{23}$. From the figure, it is found that the LFV decay rate increases rapidly as the staus become heavier than the smuons, $R>1$. When the staus are decoupled, $R\gg 1$, the mixing $\|\delta_{13}\delta_{23}\|$ in Eq. (32) is limited to be smaller than $3\times 10^{-6}$ by MEG for $a_{\mu}({\rm SUSY})=1\times 10^{-9}$. For instance, when the smuon and selectron masses are larger than $1\,{\rm TeV}$, the stau are required to be heavier than $7\,{\rm TeV}$ ($R>7$) to avoid the vacuum stability constraint, according to Fig. 9. If the lepton Yukawa matrix is related to the quark sector, e.g., by the GUT relation, it is naively expected to be $\|\delta_{13}\delta_{23}\|\sim V_{ub}V_{cb}\sim 10^{-4}$. This already exceeds the above limit. Thus, non- universal slepton mass spectra are tightly constrained by LFV. In future, if sleptons are neither discovered at LHC nor ILC, the model is expected to be probed by LFV. Otherwise, the SM Yukawa matrices are tightly limited in the model basis, or when flavor models are constructed. The flavor off-diagonal components of the Yukawa matrices are sources of the CP violations, when the slepton mass matrices are non-universal. Similarly to the correction $\kappa$ of the muon $g-2$ in Eq. (29), the electron EDM is induced as $\displaystyle\frac{d_{e}}{e}$ $\displaystyle=\frac{m_{e}}{2}\,{\rm Im}\left[A^{L}_{11}-A^{R}_{11}\right]$ $\displaystyle=(1+\delta^{\rm 2loop})\frac{\alpha_{Y}}{8\pi}M_{1}\mu\tan\beta$ $\displaystyle~{}~{}~{}\times\bigg{[}\frac{m_{\mu}}{1+\Delta_{\mu}}{\rm Im}\left[(\delta_{R})_{12}(\delta_{L})_{12}^{}\right]\left(F_{1,1}-F_{1,2}-F_{2,1}+F_{2,2}\right)$ $\displaystyle~{}~{}~{}~{}~{}~{}+\frac{m_{\tau}}{1+\Delta_{\tau}}{\rm Im}\left[(\delta_{R})_{13}(\delta_{L})_{13}^{}\right]\left(F_{1,1}-F_{1,3}-F_{3,1}+F_{3,3}\right)+\cdots\bigg{]}.$ (33) Here, we assume $\arg(M_{1}\mu\tan\beta)=0$. The omitted terms are suppressed by orders of $(\delta_{L})_{ij}$, $(\delta_{R})_{ij}$ or $m_{\mu}/m_{\tau}$. In the last term, $F_{1,1}\gg F_{1,3},F_{3,1},F_{3,3}$ is obtained when the staus are heavy. On the other hand, the right hand side vanishes when the slepton masses are universal, because the complex phases can be rotated away. Comparing Eq. (33) with Eq. (29), we obtain $\displaystyle\frac{d_{e}/e}{a_{\mu}({\rm SUSY})}\simeq\frac{1}{2m_{\mu}}{\rm{Im}}[(\delta_{R})_{13}(\delta_{L})^{\ast}_{13}]\,\frac{m_{\tau}}{m_{\mu}}\frac{1+\Delta_{\mu}}{1+\Delta_{\tau}},$ (34) for the mass spectrum (17) with $R_{L},R_{R}\gg 1$. It is independent of superparticle mass spectra except through $\Delta_{\mu}$ and $\Delta_{\tau}$. Thus, if the muon $g-2$ anomaly is solved by the SUSY contributions, EDM becomes sizable by the non-universal slepton mass. Similarly to LFV, this is valid even for large selectron and smuon masses. In the right panel of Fig. 10, contours of the electron EDM are shown. Here, the SUSY contributions to the muon $g-2$ is fixed to be $a_{\mu}({\rm SUSY})=1\times 10^{-9}$ with $M_{\rm soft}=30\,{\rm TeV}$. The result is almost independent of superparticle mass spectra except for small corrections, $\Delta_{\mu}$ and $\Delta_{\tau}$. The following data are used, * • the current limit with the YbF molecule, $\|d_{e}\|<1.05\times 10^{-27}\,e{\rm cm}$ at 90% CL [53] (solid line in the figure). * • the sensitivity with the Fr atom, $\|d_{e}\|=10^{-29}\,e{\rm cm}$ [54] (dashed line). The experiment with the ThO molecule could have a similar sensitivity by accumulating data, $\|d_{e}\|=1\times 10^{-28}/\sqrt{({\rm day})}\,e{\rm cm}$ [55, 56]. * • the sensitivity with the YbF molecule, $\|d_{e}\|=10^{-30}\,e{\rm cm}$ [57] (dotted line). The experiment with the WN ion can probe down to, $\|d_{e}\|=10^{-30}\,e{\rm cm}/{\rm day}$, where the systematic limit is at the level of $10^{-31}\,e{\rm cm}$ [58]. From the figure, it is found that EDM is sensitive to ${\rm{Im}}[(\delta_{R})_{13}(\delta_{L})^{}_{13}]$ when the staus are heavier than the selectrons. The current experimental limit puts a constraint, ${\rm{Im}}[(\delta_{R})_{13}(\delta_{L})^{}_{13}]<6\times 10^{-7}$ for $R\gg 1$ and $a_{\mu}({\rm SUSY})=1\times 10^{-9}$. The sensitivity will be improved very well. The mixing will be able to be probed at the level of $10^{-10}$. Thus, if sleptons are neither discovered at LHC nor ILC in future, the model can be sensitively probed by EDM as well as LFV. If no signal will be observed, the CP violating phase must be suppressed very tightly in order to explain the muon $g-2$ anomaly by the SUSY contributions. ## 5 Conclusion SUSY is one of the most motivated candidates of the new physics. If the muon $g-2$ anomaly is solved by SUSY, some of the superparticles are relatively light. In this paper, we focused on the Bino–smuon contribution to the muon $g-2$. It is enhanced not only by $\tan\beta$ but also by large $\mu$. Consequently, it was shown that the smuon masses can be as large as $\sim 1\,{\rm TeV}$. We examined the phenomenology of the models, in which only the superparticles that are relevant for the muon $g-2$ are light. The analyses were categorized by the slepton mass spectrum. When the mass spectrum is universal among the flavors, it was found that the vacuum stability of the stau–Higgs potential restricts the smuon masses tightly. They are predicted to be within $330~{}(460)\,{\rm GeV}$ at the $1\sigma~{}(2\sigma)$ level of the muon $g-2$. It was shown that part of the parameter region is already excluded by LHC, and argued that such sleptons are expected to be studied at LHC or ILC in future. If the staus are (much) heavier than the smuons, the vacuum stability bound of the staus is relaxed. In this mass spectrum, the smuon masses are limited by the vacuum stability condition of the smuon–Higgs potential. It was found that they are less than $1.4~{}(1.9)\,{\rm TeV}$ at the $1\sigma~{}(2\sigma)$ level of the muon $g-2$. Such slepton masses exceed the LHC/ILC reach. Instead, the non-universal slepton mass spectrum generically predicts too large LFV and EDM. They originate in the non-diagonal SM Yukawa matrices in the model basis. Although the prediction depends on the flavor models, LFV and/or EDM is very likely to be sizable in wide models. If no sleptons are discovered at LHC nor ILC, the model is expected to be probed by LFV and EDM sensitively. So far, $\tan\beta=40$ was chosen in the numerical analyses. As mentioned in Sec. 2, the observables depend on $m_{\tilde{\ell}_{LR}}^{2}$, and $\tan\beta$ contributes to them only through it. Thus, the results are independent of a choice of $\tan\beta$ up to a correction to the vacuum stability condition (see Sec. 3.2). Since the correction is small, the above conclusion does not change, even if $\tan\beta$ is varied, for instance, from 20 to 70. In conclusion, we studied the SUSY models that explain the muon $g-2$ discrepancy with the Bino–smuon contributions. 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1309.3095	# Propagation of an arbitrary vortex pair through an astigmatic optical system and determination of its net topological charge Salla Gangi Reddy Corresponding author: [email protected] Shashi Prabhakar [email protected] Aadhi A [email protected] J. Banerji [email protected] R. P. Singh [email protected] Physical Research Laboratory, Navrangpura, Ahmedabad, India - 380009 ###### Abstract We embed a pair of vortices with different topological charges in a Gaussian beam and study its evolution through an astigmatic optical system, a tilted lens. The propagation dynamics is explained by a closed-form analytical expression. Furthermore, we show that a careful examination of the intensity distribution at a predicted position past the lens can provide us with the net charge present in the beam. To the best of our knowledge, our method is the first non-interferometric technique to measure the net charge of an arbitrary vortex pair. Our theoretical results are well supported by experimental observations. ###### pacs: 050:0050, 050.4865. ## I Introduction Optical vortices have drawn considerable attention in science and engineering due to their dark core and helical wave front. An optical vortex of order $l$ centered at the origin ($r=0$) has a field distribution of the form $E(r)\exp(il\phi)$. The distribution is such that the field intensity tends to zero as $r\rightarrow 0$ whereas the phase shift in one cycle around the origin is $2l\pi$ where $l$ is an integer. The azimuthal mode index $l$, also called topological charge of the vortex, has a physical meaning in that the vortex carries an orbital angular momentum (OAM) of $l\hbar$ per photon allen . This angular momentum can be imparted to microscopic particles in order to manipulate them optically particle ; particle1 ; grier . In recent years, the OAM of light has also found applications in classical siddu as well as quantum communication OAM ; torner . These applications have led to considerable interest in the generation and study of optical vortices both in free space free ; spiral ; cgh and in guided media guide ; guide1 . The propagation of a pair of vortices has gained a lot of interest in research since the last decade dipole1 ; dipole2 ; dipole3 ; dipole4 ; dipole5 ; dipole6 . Indebetouw studied the propagation of an array of vortices through free space and showed that the relative separation between the vortices is invariant during the propagation in the case of same type (sign) of charges whereas they will attract and annihilate each other in the case of oppositely charged vortices dipole1 . Chen and Roux have studied the annihilation of dipole vortices during their propagation. They found that the background phase function at a point where two dipoles annihilate, have a continuous potential which causes the annihilation. They have used the same background phase function to accelerate the annihilation process dipole4 . Recently, the tight focusing properties of a pair of vortices have been investigated theoretically dipole5 ; dipole6 . However, it dealt only with isopolar and dipole vortices of first order. Here, we present a theoretical analysis of propagation of an arbitrary vortex pair passing through an astigmatic optical system and verify the results with experiments. We suggest that the vortices being generic to all the waves berry , the present study can be useful to acoustic thomas and matter waves angom also. Since the topological charge of a vortex determines its OAM, an accurate measurement of the topological charge is an essential and important task. There are a number of methods to determine the charge of an optical vortex and its sign inter ; inter1 ; inter2 ; pravin1 ; shashi ; hick ; mourka ; pravin2 . For a multi-singular beam, however, there is no method to measure the net charge which determines the torque imparted by the field. We show that the intensity distribution of a multi-singular beam at a predicted position beyond a tilted lens can provide information about the net charge present in it. ## II Theory Consider a pair of optical vortices embedded in a Gaussian beam, one with topological charge $\epsilon_{1}m\,(\epsilon_{1}=\pm 1)$ located at $x_{1}=-x_{0}$, $y_{1}=0$ and another with topological charge $\epsilon_{2}n\,(\epsilon_{2}=\pm 1)$ at $x_{1}=x_{0}$, $y_{1}=0$. The complex field distribution of the vortex pair at the waist plane of the host Gaussian beam, with waist size $w_{0}$, is given by $\begin{split}E_{1}(x_{1},y_{1})=(x_{1}+x_{0}+&i\epsilon_{1}y_{1})^{m}(x_{1}-x_{0}+i\epsilon_{2}y_{1})^{n}\\\ &\times\exp\left[-\left(\frac{x_{1}^{2}+y_{1}^{2}}{{w_{0}^{2}}}\right)\right].\end{split}$ (1) The tilted lens is placed at a distance $z_{0}$ from the waist plane. The vortex passes through the lens and travels a further distance $z$. The overall ray transfer matrix ${\bf M_{tot}}$ is given by pravin2 ${\bf M_{tot}}=\left(\begin{array}[]{cc}{\bf A}&{\bf B}\\\ -{\bf C}/f&{\bf D}\end{array}\right)$ (2) where ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf D}$ are $2\times 2$ diagonal matrices with diagonal elements given by $a_{j}$, $b_{j}$, $c_{j}$ and $d_{j}$ respectively. Explicitly, $\displaystyle c_{1}$ $\displaystyle=$ $\displaystyle\sec\theta,\;c_{2}=\cos\theta,\;a_{j}=1-zc_{j}/f,$ $\displaystyle d_{j}$ $\displaystyle=$ $\displaystyle 1-z_{0}c_{j}/f,\qquad b_{j}=z_{0}+zd_{j},{j=1,2}.$ (3) Next, we define two column vectors ${\bf r_{1}}$, ${\bf r_{2}}$ so that their transposes are given by row vectors ${\bf r_{i}}^{T}=(x_{i},y_{i}),{i=1,2}$. The field $E_{2}(x_{2},y_{2})$ at a distance $z$ past the lens is given by the generalized Huygens-Fresnel integral sieg : $E_{2}(x_{2},y_{2})=\frac{i/\lambda}{\|B\|^{1/2}}\int\\!\\!\\!\int dx_{1}\,dy_{1}\,E_{1}(x_{1},y_{1})e^{-(i\pi/\lambda){\bf\phi(r_{1},r_{2})}}$ (4) where $\|B\|=\|b_{1}b_{2}\|$ is the determinant of ${\bf B}$ and $\displaystyle{\bf\phi(r_{1},r_{2})}$ $\displaystyle=$ $\displaystyle{\bf r_{1}}^{T}{\bf B}^{-1}{\bf A}{\bf r_{1}}+{\bf r_{2}}^{T}{\bf D}{\bf B}^{-1}{\bf A}{\bf r_{2}}-2{\bf r_{1}}^{T}{\bf B}^{-1}{\bf r_{2}}$ (5) $\displaystyle=$ $\displaystyle x_{1}^{2}a_{1}/b_{1}+y_{1}^{2}a_{2}/b_{2}+x_{2}^{2}d_{1}/b_{1}+y_{2}^{2}d_{2}/b_{2}$ $\displaystyle\mbox{}-2(x_{1}x_{2}/b_{1}+y_{1}y_{2}/b_{2}).$ The integration over $x_{1}$ and $y_{1}$ are carried out by writing $E_{1}(x_{1},y_{1})$ as $\displaystyle E_{1}(x_{1},y_{1})$ $\displaystyle=$ $\displaystyle\lim_{\begin{subarray}{c}t\rightarrow 0\\\ t^{\prime}\rightarrow 0\end{subarray}}\left[\frac{\partial^{m}}{\partial t^{m}}\frac{\partial^{n}}{\partial{t^{\prime}}^{n}}\exp\left\\{f\left(t,t^{\prime}\right)\right\\}\right],$ (6a) $\displaystyle f(t,t^{\prime})$ $\displaystyle=$ $\displaystyle t(x_{1}+x_{0}+i\epsilon_{1}y_{1})+t^{\prime}(x_{1}-x_{0}+i\epsilon_{2}y_{1})$ (6b) $\displaystyle{}-\frac{x_{1}^{2}+y_{1}^{2}}{w_{0}^{2}}.$ Using the definition of Hermite polynomial and a recurrence relation $\displaystyle H_{n}(x)$ $\displaystyle=$ $\displaystyle\frac{\partial^{n}}{\partial{t}^{n}}\exp(2xt-t^{2})\|_{t=0}$ (7a) $\displaystyle\frac{d^{j}}{dx^{j}}H_{n}(x)$ $\displaystyle=$ $\displaystyle\frac{2^{j}n!}{(n-j)!}H_{n-j}(x)$ (7b) we finally get $\begin{split}E_{2}(x_{2},y_{2})=&\frac{kw_{1}w_{2}(i/2)^{m+n+1}\gamma^{m+n}}{(b_{1}b_{2})^{1/2}}\\\ &\,\times\exp\left[-\left(\beta_{1}x_{2}^{2}+\beta_{2}y_{2}^{2}\right)\right]F_{m,n}(x_{2},y_{2}),\end{split}$ (8a) $\begin{split}F_{m,n}(x_{2},y_{2})=&\sum_{j=0}^{\min(m,n)}{m\choose j}\,{n\choose j}\Delta^{j}j!\\\ &\,H_{m-j}[f_{1}(x_{2},y_{2})]H_{n-j}[f_{2}(x_{2},y_{2})]\end{split}$ (8b) where, $k=2\pi/\lambda$, $\displaystyle\frac{1}{w_{j}^{2}}$ $\displaystyle=$ $\displaystyle\frac{1}{w_{0}^{2}}+i\frac{ka_{j}}{2b_{j}},$ (9a) $\displaystyle\gamma$ $\displaystyle=$ $\displaystyle(w_{1}^{2}-w_{2}^{2})^{1/2},$ (9b) $\displaystyle\Delta$ $\displaystyle=$ $\displaystyle-2(w_{1}^{2}-w_{2}^{2}\epsilon_{1}\epsilon_{2})/\gamma^{2},$ (9c) $\displaystyle\alpha_{j}$ $\displaystyle=$ $\displaystyle\frac{kw_{j}^{2}}{2b_{j}},$ (9d) $\displaystyle\beta_{j}$ $\displaystyle=$ $\displaystyle\left(\frac{kw_{j}}{2b_{j}}\right)^{2}+i\frac{kd_{j}}{2b_{j}},$ (9e) and $\displaystyle\left[\begin{array}[]{c}f_{1}(x_{2},y_{2})\\\ f_{2}(x_{2},y_{2})\end{array}\right]$ $\displaystyle=$ $\displaystyle\frac{1}{\gamma}\left[\begin{array}[]{c}\alpha_{1}x_{2}+i(\epsilon_{1}\alpha_{2}y_{2}-x_{0})\\\ \alpha_{1}x_{2}+i(\epsilon_{2}\alpha_{2}y_{2}+x_{0})\end{array}\right]$ (14) $\displaystyle=$ $\displaystyle\frac{1}{\gamma}\left[\begin{array}[]{c}\phi_{1}(x_{2},y_{2})\\\ \phi_{2}(x_{2},y_{2})\end{array}\right]$ (17) Eqs. (8-10) form one of our main results. It generalizes previous work dipole6 on the propagation dynamics of a vortex pair through an astigmatic system in that the topological charges $m$ and $n$ need not be the same and can have arbitrary integer values. Before proceeding further, we note that the above general result includes the following special cases: (1) For $m=n$, we get the propagation dynamics of (a) an isopolar vortex pair if $\epsilon_{1}\epsilon_{2}=1$ and (b) a vortex dipole if $\epsilon_{1}\epsilon_{2}=-1$; (2) For $n=0$, the $j$-sum reduces to the $j=0$ term only, and we get the propagation dynamics for an off-center single vortex given by $\displaystyle E_{2}(x_{2},y_{2})$ $\displaystyle=$ $\displaystyle\frac{kw_{1}w_{2}(i/2)^{m+1}}{(b_{1}b_{2})^{1/2}}\exp\left[-\left(\beta_{1}x_{2}^{2}+\beta_{2}y_{2}^{2}\right)\right]$ (18) $\displaystyle{}\times\gamma^{m}H_{m}[(\alpha_{1}x_{2}+i\epsilon_{1}\alpha_{2}y_{2}-ix_{0})/\gamma].$ (3) Setting $x_{0}=0$ in the above result, one immediately recovers our previous result pravin2 for a single vortex at the origin. The sum $F_{m,n}$ can be evaluated formally as follows. We introduce the 2-variable Hermite-Kampé de Fériet polynomials $H_{n}(x,y)$ as khan $H_{n}(x,y)=n!\sum_{r=0}^{[n/2]}\frac{x^{n-2r}y^{r}}{(n-2r)!r!}$ (19) in terms of which the classical Hermite polynomials $H_{n}(x)$ are given by $H_{n}(x)=H_{n}(2x,-1).$ (20) Next, we consider the 4-variable 2-index 1-parameter Hermite polynomials $H_{m,n}(x,z;y,w\|\tau)$ defined as khan ; khan3 $\begin{split}H_{m,n}(x,z;y,w\|\tau)=&\sum_{s=0}^{\min(m,n)}\tau^{s}s!{m\choose s}\,{n\choose s}\,\\\ &H_{m-s}(x,z)H_{n-s}(y,w).\end{split}$ (21) It is then easy to show that $F_{m,n}=H_{m,n}(2f_{1},-1;2f_{2},-1\|\Delta)$ (22) which has the following generating function $\begin{split}\exp[&-(u^{2}+v^{2})+2(f_{1}u+f_{2}v)+\Delta uv]\\\ &=\sum_{m,n=0}^{\infty}\frac{u^{m}v^{n}}{m!n!}H_{m,n}(2f_{1},-1;2f_{2},-1\|\Delta)\end{split}$ (23) ### II.1 Determination of net topological charge As noted earlier pravin2 , the modulations due to the Hermite polynomial become most prominent when $w_{2}=w_{1}^{}$. This happens at a certain value $z=z_{c}$. To determine $z_{c}$ and also the distance $z_{0}$ between the waist plane and the lens, we impose the following conditions: $\frac{ka_{1}}{2b_{1}}\|_{z=z_{c}}=-\frac{ka_{2}}{2b_{2}}\|_{z=z_{c}}=\frac{1}{w_{0}^{2}}$ (24) Solving Eqs. (17) and introducing the Rayleigh range $z_{R}=kw_{0}^{2}/2$, we get $\displaystyle z_{0}$ $\displaystyle=$ $\displaystyle z_{R}\left(1+\frac{2f\cos\theta}{z_{R}\sin^{2}\theta}\right)^{1/2}$ $\displaystyle z_{c}$ $\displaystyle=$ $\displaystyle\frac{z_{R}(1+\cos^{2}\theta)+z_{0}\sin^{2}\theta}{2(z_{R}/f)\cos\theta-\sin^{2}\theta}$ (25) The first equality in Eqs. (17) ensures that $w_{2}=w_{1}^{}$ at $z=z_{c}$ (see Eq. 9a) whereas the last equality makes many expressions appearing in Eqs. (8-10) considerably simpler at $z=z_{c}$. Thus, at $z=z_{c}$, $\displaystyle\Delta$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}-2&\mbox{if $\epsilon_{1}\epsilon_{2}=1$},\\\ -2i&\mbox{if $\epsilon_{1}\epsilon_{2}=-1$};\end{array}\right.$ (28) $\displaystyle\left(\begin{array}[]{c}w_{1}^{2}\\\ w_{2}^{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle\frac{w_{0}^{2}}{\sqrt{2}}\left(\begin{array}[]{c}\exp(-i\pi/4)\\\ \exp(i\pi/4)\end{array}\right);$ (33) $\displaystyle\gamma$ $\displaystyle=$ $\displaystyle w_{0}\exp(-i\pi/4);$ $\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle\delta_{1}x_{2}-\epsilon_{1}\delta_{2}y_{2}+(x_{0}/w_{0})\exp(-i\pi/4)$ $\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle\delta_{1}x_{2}-\epsilon_{2}\delta_{2}y_{2}-(x_{0}/w_{0})\exp(-i\pi/4)$ (34) where $\delta_{j}=\frac{kw_{0}}{2\sqrt{2}b_{i}}$ (35) #### II.1.1 Vortices with topological charges of the same sign Suppose $\epsilon_{1}=\epsilon_{2}=1$. Then, $f_{1}=\theta_{-}+\theta_{0}$ and $f_{2}=\theta_{-}-\theta_{0}$ where, $\displaystyle\theta_{-}$ $\displaystyle=$ $\displaystyle\delta_{1}x_{2}-\delta_{2}y_{2}$ $\displaystyle\theta_{0}$ $\displaystyle=$ $\displaystyle(x_{0}/w_{0})\exp(-i\pi/4).$ (36) Note that the dependence on $x_{2}$ and $y_{2}$ is in the form $\theta_{-}$ only. For a small separation between the vortices, one can expand the Hermite polynomials appearing in Eq.(8) as functions of $x_{0}/w_{0}$ by using the formula $H_{n}(x+y)=H_{n}(x)+2nyH_{n-1}(x)+O(y^{2}).$ (37) Substituting in (8) and using the summation rule magnus $\sum_{r=0}^{\min(m,n)}(-2)^{r}r!{m\choose r}\,{n\choose r}\,H_{m-r}(x)H_{n-r}(x)=H_{m+n}(x),$ (38) we get $F_{m,n}=H_{m+n}(\theta_{-})+2\theta_{0}(m-n)H_{m+n-1}(\theta_{-})+O(\theta_{0}^{2}).$ (39) For $\epsilon_{1}=\epsilon_{2}=-1$, $\theta_{-}$ will change to $\theta_{+}=\delta_{1}x_{2}+\delta_{2}y_{2}$ in the above expressions. #### II.1.2 Vortices with topological charges of opposite signs Suppose $\epsilon_{1}=1$, $\epsilon_{2}=-1$. In this case, $\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle\theta_{-}+\theta_{0}$ $\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle\theta_{+}-\theta_{0}.$ (40) Note that in this case, the dependence on $x_{2}$ and $y_{2}$ is in the form $\theta_{\pm}=\delta_{1}x_{2}\pm\delta_{2}y_{2}$. For a small separation between the vortices, we can proceed as in the previous section, to get $\displaystyle F_{m,n}=H_{m,n}(2\theta_{-},-1;2\theta_{+},-1\|-2i)$ (41) $\displaystyle{}+2\theta_{0}H_{m-1,n}(2\theta_{-},-1;2\theta_{+},-1\|-2i)$ $\displaystyle{}-2\theta_{0}H_{m,n-1}(2\theta_{-},-1;2\theta_{+},-1\|-2i)$ $\displaystyle{}+O(\theta_{0}^{2}).$ ### II.2 Propagation dynamics away from $z=z_{c}$ As $\|z-z_{c}\|$ increases, the absolute value of $\|\gamma\|$ falls off rapidly and the modulations due to the Hermite polynomials fade away quickly. Using the limiting form $\lim_{\gamma\rightarrow 0}H_{m}(x/\gamma)=(2x/\gamma)^{m}$, we can write $E_{2}(x_{2},y_{2})$ in terms of incomplete two-variable Hermite polynomials $h_{m,n}(x,y\|\tau)$, which are defined as khan ; khan4 $\displaystyle h_{m,n}(x,y\|\tau)=m!n!\sum_{j=0}^{\min(m,n)}\frac{\tau^{j}x^{m-j}y^{n-j}}{j!(m-j)!(n-j)!}$ (44) $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}m!\tau^{m}x^{n-m}L_{m}^{(n-m)}(-xy/\tau),\qquad n>m,\\\ n!\tau^{n}y^{m-n}L_{n}^{(m-n)}(-xy/\tau),\qquad m>n.\end{array}\right.$ Thus $E_{2}(x_{2},y_{2})$ reduces to $\displaystyle E_{2}(x_{2},y_{2})=\frac{kw_{1}w_{2}i^{m+n+1}}{2(b_{1}b_{2})^{1/2}}\exp\left[-\left(\beta_{1}x_{2}^{2}+\beta_{2}y_{2}^{2}\right)\right]$ $\displaystyle\times\left\\{\begin{array}[]{ll}m!\tau^{m}\phi_{1}^{n-m}L_{m}^{(n-m)}(-\phi_{1}\phi_{2}/\tau),\qquad n>m,\\\ n!\tau^{n}\phi_{2}^{m-n}L_{n}^{(m-n)}(-\phi_{1}\phi_{2}/\tau),\qquad m>n.\end{array}\right.$ (47) where $\phi_{j}$ are as in Eq. (10) and $\tau=-(w_{1}^{2}-w_{2}^{2}\epsilon_{1}\epsilon_{2})/2$. In what follows, we will experimentally demonstrate the validity of our theoretical results. ## III Experiment The experimental set up is shown in Fig. 1. Suitable phase masks for creating vortex pairs are produced by using computer generated holography (CGH) technique cgh and sent to a spatial light modulator (SLM) via a computer. The SLM is illuminated by an intensity stabilized He-Ne laser (Spectra-Physics, Model 117A) of power $1$ mW and wavelength $632.8$ nm to produce the desired vortex pair. The vortex pair is selected with an aperture (A) and passed through a spherical bi-convex lens of focal length $50$ cm which is tilted by an angle $6^{\circ}$. The tilting of the lens has been done with a rotational stage with least count of $0.1^{\circ}$. The aperture is at a distance $z_{1}=90$ cm in front of the SLM. We use the method described in sirohi , to find that the Gaussian laser beam hosting the selected vortex pair has a beam waist $0.186$ mm at a virtual point which is at a distance of $z_{2}=60.8$ cm behind the SLM. The distance between the lens and the aperture is $z_{3}=245$ cm. Thus the total distance traveled by the vortex pair from the waist plane to the lens is $z_{0}=z_{1}+z_{2}+z_{3}=395.8$ cm. The resultant intensity patterns are recorded by a CCD camera (MediaCybernetics, Evolution VF cooled Color Camera) placed at a distance $z$ past the lens. Figure 1: (Colour online) Experimental setup for the determination of the net charge of an arbitrary vortex pair embedded in a Gaussian beam ## IV Intensity pattern at $z=z_{c}$ and determination of net topological charge In this section, we determine the net topological charge of the vortex pair from its intensity distribution at $z=z_{c}$. The predicted value of $z_{c}$ from Eq. (25) is $57.2$ cm which is close to the experimentally observed value of $56.3$ cm. In the intensity patterns, with reference to Eq. (LABEL:vor), the vortex on the left ($x_{1}=-x_{0}$) has a charge $\epsilon_{1}m$ and the vortex on the right ($x_{1}=x_{0}$) has a charge $\epsilon_{2}n$. The corresponding figure is labelled as ($\epsilon_{1}m,\,\epsilon_{2}n$). ### IV.1 Vortices with topological charges of the same sign Figure 2: The theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of a vortex pair with topological charges of the same sign, at $z=z_{c}$ for $x_{0}=0.1w_{0}$. Fig. 2 shows the theoretical (first two rows) and experimental (last two rows) images for the intensity patterns of a pair of vortices with the same sign ($\epsilon_{1}=\epsilon_{2}=1$) but different magnitudes $m$ and $n$ with the separation parameter set at $x_{0}=0.1w_{0}$. For small separation $x_{0}$, these patterns can be explained by Eq. (24). Since the first term in (24) is the leading term, one can obtain the net charge $m+n$ by noting that there are $m+n+1$ bright stripes in the intensity distribution. These stripes are parallel to one another and lie along a line that is neither horizontal nor vertical, but tilted in a clockwise direction almost along a diagonal as the dependence on $x$ and $y$ is through a single variable $\theta_{-}=\delta_{1}x_{2}-\delta_{2}y_{2}$ and $\delta_{1}\sim\delta_{2}$. However, interference with the second term will lead to a slightly asymmetric distribution of brightness among the stripes. As is clear from the second term in (24), this asymmetry depends on the difference between the magnitude of charges and the separation between them. Additionally, when the vortices swap their positions as in (4,1) and (1,4), the lower half of the pattern becomes the mirror image of the upper half and vice-versa. For $m=n$ as in (4,4), the two halves have identical intensity patterns. If the charge of each vortex were negative ($\epsilon_{1}=\epsilon_{2}=-1$), then $\theta_{-}$ would be replaced by $\theta_{+}=\delta_{1}x_{2}+\delta_{2}y_{2}$ and the bright stripes would be tilted in an anti-clockwise fashion (not shown). Figure 3: The theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of an off-axis vortex of charge 2, at $z=z_{c}$ for different values of $x_{0}$ as labelled in the figures. To investigate the effect of the separation parameter $x_{0}$, we have also studied the propagation of an off-axis vortex of charge 2 through the tilted lens. The corresponding theoretical (first two rows) and experimental (last two rows) results for the intensity patterns at $z=z_{c}$ are shown in Fig. 3. From the images, it is clear that the intensity of one of the outer lobes increases as the vortex moves farther away from the center and the remaining lobes lose their intensity. In the notation of this section, this off-axis vortex can be labeled as (2,0) for $x_{0}=\|x_{0}\|$. Consequently, when $x_{0}$ becomes negative, the vortex is identically described as (0,2) with $x_{0}=\|x_{0}\|$ and the pattern flips diagonally. The situation is analogous to the case of (4,1) and (1,4) as described in the previous paragraph ### IV.2 Vortices with topological charges of opposite signs Fig. 4 shows the theoretical (first two rows) and experimental (last two rows) images corresponding to opposite singularities ($\epsilon_{1}\epsilon_{2}=-1$) for separation parameter $x_{0}=0.1w_{0}$ and topological charges as shown in the images. Figure 4: The theoretical (first two rows) and experimental (last two rows) results for the intensity patterns of a vortex pair with topological charges of opposite signs, at $z=z_{c}$ for $x_{0}=0.1w_{0}$. For small values of $m$ and $n$, these patterns can be explained by expanding the Hermite polynomials in power series. The calculation would be long and tedious. Instead, we make the following empirical observation. If $m\neq n$, the pattern has a rectangular ‘razor-blade’ structure which is tilted clockwise (anti-clockwise) if the net charge is positive (negative). On closer observation, we note that there are $m$ bright spots on two parallel sides and $n$ bright spots on the remaining two parallel sides. Thus, for vortex dipoles ($m=n$) the patten is square with its corners in the east, west, north and south directions, each side having $m=n$ bright spots. As far as we know, Fig. 4 represents the first optical realization of the 4-variable 2-index 1-parameter Hermite polynomials $H_{m,n}(x,z;y,w\|\tau)$ modulated by an elliptical Gaussian beam (see Eqs. 8, 15 and 26). Figure 5: The theoretical (top row) and experimental (bottom row) results for the intensity patterns of a dipole vortex of charge $(2,-2)$, at $z=z_{c}$ for different values of $x_{0}$ as labelled in the figures. In Fig. 5, we show the evolution of a dipole vortex of charge $(2,-2)$ as a function of separation between the two vortices. For small separation, the intensity distribution is symmetric in both the transverse directions. As the separation is increased, the pattern becomes asymmetric. When the separation parameter $x_{0}$ becomes negative, the vortex is identically described as $(-2,2)$ with $x_{0}=\|x_{0}\|$ and the pattern flips vertically. Figure 6: Theoretical intensity patterns of a vortex pair of different charges (as given on the top) at various values of the propagation distance $z$ ( as given on the left). ## V Propagation dynamics away from $z=z_{c}$ As we move away from the point $z=z_{c}$, the modulations due to the Hermite polynomials disappear quickly. The propagation dynamics is now governed by Eq. (28). The theoretical and corresponding experimental intensity patterns for various values of $z$ are shown in Figs. 6 and 7 respectively. 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Appl. 284, 447 (2003). * (35) R. S. Sirohi, A Course Of Experiments With He-Ne Lasers (New Age International, New Delhi, 1991).	arxiv-papers	2013-09-12T10:23:54	2024-09-04T02:49:50.815363	{ "license": "Public Domain", "authors": "Salla Gangi Reddy, Shashi Prabhakar, A Aadhi, J. Banerji, and R. P.\n Singh", "submitter": "Salla Gangi Reddy", "url": "https://arxiv.org/abs/1309.3095" }
1309.3163	11institutetext: Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China NUS Graduate School for Integrative Sciences and Engineering, Singapore 117456, Republic of Singapore NUS-Tongji Center for Phononics and Thermal Energy Science and Department of Physics, Tongji University, 200092 Shanghai, PR China Heat conduction Solid surfaces and solid-solid interfaces: structure and energetics Scattering by phonons, magnons, and other nonlocalized excitations # Nonlinearity enhanced interfacial thermal conductance and rectification Lifa Zhang 11 Juzar Thingna 11 Dahai He 22 Jian-Sheng Wang 11 Baowen Li 11334411223344 ###### Abstract We study the nonlinear interfacial thermal transport across atomic junctions by the quantum self-consistent mean field (QSCMF) theory based on nonequilibrium Green’s function approach; the QSCMF theory we propose is very precise and matches well with the exact results from quantum master equations. The nonlinearity at the interface is studied by effective temperature dependent interfacial coupling calculated from the QSCMF theory. We find that nonlinearity can provide an extra channel for phonon transport in addition to the phonon scattering which usually blocks heat transfer. For weak linearly coupled interface, the nonlinearity can enhance the interfacial thermal transport; with increasing nonlinearity or temperature, the thermal conductance shows nonmonotonical behavior. The interfacial nonlinearity also induces thermal rectification, which depends on the mismatch of the two leads and also the interfacial linear coupling. ###### pacs: 44.10.+i ###### pacs: 68.35.-p ###### pacs: 72.10.Di ## 1 Introduction In modern electronics, due to the rapid increasing power density, accumulation of heat becomes an obstacle for further progress of microelectronic devices; thus the heat dissipation and manipulation has been recognized to be a crucial issue in information and energy technologies [1]. Especially, as the dimensions of materials shrink into the nanoscale, interfaces dramatically affect the thermal transport [2, 3, 4, 5] making it a lucrative field to explore. At a rough interface the atomic mixing can enhance the thermal transport [6]; however, the behavior of the nonlinearity at the interface is not clear, and people do not know whether the nonlinearity can enhance the phonon transport. Recent progress in functional thermal devices [7, 8, 9, 10], makes the emerging new field – phononics very attractive [11]. In phononics, the most fundamental property of phononic devices is thermal rectification, which is known to be realized by combining the system inherent anharmonicity with structural asymmetry [12, 13]. Whether the interface itself can induce thermal rectification is still an open question; if yes, the property of the interfacial rectification is quite interesting and helpful for both theorists and experimentalists. To investigate the thermal transport across interface, the most widely applied models are the acoustic mismatch model [14] and the diffuse mismatch model [15]. Both models offer limited accuracy in nanoscale interfacial resistance predictions [16, 17] because they make simple assumptions and neglect atomic details of actual interfaces. Classical molecular dynamics simulation is another widely used method in phonon transport and has been applied to interfacial thermal transport [18, 19, 20, 21, 22]; however due to its classical nature, it is not accurate below the Debye temperature and can not capture the quantum effects. To study the nonlinear (anharmonic) thermal transport, the effective phonon theory has been recently introduced in some dynamical models [23, 24, 25]; and the quantum correction one [26] can be used to study the low temperature thermal transport. Despite their successes, such theories can not be well applied to nonlinear interfacial transport due to the inherent weak system-bath coupling assumption required for the validity of Feynman-Jensen inequality [27]. Another effective approach, the nonequilibrium Green’s function method which originates from the study of electronic transport [28], has been applied to study the quantum phonon transport [30, 31, 32], phonon Hall effect [33] and topological magnon insulator [34]. In this paper, based on the nonequilibrium Green’s function method, to avoid the perturbation approximation we develop the QSCMF theory for the nonlinear thermal transport, which can be applied to thermal transport in an arbitrary strength nonlinear interface. Then we study the interfacial thermal transport for a model as shown in Fig.1(a); thermal conductance and rectification across the interface are studied with an effective temperature-dependent-harmonic interfacial coupling calculated form the QSCMF theory. ## 2 Model and Hamiltonian We study the interfacial thermal transport with nonlinear coupling at the solid-solid interface as shown in Fig. 1(a). To manifest the effect of the interface we exclude the nonlinear phonon transport in the two materials, and there is no disorder nor defect in the whole system, thus the only thermal resistance comes from the interface. Such model really uncovers the thermal transport properties of the interface. To study the longitudinal transport, that is, the cross-plane interfacial transport, we simplify the problem further to one dimensional interface as shown in Fig. 1(b), where two linear semi-infinite atomic chains (solid-line regimes) connect each other by the interfacial linear coupling $k_{12}$ and nonlinear interaction $\lambda$, which is similar to the one-dimensional interfacial model in Ref.[5, 3] where there is only linear coupling at the interface. The Hamiltonian of the total system is Figure 1: (Color online) (a) Heat transport in a solid-solid interface. The arrow shows the heat transport from the hot side to the cold side. (b) The atomic junction model of the solid-solid interface. The solid line regions are two semi-infinite atomic chains which are coupled by a harmonic spring with strength $k_{12}$. In addition to which, the two regions also have a fourth order nonlinear coupling $\lambda$. For the two semi-infinite chains, the mass and spring constant are $m_{1}$, and $k_{1}$, $m_{2}$, and $k_{2}$, respectively. The interface model can be partitioned to three parts, the center (dashed line) and the leads (dash-dotted lines) with temperatures $T_{1}$ and $T_{2}$. $H=H_{1}+H_{2}+\frac{1}{2}k_{12}(x_{1,1}-x_{2,1})^{2}+\frac{1}{4}\lambda(x_{1,1}-x_{2,1})^{4},$ (1) with $H_{\alpha}=\sum\limits_{i=1}^{N_{\alpha}}{\frac{1}{2}m_{\alpha}\dot{x}_{\alpha,i}^{2}}+\sum\limits_{i=1}^{N_{\alpha}-1}{\frac{1}{2}k_{\alpha}(x_{\alpha,i}-x_{\alpha,i+1})^{2}},$ (2) where $N_{\alpha}\rightarrow\infty$, $\alpha=1,2$. In our model, the scattering for phonons only comes from the interface while the phonon transport in the two semi-infinite leads is ballistic. Thus we can partition the system into three parts ($L$, $C$, $R$), where the atoms at the interface are regarded as center (dashed-line part) and leads $L$ and $R$ (dash-dotted- line regimes) are harmonic as shown in Fig.1 (b). Using a mass-normalized displacement $u_{j}=\sqrt{m_{j}}x_{j,1}$, the center Hamiltonian can be written as $H_{C}=\frac{1}{2}\dot{U}_{c}^{T}\dot{U}_{c}+\frac{1}{2}U_{c}^{T}K^{C}U_{c}+\frac{1}{4}\sum\limits_{i,j,k,l=1}^{2}{T_{ijkl}}u_{i}u_{j}u_{k}u_{l},$ (3) where $U_{c}=(u_{1},u_{2})^{T}$, $T_{ijkl}=(-1)^{i+j+k+l}\lambda$ and $K^{C}=\left({\begin{array}[]{{20}c}\frac{k_{1}+k_{12}}{m_{1}}&\frac{-k_{12}}{\sqrt{m_{1}m_{2}}}\\\ \frac{-k_{12}}{\sqrt{m_{1}m_{2}}}&\frac{k_{12}+k_{2}}{m_{1}}\\\ \end{array}}\right).$ (4) For the leads, the Hamiltonian is written as $H_{\alpha}=\frac{1}{2}\dot{U}_{\alpha}^{T}\dot{U}_{\alpha}+\frac{1}{2}U_{\alpha}^{T}K^{\alpha}U_{\alpha}$ with its coupling to the center $H_{\alpha C}=U_{\alpha}^{T}V^{\alpha C}U_{C}$, $\alpha=L,R$. Here the center-lead coupling is the same as the inter-atomic spring constant in the corresponding bath. ## 3 Quantum Self-Consistent Mean Field Theory We discuss the QSCMF based on NEGF method for a general system where the center hamiltonian has a fourth-order nonlinear interaction as given in Eq. (3). The equation of motion of Green’s function [32], without the nonlinearity, is $(\frac{\partial^{2}}{{\partial\tau^{2}}}+K^{C})G_{0}(\tau\tau^{\prime})=-I\delta(\tau-\tau^{\prime})-\int d\tau^{\prime\prime}\Sigma(\tau\tau^{\prime\prime})G_{0}(\tau^{\prime\prime}\tau^{\prime}),$ where $\Sigma(\tau\tau^{\prime\prime})$ is the self energy due to the center- lead coupling. With the nonlinearity, the full Green’s function has the equation of motion as $\begin{split}\frac{{\partial^{2}}}{{\partial\tau^{2}}}G_{im}(\tau\tau^{\prime})+\sum\limits_{j}{K_{ij}^{C}}G_{jm}(\tau\tau^{\prime})+\sum\limits_{jkl}{T_{ijkl}G_{jklm}(\tau\tau\tau\tau^{\prime})}\\\ =-\delta(\tau-\tau^{\prime})\delta_{im}-\sum\limits_{j}{\int d\tau^{\prime\prime}\Sigma_{ij}(\tau\tau^{\prime\prime})G_{jm}(\tau^{\prime\prime}\tau^{\prime})}\end{split}$ (5) where $G_{jklm}(\tau_{j}\tau_{k}\tau_{l}\tau_{m})$ is the four-point Green’s function. Under the self-consistent mean field approximation, the four-point Green’s function can be represented by the two-point Green’s function, and we have $-i\frac{1}{\hbar}G(\tau_{j}\tau_{k}\tau_{l}\tau_{m})=G(\tau_{j}\tau_{k})G(\tau_{l}\tau_{m})+G(\tau_{j}\tau_{l})G(\tau_{k}\tau_{m})+G(\tau_{j}\tau_{m})G(\tau_{k}\tau_{l}).$ Thus the full Green’s function satisfies $\begin{split}\frac{{\partial^{2}}}{{\partial\tau^{2}}}G_{im}(\tau\tau^{\prime})+\sum\limits_{j}{K_{ij}^{C}}G_{jm}(\tau\tau^{\prime})\\\ +3i\hbar\sum\limits_{jkl}{T_{ijkl}G_{kl}(0)G_{jm}(\tau\tau^{\prime})}+\sum\limits_{j}{\int d\tau^{\prime\prime}{\Sigma_{ij}(\tau\tau^{\prime\prime})G_{jm}(\tau^{\prime\prime}\tau^{\prime})}}\\\ =-\delta(\tau-\tau^{\prime})\delta_{im},\end{split}$ (6) by using the symmetry of $T_{ijkl}$ with respect to the permutation of the indices. Thus we introduce an effective dynamic matrix $\tilde{K}_{ij}^{C}=K_{ij}^{C}+3i\hbar\sum\limits_{kl}{T_{ijkl}G_{kl}(0)}=K_{ij}^{C}+3\sum\limits_{kl}{T_{ijkl}\left\langle{u_{k}u_{l}}\right\rangle}.$ (7) The nonlinearity only has the effect to modulate the dynamic matrix. It is important to note that the results are independent of the partition size of the center due to $T_{ijkl}=0$ every where else except the two atoms at the interface. Equation (7) together with $G^{r}=[(\omega+i\eta)^{2}-\tilde{K}_{ij}^{C}-\Sigma^{r}]^{-1},\;G^{<}=G^{r}\Sigma^{<}G^{a},\;\left\langle{u_{k}u_{l}}\right\rangle=i\hbar G_{kl}^{<}(t=0)=2i\hbar\int_{0}^{\infty}{G_{kl}^{<}(\omega)d\omega/(2\pi)}$ can be self-consistently calculated. Since the problem is now effectively harmonic, the heat current still satisfies the Landauer formula $J=\int_{0}^{\infty}{\frac{{d\omega}}{{2\pi}}\hbar\omega T[\omega](f_{L}-f_{R})}$, $f_{\alpha}=1/(e^{\hbar\omega/(k_{B}T_{\alpha})}-1)$, and thermal conductance is defined as $\sigma=\|J/(T_{L}-T_{R})\|$, while the transmission $T[\omega]={\rm Tr}(G^{r}\Gamma_{L}G^{a}\Gamma_{R})$ is temperature dependent. In later calculation, we will set $\hbar=1,k_{B}=1$ for simplicity. For the conversion from the dimensionless unit to physical units, we take energy unit $[E]=1$ mev, length unit is $[L]=1{\rm\AA}$, then temperature unit $[T]$ is 11.6 K, thermal conductance $[\sigma]=20.9{\rm nW/mK}$, the spring constant unit $[k]=1$ mev/${\rm\AA}^{2}$ and nonlinearity unit $[\lambda]=1$ mev/${\rm\AA}^{4}$. Applying our QSCMF theory to the nonlinear interface problem of Eq. (1), we find that the nonlinearity plays a role to modulate the interfacial linear coupling $k_{12}$, the effective one is $k_{12\rm{eff}}=k_{12}+3\lambda\bigl{(}\frac{{\left\langle{u_{1}^{2}}\right\rangle}}{{m_{1}}}-2\frac{{\left\langle{u_{1}u_{2}}\right\rangle}}{{\sqrt{m_{i}m_{j}}}}+\frac{{\left\langle{u_{2}^{2}}\right\rangle}}{{m_{2}}}\bigr{)}.$ (8) Thus it is clear that all the scattering occurs only at the interface, since all other parts are harmonic. Figure 2: (Color online) Comparison of the results for thermal conductance of the interface as illustrated in Fig. 1(b) between QSCMF theory and quantum master equation method. We add a small onsite potential $0.01$ to each atom in both leads. $m_{1}=m_{2}=1.0$, $k_{1}=k_{2}=1.0$, and $K_{11}^{C}=1.5,K_{22}^{C}=1.5$, $K_{12}^{C}=-0.5,K_{21}^{C}=-0.5$. The system-bath coupling is $0.1$ . ## 4 Comparison with the Quantum Master Equation While the effective phonon theory and its quantum correction one are valid in the linear response region at weak system-bath coupling, our QSCMF theory can study the nonequilibrium thermal transport under larger temperature bias at any system-bath coupling. For anharmonic systems with arbitrary strength of anharmonicity under the weak system-bath coupling approximation, Redfield quantum master equation is very suited solution to study the thermal transport [12, 35]. In our interface problem, we partition the atoms in the interface as a center, and the center-lead coupling is the same as the inter-atomic spring constant in the corresponding bath. Thus for such system, the system-bath coupling is always strong such that the quantum master equation cannot be applied which is limited in the weak system-bath coupling limit. However, the QSCMF method can be applied. We numerically compare our theory with the quantum master equation at a weak system-bath coupling, as shown in Fig. 2. In Fig. 2, the system-bath coupling is $0.1$ while the inter-atomic spring constant in the leads are $1.0$ such that the weak-system-bath-coupling condition is well satisfied. A small onsite potential $0.01$ is added to each atom in both leads to avoid the divergency in quantum master equation method. From Fig. 2 we find that the results from QSCMF perfectly matches those from the quantum master equation method for different nonlinearity. Therefore the QSCMF method is verified by the quantum master equation method at weak system-bath coupling; however, the QSCMF is not limited in this weak coupling, which can be applied to arbitrary-system-bath- coupling systems. ## 5 Numerical Results on Interface thermal Transport ### 5.1 Nonlinearity Suppressed Thermal Transport in Homogenous Systems Using the QSCMF theory we proposed above, the interface nonlinearity can be studied for the interfacial thermal transport. For the homogeneous lattice with $k_{1}=k_{2}=k_{12}$, we calculate the interfacial thermal conductance for different nonlinearity $\lambda$ as shown in Fig. 3(a). With zero nonlinearity, the thermal conductance increases with the temperature increasing, and tends to a constant due to the saturate of phonon modes contributing to the thermal transport. However, with nonzero interfacial nonlinearity the thermal conductance decreases at high temperatures due to the dominant scattering coming from the nonlinear interface coupling. In the low- temperature regime, the thermal conductance almost coincides with the ballistic transport and the nonlinearity almost has no effect on thermal transport. With increasing temperature, more phonon modes are excited which wins the suppressing effect nonlinear scattering such that the conductance increases. At certain temperature, the conductance arrives its maximum, after which it will decrease since the nonlinear scattering effect which defeats the enhancement effect from more excited phonon modes. With increasing nonlinearity the thermal conductance decreases due to the larger phonon scattering at the interface. As shown in Fig. 3(b), the nonlinearity always decreases the thermal transport for the homogeneous systems. The larger interfacial nonlinearity makes the system more nonhomogeneous such to induce more scattering to the phonon transport. Figure 3: (Color online) Interfacial thermal conductance as a function of $T$ ($T=(T_{L}+T_{R})/2$)(a) and $\lambda$ (b). $m_{1}=m_{2}=1.0$, $k_{1}=k_{2}=1.0$ and $k_{12}=1.0$. In (b), the solid line ($T=0.1$) corresponds to left scale, and the dashed line ($T=0.5$) and dotted line ($T=2.5$) correspond to right scale. ### 5.2 Nonlinearity Enhanced Interfacial Thermal Conductance In the weak interfacial coupling regime, that is, $k_{12}<k_{1}=k_{2}$, we find that the nonlinear interaction at interface can enhance the thermal transport as shown in Fig 4. With increasing nonlinearity, the interfacial conductance increases first; after certain maximum the conductance will decrease, as shown in Fig 4(a). The maximum conductance coincides with the point where the effective coupling equals to $1$ and the whole system is homogeneous. If $\tilde{k}_{12}$ increases further the conductance decreases due to the larger scattering at interface. At a fixed linear interfacial coupling $k_{12}<1$, the interfacial nonlinearity makes the effective one $\tilde{k}_{12}$ larger than $k_{12}$. A larger $\tilde{k}_{12}$ reduces the difference between the interface and the leads, thus decreases the phonon scattering and allows more phonons to transmit through the interface. Therefore the nonlinearity introduces an extra channel to transport phonons, which enhances the thermal transport. In Fig 4 (b), the maximum of thermal conductance does not coincide with the place of $\tilde{k}_{12}=1$, this is mainly because that the increase in temperature causes more phonon modes to transport so as to delay the maximum of conductance. For a larger nonlinearity, the maximum of thermal conductance is delayed further. Figure 4: (Color online) Interfacial thermal conductance and effective interfacial coupling as functions of $\lambda$ (a) and $T$ (b). $m_{1}=m_{2}=1.0$, $k_{1}=k_{2}=1.0$ and $k_{12}=0.80$. For both (a) and (b), the solid, dashed and dotted lines correspond to left scale (interfacial thermal conductance), and the scatter points (square, circle and triangle symbols) correspond to right scale (effective interfacial coupling). The lines with two arrows are drawn to guide the eyes. For a hetero-junction, that is, the two mismatched leads, the thermal conductance can also be enhanced by interfacial nonlinearity. We know that the thermal conductance reaches a maximum when the interfacial coupling equals the harmonic average of the spring constants of the two semi-infinite chains [5]. As shown in Fig. 5(a) the conductance of two general hetero-junctions gets to the maximum when the effective coupling $k_{12\rm{eff}}$ equals to $k_{12m}=2k_{1}k_{2}/(k_{1}+k_{2})$; after which the thermal conductance decreases. The interfacial conductance can be enhanced in a wide nonlinearity range, as shown in Fig. 5(a). The enhancement of the conductance due to the nonlinearity depends on the interfacial linear coupling. In order to show this, we plot the thermal conductance as a function of the linear coupling $k_{12}$ in Fig. 5(b). For a fixed nonlinearity, the thermal conductance enhances largest at zero $k_{12}$, where the only channel to transport phonon comes from the nonlinearity. With increasing $k_{12}$ the conductance increases, but the effective coupling due to the nonlinearity $k_{12\rm{eff}}-k_{12}$ decreases. This effect causes the overall enhancement to decrease. The inset of Fig. 5(b) shows that the nonlinear interfacial conductance equals the linear thermal case just before $k_{12}=k_{12m}=1.5$, where the scattering from nonlinearity cancels the contribution from the extra channel introduced by the nonlinearity. At $k_{12}=k_{12m}=1.5$, the linear conductance maximizes and the nonlinearity only adds to the phonon scattering, thus the conductance of nonlinear interface is small. If $k_{12}>>\lambda$, the effective coupling from the nonlinearity $k_{12\rm{eff}}-k_{12}$ nearly vanishes making the interfacial nonlinearity not important anymore, thus the conductance is almost the same with the ballistic one. Figure 5: (Color online) (a) Thermal conductance (left scale) and effective interfacial coupling (right scale) as functions of $\lambda$. $k_{1}=m_{1}=1.0$, $k_{12}=0.8$, the dashed and dotted lines correspond to interfacial thermal conductance, and the scatter pionts (circle or triangle symbols) correspond to effective interfacial coupling; the lines with two arrows are drawn to guide the eyes. (b) The effective coupling from the nonlinearity $k_{12\rm{eff}}-k_{12}$ and interfacial thermal conductance $\sigma$ as functions of interfacial linear coupling. The dashed line corresponds to the left scale ($k_{12\rm{eff}}-k_{12}$) and solid and dotted lines correspond to right scale ($\sigma$). $k_{1}=m_{1}=1.0$, $k_{2}=3.0,m_{2}=2.0$, $T_{0}=0.5$; the inset is the zoom-in plot around $k_{12}=1.5$ ### 5.3 Nonlinearity Induced Interfacial Thermal Rectification From our QSCMF approach, the effective interfacial coupling is temperature dependent; if we reverse the two temperatures of the leads it would be different so that we can observe thermal rectification. If the interface coupling is linear, then Landauer formula applies and reverse temperature only changes the sign of the heat current, thus there is no rectification. The rectification is defined as [36]: $R=(J_{+}-J_{-})/{\rm max}\\{J_{+},J_{-}\\},$ (9) where $J_{+}$ is the-forward direction heat flux when $T_{L}=T_{h},T_{R}=T_{c}$, and $J_{-}$ is that of the backward direction when $T_{L}=T_{c},T_{R}=T_{h}$. Here $T_{h}$ and $T_{c}$ correspond to the temperatures of the hot and cold baths, respectively. With the asymmetric structure the nonlinear interface shows rectification, which depends on the interfacial nonlinearity $\lambda$ and the linear coupling $k_{12}$ as shown in Fig. 6. If the nonlinearity is zero, there is no rectification. With increasing the nonlinearity, the effective coupling will be different in the forward and backward transport which causes the rectification to increase. When the nonlinearity increases further, the effective coupling will monotonically increase. Therefore the scattering from the interface will play an important role to decrease the flux difference between the forward and backward ones, which causes the rectification to decrease as shown in Fig. 6(a). With larger linear interfacial coupling, the contribution to the phonon transport from the channel of $k_{12}$ is larger as compared to the extra channel provided by the nonlinearity $\lambda$. Therefore the relative nonlinear effect is weakened and the rectification is reduced as shown in Fig. 6(b). In the opposite limit, if $k_{12}<<\lambda$ the rectification almost keeps fixed, when the nonlinearity dominates the thermal transport and the difference of the effective coupling between the forward and backward flow has almost no changes. When $k_{12}\sim\lambda$, the rectification decreases fast due to the fast decrease of the value of $k_{12\rm{eff}}-k_{12}$. If $k_{12}>>\lambda$, the rectification decreases to almost zero since the effective coupling from the nonlinearity $k_{12\rm{eff}}-k_{12}$ nearly vanishes making the nonlinearity not important anymore. Figure 6: (Color online) (a) Interfacial thermal rectification as functions of $\lambda$ for different $k_{12}$. (b) Interfacial thermal rectification (left scale) and the heat current (right scale) as functions of $k_{12}$. For both (a) and (b), $k_{1}=m_{1}=1.0$, $k_{2}=3.0,m_{2}=2.0$, and $T_{+}=T_{0}(1+\Delta)$, $T_{-}=T_{0}(1-\Delta)$ with $T_{0}=0.5$, $\Delta=0.5$; for (b) $\lambda=0.1$. The inset in (b): the effective coupling from the nonlinearity $k_{12\rm{eff}}-k_{12}$ as a function of interfacial linear coupling $k_{12}$. ## 6 Discussion We use a simplified one-dimensional atomic model to investigate the underlying physics in the interfacial thermal transport; based on which we find that the nonlinearity can enhance the thermal transport and the nonlinearity at interface can induce the interfacial thermal rectification. For a general chemical bonding strength across interface, we could represent it as a linear coupling part and nonlinear coupling part; therefore our model can be applied. However, to compare with experimental results on real materials, the simplified model need to be generalized to two-dimensional or three dimensional with input of the real parameters. The QSCMF theory is a kind of mean-field approximation based on the nonequilibrium Green’s function, which is similar as the mean field approach in the literatures [28, 29]. The QSCMF is a good candidate to solve the nonlinear problem with arbitrary nonlinearity and arbitrary system-bath coupling. By comparing to the quantum master equation which is limited by weak-system-bath coupling, we find the QSCMF is a quite accurate method while it can be applied to strong system-bath coupling. For the interface problem with two-layer atoms, the QSCMF has a very high accuracy in the wide range of temperature and nonlinearity. However, if more layer-atoms includes, the accuracy decreases. If the QSCMF is generalized to two or three dimensional interface with more atoms in the interface, the self-consistent process is time consuming, and we also need to pay more attention to its convergency. The QSCMF is quite good for two-layer-atoms interface which is a reasonable approximation for short-range inter-atom coupling; however for long-range interaction, more layers need to be considered at the interface and the numerical calculation on the nonlinear thermal transport is challenging. ## 7 Conclusion Based on the NEGF approach, an efficient QSCMF theory is developed to study the nonlinear interfacial thermal transport. We find that the nonlinearity can enhance the interfacial thermal transport at weak linear interfacial coupling while the enhancement vanishes in the strong linear coupling regime. The enhancement can exist at large nonlinearity where the effective coupling is less than the harmonic average of the spring constants of the two semi- infinite chains. Although the leads are linear, the interfacial nonlinearity can induce rectification provided that the two leads are asymmetric. With increasing the nonlinear coupling, the rectification first increases then decreases. The interfacial rectification also depends on the interfacial linear coupling; it vanishes if the linear interfacial coupling increases far beyond the nonlinear coupling. ## 8 Acknowledgements L. Z. and B. L. are supported by the grant R-144-000-300-112 from Ministry of Education of Republic of Singapore. J.-S. W. and J. T. acknowledge support from a faculty research grant R-144-000-257-112 of NUS. D. H. surported by NSFC (Grant No. 11047185) and FRFCU (Grant No. 2010121009) of China. ## References [1] Cahill D. G., Ford W. K., Goodson K. E., Mahan G. D., Majumdar A., Maris H. J., Merlin R., and Phillpot S. R., J. Appl. Phys. 93, (2003) 793. * [2] Costescu R. M., Cahill D. G., Fabreguette F. H., Sechrist Z. A. and George S. M., Science, 303, (2004) 989. * [3] Hu L., Zhang L., Hu M., Wang J.-S., Li B., and Keblinski P., Phys. Rev. B 81, (2010) 235427. * [4] Hsieh W.-P., Lyons A. 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P., _Quantum Kinetics in Transport and Optics of Semiconductors_ (Springer, Berlin, 1996). * [29] Bruus H. and Flesberg K., _Many-body Quantum Theory in Condensed Matter Physics_ (Oxford University Press, Oxford, 2004). * [30] Mingo N., Phys. Rev. B 74, (2006) 125402. * [31] Lü J. T. and Wang J.-S., Phys. Rev. B 76, (2007) 165418. * [32] Wang J.-S., Wang J., and Lü J. T., Eur. Phys. J. B 62, (2008) 381. * [33] Zhang L., Wang J.-S., and Li B., New J. Phys. 11, (2009) 113038. * [34] Zhang L., Ren J., Wang J.-S., and Li B., Phys. Rev. B 87, (2013) 144101. * [35] Thingna J., García-Palacios J. L., and Wang J.-S., Phys. Rev. B 85, (2012) 195452. * [36] Zhang L., Wang J.-S., and Li B., Phys. Rev. B 81, (2010) 100301(R).	arxiv-papers	2013-09-12T14:10:26	2024-09-04T02:49:50.824192	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lifa Zhang, Juzar Thingna, Dahai He, Jian-Sheng Wang, and Baowen Li", "submitter": "Lifa Zhang", "url": "https://arxiv.org/abs/1309.3163" }
1309.3166	# Thermal transport across metal-insulator interface via electron-phonon interaction Lifa Zhang,1 Jing-Tao Lü,2 Jian-Sheng Wang,1 Baowen Li1,3,4 1 Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Republic of Singapore 2 DTU- Nanotech, Department of Micro- and Nanotechnology, Technical University of Denmark (DTU), Ørsteds Plads, Bldg. 345E, DK-2800 Lyngby, Denmark 3 NUS Graduate School for Integrative Sciences and Engineering, Singapore 117456, Republic of Singapore 4 NUS-Tongji Center for Phononics and Thermal Energy Science and Department of Physics, Tongji University, 200092 Shanghai, PR China ###### Abstract The thermal transport across metal-insulator interface can be characterized by electron-phonon interaction through which an electron lead is coupled to a phonon lead if phonon-phonon coupling at the interface is very weak. We investigate the thermal conductance and rectification flowing between the electron part and the phonon part using nonequilibrium Green’s function method. It is found that the thermal conductance has a nonmonotonic behavior as a function of average temperature or the coupling strength between the phonon leads in the metal part and the insulator one. The metal-insulator interface shows evident thermal rectification effect, which can reverse with changing of average temperature or the electron-phonon coupling. ###### pacs: 68.35.-p, 66.70.-f, 63.20.kd, 72.10.Di ## 1 Introduction With the increase of integration density, accumulation of heat becomes a bottleneck for further development of microelectronic devices. Moreover, as most of electronic devices consist of metal and insulator/semiconductor interfaces, to understand the thermal transport through the metal- insulator/semiconductor interface is indispensable for heat dissipation [1, 2, 3]. Figure 1: (a) Heat generation in a metal with an applied bias. The arrow shows the heat transport from the metal to the insulating substrate. (b) An electron bath and a phonon bath at two different temperatures connected by EPI. Energy exchanges between the two baths and there is a net thermal current $J$ from the hot bath to the cold one. (c) The lattice model of the metal-insulator interface. The one dimensional electron lead (the left upper semi-infinite chain where the dots represent electron states ) connects to the one dimensional atomic chain (the right semi-infinite chain) via electron-phonon interaction $V_{\rm ep}$. The phonon part in the metal (the left lower semi- infinite atomic chain) can also be considered to the thermal transport, which connects to the right atomic chain by a relative coupling $\lambda$. In metals electrons dominate thermal transport while phonons do in semiconductors and insulators; thus for thermal transport across metal- nonmetal interfaces, energy must transfer between electrons and phonons [4]. Thanks to many remarkable physical properties and prospects for large-area epitaxial growth, graphene is a promising material for future electronics [5]. The biased graphene on a dielectric insulator often appears in electronic devices especially in transistors. Due to the bias the electrons in metallic part have a higher effective temperature than the phonons in the insulating substrate; and the thermal transport from the electrons in metal to the phonons in insulator is one important channel of energy transfer [7, 8]. It is found in a very recent experiment on a carbon nanotube that more that 80 percent of the electrical power supplied to the nanotube is transferred directly into the substrate by the electron-phonon interaction (EPI) [9]. Therefore to understand the thermal transport across the interface through the EPI is highly desirable for the heat dissipation in electronic devices. To study the thermal transport across interface, the acoustic mismatch model [6] and the diffuse mismatch model [10, 4] are widely applied; however, both models offer limited accuracy in nanoscale interfacial resistance predictions [11] because they neglect atomic details of actual interfaces. A scattering boundary method within the lattice dynamic approach [12, 13, 14] fully considers the atomic structures in the interface; but it can only be applied to ballistic thermal transport. Classical molecular dynamics is another widely used method in phonon transport [15, 16, 17], which is not accurate below the Debye temperature, and ignores the quantum effect. Only recently the nonequilibrium Green’s function method, which originates from the study of electronic transport [18], been applied to study the quantum phonon transport [19, 20, 21, 22]. So far the study of the coupled electronic and phonon transport [23, 24, 25] is rare, especially in the metal-nonmetal interface [4, 28, 29]. In this paper, using the nonequilibrium Green’s function method, we study the thermal transport across metal-insulator interface via the EPI. Our model can also be applied to metal-semiconductor interface. We study energy flow between an electron bath and a phonon bath which are connected by electron-phonon coupling, as shown in Fig.1 (b). The electron bath, described by a semi- infinite electronic chain under tight-binding approximation, connects with a phonon lead illustrated as a semi-infinite harmonic atomic chain by a weak electron-phonon interaction, as shown in Fig.1 (c), where the phonon degrees of freedom in metal can also be considered in the thermal transport. ## 2 Model and Method We study the interfacial thermal transport at the metal-insulator interface as shown in Fig. 1(a). To manifest the effect of the interface we exclude the nonlinear electron or phonon transport in the two materials themselves, thus the only thermal resistance comes from the interface. Such model really uncovers the thermal transport properties of the interface itself. To study the longitudinal thermal transport, that is, the cross-plane interfacial transport, we can simplify the problem further to thermal transport between an electron bath and a phonon bath which are connected by an electron-phonon coupling, as shown in Fig. 1(b), and can be represented by one dimensional lattice model as shown in Fig. 1(c), where a semi-infinite electronic chain (at temperature $T_{Le}$) connects to a semi-infinite atomic chain (at temperature $T_{R}$) by an electron-phonon interaction $V_{ep}$. The phonon part in the metal can also be considered by another semi-infinite atomic chain which connects to the right atomic chain by a relative coupling $\lambda$. In our model, the scattering for both electrons and phonons only comes from the interface, and the electron and phonon transport in the corresponding semi- infinite periodic leads is ballistic. Therefore we can partition the system into three parts ($L$, $C$, $R$), the atoms at the interface are regarded as center illustrated as the dashed-line rectangle in Fig.1 (c), and the rest parts are leads. For simplicity, we only consider one dimensional case, ignoring electron spin. Thus the Hamiltonian of the whole system in Fig.1 (c) is written as $H=H^{L}_{\rm e}+H^{C}+H^{LC}_{\rm e}+\sum_{\alpha=L,R}(H^{\alpha}_{\rm p}+H^{\alpha C}_{\rm p}),$ (1) where $H^{L}_{\rm e}=\sum_{i}\varepsilon_{0}c^{\dagger}_{i}c_{i}-\sum_{\|i-j\|=1}tc^{\dagger}_{i}c_{j}$ is the Hamiltonian of the electron lead in the left part. The electron coupling between the left lead (the site 1) and the center is $H^{LC}_{\rm e}=-tc^{\dagger}_{1}c-tc^{\dagger}c_{1}$. We set atomic mass $m=1$, and the Hamiltonian of the center is $H^{C}=\frac{1}{2}\dot{u}^{2}+\frac{1}{2}(\lambda k^{L}+k^{R})u^{2}+\varepsilon_{0}c^{\dagger}c+V_{\rm ep}c^{\dagger}cu,$ (2) where $k^{L}$ and $k^{R}$ are the spring constants of the left and right atomic chains, respectively. $u$ is the atom displacement. $\lambda k^{L}$ denotes the coupling between the two atomic chains; $\lambda$ can be chosen from 0 to 1. $V_{\rm ep}$ is the electron-phonon coupling. The phonon Hamiltonian of two leads and its coupling to the center are $H^{\alpha}_{\rm p}=\frac{1}{2}\sum_{i}\dot{u}^{\alpha}_{i}\dot{u}^{\alpha}_{i}+\frac{1}{2}\sum_{\|i-j\|=0,1}u^{\alpha}_{i}K^{\alpha}_{ij}u^{\alpha}_{j}$ and $H^{\alpha C}_{\rm p}=u^{\alpha}_{1}K^{\alpha C}u$ [20, 23]. Applying the standard procedure of nonequilibrium Green’s function method [20], without the EPI, we obtain the electron retarded Green’s function $G^{r}_{0}(\varepsilon)=\left[\varepsilon-\varepsilon_{0}-\Sigma^{r}_{L}(\varepsilon)\right]^{-1}$, where $\Sigma^{r}_{L}=g^{r}_{L}t^{2}$ is the retarded self-energy with a surface Green’s function $g^{r}_{L}$, where $g^{r}=\left[(\varepsilon+i\eta)I-H^{L}_{\rm e}\right]^{-1}$. The less Green’s function can also be easily obtained. The phonon retarded Green’s functions is $D^{r}_{0}(\omega)={D^{a}_{0}}^{\dagger}(\omega)=\left[\omega^{2}-(\lambda k^{L}+k^{R})-\Pi^{r}_{L}(\omega)-\Pi^{r}_{R}(\omega)\right]^{-1}$, where $\Pi^{r}_{L}(\omega)=(\lambda k^{L})^{2}d^{r}_{L}(\omega)$ and $\Pi^{r}_{R}(\omega)=(k^{R})^{2}d^{r}_{R}(\omega)$ are the retarded self energies of left and right leads with surface Green’s function $d^{r}_{L}$ and $d^{r}_{R}$, where $d^{r}_{L,R}=\left[(\omega+i\eta)^{2}I-K^{L,R}\right]^{-1}$. The EPI is included as perturbation. The full Green’s functions are obtained from the Dyson equation, that is, $G^{r,a}=[(G_{0}^{r,a})^{-1}-\Sigma^{r,a}_{\rm ep}]^{-1}$, and $G^{<,>}=G^{r}(\Sigma^{<,>}_{\rm ep}+\Sigma^{<,>}_{L})G^{a}$, where $\Sigma^{r,a,<,>}_{\rm ep}$ is the self-energy from the EPI. For phonon, we have $D^{r,a}=[(D_{0}^{r,a})^{-1}-\Pi^{r,a}_{\rm ep}]^{-1}$, and $D^{<,>}=D^{r}(\Pi^{<,>}_{\rm ep}+\Pi^{<,>}_{L}+\Pi^{<,>}_{R})D^{a}$, where the nonlinear self energy $\Pi^{r,a,<,>}_{\rm ep}$ comes from the electron- phonon coupling. Keeping the lowest non-zero order (second order) of the self- energies, we could obtain the nonlinear self-energies $\Sigma^{r,a,<,>}_{\rm ep}$ and $\Pi^{r,a,<,>}_{\rm ep}$ up the second order, which are general for any dimensional systems. For our one-dimensional simple model as shown in Eq.2, the self-energies could be written as $\Sigma_{\rm ep}^{>,<}(\varepsilon)=iV_{\rm ep}^{2}\int{G_{0}^{>,<}}(\varepsilon-\omega){D_{0}^{>,<}}(\omega)\frac{d\omega}{2\pi},$ (3) and $\displaystyle\Sigma_{\rm ep}^{r}(\varepsilon)$ $\displaystyle=$ $\displaystyle iV_{\rm ep}^{2}\bigl{\\{}-{D_{0}^{r}}(\omega^{\prime}=0)\int{G_{0}^{<}}(\varepsilon^{\prime})\frac{d\varepsilon^{\prime}}{2\pi}$ (4) $\displaystyle+\int\frac{d\omega}{2\pi}\bigr{[}{G_{0}^{r}}(\varepsilon-\omega){D_{0}^{<}}(\omega)+{G_{0}^{<}}(\varepsilon-\omega){D_{0}^{r}}(\omega)$ $\displaystyle+{G_{0}^{r}}(\varepsilon-\omega){D_{0}^{r}}(\omega)\bigr{]}\bigr{\\}}.$ The nonlinear self-energies for the phonons are $\Pi_{\rm ep}^{>,<}(\omega)=-iV_{\rm ep}^{2}\int\frac{d\varepsilon}{2\pi}{G_{0}^{>,<}}(\varepsilon){G_{0}^{<,>}}(\varepsilon-\omega),$ (5) and $\displaystyle\Pi_{\rm ep}^{r}(\omega)$ $\displaystyle=$ $\displaystyle- iV_{\rm ep}^{2}\int\frac{d\varepsilon}{2\pi}\bigl{[}{G_{0}^{r}}(\varepsilon){G_{0}^{<}}(\varepsilon-\omega)$ (6) $\displaystyle+{G_{0}^{<}}(\varepsilon){G_{0}^{a}}(\varepsilon-\omega)\bigr{]}.$ Eqs. (3-6) are the so-called Born approximation (BA). By replacing the bare Green’s functions $G_{0}$ and $D_{0}$ with the full Green’s functions $G$ and $D$, we do iteration under self-consistent Born approximation (SCBA). While the BA fails to satisfy the energy current conservation, the SCBA fulfills it [23]. Therefore, we obtain the heat current from electron part as [18, 30, 31] $J^{\rm e}=\int\frac{d\varepsilon}{2\pi\hbar}\varepsilon~{}{\rm Tr}\\{G^{>}(\varepsilon)\Sigma^{<}_{L}(\varepsilon)-G^{<}(\varepsilon)\Sigma^{>}_{L}(\varepsilon)\\}.$ (7) The heat current from the phonon lead is [20, 23, 32] $J^{\rm p}_{\alpha}=-\int\frac{d\omega}{4\pi}\hbar\omega~{}{\rm Tr}\\{D^{>}(\omega)\Pi^{<}_{\alpha}(\omega)-D^{<}(\omega)\Pi^{>}_{\alpha}(\omega)\\}.$ (8) Due to energy conservation, $J^{\rm e}+J^{\rm p}_{L}+J^{\rm p}_{R}=0$. If the coupling $\lambda=0$, we obtain $J^{\rm e}=-J^{\rm p}_{R}$ when the energy flows across the interface only through EPI, that is, the heat generation of the metal part transfers directly to the insulator part. We set the Plank constant $\hbar=1$ and the Boltzman constant $k_{\rm B}=1$ in the following numerical calculation; and $\varepsilon_{0}=0$, $k^{L}=k^{R}=1,\;t=1$. We define the conductance $\sigma=\|J^{\rm e}\|/\Delta T$ to illustrate the thermal transport between the electron part and phonon part. ## 3 Thermal Conductance across the Interface Figure 2: (a) Thermal conductance across the metal-insulator interface vs the average temperature of the baths. $T_{L\rm e}=T_{0}+\Delta T$, $T_{L\rm p}=T_{R}=T_{0}-\Delta T$, $\Delta T=0.01$, $V_{\rm ep}=0.01$. (b) The thermal current density $J(\varepsilon)$ ($J^{\rm e}=\int d\varepsilon J(\varepsilon)$) for electrons with energy $\varepsilon$ at $\lambda=0^{+}=10^{-4}$. (c) The thermal conductance (left scale) and current (right scale) from the electron bath vs the temperature difference $\Delta T$ at $T_{0}=1.0$. As shown in Fig.1(a), if the metal part is applied by a voltage, the average energy of electrons increases, which means the electrons have a higher effective temperature than the phonons. Thus we set the left and right phonon leads to a lower temperature $T_{L\rm p}=T_{R}=T_{0}-\Delta T$, while the electron lead has a higher temperature $T_{L\rm e}=T_{0}+\Delta T$. If we don’t consider the phonon contribution in the metal part, the thermal conductance from the electron lead to the phonon lead has a nonmonotonic behavior with increasing the average temperature $T_{0}$, as shown in Fig. 2(a). In the low-temperature region, as temperature increases, the thermal conductance increases due to more contribution from electrons with energy far from Fermi surface ($\varepsilon=0$) although there is certain decrease from the contribution near Fermi surface, which can be seen from the solid line ($T_{0}=0.2$) and dashed line ($T=0.6$) in Fig. 2 (b). With the temperature further increasing, the dominating effect comes from the decrease of the contribution near the Fermi level, as shown in the dashed line and dotted line of Fig. 2 (b). This causes the thermal conductance to decrease after certain temperature. Such dependence of thermal conductance on temperature is consistent with the experiment observation of thermal contact conductance between graphene and silicon dioxide [33]. While the conductance has a nonmonotonic dependence on $T_{0}$, the heat current and conductance shows a monotonic behavior with temperature difference $\Delta T$ increasing, as shown in the inset of Fig. 2 (c). If we include the phonon contribution in the metal part, the thermal conductance has a similar temperature dependence, but with a smaller magnitude. This means that the heat generation from the electron part is affected by the coupling strength between the two phonon parts in the metal-insulator interface. In order to investigate the effect of phonon-phonon channel $\lambda$ on the thermal transport through electron-phonon channel $V_{ep}$, in Fig. 3 (a) we plot the heat current from the electrons to the phonons when we increase the coupling strength $\lambda$ between the two phonon parts. We find that in the small $\lambda$ region, with $\lambda$ increasing the heat current flows into the right phonon lead will increase due to the smaller difference between the atom ($K^{C}=(\lambda k^{L}+k^{R})$) in interface and the atoms ($K^{R}_{ii}=2k^{R}$) in the right lead, thus the scattering for phonons at the interface decreases. The heat is easier to transport to the right lead; $J^{\rm e}$ increases. At the same time, the heat can flow from the electrons to the left phonon lead; $-J^{\rm p}_{L}$ increases with $\lambda$. When the $\lambda$ increases further, the phonon scattering between left chain and the center atom decreases, more thermal energy transferred from electron is easier to flow into the left lead. Such fast increasing of $-J^{\rm p}_{L}$ causes the decreasing of the heat into the right lead $-J^{\rm p}_{R}$. With increasing of $\lambda$, the phonon scattering at the interface between the left and right atomic chains decreases, more phonons can transport coherently in the two atomic leads, and are less efficient to couple to electrons; thus the heat flowing from the electron lead begins to decrease. Due to continuous increase of the heat flowing into the left phonon lead $-J^{\rm p}_{L}$, and $J^{\rm e}+J^{\rm p}_{L}+J^{\rm p}_{R}=0$, thus the decrease of heat from electrons $J^{\rm e}$ lags behind the decrease of the heat into the right lead $-J^{\rm p}_{R}$. And if $\lambda=1$, the heat into two phonon leads has the same value due to the symmetry. As shown in Fig. 3 (b), if the non-biased metal part in a higher temperature while the insulator in a lower one, the heat flow from electron part has a similar dependence on $\lambda$; but the heat flowing from left phonon lead and the one into the right phonon lead monotonically increase with $\lambda$ since phonon transport dominates heat flow in such case and the phonon scattering decreases. We also find that, in this case the thermal transport through the phonon-phonon channel $\lambda$ is several orders of magnitude larger than that through the electron-phonon channel. And even if $\lambda=4\times 10^{-4}$ (see the inset of Fig. 3(b)), the thermal current of phonon is ten times larger than that of electron. Therefore for the thermal transport in the non-biased metal-insulator interface, such small thermal current of electron can be ignored, which is consistent with the recent finding in Ref.[34]. However, for the interface between a biased metal and a insulator, the thermal transport through the electron-phonon interaction is very important, which dominated the thermal transport even if $\lambda=1$ as shown in Fig 3 (a), which can explain the remote Joule heating via electron-phonon interaction in Ref. [9]. Figure 3: (a) Thermal current vs the coupling $\lambda$. The solid line corresponds to the thermal current $J^{\rm e}$ flowing from the electron bath. The dashed and dotted lines correspond to the thermal current ($-J^{\rm p}_{L}$ and $-J^{\rm p}_{R}$ ) flowing into the left and right phonon baths, respectively. (b) Thermal current vs the coupling $\lambda$ at $T_{L\rm e}=T_{L\rm p}=1.01$, $T_{R}=0.99$. The solid, dashed and dotted lines correspond to the thermal current flowing from the electron bath (left scale), the left and the right phonon baths (right scale), respectively. The inset is the zoom-in of (b) for small $\lambda$. For (a) and (b), $V_{\rm ep}=0.01$. ## 4 Thermal Rectification across the Interface via EPI Figure 4: Thermal rectification of the metal-insulator interface vs temperature difference. The inset: the thermal current across the interface vs temperature difference. $T_{h}=T_{0}(1+\Delta T)$, $T_{c}=T_{0}(1-\Delta T)$, $V_{\rm ep}=0.01$. Thanks to nonlinearity and asymmetry of the metal-insulator interface, we can expect the thermal rectification, which is defined as $R=(J_{+}-J_{-})/{\rm max}\\{J_{+},J_{-}\\}$, where $J_{+}$ is the forward direction heat flux, defined as $T_{L}=T_{h},T_{R}=T_{c}$, and $J_{-}$ is that of the backward direction when $T_{L}=T_{c},T_{R}=T_{h}$. Here, $T_{h}$ and $T_{c}$ correspond to the temperatures of the hot and cold baths, respectively. In the rest of the paper, we will not consider the phonon contribution in metal (we set $\lambda=10^{-4}$, $T_{Lp}=T_{R}$ in the calculation to avoid divergence), which will not change the physical properties of the rectification, and only decrease its magnitude. Figure 4 shows the dependence of thermal rectification on temperature difference for different average temperatures. The rectification shows a monotonically increasing behavior of temperature difference due to the bigger difference of heat current in larger temperature difference as shown in the inset of Fig. 4, which is consistent with all the traditional study on thermal rectification [35, 36]. With the temperature $T_{0}$ increasing, we find that the rectification can change sign. Figure 5: (a) Thermal rectification of the metal-insulator interface vs temperature $T_{0}$ for different temperature differences at $V_{\rm ep}=0.01$. (b) Thermal rectification vs the electron-phonon interaction $V_{ep}$ at different temperatures. (c) The forward thermal current (solid line) and backward thermal one (dashed line) vs $V_{\rm ep}$ at $T_{0}=0.25$. (d1)-(d4) The local available density of states (LADOS) at surface for the left electron part ($\rho_{\rm e}={\|\rm Img}(g^{<})\|$)(solid line) and the right phonon part ($\rho_{\rm p}={\|\rm Img}(d^{<})\omega\|$)(dashed line). (d1)((d2)) and (d3)((d4)) are corresponding to the forward (backward) transport at $T_{0}=0.25$ and $1.0$, respectively. (e1)-(e4) The forward (solid lines) and backward (dashed lines) thermal current density $J(\varepsilon)$ for different temperatures and different EPI. For all the curves, $T_{h}=T_{0}(1+\Delta T)$, $T_{c}=T_{0}(1-\Delta T)$; from (b) to (e) $\Delta T=0.4$. The dependence of thermal rectification on temperature and the EPI is shown in Fig. 5. We find that at a relative larger EPI ($V_{\rm ep}=0.01$), with increasing the temperature, the thermal rectification can change sign from positive to negative. However for a very weak EPI, the rectification always keeps negative, as shown in Fig 5(b). Within the range of $\lambda=10^{-4}\sim 10^{-2}$, the rectification is always negative for higher temperatures, while it can change sign from negative to positive in a lower temperature $T=0.25$. The relation of thermal currents in the forward and backward direction changes with increasing of $\lambda$ while both their magnitudes monotonically increase, as shown in Fig 5(c). From the local available density of states (LADOS) of electrons and phonons at the surface of the corresponding leads, we find that at a higher temperature $T_{0}=1.0$, going from the forward transport to the backward one, the main change is that the LADOS phonons largely increases, as shown in Fig. 5 (d3) and (d4), which causes a larger current in the backward direction than that in the forward direction, thus the rectification is negative. At a lower temperature, the change of LADOS for electrons and phonons are not obvious, as shown in Fig. 5 (d1) and (d2), and the rectification could change with the strength of EPI. Turning on the EPI, the electron Green’s functions change much more than the phonon ones; with increasing EPI, the Green’s functions of electrons $G^{>}$ and $G^{<}$ become more asymmetric, especially for the $G^{<}$ which have larger value at the energy far away from the Fermi level ($\varepsilon=0$). At a very weak EPI, the heat current comes from the contribution of electronic energies symmetrically away from the Fermi level. The heat current of the backward transport is larger than that of the forward one, as shown in Fig. 5 (e1) and (e3), due to the increase of the LADOS of phonons in the backward direction, thus the rectification are negative for all the temperatures at weak EPI as shown in Fig 5(b). With a stronger EPI, which induces larger value of $G^{<}$ at the energy range $-2\sim-1$, where the center electron has larger density of states. Thus we find that the forward current has larger value at the range of $-2\sim-1$ as shown in Fig. 5(e2). At the same time, the heat current can be affected by the temperature, in the low temperature range, as temperature increases, more electrons with energy far away from Fermi level contribute to the thermal current, as discussed in Fig. 2(b). Thus in Fig. 5(e2), the forward heat current is larger than the backward one; the rectification is positive at low temperatures with larger EPI. In the higher temperature range, the contribution from energy far away from Fermi level will decrease with temperature increasing as shown in Fig. 2(b); further more, the mainly change from LADOS of electron and phonon is the larger phonon LADOS in the backward transport than that in the forward one; thus the backward heat current is larger than the forward one as shown in Fig. 5(e4). Therefore, due to mismatch of electron LADOS and phonon one, the rectification is negative for high temperatures or weak EPI. Due to the nonlinearity of EPI and its relation with temperature, we can find the positive rectification with relative larger EPI at low temperatures. ## 5 Discussions Figure 6: Green’s functions for electrons and phonons, $T_{L\rm e}=T_{0}+\Delta T$, $T_{L\rm p}=T_{R}=T_{0}-\Delta T$, $T_{0}=0.2$, $\Delta T=0.01$. (a) The imagine part of the bare greater and less Green’s functions for phonons and electrons: $D_{0}^{>}$ (solid line),$D_{0}^{<}$ (dotted line),$G_{0}^{>}$ (dashed line),$G_{0}^{<}$ (dash-dotted line). (b) The imagine part of the full Green’s functions for phonons and electrons: $D^{>}$ (solid line),$D^{<}$ (dotted line),$G^{>}$ (dashed line),$G^{<}$ (dash-dotted line) with the EPI ($V_{\rm ep}=0.01$). ### 5.1 Full Green’s functions in the calculation Under the SCBA, we calculate the self energies of the EPI; after the iteration is convergent we obtain all the Green’s functions of $G^{>,<}$ and $D^{>,<}$ in Eq. (3) and (4) in the main text, then we can calculate the heat currents. If we do not consider the phonon contribution in the metal part, we should choose the coupling $\lambda$ be a very tiny positive value, thus the heat current to the left part will be very small and negligible; as shown in Fig. 3 in the main text, if $\lambda=10^{-4}$ the heat current flowing into the left phonon leads is almost zero. If $\lambda=0$, the iteration to calculate self energies for EPI will be divergent, but if $\lambda=0^{+}=10^{-4}$, it can be convergent. Thus here the tiny $\lambda=0^{+}$ plays a role as very small onsite potential for the center atom, which is similar as the tiny onsite in computing the phonon Hall effect [38]. We plot the bare Green’s functions $G_{0}$, $D_{0}$ and the full Green’s functions $G$, $D$ in Fig. 6. Without the EPI, the bare Green’s functions of electrons $G_{0}^{<}$ and $G_{0}^{>}$ are symmetric with respect to the origin as shown in Fig. 6 (a), but which symmetry for $G^{<}$ and $G^{>}$ will be broken and change much if we turn on the EPI, which can be seen in Fig 6 (b). However, the Green’s functions for phonons do not change much. ### 5.2 The role of Fermi energy of electrons Figure 7: The conductance (left panel) and the rectification (right panel) vs the onsite $\varepsilon_{0}$. $\lambda=10^{-4}$. For comparison, The dotted lines in (a) and (b) are copied from the corresponding line in Fig.2 (a) and Fig.5(a), respectively. We set the onsite $\varepsilon_{0}=0$ in the main text, thus the Fermi energy of the electron system is $\varepsilon=0$. However we can change the onsite $\varepsilon$, as shown in Fig. 7, the onsite potential can change the magnitude of the thermal conductance and the rectification, and the temperature where the thermal conductance and rectification have maximum value shifts a bit. However the curves of the thermal conductance and rectification have the same properties with the case at $\varepsilon_{0}=0$. Therefore, the mechanism found for the non-monotonicity of conductance and reversal of rectification still holds for the case of nonzero Fermi energy. ### 5.3 Limitation and application We use a simplified one-dimensional model to study thermal conductance and rectification across the metal-insulator interface via electron-phonon interaction. For some quasi-one dimensional metal-insulator interface systems, the thermal transport can simply be recast to one-dimensional model, then our results of the conductance and rectification can be applied. The thermal resistance of a whole interface system comes from the two materials themselves and the interface between them. Our study in this work focuses on the interface itself, which provides understanding on the interface in the thermal transport. Although the thermal transport via electron-phonon interaction can be ignored in the non-biased metal-insulator interface, it dominates the thermal transport in the remote Joule heating in a biased metal-insulator interface. For real materials, there would be more atoms in one unit cell, thus more phonon branches will contribute to the thermal transport. To compare with experiments on real materials, the simplified model need to be generalized to two-dimensional or three dimensional with input of the real parameters. For realistic three dimensional systems, the density of states of electrons and phonons are quite different from those in one dimension [39], then the analysis for the mechanism of the thermal rectification would be different and more complicated; but the general properties on the interfacial thermal transport, such as thermal conductance as functions of $T$, $\lambda$ and the rectification tuned by temperature or electron-phonon interaction, would still hold and could be verified by the future experiments. The nonequilibrium Green’s function approach is a good candidate to solve the interfacial thermal transport. If we study two or three dimensional systems with large number atoms in the interface, the self-consistent process is time consuming, and we also need to pay more attention to its convergency. If we only are interested on the thermal conductance accurate to the second order of $V_{ep}$, the Born approximation is applicable to avoid the convergent problem. However, for the thermal rectification as a function of $V_{ep}$ can only be obtained by the self consistent Born approximation. Therefore the calculation of the simple model can provide the understanding of the basic properties of thermal transport at metal-insulator interface and can be generalized to real materials with the parameters obtained from first- principle calculation. The challenges would come from the time consuming for the self-consistent calculation of the nonlinear self energies $\Sigma_{\rm ep}$ and $\Pi_{\rm ep}$. ## 6 Conclusion In summary, using nonequilibrium Green’s function method we study the thermal transport across metal-insulator interface with an EPI. The thermal conductance and rectification across the interface is thoroughly investigated under SCBA by a clean and efficient model where an electron lead couples to a phonon lead through the EPI. We find that the thermal conductance is a nonmonotonic function of average temperature, it has a maximum value at certain temperature. Considering the phonon part in metal contributing to the thermal transport, the heat flow from electron part will change while it has a similar dependence on temperature. The thermal rectification effect across the metal-insulator interface can reverse with varying of system average temperature and the EPI. Our results are very helpful to explain and guide the experiment on the thermal transport and dissipation in electronic devices and interface system; and could be verified in the biased and non-biased graphene sheet supported on insulating substrate. The reversal rectification is significant for the study of one-way heat transport and tuning the direction of thermal transport; thus it could have wide application in the energy science. ## Acknowledgements L. Z.and B. 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B, 78, 045434, [2008])	arxiv-papers	2013-09-12T14:20:03	2024-09-04T02:49:50.831822	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lifa Zhang, Jing-Tao L\\\"u, Jian-Sheng Wang, Baowen Li", "submitter": "Lifa Zhang", "url": "https://arxiv.org/abs/1309.3166" }
1309.3277	11affiliationtext: Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA22affiliationtext: Racah Institute for Physics, The Hebrew University, Jerusalem, 91904, Israel # Magnetic Fields In Relativistic Collisionless Shocks Rodolfo Santana1a1aaffiliationmark: , Rodolfo Barniol Duran2b2baffiliationmark: , & Pawan Kumar1c1caffiliationmark: (a) [email protected]; (b) [email protected] (c) [email protected] ###### Abstract We present a systematic study on magnetic fields in Gamma-Ray Burst (GRB) external forward shocks (FSs). There are 60 (35) GRBs in our X-ray (optical) sample, mostly from Swift. We use two methods to study $\epsilon_{B}$ (fraction of energy in magnetic field in the FS). 1. For the X-ray sample, we use the constraint that the observed flux at the end of the steep decline is $\geq$ X-ray FS flux. 2. For the optical sample, we use the condition that the observed flux arises from the FS (optical sample light curves decline as $\sim t^{-1}$, as expected for the FS). Making a reasonable assumption on $E$ (jet isotropic equivalent kinetic energy), we converted these conditions into an upper limit (measurement) on $\epsilon_{B}n^{2/(p+1)}$ for our X-ray (optical) sample, where $n$ is the circumburst density and $p$ is the electron index. Taking $n=1\mbox{ cm}^{-3}$, the distribution of $\epsilon_{B}$ measurements (upper limits) for our optical (X-ray) sample has a range of $\sim 10^{-8}-10^{-3}$ ($\sim 10^{-6}-10^{-3}$) and median of $\sim\mbox{ few}\times 10^{-5}$ ($\sim\mbox{ few}\times 10^{-5}$). To characterize how much amplification is needed, beyond shock compression of a seed magnetic field $\sim 10\mu\mbox{G}$, we expressed our results in terms of an amplification factor, $AF$, which is very weakly dependent on $n$ ($AF\propto n^{0.21}$). The range of $AF$ measurements (upper limits) for our optical (X-ray) sample is $\sim 1-1000$ ($\sim 10-300$) with a median of $\sim 50$ ($\sim 50$). These results suggest that some amplification, in addition to shock compression, is needed to explain the afterglow observations. ###### Subject headings: gamma-ray burst: general -radiation mechanisms: non-thermal - methods: analytical ## 1\. Introduction Gamma-Ray Bursts (GRBs) are bright explosions occurring at cosmological distances which release gamma-rays for a brief time, typically on a timescale of $\sim\mbox{ few}\times$ 10 sec (e.g. Piran 2004, Gehrels et al. 2009, Zhang 2011). This short-lived emission of gamma-rays is known as the prompt emission. After the prompt emission, long-lived emission in the X-ray, optical, and radio bands (on timescales of days, months, or even years) is also observed from what is called the “afterglow”. Although the mechanism for the prompt emission is currently being debated, the afterglow emission has a well-established model based on external shocks (Rees & Meszaros, 1992; Meszaros & Rees, 1993; Paczynski & Rhoads, 1993). In this framework, a relativistic jet emitted by the central engine interacts with the medium surrounding the GRB progenitor. This interaction produces two shocks; the external-reverse shock and the external-forward shock (Meszaros & Rees, 1997; Sari & Piran, 1999b). The external-reverse shock heats up the jet while the external-forward shock heats up the medium surrounding the explosion. The external-reverse shock is believed to be short-lived in the optical band (Sari & Piran, 1999a) and might have been observed, perhaps, in a few cases. The long-lived afterglow emission is interpreted as synchrotron radiation from the external-forward shock. This shock is taken to produce a power-law distribution of high energy electrons and to amplify the pre-existing seed magnetic field in the surrounding medium. These high energy electrons are then accelerated by the amplified magnetic field and emit radiation by the synchrotron process. One of the open questions in the field of GRB afterglows is: what is the dynamo mechanism amplifying magnetic fields in the collisionless relativistic shocks involved for GRB external shocks? The magnetic field strength downstream of the shock front is expressed in terms of $\epsilon_{B}$, which is defined as the fraction of energy that is in the magnetic field downstream of the shock front. With this definition, the explicit expression for $\epsilon_{B}$ is $\epsilon_{B}=\frac{B^{2}}{32\pi nm_{p}c^{2}\Gamma^{2}},$ (1) where $B$ is the co-moving magnetic field downstream of the shock front, $n$ is the density surrounding the GRB progenitor, $m_{p}$ is the proton mass, $c$ is the speed of light, and $\Gamma$ is the Lorentz factor of the shocked fluid downstream of the shock front (e.g. Sari et al., 1998; Wijers & Galama, 1999; Panaitescu & Kumar, 2000). If shock compression is the only mechanism amplifying the magnetic field downstream of the shock front, then $B$ is given by $B=4\Gamma B_{0}$ (e.g. Achterberg et al., 2001), where $B_{0}$ is the seed magnetic field in the medium surrounding the GRB progenitor. Using this expression for $B$, $\epsilon_{B}$ simplifies to $\epsilon_{B}=B_{0}^{2}/2\pi nm_{p}c^{2}$. Using the value for the ambient magnetic field of the Milky Way galaxy $B_{0}\sim\mbox{ few }\mu\mbox{G}$ and a density for the surrounding medium of $n=1\mbox{ cm}^{-3}$, $\epsilon_{B}$ is expected to be $\sim 10^{-9}$. Several studies have modeled afterglow data to determine what values of the afterglow parameters best describe the observations (e.g. Wijers & Galama, 1999; Panaitescu & Kumar, 2002; Yost et al., 2003; Panaitescu, 2005). The results from previous studies show that $\epsilon_{B}$ ranges from $\epsilon_{B}\sim 10^{-5}-10^{-1}$. These values for $\epsilon_{B}$ are much larger than the $\epsilon_{B}\sim 10^{-9}$ expected from shock compression alone and suggest that some additional amplification is needed to explain the observations. There have been several theoretical and numerical studies that have considered possible mechanisms, operating in the plasma in the medium surrounding the GRB, that can generate extra amplification for the magnetic field. The mechanisms that have been proposed are the two-stream Weibel instability (Weibel, 1959; Medvedev & Loeb, 1999; Gruzinov & Waxman, 1999; Silva et al., 2003; Medvedev et al., 2005) and dynamo generated by turbulence (Milosavljević & Nakar, 2006; Milosavljevic et al., 2007; Sironi & Goodman, 2007; Goodman & MacFadyen, 2008; Couch et al., 2008; Zhang et al., 2009; Mizuno et al., 2011; Inoue et al., 2011). Recent results (Kumar & Barniol Duran, 2009, 2010) found surprisingly small values of $\epsilon_{B}\sim 10^{-7}$ for 3 bright GRBs with Fermi$/$LAT detections. These values of $\epsilon_{B}$ are $\sim 2$ orders of magnitude smaller than the smallest previously reported $\epsilon_{B}$ value and they can be explained with the only amplification coming from shock compression of a seed magnetic field of a few $10\mu\mbox{G}$ 111The values given above of $\epsilon_{B}\sim 10^{-7}$ are under the assumption of $n=1\mbox{ cm}^{-3}$. It is important to note that when reaching the conclusion that shock compression provides enough amplification to explain the afterglow data, Kumar & Barniol Duran (2009, 2010) did not assume a value for $n$. Also, the results of small $\epsilon_{B}\sim 10^{-7}$ values do not depend on whether or not the LAT emission is produced by the external-forward shock. These small $\epsilon_{B}$ values were inferred from the late time X-ray and optical data and from the constraint that the external-forward shock does not produce flux at 150 keV that exceeds the observed prompt emission flux at 50 seconds.. Although this seed magnetic field is stronger than the one of the Milky Way galaxy by about a factor $\sim 10$, seed magnetic fields of a few $10\mu\mbox{G}$ have been measured before. The seed magnetic fields in the spiral arms of some gas-rich spiral galaxies with high star formation rates have been measured to be 20-30 $\mu\mbox{G}$ (Beck, 2011). Seed magnetic fields as high as $0.5-18$ mG were measured in starburst galaxies by measuring the Zeeman splitting of the OH megamaser emission line at 1667 MHz (Robishaw et al., 2008). Given this disagreement between the recent and previous results, the question regarding the amplification of magnetic fields in GRB external relativistic collisionless shocks remains unanswered. The first goal of this study is to provide a systematic determination of $\epsilon_{B}$ for a large sample of GRBs by using the same method to determine $\epsilon_{B}$ for each burst in our X-ray or optical sample. This is the first time such a large and systematic study has been carried out for $\epsilon_{B}$. Knowing the value of $\epsilon_{B}$ for large samples will help us determine how much amplification of the magnetic field is needed to explain the afterglow observations. We mostly limit our samples to GRBs detected by the Swift satellite, with measured redshift. In this study, we determine an upper limit on $\epsilon_{B}$ for our X-ray sample and a measurement of $\epsilon_{B}$ for our optical sample. We use a new method to determine an upper limit on $\epsilon_{B}$ with X-ray data, which relies on using the steep decline observed by Swift in many X-ray light curves. We expect that the observed flux at the end of the steep decline is larger than the predicted flux from the external-forward shock. Making reasonable assumptions about the other afterglow parameters, we are able to convert this constraint into an upper limit on $\epsilon_{B}$. To determine a measurement of $\epsilon_{B}$ for our optical sample, we restrict our sample to light curves that show a power law decline with a temporal decay $\sim 1$ at early times, $\sim 10^{2}-10^{3}$ seconds, as expected for external-forward shock emission. We choose this selection criteria so that the optical emission is most likely dominated by external-forward shock emission. Making the same reasonable assumptions for the other afterglow parameters and using the condition that the observed flux from the optical light curve is equal to the external-forward shock flux, we are able to convert this condition into a measurement of $\epsilon_{B}$. We also applied a consistency check for the bursts that are in common to our X-ray and optical samples to make sure our results for $\epsilon_{B}$ are correct. The second goal of this study is to determine how much amplification, in addition to shock compression, is needed to explain the results for the $\epsilon_{B}$ upper limits$/$measurements. To quantify how much amplification beyond shock compression is required by the observations, we also express the results we found for the $\epsilon_{B}$ upper limits (measurements) for our X-ray (optical) sample in terms of an amplification factor. This paper is organized as follows. We begin in Section 2 by presenting a review of the values previous studies have found for the microphysical afterglow parameters $\epsilon_{e}$ (the fraction of energy in electrons in the shocked plasma) and $\epsilon_{B}$. In Section 3 (Section 4), we present the method we use to determine an upper limit (measurement) on $\epsilon_{B}$ and apply it to our X-ray (optical) sample of GRBs. In Section 5, we use the GRBs that are in common to both samples to perform a consistency check. We search for a possible correlation between the kinetic energy of the blastwave and $\epsilon_{B}$ in Section 6. In Section 7, we write our results for $\epsilon_{B}$ for our X-ray and optical samples in terms of an amplification factor, which quantifies how much amplification – beyond shock compression – is required by the observations. Lastly, in Section 8, we discuss our results and present our conclusions. The convention we use for the specific flux $f_{\nu}$, the flux per unit frequency $\nu$, is $f_{\nu}\propto\nu^{-\beta}t^{-\alpha}$. In this convention, $\beta$ is the spectral decay index and $\alpha$ is the temporal decay index. For a GRB at a given redshift $z$, when calculating the luminosity distance to the GRB, $d_{L}$, we used the Cosmological parameters $H_{0}=71\mbox{ km}/\mbox{sec}/\mbox{Mpc}$, $\Omega_{m}=0.27$, and $\Omega_{\Lambda}=0.73$. ## 2\. Literature Review Of Values Of $\epsilon_{e}$ and $\epsilon_{B}$ The flux observed from the external-forward shock depends on 6 parameters. These parameters are $E$, $n$, $s$, $\epsilon_{e}$, $\epsilon_{B}$, and $p$. $E$ is the isotropic equivalent kinetic energy of the jet and $n$ is the number density of the surrounding medium. The density is taken to be spherically symmetric and to decrease with $r$ as $n(r)\propto r^{-s}$, where $s$ is a constant determining the density profile of the surrounding medium and $r$ is the distance from the center of the explosion. Two cases are usually considered for the density profile: $s=0$ and $s=2$, which respectively correspond to a constant density medium and a wind medium. The microphysical parameters are $\epsilon_{e}$ and $\epsilon_{B}$, where $\epsilon_{e}$ ($\epsilon_{B}$) is the fraction of energy in the electrons (magnetic field) in the shocked fluid. The microphysical parameters are taken to be constant throughout the afterglow emission. A power law distribution of electrons, $dN_{e}/d\gamma_{e}\propto\gamma_{e}^{-p}$ with $\gamma_{e}\geq\gamma_{i}$, where $\gamma_{e}$ is the Lorentz factor of the electrons, $\gamma_{i}$ is the minimum Lorentz factor of the electrons, and $N_{e}$ is the number of electrons, is assumed to be produced when the external-forward shock interacts with the surrounding medium. The power-law index of the electron distribution, $p$, is a constant known as the electron index. In practice, it is very difficult to determine the values of the 6 afterglow parameters. The value of $p$ and the density profile of the surrounding medium (whether we have a $s=0$ or $s=2$ medium) can be determined from observations of the afterglow spectral decay and temporal decay of the light curve with the so-called “closure” relations (e.g. Zhang et al., 2006). The remaining 4 afterglow parameters are more difficult to determine. What is needed to determine these 4 parameters is observations of the afterglow emission in the 4 different spectral regimes of the synchrotron afterglow spectrum (we will discuss the afterglow synchrotron spectrum in more detail in Section 3.4). In practice, most GRBs do not have this wealth of observations. In order to determine these 4 parameters, previous works have either focused only on determining the afterglow parameters for bursts with high quality data, spanning all portions of the synchrotron spectrum, or have applied various simplifying assumptions. We performed a literature search for papers that determine values for $\epsilon_{e}$ and $\epsilon_{B}$ to get an idea of what typical values previous works have found. Different authors applied different techniques for finding $\epsilon_{e}$ and $\epsilon_{B}$. When displaying the results from the literature, we did not discriminate against any method and simply plotted every value we found. However, we did not consider works that made simplifying assumptions when determining $\epsilon_{e}$ and $\epsilon_{B}$, such as equipartition of proton and electron energy ($\epsilon_{e}$). The GRBs for which we found $\epsilon_{e}$ and $\epsilon_{B}$ values are shown in Table LABEL:epsilon_e_and_epsilon_B_table in Appendix A. Except for GRB 080928, all the GRBs in our sample have radio, optical, and X-ray observations, allowing for a determination of all the afterglow parameters. We included GRB 080928 in our sample because $\epsilon_{e}$ and $\epsilon_{B}$ were able to be uniquely determined from optical and X-ray observations (Rossi et al., 2011). The $\epsilon_{e}$ values we found in the literature for 29 GRBs are shown in the histogram in the left panel of Figure 1. There is a narrow distribution for $\epsilon_{e}$; it only varies over one order of magnitude, from $\sim 0.02-0.6$, with very few GRBs reported to have $\epsilon_{e}<0.1$. The mean of this distribution is 0.24 and the median is 0.22. About 62% of the GRBs in this sample have $\epsilon_{e}\sim 0.1-0.3$. These results for $\epsilon_{e}$ are also supported by recent simulations of relativistic magnetized collisionless electron-ion shocks presented in Sironi & Spitkovsky 2011, where they found $\epsilon_{e}\sim 0.2$. The narrow distribution of $\epsilon_{e}$ values from the literature and the results from recent simulations both show that $\epsilon_{e}$ does not change by much from GRB to GRB. The $\epsilon_{B}$ values we found in the literature for 30 GRBs are show in the histogram in the right panel of Figure 1. Comparing the two histograms in Figure 1, we can immediately see that there is a much wider range in the distribution of $\epsilon_{B}$, with $\epsilon_{B}$ ranging from $\sim 3.5\times 10^{-5}-0.33$. A noticeable peak, containing about 24% of the bursts, is seen in the bin with $-1<\mbox{log}_{10}(\epsilon_{B})<-0.5$. Two other peaks, each containing about 17% of the GRBs, are seen in the bins with $-2<\mbox{log}_{10}(\epsilon_{B})<-1.5$ and $-4<\mbox{log}_{10}(\epsilon_{B})<-3.5$. The mean of this distribution is $6.3\times 10^{-2}$ and the median is $1.0\times 10^{-2}$. The important point of the $\epsilon_{B}$ histogram is that $\epsilon_{B}$ varies over 4 orders of magnitude, showing that $\epsilon_{B}$ has a wide distribution and is an uncertain parameter. $\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{epsilon_e_histogram.eps}&\includegraphics[scale={0.6}]{epsilon_B_histogram.eps}\end{array}\end{array}$ Figure 1.— Left Panel: Histogram of the distribution of $\epsilon_{e}$ values we found in the literature (Table LABEL:epsilon_e_and_epsilon_B_table in Appendix A). Right Panel: Histogram of the distribution of $\epsilon_{B}$ values we found in the literature (Table LABEL:epsilon_e_and_epsilon_B_table in Appendix A) ## 3\. Upper Limit On $\epsilon_{B}$ With Swift X-ray Light Curves ### 3.1. Constraining $\epsilon_{B}$ With The X-ray Light Curve Steep Decline One interesting property found by Swift (Gehrels et al., 2004) is that at early times, about 50% of the light curves detected by the XRT (X-ray Telescope, Burrows et al., 2005) display a very rapid decline in flux, known as the steep decline (Gehrels et al., 2009). The flux during the steep decline typically decays as $t^{-3}$ and it usually lasts $\sim 10^{2}-10^{3}$ sec. By extrapolating the BAT (Burst Alert Telescope, Barthelmy et al., 2005) emission to the X-ray band, O’Brien et al. 2006 showed that there is a continuous transition between the end of the prompt emission and the start of the steep decline phase. This important conclusion lead to the interpretation that the X-ray steep decline has an origin associated with the end of the prompt emission. The favored interpretation for the origin of the steep decline is high latitude emission (Kumar & Panaitescu, 2000). Although high latitude emission is able to explain most of the steep decline observations, some GRBs display spectral evolution during the steep decline (Zhang et al., 2007b), which is not expected. In any case, the steep decline cannot be produced by the external-forward shock. Therefore, the observed flux during the steep decline should be larger than or equal to the flux produced by the external-forward shock. We do however assume that the time at which the steep decline typically ends, at about $10^{2}-10^{3}$ seconds, is past the deceleration time222The deceleration time marks the time when about half of the kinetic energy of the blastwave has been transferred to the surrounding medium.. For our upper limit on $\epsilon_{B}$ with X-ray data, we will use the expression for the flux from the external-forward shock, which uses the kinetic energy of the blastwave given by the Blandford and McKee solution (Blandford & McKee, 1976). Since this solution is only valid for a decelerating blastwave, we need to be past the deceleration time for it to be applicable. Theoretically, it is expected that that the deceleration time occurs before the end of of the steep decline. Depending on the density profile of the surrounding medium, the deceleration time $t_{\mathrm{dec}}$ is given by $t_{\mathrm{dec}}=\left\\{\begin{array}[]{ll}\hskip-7.0pt(220\mbox{ sec})E_{53}^{1/3}n_{0}^{-1/3}\Gamma_{2}^{-8/3}(1+z)&s=0\\\ &\\\ \hskip-7.0pt(67\mbox{ sec})E_{53}A_{,-1}^{-1}\Gamma_{2}^{-4}(1+z)&s=2\end{array}\right.$ (2) (e.g. Panaitescu & Kumar, 2000). In these expressions, $\Gamma$ is the Lorentz factor of the shocked fluid, $z$ is the redshift, and we have adopted the usual notation $Q_{n}\equiv Q/10^{n}$. For $s=2$, the proportionality constant of the density, $A$, is normalized to the typical mass loss rate and stellar wind velocity of a Wolf-Rayet star, which is denoted by $A_{}$ and is defined as $A_{}\equiv A/(5\times 10^{11}\mbox{ g}\mbox{ cm}^{-1})$ (Chevalier & Li, 2000). For typical GRB afterglow parameters of $E_{53}=1$, $n_{0}=1$ (or $A_{}=0.1$ for $s=2$), $\Gamma_{2}=3$, and $z=2.5$, the deceleration time is under $100$ seconds for both $s=0$ and $s=2$. Although there can be a large uncertainty in the afterglow parameters $E$ and $n$, $\Gamma$ is the most important parameter when calculating $t_{\mathrm{dec}}$ since $t_{\mathrm{dec}}$ has a very strong dependence on $\Gamma$. For $s=0$ ($s=2$), even if we take extreme parameters for $E$ and $n$, such as a high kinetic energy of $E_{53}=100$ and a low density of $n=10^{-3}\mbox{ cm}^{-3}$ ($A_{}=10^{-2}$), with a typical $\Gamma_{2}=3-4$ (e.g. Molinari et al., 2007; Xue et al., 2009; Liang et al., 2010), $t_{\mathrm{dec}}$ is still a few hundred seconds. Thus, since the deceleration time is less than the typical time at which the steep decline ends, the onset of the external-forward shock emission occurs before the end of the steep decline. Observationally, the deceleration time is also seen to occur before the end of the steep decline for many GRBs. If the dominant contribution to the light curves at early times is the external-forward shock, the light curve is expected to rise as a power law, reach a peak, and then decline as a power law, with the peak signifying the deceleration time. In Liang et al. 2010, a sample of optical light curves that display this peak is studied. In their Figure 1, for each GRB, they display both the optical light curve and the X-ray light curve. For all their bursts that display a steep decline in the X-ray light curve (except for GRB 080303 and GRB 081203A), it can be seen that the peak of the optical light curve occurs before the end of the X-ray steep decline. We did not include GRB 080330 and GRB 081203A in our samples. Since an increase in $\epsilon_{B}$ increases $f_{\mathrm{ES}}$ (external- forward shock flux), the condition that the X-ray flux during the steep decline should be larger than or equal to $f_{\mathrm{ES}}$ gives an upper limit on $\epsilon_{B}$. Since our goal is to attain the most stringent upper limit on $\epsilon_{B}$, we take this constraint at the end of the steep decline. Explicitly, the constraint we will use to find an upper limit on $\epsilon_{B}$ with X-ray data is $f_{\mathrm{EoSD}}\geq f_{\mathrm{ES}}(E,n,s,\epsilon_{e},\epsilon_{B},p).$ (3) In this inequality, $f_{\mathrm{EoSD}}$ represents the observed flux at the end of the steep decline (EoSD). We have also explicitly shown the dependence of $f_{\mathrm{ES}}$ on the afterglow parameters. We will now discuss the assumptions we make on the other afterglow parameters, which will allow us to calculate an upper limit on $\epsilon_{B}$. ### 3.2. The Other Afterglow Parameters #### 3.2.1 $E$ and $\epsilon_{e}$ Although we do not know the isotropic equivalent kinetic energy of the blastwave $E$, we can calculate the isotropic energy released in gamma-rays during the prompt emission: $E_{iso}^{\gamma}=\frac{\mbox{fluence}\times 4\pi d_{L}^{2}}{1+z}.$ (4) In this equation, the fluence has units of ergs/$\mbox{cm}^{2}$ and it represents the flux detected in gamma-rays, integrated over the duration of the prompt emission. $d_{L}$ is the luminosity distance. Since we are interested in the fluence radiated in gamma-rays, we use the fluence detected in the BAT band, ranging from 15-150 keV (Barthelmy et al., 2005). The fluences detected by BAT for each GRB can be found in NASA’s Swift GRB Table and Lookup website333 http://heasarc.gsfc.nasa.gov/docs/swift/archive/grb_table/ . To convert $E_{iso}^{\gamma}$ to $E$, we need to know the efficiency in the conversion of kinetic energy of the jet to prompt gamma-ray emission. Recent studies on the prompt emission efficiency, using X-ray light curves with plateaus detected by Swift, were presented in Granot et al. (2006) and Zhang et al. (2007a). For the 23 GRBs for which Zhang et al. (2007a) presented results for the efficiency (see their Table 3), more than half of them were found to have a high efficiency $\gtrsim$ 30% 444In Table 3 of Zhang et al. (2007a), they present two different estimates for the efficiency. If the shallow decay of plateaus seen in X-ray light curves is due to energy injection, the more appropriate of the two estimates for the efficiency is denoted as $\eta_{\gamma}(t_{\mathrm{dec}})$. $\eta_{\gamma}(t_{\mathrm{dec}})$ represents the efficiency in gamma-ray radiation ($\eta_{\gamma}$) calculated at the deceleration time $t_{\mathrm{dec}}$., with a few being estimated to have an efficiency as high as 90%. A high efficiency of $\sim$90% was also found in Granot et al. (2006). However, Fan & Piran 2006 argue that bursts with X-ray plateaus should have more moderate efficiencies $\sim 10\%$. In addition, Zhang et al. (2007a) mention that the efficiencies they calculate for some bursts have large errors due to the uncertainty in the microphysical parameters $\epsilon_{e}$ and $\epsilon_{B}$. In this work, we will take a standard choice and calculate $E$ with the expression $E=5E_{\mathrm{iso}}^{\gamma}.$ (5) From the definition of the efficiency, $\eta=E^{\gamma}_{\mathrm{iso}}/(E^{\gamma}_{\mathrm{iso}}+E)$, Equation 5 corresponds to an efficiency of $\sim 20$%. At the end of Section 3.5, we will discuss in more detail how the uncertainty in the efficiency affects our results. For our X-ray sample (see Section 3.3), from Equation 5, we found values for $E$ in the range of $10^{51}-10^{54}$ ergs, with a typical value $\sim 10^{53}$ ergs. The average value of $E_{53}$ for our X-ray sample is 2.8 and the median is 1.6. The fluence detected in the BAT band, $E^{\gamma}_{\mathrm{iso,}52}$, $z$, and $d_{L28}$ for each GRB in our X-ray sample are shown in Table LABEL:x-ray_properties_table 555Except for GRBs 060708, 060906, 061021, 061222A, 080906, and 081230, we obtained all the redshifts from NASA’s Swift GRB Table and Lookup website. For the exceptions, we obtained the redshifts from the website on GRB redshifts maintained by J. Greiner, http://www.mpe.mpg.de/ jcg/grbgen.html http://www.mpe.mpg.de/$\sim$jcg/grbgen.html. For $\epsilon_{e}$, we assumed a value of 0.2 for all of the GRBs in our sample. This choice for $\epsilon_{e}$ is justified from the results of $\epsilon_{e}$ with previous afterglow studies (Figure 1) and with recent results from simulations (Sironi & Spitkovsky, 2011), as discussed in Section 2. #### 3.2.2 Electron power-law index and density profile In afterglow studies, $s$ and $p$ can be obtained by determining which closure relation the observed temporal and spectral decay indices satisfy. We cannot use this strategy to determine $p$ and $s$ for our X-ray sample since the external-forward shock flux is below the observed steep decline emission. Instead, we use a fixed $p$ for all GRBs in our X-ray sample. We consider a small value of $p=2.2$, a typical value of $p=2.4$ for Swift GRBs (Curran et al., 2010), and a larger value of $p=2.8$. Previous afterglow studies have found that the majority of afterglow observations are better described by a constant density medium (e.g. Panaitescu & Kumar, 2002; Schulze et al., 2011). However, there are still a number of cases where the wind medium is a better model for the afterglow observations. Therefore, we will consider both $s=0$ and $s=2$ when displaying the results for the upper limit on $\epsilon_{B}$. #### 3.2.3 Density The density of the medium in the vicinity of GRBs is a highly uncertain parameter. A histogram of values of $n$ determined by previous afterglow modelling studies can be found in Figure 9 of Soderberg et al. (2006), which shows $n$ to vary over 5 orders of magnitude, ranging from $\sim 10^{-3}\mbox{ cm}^{-3}$ to $\sim 10^{2}\mbox{ cm}^{-3}$. In Section 3.5, we will discuss in more detail how the uncertainty in the density affects our results for the $\epsilon_{B}$ upper limits. ### 3.3. The X-ray Sample For our constraint on $\epsilon_{B}$ with X-ray data, we only consider X-ray data detected by the XRT on board Swift. We used the X-ray light curves presented in Butler & Kocevski (2007)666 http://butler.lab.asu.edu/swift/ http://butler.lab.asu.edu/swift/. With the exception of two cases, we only consider bursts that display a steep decline in their X-ray light curve (see below). After the end of the steep decline, GRBs display a variety of temporal decays (Evans et al., 2009). Our sample can be divided into 4 different subgroups, based on the temporal decay after the steep decline: 1. 1. Steep Decline To Plateau: In this subgroup, GRBs display a plateau after the steep decline ($\sim$ 73% of our sample). In Table LABEL:x-ray_properties_table, we display the time at the end of the steep decline in units of $10^{2}$ sec, $t_{2,\mathrm{EoSD}}$, and the observed flux at the end of the steep decline at 1 keV, $f_{1\mathrm{keV},\mathrm{EoSD}}$, in $\mu$Jy. 2. 2. Steep Decline To Normal Decline: In this subgroup, GRBs display a temporal decay of $\alpha\sim 1$ after the steep decline ($\sim$ 18% of our sample). In Table LABEL:x-ray_properties_table, we show the time and the flux at the end of the steep decline. 3. 3. Clear Steep Decline But Not A Clear End To The Steep Decline: In this subgroup, it cannot be determined where the steep decline ends (the XRT observations end before the steep decline ends). The following GRBs fall into this subgroup: 050315, 060202, 070419A, 071122, and 090516. For these GRBs, in Table LABEL:x-ray_properties_table, we show the flux and the time of the last steep decline point observed. For these GRBs, to be sure that the last steep decline point observed is past the deceleration time, we made sure that it is at a few hundred seconds. 4. 4. No Clear Steep Decline Seen, Just Plateau: Two of the GRBs in our X-ray sample, 050401 and 060927, do not display a steep decline. The first observation of the X-ray light curve for these bursts is during the plateau. We did not remove these GRBs from our X-ray sample because they are also part of our optical sample (bursts with both X-ray and optical data are important because they allow us to cross-check our results, see Section 5). For these 2 bursts, we considered the first observation in the X-ray light curve for our constraint so that we have the least amount of energy injection. We made sure that this point is at least at a few hundred seconds so that we can be confident that the onset of the external-forward shock emission has occurred. For these two bursts, we show the time and the flux of the first X-ray observation of the plateau in Table LABEL:x-ray_properties_table. In addition, 25% of the GRBs in our X-ray sample display X-ray flares during the steep decline. We only consider bursts where the X-ray flare ends before the end of the steep decline because it is difficult to determine the flux and time at the end of the steep decline for bursts that show X-ray flares near the end of the steep decline. It is fine to consider these bursts because observationally, after the X-ray flare, the X-ray light curve is seen to return to the same temporal decay prior to the X-ray flare (e.g. Chincarini et al. 2007). Lastly, GRB 051221A is the only short GRB in our X-ray sample; all the other bursts in our X-ray sample are long GRBs. Table 1Properties of X-ray Sample GRB \| $z$ \| $d_{L28}$ \| Fluence \| $E^{\gamma}_{\mathrm{iso,}52}$ \| $t_{2,\mathrm{EoSD}}$ \| $f_{1\mathrm{keV,EoSD}}$ \| $\mbox{log}_{10}(\epsilon_{B})$ \| $\mbox{log}_{10}(\epsilon_{B})$ ---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| $[\times 10^{-6}\mbox{ ergs}/\mbox{cm}^{2}$] \| \| \| [$\mu$Jy] \| ($s=0$) \| ($s=2$) 050315 \| 1.949 \| 4.71 \| 3.22 \| 3.04 \| 4 \| 3 \| -5.2 \| -6.0 050401 \| 2.9 \| 7.67 \| 8.22 \| 15.56 \| 2 \| 80 \| -4.7 \| -5.4 050721 \| 2.5 \| 6.40 \| 3.62 \| 5.32 \| 4 \| 30 \| -4.2 \| -5.0 050803 \| 0.422 \| 0.71 \| 2.15 \| 0.10 \| 3 \| 10 \| -3.8 \| -5.4 050814 \| 5.3 \| 15.76 \| 2.01 \| 9.95 \| 9 \| 1 \| -5.4 \| -6.0 051221A \| 0.547 \| 0.97 \| 1.15 \| 0.09 \| 6 \| 5 \| -3.5 \| -4.9 060108 \| 2.03 \| 4.95 \| 0.37 \| 0.38 \| 6 \| 0.8 \| -4.2 \| -5.5 060111B \| 2 \| 4.86 \| 1.60 \| 1.58 \| 2 \| 8 \| -4.6 \| -5.8 060115 \| 3.53 \| 9.72 \| 1.71 \| 4.48 \| 8 \| 1 \| -5.2 \| -5.9 060210 \| 3.91 \| 10.99 \| 7.66 \| 23.66 \| 8 \| 40 \| -4.4 \| -4.7 060418 \| 1.49 \| 3.37 \| 8.33 \| 4.79 \| 5 \| 40 \| -4.3 \| -4.9 060502A \| 1.51 \| 3.43 \| 2.31 \| 1.36 \| 3 \| 7 \| -4.6 \| -5.7 060607A \| 3.082 \| 8.25 \| 2.55 \| 5.35 \| 5 \| 60 \| -3.6 \| -4.3 060707 \| 3.425 \| 9.37 \| 1.60 \| 3.99 \| 9 \| 3 \| -4.5 \| -5.2 060708 \| 1.92 \| 4.62 \| 0.49 \| 0.45 \| 2 \| 10 \| -3.7 \| -5.2 060714 \| 2.71 \| 7.06 \| 2.83 \| 4.78 \| 3 \| 10 \| -4.8 \| -5.7 060729 \| 0.54 \| 0.96 \| 2.61 \| 0.20 \| 6 \| 7 \| -3.8 \| -5.1 060814 \| 0.84 \| 1.65 \| 14.60 \| 2.71 \| 8 \| 8 \| -5.0 \| -5.6 060906 \| 3.686 \| 10.24 \| 2.21 \| 6.21 \| 4 \| 0.8 \| -5.9 \| -6.7 060926 \| 3.208 \| 8.66 \| 0.22 \| 0.49 \| 2 \| 4 \| -3.8 \| -5.4 060927 \| 5.6 \| 16.81 \| 1.13 \| 6.08 \| 0.8 \| 8 \| -5.3 \| -6.6 061021 \| 0.3463 \| 0.56 \| 2.96 \| 0.09 \| 3 \| 10 \| -3.9 \| -5.5 061110A \| 0.758 \| 1.45 \| 1.06 \| 0.16 \| 5 \| 5 \| -3.6 \| -6.0 061121 \| 1.314 \| 2.88 \| 13.70 \| 6.19 \| 2 \| 40 \| -5.1 \| -5.8 061222A \| 2.088 \| 5.13 \| 7.99 \| 8.55 \| 2 \| 40 \| -4.9 \| -5.7 070110 \| 2.352 \| 5.94 \| 1.62 \| 2.14 \| 4 \| 8 \| -4.3 \| -5.3 070306 \| 1.497 \| 3.39 \| 5.38 \| 3.12 \| 7 \| 4 \| -5.0 \| -5.6 070714B \| 0.92 \| 1.85 \| 0.72 \| 0.16 \| 4 \| 6 \| -3.5 \| -5.0 070802 \| 2.45 \| 6.24 \| 0.28 \| 0.40 \| 5 \| 0.8 \| -4.2 \| -5.5 071122 \| 1.14 \| 2.41 \| 0.58 \| 0.20 \| 8 \| 0.8 \| -4.1 \| -5.4 080310 \| 2.43 \| 6.18 \| 2.30 \| 3.22 \| 10 \| 2 \| -4.8 \| -5.4 080413A \| 2.433 \| 6.19 \| 3.50 \| 4.91 \| 2 \| 10 \| -5.1 \| -6.0 080430 \| 0.767 \| 1.47 \| 1.20 \| 0.19 \| 2 \| 10 \| -3.9 \| -5.5 080607 \| 3.036 \| 8.10 \| 24.00 \| 49.07 \| 3 \| 90 \| -5.2 \| -5.5 080721 \| 2.591 \| 6.68 \| 12.00 \| 18.75 \| 0.7 \| 900 \| -4.3 \| -5.1 080905B \| 2.374 \| 6.00 \| 1.80 \| 2.42 \| 2 \| 20 \| -4.3 \| -5.4 080906 \| 2.1 \| 5.16 \| 3.50 \| 3.78 \| 7 \| 20 \| -4.0 \| -4.7 080916A \| 0.689 \| 1.29 \| 4.00 \| 0.50 \| 3 \| 10 \| -4.4 \| -5.7 081007 \| 0.5295 \| 0.93 \| 0.71 \| 0.05 \| 2 \| 8 \| -3.5 \| -5.3 081008 \| 1.9685 \| 4.77 \| 4.30 \| 4.14 \| 6 \| 20 \| -4.2 \| -4.9 081230 \| 2 \| 4.86 \| 0.82 \| 0.81 \| 3 \| 6 \| -4.1 \| -5.3 090418A \| 1.608 \| 3.71 \| 4.60 \| 3.05 \| 2 \| 20 \| -4.8 \| -5.8 090516A \| 4.109 \| 11.66 \| 9.00 \| 30.08 \| 6 \| 10 \| -5.4 \| -5.7 090519 \| 3.85 \| 10.79 \| 1.20 \| 3.62 \| 5 \| 0.8 \| -5.4 \| -6.2 090529 \| 2.625 \| 6.79 \| 0.68 \| 1.09 \| 20 \| 0.5 \| -4.3 \| -5.1 090618 \| 0.54 \| 0.96 \| 105.00 \| 7.86 \| 4 \| 200 \| -4.9 \| -5.3 090926B \| 1.24 \| 2.68 \| 7.30 \| 2.95 \| 5 \| 10 \| -4.8 \| -5.6 091029 \| 2.752 \| 7.19 \| 2.40 \| 4.16 \| 6 \| 1 \| -5.5 \| -6.2 091109A \| 3.076 \| 8.25 \| 1.60 \| 3.35 \| 5 \| 2 \| -5.0 \| -5.8 100302A \| 4.813 \| 14.06 \| 0.31 \| 1.33 \| 8 \| 1 \| -4.2 \| -5.2 100425A \| 1.755 \| 4.14 \| 0.47 \| 0.37 \| 3 \| 7 \| -3.6 \| -5.0 100513A \| 4.772 \| 13.92 \| 1.40 \| 5.91 \| 9 \| 7 \| -4.1 \| -4.8 100621A \| 0.542 \| 0.96 \| 21.00 \| 1.58 \| 4 \| 20 \| -5.0 \| -5.8 100704A \| 3.6 \| 9.95 \| 6.00 \| 16.23 \| 6 \| 9 \| -5.1 \| -5.5 100814A \| 1.44 \| 3.23 \| 9.00 \| 4.85 \| 6 \| 9 \| -5.0 \| -5.6 100906A \| 1.727 \| 4.05 \| 12.00 \| 9.09 \| 3 \| 20 \| -5.2 \| -5.9 110808A \| 1.348 \| 2.98 \| 0.33 \| 0.16 \| 3 \| 3 \| -3.6 \| -5.2 110818A \| 3.36 \| 9.16 \| 4.00 \| 9.67 \| 20 \| 1 \| -5.3 \| -5.5 111008A \| 4.9898 \| 14.68 \| 5.30 \| 23.95 \| 3 \| 9 \| -5.5 \| -6.1 111228A \| 0.72 \| 1.36 \| 8.50 \| 1.15 \| 5 \| 8 \| -4.8 \| -5.7 Table 1Properties of X-ray Sample (Continued) ### 3.4. Expected External-Forward Shock Emission At The End Of The Steep Decline The synchrotron afterglow spectrum consists of four power-law segments that are smoothly joined together at three characteristic frequencies of synchrotron emission (e.g. Sari, Piran, & Narayan, 1998; Granot & Sari, 2002). These three characteristic frequencies are: $\nu_{a}$, the synchrotron self- absorption frequency, $\nu_{i}$ (also commonly referred to as $\nu_{m}$), the frequency of the photons emitted by the power-law distribution of injected electrons with the smallest energy, and $\nu_{c}$, the cooling frequency corresponding to electrons cooling on a dynamical time. For this work, we will only consider the standard case for the ordering of the characteristic frequencies, the slow cooling case, where $\nu_{a}<\nu_{i}<\nu_{c}$. One argument against the fast cooling case ($\nu_{a}<\nu_{c}<\nu_{i}$) is that if the observing frequency is between $\nu_{c}$ and $\nu_{i}$, the spectrum should be $f_{\nu}\propto\nu^{-1/2}$; however, the spectral index $\beta=0.5$ disagrees with the typical observed afterglow spectral index $\beta\approx 0.9$ (e.g. Piro, 2001). The next question we need to consider is where the X-ray band lies at the end of the steep decline (here, we consider 1 keV for the X-ray band because the light curves we used are plotted at this energy). The two possibilities for the spectral regime of the X-ray band are $\nu_{i}<1\mbox{ keV}<\nu_{c}$ or $\nu_{c}<1\mbox{ keV}$. We rule out $\nu_{c}<1\mbox{ keV}$ with the following two arguments. First, we compare the observed flux at the end of the steep decline, $f_{\mathrm{1keV,EoSD}}$, to the flux predicted by the external-forward shock at the same time, if $\nu_{c}<1\mbox{ keV}$ (defined as $f_{\mathrm{pred}}$). For $s=0$ , $f_{\mathrm{pred}}$ is given by (Granot & Sari, 2002) $\displaystyle f_{\mathrm{pred}}=0.855(p-0.98)10^{\frac{3p+2}{4}}8.64^{\frac{3p-2}{4}}e^{1.95p}$ $\displaystyle\times(1+z)^{\frac{p+2}{4}}(\bar{\epsilon}_{e,-1})^{p-1}\epsilon_{B}^{\frac{p-2}{4}}E_{53}^{\frac{p+2}{4}}(t_{\mathrm{2,EoSD}})^{-\frac{(3p-2)}{4}}$ $\displaystyle\times d_{L28}^{-2}\nu_{14}^{-\frac{p}{2}}\mbox{ mJy ,}$ (6) where $\bar{\epsilon}_{e,-1}\equiv(p-2)/(p-1)\epsilon_{e,-1}$. For $\nu_{c}<1\mbox{ keV}$, the external-forward shock flux is independent of the density and the $s=2$ expression is almost identical. When calculating $f_{\mathrm{pred}}$, for each of the bursts in our X-ray sample, we assumed a standard $p=2.4$, $\epsilon_{e}=0.2$, and $\nu_{14}=2.4\times 10^{3}$, the frequency corresponding to 1 keV. For the parameters $E$, $t$, $z$, and $d_{L}$, we used the values given in Table LABEL:x-ray_properties_table for each burst ($E=5E_{iso}^{\gamma}$). The remaining parameter we need to compute $f_{\mathrm{pred}}$ is $\epsilon_{B}$. Since $\epsilon_{B}$ is raised to the power of $(p-2)/4$, for a typical $p\sim 2-3$, the dependence on $\epsilon_{B}$ is very weak. When calculating $f_{\mathrm{pred}}$, we assumed a low value of $\epsilon_{B}=10^{-3}$. We computed the ratio $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}$ for all the GRBs in our X-ray sample and found that $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}>1$ for all the bursts and $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}>10$ for $54/60$ bursts. The mean value of $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}$ is 50 and the median value is 34. This means that the predicted flux from the external-forward shock, when $\nu_{c}<\mbox{1 keV}$, over-predicts the observed flux at the end of the steep decline by a factor that is larger than 10 for the majority of the bursts. Therefore, the assumption that $\nu_{c}<1$ keV is incorrect. This is a robust conclusion because the X-ray flux from the external-forward shock, when $\nu_{c}<1$ keV, basically only depends on $\sim\epsilon_{e}E$ (see Equation 6), which cannot be decreased by a factor of $>10$ without introducing serious efficiency problems in producing the prompt gamma-rays. Even if we allow for an uncertainty of a factor of $\sim 2-3$ in both $\epsilon_{e}$ and $E$, this is not enough to decrease $f_{\mathrm{pred}}/f_{\mathrm{1keV,EoSD}}$ below 1 for the majority of bursts in our X-ray sample. Before continuing, we want to add that $f_{\mathrm{pred}}$ (Equation 6) also has a dependence on the Compton-$Y$ parameter: $f_{\mathrm{pred}}\propto(1+Y)^{-1}$. With this dependence, if the Compton-$Y$ parameter is large, then it is possible for $f_{\mathrm{pred}}$ to decrease below $f_{\mathrm{1keV,EoSD}}$. For a few bursts in our X-ray sample, we performed a detailed numerical calculation of the external-forward shock flux with the formalism presented in Barniol Duran & Kumar (2011), which includes a detailed treatment of Compton-$Y$ with Klein-Nishina effects. From this calculation, we also found that $f_{\mathrm{pred}}$ over-predicts $f_{\mathrm{1keV,EoSD}}$ by a factor larger than 10, which means that the Inverse Compton (IC) cooling of electrons producing 1 keV synchrotron photons is a weak effect (even when $\epsilon_{B}$ is small). Numerically, we also found that without making any assumption about the location of $\nu_{c}$, solutions to the constraint $f_{\mathrm{ES,1keV}}\leq f_{1keV,EoSD}$ were only found when 1 keV $<\nu_{c}$ (when $\epsilon_{B}$ is small such that 1 keV $<\nu_{c}$, it turns out that IC cooling of electrons producing 1 keV photons takes place in the Klein-Nishina regime at these early times of $\sim$ few $\times 100$ sec). Another argument in favor of the spectral regime of the X-ray band being $\nu_{i}<\mbox{1 keV}<\nu_{c}$ at the end of the steep decline comes from the extrapolation of $\nu_{c}$ at late times to the end of the steep decline. In Liang et al. (2008), they made fits to late time XRT light curves during the normal decline phase and they also provided the value of $\nu_{c}$ during the geometrical midpoint of the normal decline phase777The geometrical midpoint of the normal decline is defined by $\mbox{log}_{10}t=(\mbox{log}_{10}t_{1}+\mbox{log}_{10}t_{2})/2$ , where $t_{1}$ represents the time of the first observation of the normal decline and $t_{2}$ represents the time of the last observation of the normal decline.. For this argument, we focus on GRBs that are in common to our X-ray sample and to the sample of Liang et al. (2008). For these bursts, we extrapolate $\nu_{c}$ at late times to the end of the steep decline. In Liang et al. (2008), they only considered a constant density medium, where $\nu_{c}\propto t^{-1/2}$. The results of the extrapolation of $\nu_{c}$ are shown in Table 2. GRB \| $t_{\mathrm{late},4}$ \| $t_{\mathrm{EoSD},2}$ \| $\nu_{\mathrm{c,late}}$ (keV) \| $\nu_{\mathrm{c,EoSD}}$ (keV) ---\|---\|---\|---\|--- 050315 \| $3.5$ \| 4 \| 0.17 \| 1.6 050401 \| $6.9$ \| 2 \| 4.1 \| 77 051221A \| $1.7$ \| 6 \| 4.1 \| 22 060210 \| $3.2$ \| 8 \| 0.97 \| 6.1 060502A \| $0.42$ \| 3 \| 4.1 \| 15 060714 \| $3.5$ \| 3 \| 3.6 \| 39 060729 \| $40$ \| 6 \| 1.4 \| 36 060814 \| $2.0$ \| 8 \| 4.1 \| 21 060926 \| $0.26$ \| 2 \| 6.7 \| 24 061121 \| $1.1$ \| 2 \| 4.2 \| 31 070110 \| $14$ \| 4 \| 4 \| 74 Table 2This table shows the extrapolation of $\nu_{c}$ from late times to the end of the steep decline for GRBs in common to our sample and to the sample in Liang et al. (2008) (first column). The second column shows $t_{\mathrm{late},4}$, the late time in units of $10^{4}$ seconds at which Liang et al. (2008) determined $\nu_{c}$. The third column shows $t_{\mathrm{EoSD},2}$, the time at the end of the steep decline in units of $10^{2}$ seconds. $\nu_{\mathrm{c,late}}$, given in keV, is the value found in Liang et al. (2008) for $\nu_{c}$ at $t_{\mathrm{late}}$. The last column shows $\nu_{\mathrm{c,EoSD}}$ in keV. $\nu_{\mathrm{c,EoSD}}$ is found by extrapolating $\nu_{\mathrm{c,late}}$ to $t_{\mathrm{EoSD}}$. Since Liang et al. (2008) assume a constant density medium, we take a constant density medium for all the GRBs in this sample when making the extrapolation of $\nu_{c}$ to $t_{\mathrm{EoSD}}$. In Table 2, we find that at the end of the steep decline, 1 keV $<\nu_{c,\mathrm{EoSD}}$ for all GRBs. This further confirms our choice that $\nu_{i}<\mbox{ 1 keV}<\nu_{c}$ at the end of the steep decline888Including energy injection will make the values of $\nu_{\mathrm{c,EoSD}}$ in Table 2 larger, making the conclusion that $\nu_{i}<\mbox{ 1 keV}<\nu_{c}$ at the end of the steep decline more robust. For the bursts that have plateaus in their X-ray light curve, energy injection needs to be considered. During the energy injection episode, $E$ increases as $E\propto t^{1-q}$ (Zhang et al., 2006), where $q$ is a positive constant that satisfies $0\leq q\leq 1$. Therefore, between the end of the plateau and the end of the steep decline, since $\nu_{c}\propto E^{-1/2}t^{-1/2}$, $\nu_{c}\propto t^{-1+(q/2)}$. This time evolution of $\nu_{c}$ is steeper than $\nu_{c}\propto t^{-1/2}$ without energy injection for $s=0$ (Liang et al. (2008) only considered $s=0)$.. The knowledge of the spectral regime regime at the end of the steep decline allows us to write in an explicit expression for $f_{\mathrm{ES}}$ in Equation 3: $\frac{f_{\mathrm{1keV,EoSD}}}{\mbox{mJy}}\geq\left\\{\begin{array}[]{ll}\hskip-7.0pt0.461(p-0.04)10^{\frac{3p+1}{4}}8.64^{\frac{3(p-1)}{4}}e^{2.53p}&\\\ \times(1+z)^{\frac{p+3}{4}}(\bar{\epsilon}_{e,-1})^{p-1}\epsilon_{B}^{\frac{p+1}{4}}n_{0}^{\frac{1}{2}}E_{53}^{\frac{p+3}{4}}&\\\ \times(t_{\rm 2,EoSD})^{-\frac{3(p-1)}{4}}d_{L28}^{-2}\nu_{14}^{-\frac{(p-1)}{2}}\mbox{ $s=0$}\\\ &\\\ \hskip-7.0pt3.82(p-0.18)10^{\frac{3p-1}{4}}8.64^{\frac{3p-1}{4}}e^{2.54p}&\\\ \times(1+z)^{\frac{p+5}{4}}(\bar{\epsilon}_{e,-1})^{p-1}\epsilon_{B}^{\frac{p+1}{4}}A_{,-1}E_{53}^{(p+1)/4}&\\\ \times(t_{\rm 2,EoSD})^{-\frac{(3p-1)}{4}}d_{L28}^{-2}\nu_{14}^{-\frac{(p-1)}{2}}\mbox{ $s=2$}.\end{array}\right.$ (7) On the left hand side of this inequality, we have the observed X-ray flux at the end of the steep decline and on the right hand side we have the expression for the external-forward shock flux when $\nu_{i}<\mbox{ 1 keV}<\nu_{c}$ (Granot & Sari, 2002). The notation used for $\bar{\epsilon}_{e}$ and $A_{}$ is defined as $\bar{\epsilon}_{e,-1}\equiv\bar{\epsilon}_{e}/10^{-1}$ and $A_{,-1}\equiv A_{}/10^{-1}$. The expressions in Equation 7 are only valid for $p>2$, which we consider in this work (for $p<2$, see Bhattacharya 2001 and Resmi & Bhattacharya 2008). Before displaying our results for the $\epsilon_{B}$ upper limit for our entire X-ray sample, we show a simple calculation to get an idea of what values to expect for the $\epsilon_{B}$ upper limits from the X-ray constraint given in Equation 7. For a standard $p=2.4$, the X-ray constraint is $\frac{f_{\mathrm{1keV,EoSD}}}{\mbox{mJy}}\geq\left\\{\begin{array}[]{ll}\hskip-7.0pt5.0\times 10^{1}t_{2,\mathrm{EoSD}}^{-1.05}\epsilon_{e,-1}^{1.4}E_{53}^{1.35}n_{0}^{0.5}\epsilon_{B}^{0.85}&s=0\\\ &\\\ \hskip-7.0pt7.0\times 10^{2}t_{2,\mathrm{EoSD}}^{-1.55}\epsilon_{e,-1}^{1.4}E_{53}^{0.85}A_{,-1}\epsilon_{B}^{0.85}&s=2.\end{array}\right.$ (8) For this calculation, we used the average $z=2.5$ for Swift GRBs (Gehrels et al., 2009) (with a corresponding $d_{L28}=6.4$) and $\nu_{14}$ corresponding to 1 keV. Solving for $\epsilon_{B}$, the upper limit depends on the afterglow parameters as999For bursts with plateaus in their X-ray light curve, it is possible that energy injection begins before the steep decline ends. However, even with energy injection, there still exists a self-similar solution for the energy (Equation 52 of Blandford & McKee 1976). For both $s=0$ and $s=2$, we calculated the external-forward shock synchrotron flux with this new self- similar solution and then calculated the upper limit on $\epsilon_{B}$ as shown in Equation 9. We found that the $\epsilon_{B}$ upper limits are affected by less than a factor of $\sim 2$. Thus, even if energy injection begins before the end of the steep decline, it has very little to no effect on our $\epsilon_{B}$ upper limit results. $\epsilon_{B}\leq\left\\{\begin{array}[]{ll}\hskip-7.0pt1.0\times 10^{-2}\left(\frac{f_{1keV,\mathrm{EoSD}}}{\mbox{mJy}}\right)^{1.18}t_{2,\mathrm{EoSD}}^{1.24}&\\\ \times\epsilon_{e,-1}^{-1.65}E_{53}^{-1.59}n_{0}^{-0.59}&s=0\\\ &\\\ \hskip-7.0pt4.5\times 10^{-4}\left(\frac{f_{1keV,\mathrm{EoSD}}}{\mbox{mJy}}\right)^{1.18}t_{2,\mathrm{EoSD}}^{1.82}&\\\ \times\epsilon_{e,-1}^{-1.65}E_{53}^{-1}A_{,-1}^{-1.18}&s=2.\end{array}\right.$ (9) The median values for our X-ray sample for the parameters $f_{\mathrm{1keV,EoSD}}$, $t_{2,\mathrm{EoSD}}$, and $E_{53}$ are $8\times 10^{-3}$ mJy, 4, and 1.6, respectively. Using these median values and $\epsilon_{e,-1}=2$, the upper limit on $\epsilon_{B}$ becomes $\epsilon_{B}\leq\left\\{\begin{array}[]{ll}\hskip-7.0pt2.8\times 10^{-5}\times n_{0}^{-0.59}&s=0\\\ &\\\ \hskip-7.0pt3.7\times 10^{-6}\times A_{-1}^{-1.18}&s=2.\end{array}\right.$ (10) For a standard $n_{0}=1$ and $A_{,-1}=1$, it can be seen that the $\epsilon_{B}$ upper limit is lower for $s=2$. This is expected because for $A_{,-1}=1$, there is a larger density for the surrounding medium within a typical deceleration radius of $10^{17}\mbox{ cm}$. In Equation 10, the explicit dependence of the $\epsilon_{B}$ upper limit on the density is shown for $p=2.4$. In the next subsection, we will display the results of the $\epsilon_{B}$ upper limits for our entire X-ray sample. To keep the density-dependence, we will display histograms of upper limits on the quantity $\epsilon_{B}n_{0}^{0.59}$ ($\epsilon_{B}A_{-1}^{1.18}$) for $s=0$ ($s=2$) for $p=2.4$, or $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) for $s=0$ ($s=2$) for a general $p$ (see Equation 7). ### 3.5. $\epsilon_{B}$ Upper Limits For Our X-ray Sample We display the results for the upper limits (from Equation 7) on the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) for $s=0$ ($s=2$) assuming all GRBs in our X-ray sample have $p=2.2$, 2.4, and 2.8 in the Top-Left, Top-Right, and Bottom panels of Figure 2, respectively. Two histograms are shown in each panel, one for $s=0$ and the other for $s=2$. Table 3 shows the mean and median upper limits on the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) for $s=0$ ($s=2$) for each histogram. X-ray \| \| \| \| ---\|---\|---\|---\|--- $(s=0)$ \| $p=2.2$ \| $p=2.4$ \| $p=2.8$ \| Mean \| $1.1\times 10^{-4}$ \| $7.2\times 10^{-5}$ \| $6.1\times 10^{-5}$ \| Median \| $4.2\times 10^{-5}$ \| $2.8\times 10^{-5}$ \| $2.0\times 10^{-5}$ \| X-ray \| \| \| \| $(s=2)$ \| $p=2.2$ \| $p=2.4$ \| $p=2.8$ \| Mean \| $7.9\times 10^{-6}$ \| $5.5\times 10^{-6}$ \| $5.6\times 10^{-6}$ \| Median \| $4.0\times 10^{-6}$ \| $3.2\times 10^{-6}$ \| $3.4\times 10^{-6}$ \| Opt. \| \| \| \| $(s=0)$ \| $p=2.2$ \| $p=2.4$ \| $p=2.8$ \| $p$ from $\alpha_{\mathrm{O}}$ Mean \| $3.5\times 10^{-4}$ \| $1.0\times 10^{-4}$ \| $2.9\times 10^{-5}$ \| $9.5\times 10^{-5}$ Median \| $1.1\times 10^{-4}$ \| $3.3\times 10^{-5}$ \| $5.5\times 10^{-6}$ \| $2.4\times 10^{-5}$ Opt. \| \| \| \| $(s=2)$ \| $p=2.2$ \| $p=2.4$ \| $p=2.8$ \| Mean \| $7.2\times 10^{-5}$ \| $2.4\times 10^{-5}$ \| $7.5\times 10^{-6}$ \| Median \| $1.2\times 10^{-5}$ \| $3.9\times 10^{-6}$ \| $8.8\times 10^{-7}$ \| Table 3Mean and median $\epsilon_{B}$ values for the X-ray (upper limits on $\epsilon_{B}$) and optical (measurements of $\epsilon_{B}$) histograms shown in Figures 2 and 3. The section labeled “X-ray $(s=0)$” (“X-ray $(s=2)$”) shows the mean and median $\epsilon_{B}$ upper limits assuming a constant density (wind) medium with a standard $n=1\mbox{ cm}^{-3}$ ($A_{}=0.1$). The columns show the value of $p$ that was assumed. The section labeled “Opt. $(s=0)$” (“Opt. $(s=2)$”) shows the mean and median $\epsilon_{B}$ measurements assuming a constant density (wind) medium with a standard $n=1\mbox{ cm}^{-3}$ ($A_{}=0.1$). The columns show the value of $p$ that was assumed. The column labeled “$p$ from $\alpha_{O}$” shows the mean and median $\epsilon_{B}$ measurements with $p$ determined from $\alpha_{O}$. There are 60 GRBs in our X-ray sample and 35 GRBs in our optical sample. For the remainder of this section, we assume a standard $n_{0}=1$ ($A_{-1}=1$) for $s=0$ ($s=2$) when discussing our results for the $\epsilon_{B}$ upper limits. The $\epsilon_{B}$ upper limit histograms show a wide distribution. For a constant density (wind) medium, all the histograms show a distribution ranging from $\sim 10^{-6}-10^{-3}$ ($\sim 10^{-7}-10^{-4}$). For a constant density (wind) medium, the mean and median $\epsilon_{B}$ upper limit values are $\sim\mbox{ few}\times 10^{-5}$ ($\sim\mbox{ few}\times 10^{-6}$). Assuming a different value of $p$ does not have a significant effect on the distributions of the $\epsilon_{B}$ upper limits for our X-ray sample. For both the $s=0$ and $s=2$ cases, when changing $p$, the mean and median $\epsilon_{B}$ upper limit values change by less than a factor $\sim 2$. Although previous afterglow studies also showed a wide distribution for $\epsilon_{B}$ (Figure 1), our distribution of $\epsilon_{B}$ upper limits is shifted towards lower values. Unlike Figure 1, which shows that many GRBs have been reported to have $\epsilon_{B}\sim 10^{-3}-10^{-1}$, none of our histograms of $\epsilon_{B}$ upper limits show an $\epsilon_{B}$ upper limit larger than $10^{-3}$. We now discuss how our assumptions on the afterglow parameters can affect the distribution of $\epsilon_{B}$ upper limits. For this discussion, we will take a typical $p=2.4$; Equation 9 shows how the $\epsilon_{B}$ upper limit depends on the other afterglow parameters. The strongest dependence is on $\epsilon_{e}$, which is raised to the power of $-1.65$. However, as we displayed in Figure 1, according to previous studies, the distribution of $\epsilon_{e}$ values is narrow, with the $\epsilon_{e}$ values ranging only over one order of magnitude. In addition, $\sim 62\%$ of the bursts have $\epsilon_{e}\sim 0.1-0.3$. From Figure 1, a likely error in $\epsilon_{e}$ from our assumed $\epsilon_{e}=0.2$ is a factor $\sim 2$. From Equation 9, an error in $\epsilon_{e}$ by a factor $\sim 2$ will only lead to an error in the $\epsilon_{B}$ upper limit by a factor $\sim 3$. For a constant density (wind) medium, the $\epsilon_{B}$ upper limit depends on $E$ as $E^{-1.59}$ ($E^{-1}$). We assumed an efficiency of $\sim 20\%$ in the conversion of kinetic energy to prompt gamma-ray radiation. Recent studies have found higher values for the efficiency (Granot et al. 2006, Zhang et al. 2007a, see however, Fan & Piran 2006). In Zhang et al. 2007a, the mean (median) efficiency they reported is $\sim 37\%$ ($\sim 32\%$). Taking the efficiency to be $\sim 30\%-40\%$ instead of $\sim 20\%$ would lead to an error in $E$ by a factor $\sim 2-3$. From Equation 9, an error in $E$ by a factor $\sim 2-3$ would lead to an error in the $\epsilon_{B}$ upper limit by a factor $\sim 3-6$ ($\sim 2-3$) for a constant density (wind) medium. Lastly, the largest source of uncertainty for the $\epsilon_{B}$ upper limits is the density, since it has been reported to have a range $\sim 10^{-3}\mbox{ cm}^{-3}-10^{2}\mbox{ cm}^{-3}$. For a constant density (wind) medium, the $\epsilon_{B}$ upper limit depends on the density as $n^{-0.59}$ ($A_{}^{-1.18}$). For $s=0$ ($s=2$), we assumed a standard $n=1\mbox{ cm}^{-3}$ ($A_{}=0.1$). An error in the density by a factor $\sim 10^{3}$ ($\sim 10^{2}$) will lead to an error in the $\epsilon_{B}$ upper limit by a factor $\sim 60$ ($\sim 230$). In summary, the expected errors in $\epsilon_{e}$ and $E$ of a factor $\sim 2-3$ will not change the $\epsilon_{B}$ upper limits by an order of magnitude. On the other hand, the density is a very uncertain parameter and an error in the density by $\sim 2-3$ orders of magnitude will lead to an error in the $\epsilon_{B}$ upper limits by $\sim 1-2$ orders of magnitude. $\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{x_ray_p_2_2_histogram.eps}&\includegraphics[scale={0.6}]{x_ray_p_2_4_histogram.eps}\end{array}\\\ \includegraphics[scale={0.6}]{x_ray_p_2_8_histogram.eps}\end{array}$ Figure 2.— The Top-Left, Top-Right, and Bottom panels show the histograms of upper limits assuming $p=2.2$, $p=2.4$, and $p=2.8$ respectively, for all of the GRBs in our X-ray sample (obtained with Equation 7). The filled-in (un- filled) histograms show upper limits on the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) for $s=0$ ($s=2$) assuming all the GRBs in our X-ray sample are described by a constant density (wind) medium. Two additional parameters that can affect our $\epsilon_{B}$ upper limits are: 1. $\xi$, the fraction of electrons accelerated to a power-law distribution and 2. $f$, which is a factor that takes into account the degeneracy for a set of afterglow parameters. For a set of parameters $E$, $n$, $\epsilon_{e}$, $\epsilon_{B}$, $\xi$ producing the observed external-forward shock flux, another set of primed parameters $E^{\prime}=E/f$, $n^{\prime}=n/f$, $\epsilon_{e}^{\prime}=f\epsilon_{e}$, $\epsilon_{B}^{\prime}=f\epsilon_{B}$, $\xi^{\prime}=f\xi$ can also produce the observed external-forward shock flux (Eichler & Waxman, 2005). Afterglow studies usually assume $\xi=f=1$ for simplicity but $\xi\leq 1$ and $m_{e}/m_{p}\leq f\leq 1$ (Eichler & Waxman, 2005), where $m_{e}$ ($m_{p}$) is the electron (proton) mass. The external- forward shock flux depends on $\xi$ and $\epsilon_{B}$ as $f_{\nu}\propto\xi^{2-p}\epsilon_{B}^{(p+1)/4}$ (Leventis et al., 2012). From this dependence, we find that the $\epsilon_{B}$ upper limit $\propto\xi^{4(p-2)/(p+1)}$. Thus, including $\xi$ will decrease the values of our $\epsilon_{B}$ upper limits. Values of $\xi$ have not been determined for GRB external-forward shocks so we cannot quantify by how much the $\epsilon_{B}$ upper limit values will decrease. Including $f$ will also decrease the values of the $\epsilon_{B}$ upper limits since $\epsilon_{B}^{\prime}=f\epsilon_{B}$ and $f<1$. Like $\xi$, values of $f$ have also not been determined from afterglow observations. The largest effect $f$ can have on the $\epsilon_{B}$ upper limit values is decrease them by a factor of $m_{p}/m_{e}\sim 2000$. For the remainer of this paper, we will be conservative and continue to assume $\xi=f=1$, but we should keep in mind that considering $\xi$ and $f$ will decrease the values of the $\epsilon_{B}$ upper limits. ## 4\. Measurement Of $\epsilon_{B}$ With Optical Light Curves ### 4.1. $\epsilon_{B}$ Determination With Optical Data The light curves we consider for our optical sample decline with a temporal decay index $\alpha\sim 1$ from early times, $\sim 10^{2}-10^{3}\mbox{ sec}$, as expected for the external-forward shock emission (see Section 4.2). Since the light curves of these bursts are likely dominated by the external-forward shock, this means that the observed optical flux is an actual measurement of the external-forward shock flux, that is101010At late times, $\sim 10^{3}-10^{5}$ sec, many X-ray light curves decline with $\alpha\sim 1$: the “normal” decline. If this segment arises from the external-forward shock, then $\epsilon_{B}$ can be determined as described in this Section for our optical sample. However, this is not straightforward since energy injection (as evidenced by the plateau phase) should be considered. Also, one can attempt to use the upper limit on $\epsilon_{B}$, found in Section 3, to calculate a lower limit on $\nu_{c}$ during the normal decline and compare this to the observed spectral regime ($\mbox{1 keV}<\nu_{c}$ or $\mbox{1 keV}>\nu_{c}$). However, there are two difficulties with this $\nu_{c}$ consistency check: 1. several “closure relations” are simultaneously satisfied within 2-$\sigma$ for most of our sample (due to large errors in the spectral and temporal indices at late times, which can be found in Evans et al. (2007, 2009)). 2\. for the bursts that uniquely satisfy one closure relation, $\nu_{c}$ cannot be determined precisely since Klein-Nishina suppression is weaker at late times and $\nu_{c}$ depends strongly on $n$ when Compton-$Y$ is considered. $f_{\mathrm{obs}}=f_{\mathrm{ES}}(E,n,\epsilon_{e},\epsilon_{B},p,s).$ (11) Later in this section we will use this condition to determine $\epsilon_{B}$ for the bursts in our optical sample. We want to stress that we determine $\epsilon_{B}$ for the optical sample. This is in contrast to the X-ray sample, which only allowed us to determine an upper limit on $\epsilon_{B}$. ### 4.2. The Optical Sample Our optical sample consists of 35 GRBs. $33/35$ of the bursts triggered Swift and the remaining two bursts, 050502A and 080603A, were detected by INTEGRAL (INTErnational Gamma-Ray Astrophysics Laboratory, Winkler et al. 2003). Table LABEL:optical_properties_table shows properties of our optical sample. With a few exceptions, most of the GRBs in our optical sample have a known redshift111111The redshifts were taken from NASA’s Swift GRB Table and Lookup website (exceptions are GRB 071003 (Perley et al., 2008) and GRBs 050502A and 071025 (J. Greiner’s website)). Three GRBs (050721, 070420, 060111B) in our optical sample do not have a known redshift. For GRB 050721 and GRB 070420, we assumed the average redshift of 2.5 for Swift GRBs (Gehrels et al., 2009). The redshift for GRB 060111B was approximated as 2 in Stratta et al. (2009).. Table 4Optical Sample Properties GRB \| $z$ \| $d_{L28}$ \| Fluence \| $E^{\gamma}_{\mathrm{iso,}52}$ \| $\alpha_{O}$ \| Ref. \| $t_{2}$ \| $f_{2\mathrm{eV}}$ \| $\mbox{log}_{10}(\epsilon_{B})$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| $[\times 10^{-6}\mbox{ ergs}/\mbox{cm}^{2}]$ \| \| \| \| \| [mJy] \| ($s=0$) 050401 \| 2.9 \| 7.67 \| 8.22 \| 15.55 \| $0.80\pm 0.03$ \| [1] \| 0.72 \| 0.3 \| -5.5 050502A \| 3.793 \| 10.59 \| 1.4 \| 4.12 \| $1.16\pm 0.03$ \| [2] \| 1 \| 5 \| -4.5 050525A \| 0.606 \| 1.10 \| 15.3 \| 1.45 \| $1.12\pm 0.05$ \| [3] \| 34.56 \| 0.5 \| -4.3 050721 \| 2.5 \| 6.40 \| 3.62 \| 5.32 \| $1.29\pm 0.06$ \| [4] \| 20 \| 0.2 \| -5.0 050730 \| 3.97 \| 11.19 \| 2.38 \| 7.53 \| $0.89\pm 0.05$ \| [17] \| 7.5 \| 0.57 \| -3.9 050802 \| 1.71 \| 4.00 \| 2.00 \| 1.49 \| $0.82\pm 0.03$ \| [1] \| 3.6 \| 0.5 \| -3.5 051111 \| 1.55 \| 3.54 \| 4.08 \| 2.52 \| $1.00\pm 0.02$ \| [2] \| 30 \| 0.4 \| -3.8 051221A \| 0.5465 \| 0.97 \| 1.15 \| 0.09 \| $0.96\pm 0.03$ \| [5] \| 100 \| 0.02 \| -3.4 060111B \| 2 \| 4.86 \| 1.60 \| 1.58 \| $1.18\pm 0.05$ \| [6] \| 2 \| 0.4 \| -5.2 060210 \| 3.91 \| 10.99 \| 7.66 \| 23.65 \| $1.03\pm 0.06$ \| [2] \| 10 \| 0.1 \| -6.0 060418 \| 1.49 \| 3.37 \| 8.33 \| 4.78 \| $1.13\pm 0.02$ \| [7] \| 2 \| 8 \| -4.6 060607A \| 3.082 \| 8.25 \| 2.55 \| 5.34 \| $1.20\pm 0.03$ \| [7] \| 2 \| 10 \| -4.2 060904B \| 0.703 \| 1.32 \| 1.62 \| 0.21 \| $1.00\pm 0.18$ \| [17] \| 5.5 \| 0.58 \| -3.5 060908 \| 2.43 \| 6.18 \| 2.80 \| 3.91 \| $1.05^{+0.03}_{-0.03}$ \| [8] \| 2 \| 2 \| -4.5 060927 \| 5.6 \| 16.81 \| 1.13 \| 6.08 \| $1.21\pm 0.06$ \| [2] \| 0.5 \| 2 \| -5.5 061007 \| 1.26 \| 2.74 \| 44.4 \| 18.48 \| $1.70\pm 0.02$ \| [7] \| 2 \| 50 \| -6.0 061110B \| 3.44 \| 9.42 \| 1.33 \| 3.34 \| $1.64\pm 0.08$ \| [2] \| 20 \| 0.02 \| -5.9 061121 \| 1.314 \| 2.88 \| 13.7 \| 6.19 \| $0.82\pm 0.02$ \| [7] \| 4 \| 0.5 \| -4.7 061126 \| 1.159 \| 2.47 \| 6.77 \| 2.39 \| $0.89\pm 0.02$ \| [2] \| 10 \| 0.2 \| -4.6 070318 \| 0.84 \| 1.65 \| 2.48 \| 0.46 \| $0.96\pm 0.03$ \| [7] \| 20 \| 0.2 \| -3.7 070411 \| 2.954 \| 7.84 \| 2.70 \| 5.27 \| $0.92\pm 0.04$ \| [2] \| 20 \| 0.07 \| -4.7 070420 \| 2.5 \| 6.40 \| 14.0 \| 20.56 \| $0.81\pm 0.04$ \| [2] \| 3 \| 0.8 \| -4.7 070714B \| 0.92 \| 1.85 \| 0.72 \| 0.16 \| $0.83\pm 0.04$ \| [2] \| 10 \| 0.03 \| -3.7 071003 \| 1.6 \| 3.69 \| 8.3 \| 5.45 \| $1.466\pm 0.006$ \| [9] \| 0.6 \| 20 \| -5.7 071025 \| 5.2 \| 15.41 \| 6.5 \| 31.26 \| $1.27\pm 0.04$ \| [10] \| 20 \| 0.02 \| -6.8 071031 \| 2.692 \| 7.00 \| 0.9 \| 1.5 \| $0.97\pm 0.06$ \| [11] \| 10.5 \| 0.4 \| -3.4 071112C \| 0.823 \| 1.61 \| 3.00 \| 0.53 \| $0.95\pm 0.02$ \| [12] \| 10.5 \| 0.003 \| -6.3 080603A \| 1.688 \| 3.94 \| 1.1 \| 0.80 \| $0.99\pm 0.07$ \| [13] \| 30 \| 0.1 \| -3.6 080607 \| 3.036 \| 8.10 \| 24.0 \| 49.04 \| $1.65$ \| [14] \| 3 \| 0.2 \| -8.0 080721 \| 2.591 \| 6.68 \| 12.0 \| 18.74 \| $1.22\pm 0.01$ \| [5] \| 3 \| 10 \| -5.0 080810 \| 3.35 \| 9.13 \| 4.60 \| 11.06 \| $1.23\pm 0.01$ \| [7] \| 3 \| 30 \| -3.9 080913 \| 6.7 \| 20.72 \| 0.56 \| 3.92 \| $1.03\pm 0.02$ \| [15] \| 10 \| 0.02 \| -5.2 081008 \| 1.967 \| 4.76 \| 4.30 \| 4.13 \| $0.96\pm 0.03$ \| [16] \| 2 \| 3 \| -4.1 090313 \| 3.375 \| 9.21 \| 1.40 \| 3.41 \| $1.25\pm 0.08$ \| [17] \| 20 \| 2 \| -3.4 090418A \| 1.608 \| 3.71 \| 4.60 \| 3.05 \| $1.21\pm 0.04$ \| [7] \| 2 \| 0.8 \| -5.5 Table 4Optical Sample Properties (Continued) With the exception of the only short GRB in our optical sample, GRB 051221A, all the optical light curves in our sample decline before 3500 seconds. Considering early times has the advantage of minimizing possible energy injection. Our optical sample can be separated into 4 different subgroups, depending on the temporal behavior of the light curve before the $\alpha\sim 1$ decay as follows. 1. 1. Light Curves With A Peak At Early Times: The light curves of this subgroup are characterized by a power law rise, reaching a peak, and then a power law decline with $\alpha_{O}\sim 1$ ($\sim$43% of our sample). The peak of the light curve is believed to be due to the deceleration time. For the bursts in this subgroup, we show the temporal decay of the optical light curve after the peak and the flux and the time of the second data point after the peak in Table LABEL:optical_properties_table. We take the second data point to be confident that the optical light curve is declining. 2. 2. Single Power Law Decay From Early Times: In this subgroup, the optical light curve shows a decline as a single power law with $\alpha\sim 1$ from the beginning of the observations ($\sim$ 40% of our sample). We display the temporal decay of the optical light curve and the time and the flux of the second data point observed in Table LABEL:optical_properties_table. 3. 3. Optical Light Curves With Plateaus At Early Times: The optical light curves of 3 bursts in our optical sample (GRBs 050525A, 060210, and 070411) display plateaus at early times. The plateaus in our optical sample are short, with the longest plateau lasting under 3500 seconds. After the plateau ends, the light curves of these 3 bursts show a decay $\alpha_{O}\sim 1$, as expected for the external forward-shock emission. In Table LABEL:optical_properties_table, for these 3 bursts, we show the temporal decay after the plateau and the time and the flux of the second data point after the plateau. 4. 4. Light Curves With Possible Reverse Shock Emission At Early Times: 3 GRBs in our optical sample (060111B, 060908, and 061126) show possible emission from the reverse shock. The light curves in this subgroup show an initial steep decline at early times, characteristic of the reverse shock, and then transition to a more shallow decay of $\alpha\sim 1$ that is more typical for the external-forward shock emission. For these GRBs, in Table LABEL:optical_properties_table we show the temporal decay of the light curve and the time and flux of the second data point after the possible reverse shock emission. ### 4.3. Optical External-Forward Shock Spectral Regime and Afterglow Parameter Assumptions When referring to the optical band, we will use 2 eV since most of the light curves in our optical sample are either plotted at 2eV or were observed in the R filter. As we did with the X-ray sample, we will only consider the slow cooling ordering of the synchrotron characteristic frequencies, $\nu_{a}<\nu_{i}<\nu_{c}$. Because the optical light curve is declining at the time we are considering, the optical band must be above $\nu_{i}$ at this time. In Section 3.4, we argued that the X-ray band is between $\nu_{i}$ and $\nu_{c}$ at the end of the steep decline at a few 100 sec; therefore, the optical band must also be in this spectral regime at the early times $(\sim 10^{2}-10^{3})$ sec we are considering. The expression we will use to determine the optical external-forward shock flux is also Equation 7; however, we will have an equality (instead of an inequality), we replace $f_{\mathrm{1keV,EoSD}}$ with $f_{\mathrm{2eV}}$ (which represents the specific flux observed at 2 eV), and use $\nu_{14}$ corresponding to 2 eV. The other afterglow parameters are determined as in Section 3.2: $\epsilon_{e}=0.2$ and with $z$ and the fluence121212For the two bursts detected by INTEGRAL, the fluence in Table LABEL:optical_properties_table is in the 20-200 keV band of the instrument IBIS (Imager on-Board the INTEGRAL Satellite, Ubertini et al. (2003)): GRB 050502A (Gotz & Mereghetti, 2005) and GRB 080603A (Guidorzi et al., 2011)., we obtain $E^{\gamma}_{\mathrm{iso}}$ and use $E=5E^{\gamma}_{\mathrm{iso}}$. As with our X-ray sample, we will display our $\epsilon_{B}$ results with $p=2.2$, 2.4, and 2.8. We can also determine $p$ by using the temporal decay of the optical light curve, $\alpha_{O}$, which is shown in Table LABEL:optical_properties_table for each burst (optical spectrum is not always available, so we cannot use the closure relations for the optical sample). In order to have $p>2$ for all of the bursts in our optical sample, we only consider a constant density medium when determining $p$ with $\alpha_{O}$ ($\alpha_{O}=3(p-1)/4$ for $s=0$ and $\alpha_{O}=(3p-1)/4$ for $s=2$). Lastly, as we did for our X-ray sample, to keep the density-dependence, we will plot the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) for $s=0$ ($s=2$). ### 4.4. $\epsilon_{B}$ Results For Optical Sample $\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{optical_p_2_2_histogram.eps}&\includegraphics[scale={0.6}]{optical_p_2_4_histogram.eps}\end{array}\\\ \begin{array}[]{cc}\includegraphics[scale={0.6}]{optical_p_2_8_histogram.eps}&\includegraphics[scale={0.6}]{s_0_optical_histogram_with_temporal_p.eps}\end{array}\end{array}$ Figure 3.— The Top-Left, Top-Right, and Bottom-Left panels show the histograms of measurements assuming $p=2.2$, $p=2.4$, and $p=2.8$ respectively for all of the GRBs in our optical sample. The filled-in (un-filled) histograms show measurements of the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) assuming all the GRBs in our optical sample are described by a constant density (wind) medium. Bottom-Right Panel: The filled-in histogram shows the measurements on the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ with $p$ determined from $\alpha_{O}$. The un- filled histogram shows measurements on the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$, assuming $p=2.4$ for all of the bursts in our optical sample (this histogram was also shown in the Top-Right panel). We display the results for the measurements (from Equation 7) on the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) for $s=0$ ($s=2$) assuming all the GRBs in our optical sample have $p=2.2$, 2.4, and 2.8 in the Top-Left, Top-Right, and Bottom-Left panels of Figure 3, respectively. Two histograms are shown in each panel, one for $s=0$ and the other for $s=2$. We also use $\alpha_{O}$ to determine $p$ (assuming $s=0$) and compare the results to the ones obtained with $p=2.4$ and $s=0$ (Bottom-Right panel of Figure 3). In Table 3, we display a summary of the mean and median values of the measurements of the quantity $\epsilon_{B}n_{0}^{2/(p+1)}$ ($\epsilon_{B}A_{-1}^{4/(p+1)}$) for $s=0$ ($s=2$) for each histogram. For the remainder of this section, we assume a standard $n_{0}=1$ ($A_{-1}=1$) for $s=0$ ($s=2$) when discussing our results for the $\epsilon_{B}$ measurements for our optical sample. For a constant density (wind) medium, the mean and median $\epsilon_{B}$ measurements are $\sim\mbox{ few}\times 10^{-5}$ ($\sim\mbox{ few}\times 10^{-6}$) The mean and median $\epsilon_{B}$ measurements only change by a factor of a few when assuming a different value of $p$. To determine if assuming a standard $p=2.4$, as opposed to determining $p$ for each burst from $\alpha_{O}$, significantly affects the distribution of $\epsilon_{B}$ measurements, we compared the two histograms in the Bottom-Right panel of Figure 3 with a Kolmogorov-Smirnov (KS) test. The null hypothesis of the KS test is that the two histograms are drawn from the same distribution. We test this null hypothesis at the 5% significance level. The KS test confirmed the null hypothesis that the two histograms are consistent with being drawn from the same distribution. As with the $\epsilon_{B}$ upper limits from X-ray data, the mean and median $\epsilon_{B}$ measurements decrease by about an order of magnitude when assuming a wind medium as opposed to a constant density medium. Compared to the distribution of $\epsilon_{B}$ upper limits we attained from X-ray data, the $\epsilon_{B}$ measurements from optical data show a much wider distribution. For a constant density (wind) medium, the $\epsilon_{B}$ measurements range from $\epsilon_{B}\sim 10^{-8}-10^{-3}$ ($\epsilon_{B}\sim 10^{-9}-10^{-3}$). Also, since we used the same equation (Equation 7) to find both the upper limits on $\epsilon_{B}$ with X-ray data and the $\epsilon_{B}$ measurements with optical data, the discussion at the end of Section 3.5 on how the uncertainty in the afterglow parameters and the parameters $\xi$ and $f$ can affect the distribution of $\epsilon_{B}$ upper limits also applies to the distributions of $\epsilon_{B}$ measurements we presented in this section. In addition, since $\xi$ and $f$ are less than unity, including these two parameters will mean that our $\epsilon_{B}$ measurements are effectively upper limits on $\epsilon_{B}$. ### 4.5. Comparison Of Our Results On $\epsilon_{B}$ To Previous Studies We performed a KS test between our optical $\epsilon_{B}$ measurements and the results from previous studies on $\epsilon_{B}$ (Figure 1). For our $\epsilon_{B}$ results, we used the optical $\epsilon_{B}$ measurements with $n=1\mbox{ cm}^{-3}$ and $p$ determined from $\alpha_{O}$ (filled-in histogram in Bottom-Right panel of Figure 3; the $\epsilon_{B}$ values are shown in Table LABEL:optical_properties_table). The result of the KS test is that the null hypothesis is rejected. The P-value, which measures the probability that the null hypothesis is still true, is $2.1\times 10^{-9}$. This result shows that the rejection of the null hypothesis is statistically significant. It is not surprising that the null hypothesis was rejected. The distribution from the previous studies is very inhomogeneous, with the values for $\epsilon_{B}$ being drawn from many different studies with different methodologies. Also, comparing the histogram in Figure 1 to the filled-in histogram in the Bottom- Right panel of Figure 3, a couple of significant differences can be seen. The range for the histogram of $\epsilon_{B}$ values found in the literature is $\sim 10^{-5}-10^{-1}$, whereas the range for our $\epsilon_{B}$ results is $\sim 10^{-8}-10^{-3}$. The mean and median values for these two histograms are also significantly different. The mean (median) value for the $\epsilon_{B}$ histogram from the literature, $6.3\times 10^{-2}$ ($1.4\times 10^{-2}$), is a factor $\sim 700$ ($\sim 600$) times larger than the mean (median) $\epsilon_{B}$ value of the histogram with our results, which is $9.5\times 10^{-5}$ ($2.4\times 10^{-5}$). One assumption that is commonly made in afterglow modeling studies is equipartition between $\epsilon_{e}$ and $\epsilon_{B}$. As we discussed in Section 2, the results for $\epsilon_{e}$ from the literature and the results from recent simulations of relativistic collisionless shocks support $\epsilon_{e}\sim 0.2$. From this result, many works assume $\epsilon_{B}\sim 10^{-2}-10^{-1}$. However, there is no physical argument to expect equipartition. Our distribution of $\epsilon_{B}$ upper limits and measurements, although wide, supports that there is no equipartition between electron and magnetic energies because none of the $\epsilon_{B}$ upper limits or measurements in our samples has a value larger than $\epsilon_{B}\sim\mbox{ few }\times 10^{-3}$. Another source of error that can lead to differences in $\epsilon_{B}$ values is differences in the determination of the spectral regime for the optical band. We took it to be between $\nu_{i}$ and $\nu_{c}$, but it is also possible for the optical band to be above $\nu_{c}$ at late times (e.g. Panaitescu & Kumar, 2002; Cenko et al., 2010). Another source for error is energy injection. We did not consider energy injection as a source of error because only $3/35$ of the bursts in our optical sample show plateaus (and these plateaus are short). The X-ray and optical light curves of many bursts show plateaus and in these cases energy injection needs to be considered. Also, errors in our determination of fluxes and times from X-ray and optical light curves can also lead to small errors in $\epsilon_{B}$. In summary, the main assumption we made when determining $\epsilon_{B}$ is an efficiency of $\sim 20\%$ in the conversion of kinetic energy to gamma-ray energy, and we did not assume equipartition between $\epsilon_{e}$ and $\epsilon_{B}$. Different authors have made different assumptions that can have a large effect on the results for $\epsilon_{B}$. Lastly, for a few bursts, we checked if our method of determining $\epsilon_{B}$ is consistent with the values determined for $\epsilon_{B}$ with other techniques. GRBs 980519 and 990123, discussed in the afterglow modelling study of Panaitescu & Kumar 2002, have optical light curves that decline as a power-law before the jet-break. The optical band for both of these bursts was determined to be in the spectral regime $\nu_{i}<\mbox{ 2 eV}<\nu_{c}$. Applying our technique to find a $\epsilon_{B}$ measurement for both of these bursts and using the value of $n$ reported in Panaitescu & Kumar 2002 for both of these bursts, we find that these bursts have $\epsilon_{B}\sim 10^{-5}$, consistent with the results reported in Panaitescu & Kumar 2002 for both of these bursts within a factor of a few. The small differences in $\epsilon_{B}$ values can be accounted for by differences in the coefficients used for the external-forward shock flux. ## 5\. GRBs In Common To Both Our X-ray And Optical Samples 14 bursts we considered are both in our X-ray and in our optical sample (GRB number is in bold in the first columns of Tables LABEL:x-ray_properties_table and LABEL:optical_properties_table). In this section, we verify for these bursts that the X-ray $\epsilon_{B}$ upper limit is above the optical $\epsilon_{B}$ measurement. For the optical data, we will use the $\epsilon_{B}$ measurements with $p$ determined from $\alpha_{O}$ and $n=1\mbox{ cm}^{-3}$ (filled-in histogram in the Bottom-Right panel of Figure 3). For the X-ray data, in this section, we will also consider $n=1\mbox{ cm}^{-3}$ ($s=0$) and we will use the value of $p$ determined from $\alpha_{O}$ 13131310 out of 14 of these bursts have optical data before 1000 seconds. For these 10 bursts, we can use the optical data to check if they satisfy the assumption we made in Section 3.1, $t_{\mathrm{dec}}<t_{\mathrm{EoSD}}$. 9 of these 10 GRBs do satisfy this assumption; for the remaining GRB (080721), we are not able to check this assumption because the first optical observation (at 100 sec.) is after the end of the steep decline ($t_{\mathrm{EoSD}}=70$ sec).. The comparison between the X-ray $\epsilon_{B}$ upper limits and the optical $\epsilon_{B}$ measurements is shown in Figure 4. This plot shows that all the X-ray $\epsilon_{B}$ upper limits are above the optical $\epsilon_{B}$ measurements. Figure 4.— Comparison of the $\epsilon_{B}$ upper limits from X-ray data to the $\epsilon_{B}$ measurements from optical data. The 14 dots correspond to the 14 GRBs that are both in our X-ray and optical samples. The straight line indicates where the $\epsilon_{B}$ measurements are equal to the $\epsilon_{B}$ upper limits. ## 6\. $E$ vs. $\epsilon_{B}$ Correlation? We now use the $\epsilon_{B}$ measurements from our optical sample to determine if there is a correlation between $E$ and $\epsilon_{B}$. The 3 bursts studied in Kumar & Barniol Duran 2009, 2010 have small values of $\epsilon_{B}$, consistent with shock compression of a seed magnetic field $B_{0}\sim\mbox{ few}\times 10\mu\mbox{G}$. One property that distinguishes these 3 GRBs is that they were particularly energetic, with $E_{\mathrm{iso}}^{\gamma}\sim 10^{53}-10^{54}\mbox{ ergs}$. Could the large energy intrinsic to these 3 bursts explain why these 3 bursts have low $\epsilon_{B}$ values? We investigate this possibility in Figure 5 by plotting the values of $E$ and $\epsilon_{B}$ for all the bursts in our optical sample. Figure 5.— We plot the values of $E$ and the measurements of $\epsilon_{B}$ to determine if they are correlated. The 35 points represent the GRBs in our optical sample and the straight line is the best fit line: $\mbox{log}_{10}(\epsilon_{B})=-1.02\mbox{log}_{10}(E_{53})-4.51$, with the slope of the line being $-1.02\pm 0.23$ and the y-intercept of the line being $-4.51\pm 0.16$. The correlation coefficient of the fit is 0.62 and the P-value of the correlation is $1.2\times 10^{-4}$ (3.8$\sigma$ significance). The $\epsilon_{B}$ measurements are for $n=1\mbox{ cm}^{-3}$ and $p$ determined from $\alpha_{O}$ (shown in the filled-in histogram in the Bottom- Right panel of Figure 3 and Table LABEL:optical_properties_table) and the values of $E$ were determined by assuming an efficiency $\sim 20\%$ for all the GRBs in our optical samples. For the energy of each burst, we assumed an efficiency of $\sim 20\%$ ($E=5E^{\gamma}_{\mathrm{iso}}$) in the conversion of the kinetic energy of the jet to gamma-ray radiation; the observables involved in calculating $E$ are the gamma-ray fluence and $z$. For the $\epsilon_{B}$ measurements, we used the values with $p$ determined from $\alpha_{O}$ and $n=1\mbox{ cm}^{-3}$. The observables involved in determining $\epsilon_{B}$ are the observed specific optical flux and the time. In Figure 5, we also show the best fit line141414From Equation 7, it may be expected that $E$ and $\epsilon_{B}$ are correlated. This is not necessarily true because each burst has a different value for the observed quantities $f_{\nu}$ and $t$. In addition, since $p$ was determined from $\alpha_{O}$, each burst has a different $p$. We also checked if $E$ and $\epsilon_{B}$ were correlated when assuming a fixed $p=2.4$. From Equation 9, we would expect the slope of the line to be $-1.59$. The best fitting line for $p=2.4$ is $\mbox{log}_{10}(\epsilon_{B})=-0.77\mbox{log}_{10}(E_{53})-4.53$, with the slope of the line being $-0.77\pm 0.24$ and the y-intercept being $-4.53\pm 0.16$. The slope of this line is more than $3\sigma$ away from $-1.59$, showing that there is no expected correlation between $E$ and $\epsilon_{B}$ for our methodology of determining $\epsilon_{B}$.. This fit shows that an increase in $E$ leads to a decrease in $\epsilon_{B}$. The correlation coefficient of the fit is $0.62$, indicating that there is a weak correlation between $E$ and $\epsilon_{B}$. A value of the correlation coefficient close to 1 would indicate a strong correlation. The P-value of the correlation is $1.2\times 10^{-4}$ (3.8$\sigma$ significance), indicating that there is a small probability that the correlation occurred by chance. Although many points show large deviations from the best fit line, the scatter of the points may be reduced or increased by the uncertainty in the afterglow parameters. An error in the efficiency would affect the values of $E$ and an error in $\epsilon_{e}$, the efficiency, or $n$ would affect the $\epsilon_{B}$ measurements (see Section 3.5 for a discussion on how the $\epsilon_{B}$ measurements would be affected when an error in a afterglow parameter is made). It is possible that the uncertainty in $E$ and $\epsilon_{B}$ can reduce or increase the scatter and make the correlation between $E$ and $\epsilon_{B}$ stronger or weaker. ## 7\. Magnetic Field Amplification Factor For X-ray And Optical Results In Sections 3-5, we presented our results for the strength of the magnetic field downstream of the shock front in terms of the afterglow parameter $\epsilon_{B}$. If shock compression was the only mechanism amplifying the ambient magnetic field (assuming a standard $B_{0}\sim\mbox{few }\mu\mbox{G}$ and a standard $n=1\mbox{ cm}^{-3}$), then $\epsilon_{B}\sim 10^{-9}$ is expected. Most of the bursts in our distributions of $\epsilon_{B}$ upper limits and measurements have values larger than $\epsilon_{B}\sim 10^{-9}$. These results suggest that amplification of the magnetic field, in addition to shock compression, is needed to explain the afterglow observations. In this section, we will present our results in terms of an amplification factor, which quantifies the amplification that is needed, beyond shock compression, to explain the observations. If shock compression were the only mechanism amplifying the seed magnetic field $B_{0}$, then $B=4\Gamma B_{0}$. To quantify how much additional amplification of the ambient magnetic field is needed, beyond shock compression, we define the amplification factor, $AF$, as $AF\equiv\frac{B}{4B_{0}\Gamma}.$ (12) $AF$ is a constant that satisfies $AF\geq 1$ since $B\geq 4B_{0}\Gamma$. $AF=1$ means than the observed $B$ is consistent with the only amplification arising from seed magnetic field shock compression. The expression for $\epsilon_{B}$ is $\epsilon_{B}=B^{2}/32\pi m_{p}c^{2}n\Gamma^{2}$. With the definition for $AF$, $\epsilon_{B}$ is $\epsilon_{B}=(AF)^{2}\times\frac{B_{0}^{2}}{2\pi nm_{p}c^{2}}.$ (13) We note that $\epsilon_{B}$ is given by $(AF)^{2}$ times the $\epsilon_{B}$ we would get if shock compression were the only mechanism amplifying the magnetic field. We will now use Equation 13 and our previous results for the X-ray $\epsilon_{B}$ upper limits to determine an upper limit on $AF$. In Section 3.5, if we assumed a standard $n=1\mbox{ cm}^{-3}$, we were able to attain an upper limit on $\epsilon_{B}$ for a constant density medium 151515We will only consider a constant density medium when displaying the results for the amplification factor. We will show in Equation 15 that $AF$ has a weak dependence on the density.. We will refer to these $\epsilon_{B}$ upper limits as $\overline{\epsilon_{B}}(p,n_{0}=1)$. In the notation $\overline{\epsilon_{B}}(p,n_{0}=1)$, the bar over $\epsilon_{B}$ signifies that this is an upper limit on $\epsilon_{B}$, the $p$ in the parenthesis shows that the $\epsilon_{B}$ upper limit depends on the value of $p$ we used, and the $n_{0}=1$ shows that we assumed $n=1\mbox{ cm}^{-3}$. With this notation, we can keep the dependence of the $\epsilon_{B}$ upper limit on $n$ (see Equation 7): $\epsilon_{B}<\frac{\overline{\epsilon_{B}}(p,n_{0}=1)}{n_{0}^{2/(p+1)}}.$ (14) Combining Equation 13 and Equation 14, the expression for the upper limit on $AF$ is $AF<\frac{1.0\times 10^{4}[\overline{\epsilon_{B}}(p,n_{0}=1)]^{1/2}}{B_{0,10\mu\mathrm{G}}n_{0}^{(1-p)/(2p+2)}},$ (15) where $B_{0,10\mu\mathrm{G}}\equiv B_{0}/10\mu\mbox{G}$. One advantage to expressing the results of the magnetic field downstream of the shock front in terms of $AF$ is that $AF$ depends weakly on $n$. For a standard $p=2.4$, $AF\propto n^{0.21}$. On the other hand, there is a strong dependence on $B_{0}$, $AF\propto B_{0}^{-1}$. ### 7.1. Amplification Factor Upper Limit For Our X-ray Sample We will now show the results for the $AF$ upper limits we obtained from Equation 15. Since the amplification factor has a weak dependence on the density, we will assume a standard $n_{0}=1$ when displaying the results for the $AF$ upper limits. When plotting the results for $AF$, we will keep the dependence on $B_{0}$ and plot the quantity $(AF)B_{0,10\mu\mathrm{G}}$. In the left and right panels of Figure 6, we show the upper limits on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for a fixed $p=2.4$ and $p=2.2$, 2.8, where the values of $\overline{\epsilon_{B}}(p,n_{0}=1)$ used in Equation 15 were shown in Top-Right, Top-Left, and Bottom panels of Figure 2, respectively. For the remainder of this section, we will assume $B_{0}=10\mu\mbox{G}$ when discussing the results for the $AF$ upper limits. The mean and median values of the $AF$ upper limits are summarized in Table 5. The mean (median) $AF$ upper limits range from $AF\sim 60-80$ ($AF\sim 40-60$). The $AF$ upper limit histograms show a wide distribution, with a range of $\sim 10$ to $\sim 300$. To determine if assuming a different value of $p$ has a significant effect on the distribution of $AF$ upper limits, we performed a KS test between the histograms in the right panel of Figure 6. The KS test confirmed the null hypothesis, leading us to conclude that the $AF$ upper limit results are not sensitive to the value of $p$ we assume. X-ray \| \| \| \| ---\|---\|---\|---\|--- $(s=0)$ \| $p=2.2$ \| $p=2.4$ \| $p=2.8$ \| Mean \| $84$ \| $67$ \| $62$ \| Median \| $63$ \| $51$ \| $44$ \| Opt. \| \| \| \| $(s=0)$ \| $p=2.2$ \| $p=2.4$ \| $p=2.8$ \| $p$ from $\alpha_{\mathrm{O}}$ Mean \| $130$ \| $71$ \| $36$ \| $71$ Median \| $100$ \| $56$ \| $23$ \| $48$ Table 5Mean and median $AF$ values for the X-ray (upper limits on $AF$) and optical (measurements of $AF$) histograms shown in Figures 6 and 7. All the labels are the same as in Table 3. A constant density medium with $n=1\mbox{ cm}^{-3}$ (the amplification factor has a weak dependence on the density, see Section 7 and Equation 15) and a seed magnetic field $B_{0}=10\mu\mbox{G}$ were assumed for all the bursts in our X-ray and optical samples. $\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{amplification_factor_x_ray_p_2_4.eps}&\includegraphics[scale={0.6}]{amplification_factor_x_ray_p_2_2_p_2_8.eps}\end{array}\end{array}$ Figure 6.— Left: Upper limits on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for our X-ray sample assuming $p=2.4$. Right: Upper limits on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for our X-ray sample assuming $p=2.2$ and $p=2.8$. A fixed $n=1\mbox{ cm}^{-3}$ was assumed for all of the histograms (the precise value of $n$ is unimportant since $AF$ has a weak dependence on $n$, see Section 7 and Equation 15). We now discuss how an error in each of the afterglow parameters can affect our results for the $AF$ upper limits. For this discussion, we will assume $p=2.4$. From Equation 15, $\mbox{AF}\propto\sqrt{\overline{\epsilon_{B}}(p=2.4,n_{0}=1)}B_{0}^{-1}n^{0.2}$ and from Equation 9, $\epsilon_{B}\propto\epsilon_{e}^{-1.6}E^{-1.6}n^{-0.6}$. From these two expressions, $\mbox{AF}\propto\epsilon_{e}^{-0.8}E^{-0.8}n^{0.2}B_{0}^{-1}$. We note that compared to the $\epsilon_{B}$ upper limit (Equation 9), the $AF$ upper limit has a weaker dependence on $\epsilon_{e}$, $E$, and $n$. As we discussed at the end of Section 3.5, a likely error in $\epsilon_{e}$ is a factor of $\sim 2$; this error in $\epsilon_{e}$ will translate into an error in the $AF$ upper limits by a factor of only $\sim 2$. For the energy, we assumed an efficiency $\sim 20\%$ and a likely error in the efficiency is a factor $\sim 2-3$ (see Section 3.5); this error in the efficiency would lead to an error in the $AF$ upper limits by only a factor $\sim 2$. One advantage to expressing the results of the magnetic field downstream of the shock front in terms of $AF$ is that $AF$ has a very weak dependence on $n$. An error in $n$ by a factor $\sim 10^{3}$ (see Section 3.5) from our assumed $n=1\mbox{ cm}^{-3}$ will only lead to an error in the $AF$ upper limits by a factor $\sim 4$. The price to pay for a weak $n$ dependence is a linear dependence on $B_{0}$, with $\mbox{AF}\propto B_{0}^{-1}$. $B_{0}$ is an uncertain parameter that likely varies from GRB environment to GRB environment and it is the largest source of uncertainty for $AF$. ### 7.2. Amplification Factor Measurement For Our Optical Sample As we discussed in Section 4.2, for our optical sample, we found a measurement for $\epsilon_{B}$ instead of an upper limit. This will allow us to determine a measurement for $AF$. To do this, we will use Equation 15, but in this case we have an equality instead and we have $\epsilon_{B}(p,n_{0}=1)$ instead of $\overline{\epsilon_{B}}(p,n_{0}=1)$. The notation $\epsilon_{B}(p,n_{0}=1)$ denotes the $\epsilon_{B}$ measurements for our optical sample from Section 4.4 if we assume a standard $n=1\mbox{ cm}^{-3}$. Also, as with the X-ray sample, we only consider $s=0$ when calculating the $AF$ measurements and assume a fixed $n=1\mbox{ cm}^{-3}$. In the left panel of Figure 7, we show the results for the measurements on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for $p$ determined from $\alpha_{O}$ and also assuming $p=2.4$. In the right panel of Figure 7, we show the measurements on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for $p=2.2$ and $p=2.8$. For the remainder of this section, we will assume $B_{0}=10\mu\mbox{G}$ when discussing the results for the $AF$ measurements. A summary of the mean and median $AF$ measurements for our optical sample is shown in Table 5. To determine if assuming a standard $p=2.4$, as opposed to determining $p$ from $\alpha_{O}$ for each burst, has a statistically significant effect on the distribution of $AF$ measurements, we performed a KS test between the two histograms in the left panel of Figure 7. The KS test confirmed the null hypothesis. The mean (median) $AF$ measurements for the optical histograms range from $\sim 40$ to $\sim 130$ ($\sim 20$ to $\sim 100$). Compared to the $AF$ upper limit histograms, the $AF$ measurement histograms show a wider distribution, ranging from $AF\sim 1$ to $AF\sim 1000$. Also, since we used the same expression to determine the $AF$ upper limits and measurements (Equation 15), the discussion at the end of Section 7.1 on how an error in one of the afterglow parameters can affect the $AF$ upper limits also applies to the $AF$ measurements161616$\xi$ and $f$ also affect our $AF$ results. To account for $\xi$, since the $\epsilon_{B}$ upper limit$/$measurement is $\propto\xi^{4(p-2)/(p+1)}$ and the $AF$ upper limit$/$measurement is $\propto(\overline{\epsilon_{B}}(p,n_{0}=1))^{1/2}$, $AF\propto\xi^{2(p-2)/(p+1)}$. To account for $f$, since $AF\propto(\overline{\epsilon_{B}}(p,n_{0}=1))^{1/2}$, $AF\propto f^{1/2}$. Thus, including $\xi$ and $f$ will decrease the values of the $AF$ upper limits$/$measurements. Taking the lowest possible value for $f$, the $AF$ upper limits$/$measurements can decrease by up to a factor $\sim 40$. This would make $\sim 50\%$($\sim 60\%$) of the bursts in our X-ray (optical) sample consistent with shock compression. In addition, as with the $\epsilon_{B}$ measurements, since $\xi$ and $f$ are less than unity, including these two parameters will mean that our $AF$ measurements will become upper limits on $AF$.. $\begin{array}[]{c}\begin{array}[]{cc}\includegraphics[scale={0.6}]{amplification_factor_optical_p_temporal_p_2_4.eps}&\includegraphics[scale={0.6}]{amplification_factor_optical_p_2_2_p_2_8.eps}\end{array}\end{array}$ Figure 7.— Left Panel: The filled-in (un-filled) histogram shows the measurements on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for $p$ calculated from $\alpha_{O}$ ($p=2.4$). Right Panel: The filled-in (un-filled) histogram shows the measurements on the quantity $(AF)B_{0,10\mu\mathrm{G}}$ for $p=2.2$ ($p=2.8$). A fixed $n=1\mbox{ cm}^{-3}$ was assumed for all of the histograms (the precise value of $n$ is unimportant since $AF$ has a weak dependence on $n$, see Section 7 and Equation 15). ## 8\. Discussion And Conclusions In this work, we presented a systematic study on the magnetic field downstream of the shock front for large samples of GRBs (60 in our X-ray sample and 35 in our optical sample). We expressed the strength of the downstream magnetic field in terms of both the afterglow parameter $\epsilon_{B}$ and a amplification factor, denoted by $AF$, which quantifies how much amplification, beyond shock compression of the seed magnetic field, is needed to explain the downstream magnetic field. This is the first time a large and systematic study has been carried out to study $\epsilon_{B}$ and to determine how much amplification of the seed magnetic field is required by the observations. For our X-ray (optical) sample, we determined an upper limit (measurement) for both $\epsilon_{B}$ and $AF$. The upper limits on $\epsilon_{B}$ and $AF$ for our X-ray sample were found from the constraint that the observed flux at the end of the steep decline is greater than or equal to the external-forward shock flux. This is a new method to constrain $\epsilon_{B}$ that relies on the steep decline emission, which has been observed by Swift for many GRBs. Our optical sample was restricted to light curves that decline with $\alpha\sim 1$ from the early times $\sim 10^{2}-10^{3}$ sec, as expected for the external-forward shock emission. We found the measurements for $\epsilon_{B}$ and $AF$ for our optical sample from the condition that the observed flux is equal to the external-forward shock flux. The condition used for our X-ray (optical) sample was converted into an upper limit (measurement) on the quantity $\epsilon_{B}n^{2/(p+1)}$ for $s=0$ or $\epsilon_{B}A_{-1}^{4/(p+1)}$ for $s=2$ by assuming a $\sim 20\%$ efficiency in the conversion of kinetic energy to prompt gamma-ray radiation. To find an upper limit (measurement) on $\epsilon_{B}$ for our X-ray (optical) sample, we assumed a standard $n=1\mbox{ cm}^{-3}$ ($A_{}=0.1$) for the density for a constant density (wind) medium. A discussion on how the uncertainty in the afterglow parameters affects our results for $\epsilon_{B}$ can be found at the end of Section 3.5. The largest source of uncertainty for our results on $\epsilon_{B}$ is the density, since the value of the density has been observed to vary over many orders of magnitude and its precise value is not known for each GRB (see Section 3.2). For the bursts that are both in our X-ray and optical samples, we also applied a consistency check to make sure our results for $\epsilon_{B}$ are correct (see Section 5). From Table 3, for a constant density (wind) medium, most of the $\epsilon_{B}$ upper limit and measurement histograms have a median value $\sim\mbox{ few}\times 10^{-5}$ ($\sim\mbox{ few}\times 10^{-6}$). These results imply that half of the bursts in both our X-ray and optical samples have a $\epsilon_{B}$ value $\sim\mbox{ few}\times 10^{-5}$ or lower. Assuming $n=1\mbox{ cm}^{-3}$ and $B_{0}\sim\mbox{few}\times\mu$G, shock compression is only able to produce $\epsilon_{B}\sim 10^{-9}$. Although $\epsilon_{B}\sim 10^{-9}$ is 4 orders of magnitude lower than $\epsilon_{B}\sim 10^{-5}$, $\epsilon_{B}\sim 10^{-5}$ is smaller by $\sim 2-4$ orders of magnitude compared to the majority of previously reported $\epsilon_{B}$ values (Figure 1), which are $\epsilon_{B}\sim 10^{-3}-10^{-1}$. Assuming $B_{0}\sim 10\mu\mbox{G}$, $\epsilon_{B}\sim\mbox{ few}\times 10^{-5}$ corresponds to $AF\sim 50$ (Equation 15). Our result of a median $\epsilon_{B}\sim\mbox{ few}\times 10^{-5}$ shows that the majority of the bursts in our X-ray and optical samples only require a weak amplification beyond shock compression, by a factor $\sim 50$ or lower. The near equipartition $\epsilon_{B}\sim 0.01-0.1$ determined near the shock front by theoretical studies and Particle-In-Cell (PIC) simulations (Medvedev & Loeb, 1999; Chang et al., 2008; Martins et al., 2009; Keshet et al., 2009; Lemoine, 2013; Sironi et al., 2013) stands in contrast with our median results of $\epsilon_{B}\sim\mbox{ few}\times 10^{-5}$. PIC simulations of relativistic collisionless shocks performed by Chang et al. 2008 and Keshet et al. 2009 found that the magnetic field generated near the shock front decays with distance downstream of the shock front. Lemoine 2013 and Lemoine et al. 2013 studied the effects that this decaying magnetic field has on the shock accelerated electrons radiating afterglow emission downstream of the shock front. The main effect is that electrons with different Lorentz factors cool in regions with different magnetic fields, with the higher (lower) energy photons being emitted by electrons that are closer (further) from the shock front (Lemoine et al., 2013). Considering the decay of the downstream magnetic field, Lemoine et al. (2013) modelled the afterglow data of 4 GRBs that have extended emission at energies $>100\mbox{ MeV}$ (detected by Fermi-LAT) and also X-ray, optical, and radio data. Their afterglow modelling results for the X-ray, optical, and radio data found $\epsilon_{B}\sim 10^{-6}-10^{-4}$, consistent with our results for the median $\epsilon_{B}$ upper limits and measurements attained from X-ray and optical data. Our next main result relates to the distribution of $\epsilon_{B}$ values. One property the $\epsilon_{B}$ values from the literature shared with our optical $\epsilon_{B}$ measurements is that they both show a wide distribution. The literature compilation (Figure 1) showed $\epsilon_{B}\sim 10^{-5}-10^{-1}$ and our optical $\epsilon_{B}$ measurement histograms showed an even wider distribution, ranging from $\epsilon_{B}\sim 10^{-8}-10^{-3}$ ($\epsilon_{B}\sim 10^{-9}-10^{-3}$) for a constant density (wind) medium. One possibility we investigated to explain the wide distribution of $\epsilon_{B}$ values is whether bursts with smaller $\epsilon_{B}$ values are more energetic than bursts with larger $\epsilon_{B}$ values (Figure 5). Although the bursts in our optical sample did show the trend that bursts with larger $E$ have a smaller $\epsilon_{B}$, the correlation was weak, with a correlation coefficient of $0.62$ and a P-value of $1.2\times 10^{-4}$ (3.8$\sigma$ significance). Another possibility to explain the wide distribution of $\epsilon_{B}$ relates to the uncertainty in the environmental parameters ($B_{0}$ and $n$) in the medium surrounding GRBs. One possibility to explain bursts with values of $\epsilon_{B}\sim 10^{-5}-10^{-3}$, under the interpretation of weak amplification beyond shock compression, is that they occurred in environments with particularly high seed magnetic fields. Since $\epsilon_{B}\propto B_{0}^{2}$, an increase in $B_{0}$ by an order of magnitude will lead to an increase in $\epsilon_{B}$ by two orders of magnitude. A weak amplification beyond shock compression of $AF\sim 50$ (as inferred for the bursts with $\epsilon_{B}\sim\mbox{ few}\times 10^{-5}$) and a strong $B_{0}\sim 10\mu\mbox{G}-100\mu\mbox{G}$ yields $\epsilon_{B}\sim 10^{-5}-10^{-3}$ (see Equation 15). We note that for this estimate we assumed $n=1\mbox{ cm}^{-3}$, but as we discussed in Section 7, the amplification factor has a weak dependence on the density. This possibility, that the larger values of $\epsilon_{B}$ may be explained by bursts going off in environments with particularly strong $B_{0}$, will be discussed further in Barniol Duran 2013 (in preparation). In addition to many bursts in our optical sample having particularly large $\epsilon_{B}\sim 10^{-5}-10^{-3}$, there were also some bursts in our optical sample that have particularly low $\epsilon_{B}$ values. For $p$ determined from $\alpha_{O}$ and $n=1\mbox{ cm}^{-3}$ (Bottom-right panel of Figure 3), GRBs 071025 and 071112C have $\epsilon_{B}\sim 10^{-7}$ and GRB 080607 has $\epsilon_{B}\sim 10^{-8}$. Assuming $n=1\mbox{ cm}^{-3}$ and $B_{0}=10\mu\mbox{G}$, these bursts with $\epsilon_{B}\sim 10^{-8}-10^{-7}$ are consistent with shock compression of a seed magnetic field of a few 10 $\mu\mbox{G}$ being the only amplification necessary to explain the observations. Lastly, we mention that a similar conclusion for radio observations of supernova remnants (SNRs) was reached in Thompson et al. (2009). Like GRB afterglow emission, SNR emission arises from a blastwave interacting with the surrounding medium, but at non-relativistic speeds. Thompson et al. (2009) found that starburst galaxies have strong ambient magnetic fields $\sim 1$ mG and they concluded that shock compression of this strong seed magnetic field is enough to explain the radio emission from SNRs. On the other hand, for normal spiral galaxies with ambient magnetic fields $\sim 5-10\mu$G, they concluded that additional amplification beyond shock compression by a factor $\sim 3-7$ was necessary to explain the radio emission from SNRs. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. 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For the first GRBs with high quality afterglow data, different works (e.g. Wijers & Galama 1999; Chevalier & Li 1999, 2000; Li & Chevalier 2001; Harrison et al. 2001; Yost et al. 2003) have presented afterglow modeling studies on the same GRBs. In Table LABEL:epsilon_e_and_epsilon_B_table, for these early afterglow studies, we show the results from Panaitescu & Kumar 2001a, b; Panaitescu & Kumar 2002 since they have the largest afterglow modeling compilations. It is important to note that in some cases, different groups determined significantly different values for the afterglow parameters (e.g. GRB 970508 Wijers & Galama 1999; Chevalier & Li 2000; Frail et al. 2000; Panaitescu & Kumar 2002, GRB 000418 Berger et al. 2001; Panaitescu & Kumar 2002). Table 6$\epsilon_{e}$ and $\epsilon_{B}$ Values From The Literature GRB \| $\epsilon_{e}$ \| $\epsilon_{B}$ \| Ref. ---\|---\|---\|--- 970508 \| $0.62$ \| $0.10$ \| [1] 980329 \| $0.12$ \| $0.17$ \| [2] 980519 \| $0.25$ \| $(3.5^{+32}_{-2.3})\times 10^{-5}$ \| [1] 980703 \| $0.14$ \| $4.6\times 10^{-4}$ \| [3] 990123 \| $0.59$ \| $(7.4^{+23}_{-5.9})\times 10^{-4}$ \| [1] 990510 \| $>0.3$ \| $6\times 10^{-3}$ \| [4] 991208 \| $0.32$ \| $2.1\times 10^{-2}$ \| [1] 991216 \| $0.4$ \| $2\times 10^{-2}$ \| [4] 000301C \| $0.4$ \| $7\times 10^{-2}$ \| [4] 000926 \| $0.35$ \| $(6.5^{+1.5}_{-1.1})\times 10^{-2}$ \| [1] 010222 \| $>0.3$ \| $2\times 10^{-4}$ \| [4] 011211 \| $0.22$ \| $5.0\times 10^{-4}$ \| [5] 020405 \| $0.1$ \| $0.3$ \| [6] 020813 \| \| $4.0\times 10^{-4}$ \| [5] 021004 \| $0.21$ \| $2\times 10^{-4}$ \| [7] 030226 \| $0.11$ \| $2.5\times 10^{-4}$ \| [5] 030329 \| $0.16$ \| $0.10$ \| [8] 050416A \| $0.2-0.333$ \| $0.072-0.333$ \| [9] 050820A \| $0.14^{+0.02}_{-0.01}$ \| $0.013^{+0.003}_{-0.001}$ \| [10] 050904 \| $0.02$ \| $0.015$ \| [11] 051022 \| $0.0681^{+0.3951}_{-0.0348}$ \| $(8.02^{+28.18}_{-7.17})\times 10^{-3}$ \| [12] 051221A \| $0.24-0.333$ \| $0.12-0.333$ \| [13] 060418 \| $0.06^{+0.01}_{-0.02}$ \| $0.15^{+0.14}_{-0.01}$ \| [10] 070125 \| $0.27^{+0.03}_{-0.01}$ \| $0.0277^{+0.0044}_{-0.0075}$ \| [14] 080129 \| $0.4$ \| $5\times 10^{-2}$ \| [15] 080319B \| $0.11\pm 0.01$ \| $0.33$ \| [10] 080928 \| $0.165$ \| $(2.5^{+16}_{-2.4})\times 10^{-4}$ \| [16] 090323 \| $0.070^{+0.005}_{-0.005}$ \| $0.0089^{+0.0007}_{-0.0018}$ \| [17] 090328 \| $0.11^{+0.06}_{-0.01}$ \| $0.0019^{+0.0004}_{-0.0008}$ \| [17] 090423 \| $0.28$ \| $1.6\times 10^{-4}$ \| [18] Table 6$\epsilon_{e}$ and $\epsilon_{B}$ Values From The Lit. (Continued)	arxiv-papers	2013-09-12T20:00:03	2024-09-04T02:49:50.843720	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rodolfo Santana, Rodolfo Barniol Duran, Pawan Kumar", "submitter": "Rodolfo Santana", "url": "https://arxiv.org/abs/1309.3277" }
1309.3294	# A Short Proof that Minimal Sets of Planar Ordinary Differential Equations are Trivial Ido Bright Department of Applied Mathematics, University of Washington. ###### Abstract. We present a short proof, relaying on the divergence theorem, verifying that minimal sets in the plane are trivial. ‘ ## 1\. Introduction We consider the ordinary differential equation in the plane defined by (1.1) $\frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x}),$ where $\mathbf{f}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is a locally Lipschitz function. (Although, we only use uniqueness with respect to initial conditions of (1.1) and the continuity of $\mathbf{f}$.). A minimal set is a nonempty closed invariant set, which is minimal with respect inclusions. A trivial minimal set is a set that is the image if either a stationary solution or a periodic solution. We present a new short proof of the following well known result. ###### Theorem 1. Any minimal set of (1.1) either corresponds to a stationary solution or to the image of a periodic solution, namely, all minimal sets are trivial. The text-book proof of this theorem relays on the Poincaré–Bendixson theorem, and employs dynamical arguments. The proof presented in this paper relays on a different argument, relaying on a property of the velocity of Jordan curves. This idea was introduced in [1] and further developed in [2, 3]. ## 2\. Proof of Main Result In the proof of the main result we use the following notation. The 2-dimensional euclidean space is denoted by $\mathbb{R}^{2}$, and the norm of a vector $\mathbf{y}\in\mathbb{R}^{2}$ is denoted by $\left\|\mathbf{y}\right\|$. The open ball in $\mathbb{R}^{2}$, centered at $\mathbf{y}$ with radius $r$, is denoted by $B\left(\mathbf{y},r\right)$. The closure of an open set $O\subset\mathbb{R}^{2}$ is denoted by $\bar{O}$, its boundary by $\partial O$, and its exterior normal and tangent vector at the point $\mathbf{y}\in\partial O$ are denoted by $\mathbf{N}_{\partial O}\left(\mathbf{y}\right)$ and $\mathbf{T}_{\partial O}\left(\mathbf{y}\right)$, respectively. We shall use the following results that are well known in the smooth case. ###### Lemma 2. Let $O\subset\mathbb{R}^{2}$ be a bounded open set with rectifiable boundary. Then $\mathbf{v}=\int_{\partial O}\mathbf{N}_{\partial O}\left(\mathbf{y}\right)d\mathbf{y}=\mathbf{0}\in\mathbb{R}^{2}.$ ###### Proof. Assume in contradiction that $\mathbf{v}\neq\mathbf{0}$, and set $\mathbf{g}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ by $\mathbf{g}\equiv\frac{\mathbf{v}}{\left\|\mathbf{v}\right\|}$. The divergence theorem for sets of finite perimeter (see, e.g., [4]) implies that $\left\|\mathbf{v}\right\|=\frac{\mathbf{v}}{\left\|\mathbf{v}\right\|}\mathbf{v}=\left\|\int_{\partial O}\mathbf{g}\left(\mathbf{y}\right)\mathbf{N}_{\partial O}\left(\mathbf{y}\right)d\mathbf{y}\right\|=\left\|\int_{O}\nabla\cdot\mathbf{g}\left(\mathbf{y}\right)d\mathbf{y}\right\|=0,$ in contradiction. ∎ The following lemma appears in [5]. ###### Lemma 3. Suppose $I\subset\mathbb{R}$ is a bounded interval and $g:I\rightarrow\mathbb{R}$ is a Lipschitz function. Then for almost every $r\in\mathbb{R}$ the set $g^{-1}\left(r\right)=\left\\{t\in I\|g\left(t\right)=r\right\\}$ is finite. To prove the main theorem, let us now fix a minimal set $\Omega\subset\mathbb{R}^{2}$ and a solution $\mathbf{x}^{}\left(\cdot\right)$ of (1.1), defined on $[0,\infty)$, with trajectory contained in $\Omega$. We shall also use the following well known fact. ###### Lemma 4. For every $\mathbf{y_{0}}\in\Omega$ and $\delta>0$ there exists $t>s$ such that $\left\|\mathbf{x}^{}\left(t\right)-\mathbf{y_{0}}\right\|<\delta$. ###### Proof. Otherwise, suppose that the lamma does not hold for some $\mathbf{y_{0}},\delta$ and $s$. Then the curve $\mathbf{y}^{}\left(t\right)=\mathbf{x}^{}\left(s+t\right)$ is a solution of (1.1) with trajectory contained in $\Omega\backslash B\left(\mathbf{y_{0}},\delta\right)$ for a suitable $\delta>0$, in contradiction to the minimality of $\Omega$. ∎ If $\Omega$ is not a singleton we choose $D>0$ such that $\Omega\backslash B\left(\mathbf{x}^{}\left(0\right),3D\right)\neq\emptyset$ and apply the following construction: ###### Construction 5. Set $\delta_{0}=D$ and $t_{0}$ as the first time point where $\mathbf{x}^{}\left(\cdot\right)$ meets $\partial B\left(\mathbf{x}^{}\left(0\right),\delta_{0}\right)$. For $i=1,2,\dots$ do the following: 1. (1) Choose $\delta_{i}<\delta_{i-1}/2$ small enough, such that $\left\|\mathbf{x}^{}\left(0\right)-\mathbf{x}^{}\left(t\right)\right\|>\delta_{i}$ for all $t\in\left[t_{0},t_{i-1}\right]$. 2. (2) Set $t_{i}$ as the first time point after $t_{0}$ where the curve $\mathbf{x}^{}\left(\cdot\right)$ meets $\partial B\left(\mathbf{x}^{}\left(0\right),\delta_{i}\right)$. (Here we use Lemma 4). 3. (3) Starting from $\mathbf{x}^{}\left(t_{i}\right)$ follow the line connecting it to $\mathbf{x}^{}\left(0\right)$, until first meeting a point in $\mathbf{x}^{}\left(\left[0,t_{0}\right]\right)$. Let $\mathbf{x}^{}\left(s_{i}\right)$ be this point. 4. (4) Let $\gamma_{i}$ be the parametrized Jordan curve obtained by following the curve $\mathbf{x}^{}\left(\cdot\right)$ in the interval $\left[s_{i},t_{i}\right]$ and then the line connecting its endpoints, with velocity of norm $1$. ###### Lemma 6. If $t_{i}\rightarrow t^{}$ then $\mathbf{x}^{}\left(0\right)=\mathbf{x}^{}\left(t^{}\right)$, and $\mathbf{x}^{}\left(\cdot\right)$ is a periodic solution with image $\Omega$. ###### Proof. According to our construction $\left\|\mathbf{x}^{}\left(0\right)-\mathbf{x}^{}\left(t_{i}\right)\right\|=\delta_{i}<2^{-i}D$ for every $i$. Hence, by continuity $\mathbf{x}^{}\left(t^{}\right)=\mathbf{x}^{}\left(0\right)$, and $\mathbf{x}^{}\left(\cdot\right)$ is periodic. By the minimality of $\Omega$, the image of $\mathbf{x}^{}\left(\cdot\right)$ is $\Omega$. ∎ ###### Proof of Theorem 1. Clearly, $\Omega$ is a singleton if and only if it contains a point $\mathbf{y}\in\Omega$ such that $\mathbf{f}\left(\mathbf{y}\right)=\mathbf{0}$. In this case and when the condition of Lemma 6 holds, we are done. Thus, we assume that $\mathbf{f}$ does not vanish in $\Omega$ and that $t_{i}\rightarrow\infty$. Fix $\mathbf{y}_{0}\in\Omega$ such that $\left\|\mathbf{y}_{0}-\mathbf{x}^{}\left(0\right)\right\|>2D$. Using Lemma 3 we fix an arbitrary small ball $B=B\left(\mathbf{y}_{0},r_{0}\right)$, such that $r_{0}<D$, and that $\left\\{0\leq t\leq s\|\left\|\mathbf{x}^{}\left(t\right)-\mathbf{y_{0}}\right\|=r_{0}\right\\}$ is finite for every $s>0$. Note that this implies that for every $i$ the Jordan curve $\gamma_{i}$ intersects $\partial B$ at a finite number of points, and that the portion of $\gamma_{i}$ in $B$ corresponds to the trajectory $\mathbf{x}^{}\left(\cdot\right)$. For every $i$ we denote the interior of $\gamma_{i}$ by $O_{i}$, and, using the identity $\partial\left(O_{i}\cap B\right)\subset\left(\partial O_{i}\cap B\right)\cup\left(O_{i}\cap\partial B\right)\cup\left(\partial O_{i}\cap\partial B\right),$ we obtain, by Lemma 2, that $\mathbf{0}=\int_{\partial\left(O_{i}\cap B\right)}\mathbf{N}_{\partial\left(O_{i}\cap B\right)}\left(\mathbf{y}\right)d\mathbf{y}=\int_{\partial O_{i}\cap B}\mathbf{N}_{\partial O_{i}}\left(\mathbf{y}\right)d\mathbf{y}-\int_{O_{i}\cap\partial B}\mathbf{N}_{\partial B}\left(\mathbf{y}\right)d\mathbf{y},$ since $\partial O_{i}\cap\partial B$ has zero measure. This bounds (2.1) $\left\|\int_{\partial O_{i}\cap B}\mathbf{N}_{\partial O_{i}}\left(\mathbf{y}\right)d\mathbf{y}\right\|=\left\|\int_{O_{i}\cap\partial B}\mathbf{N}_{\partial B}\left(\mathbf{y}\right)d\mathbf{y}\right\|\leq\left\|\int_{O_{i}\cap\partial B}\left\|\mathbf{N}_{\partial B}\left(\mathbf{y}\right)\right\|d\mathbf{y}\right\|\leq 2\pi r_{0}.$ For each $i$ the set $\partial O_{i}\cap B$ contains a finite number of arcs, and applying a change of variable it is easy to see that $\int_{\partial O_{i}\cap B}\mathbf{N}_{\partial O_{i}}\left(\mathbf{y}\right)d\mathbf{y}=P_{i}\int_{\partial O_{i}\cap B}\mathbf{T}_{\partial O_{i}}\left(\mathbf{y}\right)d\mathbf{y}=P_{i}\int_{\left\\{t\leq t_{i}\|\mathbf{x}^{}\left(t\right)\in B\right\\}}\frac{d}{dt}\mathbf{x}^{}\left(t\right)dt,$ where $\mathbf{T}_{\partial O_{i}}$ is chosen to agree with the direction of $\gamma_{i}$, and $P_{i}$ is a $\frac{\pi}{2}$-rotation matrix. Here we use the fact that the portion of $\gamma_{i}$ in $B$ corresponds to the original trajectory $\mathbf{x}^{}\left(\cdot\right)$. Combined with (2.1) we conclude that for every $i$ $\left\|\int_{\left\\{t\leq t_{i}\|\mathbf{x}^{}\left(t\right)\in B\right\\}}\mathbf{f}\left(\mathbf{x}^{}\left(t\right)\right)dt\right\|\leq 2\pi r_{0}.$ The minimality of $\Omega$ and Lemma 4 implies that the set $\left\\{t\|\mathbf{x}^{}\left(t\right)\in B\right\\}$ has infinite measure. This implies that $\mathbf{0}$ is contained in the convex hull of $\left\\{\mathbf{f}\left(\mathbf{y}\right)\|\mathbf{y}\in\bar{B}\right\\}$. The radius $r_{0}$ can be chosen arbitrary small, thus, the continuity of $\mathbf{f}$ implies that $\mathbf{f}\left(\mathbf{y}_{0}\right)=\mathbf{0}$, in contradiction. ## References [1] Z. Artsein and I. Bright. Periodic optimization suffices for infinite horizon planar optimal control. SIAM Journal on Control and Optimization, 48:4963–4986, 2010. * [2] I. Bright. A reduction of topological infinite-horizon optimization to periodic optimization in a class of compact 2-manifolds. Journal of Mathematical Analysis and Applications, 394(1):84 – 101, 2012. * [3] I. Bright and J.M. Lee. Bounds for flux integrals over regular domains. Submitted. * [4] W.F. Pfeffer. The Divergence Theorem and the Sets of Finite Perimeter, volume 303\. CRC Press, 2012. * [5] J.C. Wells. A note on a corollary of sardś theorem. Proceedings of the American Mathematical Society, 48(2):513–514, 1975. ∎	arxiv-papers	2013-09-12T20:36:26	2024-09-04T02:49:50.862951	{ "license": "Public Domain", "authors": "Ido Bright", "submitter": "Ido Bright", "url": "https://arxiv.org/abs/1309.3294" }
1309.3524	Gakuto International Series Mathematical Sciences and Applications Vol.XX (2011) International Symposium on Computational Science 2011 pp.XXXXX Stochastic moving particle semi-implicit for inviscid fluid wave simulation Christian Fredy Naa Institute of Science and Engineering, School of Mathematics and Physics, Kanazawa University Faculty of Mathematics and Natural Science, Institut Teknologi Bandung Seiro Omata Institute of Science and Engineering, School of Mathematics and Physics, Kanazawa University Masaki Kazama Fujitsu Limited Abstract. The present paper introduces stochastic velocity as improvement for moving particle semi-implicit (MPS) method. This improvement is to overcome energy loss caused by numerical dissipation in the basic MPS that brings about rapid decay of waves. Stochastic velocity is added in the explicit step of the basic MPS method. MPS with stochastic improvement is compared with the basic method in the case of linear water waves, in particular dam break problem and standing wave in a rectangular tank. Surface detection and curve fitting are used to analyze the parameters of wave on the standing wave case. The surface detection and curved fitting was efficient to determine parameters of the wave and it was found that the stochastic improvement made the waves survived longer than in the basis method. ——————————————————————– ———- Received xxxxxxxxxx, 2011. This work is supported by xxxxxxxxxxxxxxxxxxxx. AMS Subject Classification xxxxxxxxxx, xxxxxxxxxx ## 1 Introduction Moving particle semi-implicit (MPS) [1] method is a particle method for simulating incompressible fluids. MPS method was used for example to analyze breaking waves [2], droplet breakup behavior [3] and to predict wave impact pressure [4]. Since MPS method is based on Lagrangian system, computational grids are not necessary. Governing equations are discretized based on particle interaction models representing density, gradient, Laplace operator and free surface. However, MPS has a weakness in the energy conservation. The waves calculated by the basic MPS decay rapidly since the mechanical energy is not fully conserved. This lack of conservation of energy is caused by numerical dissipation. A work has been introduced to recover this weakness: Suzuki [5] introduced Hamiltonian moving particle semi-implicit (HMPS). Even then, the mechanical energy was not fully conserved, but the HMPS was able to make the waves survive longer. The objective of this paper is to overcome the loss of energy in the basic MPS using stochastic modification of velocity. The purpose of this stochastic concept is to add extra kinetic energy to the particles so that the kinetic energy gained by the stochastic velocity recovers the loss of energy caused by numerical dissipation. The continuity equation and Euler’s equation are used as governing equation. Stochastic and basic MPS were compared in the cases of dam break problem and standing wave in a rectangular tank. Surface detection algorithm is used to determine amplitude in the standing wave case. This kind of algorithm is used because it is hard to judge the important parameters of the wave only from the distribution of particles. The paper is organized in the following way. A brief explanation of standard MPS method is presented in second section. The stochastic improvement of velocity is introduced in third section. The fourth section describes the surface detection algorithm. Finally, the improved MPS method compared with the standard MPS method in the case of dam break problem and standing wave in a rectangular tank. ## 2 Standard MPS method In this section, the MPS method is briefly explained based on description which provided by Koshizuka [1]. In the MPS method, the fluid is modeled using interaction of particles according to equations of motion. Governing equations for inviscid fluid motion are continuity equation and Euler’s equation: $\frac{1}{\rho}\frac{D\rho}{Dt}+\nabla\cdot{\bf{u}}=0$ (1) $\frac{D\bf{{u}}}{Dt}=-\frac{1}{\rho}\nabla P+{\bf{g}},$ (2) where ${\bf{u}}$ denotes particle velocity vector, $t$ denotes time, $\rho$ denotes density, $P$ denotes pressure and ${\bf{g}}$ denotes gravity acceleration vector. It should be noted that Eq. (1) is written in the form of a compressible flow. In the MPS method, incompressibility is enforced by the way setting $\frac{D\rho}{Dt}=0$ at each particle at each calculation time step. According to [4] the left hand side of Eq. (2) denotes the material derivative $\frac{D\bf{{u}}}{Dt}$ involving the advection term. In the particle methods, including the MPS method, the advection term is automatically calculated through the tracking of particle motion; hence, the numerical diffusion arising from the successive interpolation of the advection function in Eulerian grid based methods is controlled without the need for a sophisticated algorithm. The main idea of MPS method is to divide Eq. (2) into two parts to calculate the change of velocity, as follows $\left(\frac{d{\bf{u}}}{{dt}}\right)^{\text{explicit}}={\bf{g}},$ (3) $\left(\frac{d{\bf{u}}}{dt}\right)^{\text{implicit}}=-\frac{1}{\rho}\nabla P.$ (4) Particle interaction is described in terms of weight function. Weight function $w(r)$ in MPS method is defined as [1] $w(r)=\begin{cases}\frac{r_{e}}{r}-1&\text{if }r<r_{e}\\\ 0&\text{if }r\geq r_{e}.\end{cases}$ (5) Here, $r$ will have the meaning of distance between particles and $r_{e}$ is the _cut off_ distance. In this paper, $r_{e}$ equals to $2.1l_{0}$, where $l_{0}$ is the initial distance between particles. The particle number density for particle $i$ ($n_{i}$) is calculated by $n_{i}=\sum_{j\neq i}^{N}w(r),$ (6) where $N$ denotes the total number of particles. The Laplace operator of a scalar quantity $\phi$ for particle $i$ is evaluated using $\left<\nabla^{2}\phi\right>_{i}=\frac{2d}{\lambda n_{0}}\sum_{j\neq i}^{N}(\phi_{j}-\phi_{i})w(\|{\bf{r}}_{j}-{\bf{r}}_{i}\|).$ (7) Here, ${\bf{r_{i}}}$ is position vector of particle $i$ and $d$ is the space dimension. In this paper, the constant $n_{0}$ is defined as the maximum value of particle number density $n_{0}=\max{n_{i}}.$ (8) The parameter $\lambda$ is defined as [4] $\lambda=\frac{\sum_{i\neq j}^{N}\|{\bf{r}}_{j}-{\bf{r}}_{i}\|^{2}w(\|{\bf{r}}_{j}-{\bf{r}}_{i}\|)}{\sum_{i\neq j}^{N}w(\|{\bf{r}}_{j}-{\bf{r}}_{i}\|)}.$ (9) The gradient of pressure $\nabla P$ for particle $i$ is defined as $\left<\nabla P\right>_{i}=\frac{d}{n_{0}}\sum_{j\neq i}^{N}\frac{P_{j}-\hat{P}_{i}}{\|{\bf{r}}_{j}-{\bf{r}}_{i}\|^{2}}w(\|{\bf{r}}_{j}-{\bf{r}}_{i}\|)({\bf{r}}_{j}-{\bf{r}}_{i}),$ (10) where $\hat{P}_{i}$ denotes the minimum pressure among particles within certain cut-off distance. For modeling the incompressibility, the number of densities $n^{}$ that are calculated at the end of explicit step deviate from the constant number of density $n_{0}$; hence, a second corrective process is required to adjust the number of densities to initial values prior to the time step. In the implicit step, the intermediate particle velocities are updated implicitply through solving the Poisson Pressure Equation (PPE) derived as [1] $\left<\nabla^{2}P_{k+1}\right>_{i}=\frac{\rho}{\Delta t^{2}}\frac{n_{0}-(n_{k}^{})_{i}}{n_{0}},$ (11) where $\Delta t$ denotes calculation time step and $k$ denotes the step of calculation. To set the Dirichlet boundary condition for the Poisson’s equation of pressure, particles satisfying [3] $n_{i}<\beta n_{0},$ (12) are judged as surface particles and their pressure is fixed to zero or to atmospheric pressure value. Here, $\beta$ is a value between $0.8$ to $0.98$. The time step is important for numerical stability. According to [6] the time step should follow the CFL condition $\Delta t\leq 0.2\frac{l_{0}}{u_{\text{max}}},$ (13) here, $u_{\text{max}}$ is the maximum velocity among the particles. ## 3 Modification of MPS method As described before that the standard MPS method has a weakness in the energy conservation. The mechanical energy is not fully conserved caused by numerical dissipation. To overcome this problem, the stochastic improvement of velocity is introduced. The stochastic velocity is added in standard MPS method. To add such stochastic velocity, after the particle’s velocity in the explicit step (Eq. (3)) is calculated, the stochastic velocity ${\bf{u}}^{\text{stoc}}$ is added. The direction of stochastic velocity ${\bf{u}}^{\text{stoc}}$ should be determined so that it does not reduce the particle velocity in the particular time step. To generate such direction, first the unit vector of particle’s velocity u is calculated by $\begin{cases}\hat{u}_{x}=\frac{u_{x}}{\|{\bf{u}}\|}\\\ \hat{u}_{y}=\frac{u_{y}}{\|{\bf{u}}\|},\end{cases}$ (14) where $\|{\bf{u}}\|$ is the length of the vector. Next, random angle $\theta$ is generated from the interval ($-\pi/2$,$\pi/2$) by choosing a random number $r$ from a uniform distribution $U(0,1)$: $\theta=r\left(\pi/2-(-\pi/2)\right)-\pi/2.$ (15) Then, rotating the vector $\hat{u}_{x}$ and $\hat{u}_{y}$ by the angle $\theta$ counter-clockwise about the origin, the unit vector of the stochastic velocity is determined by $\left(\begin{array}[]{cc}u_{x}^{\text{stoc}}\\\ u_{y}^{\text{stoc}}\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\ \sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{cc}{\hat{u}_{x}}\\\ {\hat{u}_{y}}\end{array}\right)$ (16) Finally, the unit vector determined by Eq. (16) is multiplied by the magnitude of the stochastic velocity $m$ to get the final vector form ${\bf{u}}^{\text{res}}={\bf{u}}+m{\bf{u}}^{\text{stoc}}.$ (17) Here, u is the velocity calculated by basic MPS method (after the explicit step). ${\bf{u}}^{\text{res}}$ is the resultant velocity due to addition ${\bf{u}}$ and ${\bf{u}}^{stoc}$ This process is illustrated in Fig. 1. Figure 1: Stochastic velocity The magnitude of the stochastic velocity $m$ is also given by choosing a random number $r$ from a uniform distribution $U(0,1)$: $m=r\left(\text{max}-\text{min}\right)+\text{min},$ (18) where min is the minimum range of the magnitude which is defined to be $0$, max is the maximum range which defined as $\text{max}=\alpha u_{\text{loss}}.$ (19) Here, $\alpha$ is a positive constant which determines the strength of the stochastic velocity, while the term $u_{\text{loss}}$ is defined as the amount of velocity that is lost due to energy difference on subsequent time steps. The value of $u_{\text{loss}}$ is determined based on the idea that the energy loss on two neighboring time steps corresponds to the loss of kinetic energy $\Delta EK$ $\Delta EK=E_{\text{total}}^{k-1}-E_{\text{total}}^{k},$ (20) where $E_{\text{total}}$ denotes total energy and $k$ denotes step number. Since kinetic energy per unit mass is given by $EK=\frac{{\bf{\|u\|}}^{2}}{2}$, then $u_{\text{loss}}=\sqrt{2\Delta EK}.$ (21) The addition of stochastic velocity follows several constraints: 1. 1. From experience of several simulations, the constant $\alpha$ is usually much less than $0.1$. If its more than $0.1$, the particles will get extra high velocity that leads to excessive energy making the particles jump off. 2. 2. It may happen that $\Delta EK$ in Eq. (20) equals to zero or even to a negative number (which means the energy is larger than in the previous step). In this case, the explicit step will be calculated without stochastic velocity. The complete algorithm of stochastic MPS in some step $k$ is described below: Algorithm 1 Stochastic MPS method algorithm 1:Total energy $E^{k-1}_{total}$ 2:Calculate total energy $E^{k}_{\text{total}}$. 3:Determine direction and magnitude ${\bf{u}}^{\text{stoc}}$ 4:if $E^{j-1}_{\text{total}}\leq E^{k}_{\text{total}}$ then 5: ${\bf{u}}^{\text{res}}={\bf{u}}+m{\bf{u}}^{\text{stoc}}$ 6:else 7: ${\bf{u}}^{\text{res}}={\bf{u}}$ 8:end if 9:$E^{k-1}_{\text{total}}=E^{k}_{\text{total}}$. 10:check termination. This paper concerns the addition of stochastic velocity in 2D case; similar idea can be adopted for 3D case. Since this stochastic concept is a new improvement in MPS method, it is not yet verified by any other research. ## 4 Surface Detection In order to compare the result from the simulations of basic MPS method and the stochastic improvement, a surface detection algorithm is needed. It is hard to judge the important parameters of the wave only from the distribution of particles. The algorithm begins with finding the upper left and right particles. Then the algorithm starts to trace the surface particles from the left to the right. From the left particle, the vertical reference axis is used to detect the second surface particle. Figure 2: Detection by using vertical imaginary axis. The solid circles denote the surface solution. The mechanism to detect the second surface particles is shown in Fig. 2. The radius of search area is bounded by constant $r_{e}$ which is the same as the cut-off radius in the above described theory of MPS. Angle from the vertical reference axis to each particle within the search radius is measured. Once the angles are measured, they are converted into quadrant value. The second surface particle is then determined as the particle that has the minimum angle with the vertical reference axis. After the second surface particle is determined, this surface particle is called the _current point_ and it is used to get the third surface particle. Figure 3: Detection using dot product relation. The solid circles denote the surface solution. The third surface particle and so on are determined using different algorithm. This algorithm is shown in Fig. 3, the surface particle detected before the current point is used (in this case: the third surface particle is determined using the first and the second particles). First, the reference vector that connects the current point and the previous surface particle is measured. Then the normal vector perpendicular to the reference vector which direct out from the particle distribution is determined using geometry transformation matrix. Candidate vectors from current point to all the particles inside the search area are also measured. Using two dot products relation (the product of between the reference vector and the candidate vectors, and the product of normal vector and the candidate vectors), the angles $\theta$ from the reference vector to the candidate vectors as shown in Fig. 3 can be determined. The next surface particle is the one corresponding to the smallest angle $\theta$. Figure 4: Surface detection algorithm Once the next surface particle is determined, this particle becomes the current point and the dot product algorithm is used to get the next surface particle. This mechanism is repeated until the upper right particle is detected. The flow chat of the algorithm is shown in Fig. 4. In some simulations, particles jump off. The algorithm should consider a constraint that would exclude this kind of particles because these particles are actually not part of the wave. The value of particle number density is used as a constraint. Namely If the particle number density is less than $0.4n_{0}$, the algorithm will exclude these particles from the surface detection process. ## 5 Test Cases ### 5.1 Dam break problem Two-dimensional dam break problem were simulated. This kind of simulation has been the most common test case in fluid dynamics. #### 5.1.1 Conditions of computation The computation domain was set to be $((x,y)\|0\leq x\leq l,0\leq y\leq l)$, where $l$ was taken to be $2$ m. The initial water column was set to be $((x,y)\|0\leq x\leq 0.5,0\leq y\leq 1)$. Rigid boundary condition was applied. The initial velocity and pressure was set to zero for each particle. The distance between particles was set to be $0.02$ m and $2.171$ particles were used in the computation while the simulation is taken until 15 seconds. The initial configuration of the particles is illustrated in Fig. 5. Figure 5: Initial condition of dam break problem #### 5.1.2 Results In this implementation, the basic MPS and stochastic MPS were simulated. Stochastic MPS used several $\alpha$ values (see (19)). Namely, $\alpha=0.003$, $\alpha=0.001$ and $\alpha=0.0009$ were simulated. The results of the simulation are shown in Fig. 6. Since the time step on each simulation is different, then the comparison was taken on the slightly different time and the main focus of the comparison is the time when wave disappeared . As shown on Fig. 6, at time $12.76s$ It is shown that the wave from the basic MPS already disappeared while the stochastic MPS still survived until several cycle. Thus it can be concluded that the wave which performed by stochastic improvement stronger and survived longer than the basic MPS. $\begin{array}[]{cc}\includegraphics[scale={0.32}]{dam0_2eps}&\includegraphics[scale={0.32}]{dam1_2eps}\\\ \includegraphics[scale={0.32}]{dam0_6eps}&\includegraphics[scale={0.32}]{dam1_6eps}\\\ \includegraphics[scale={0.32}]{dam0_10eps}&\includegraphics[scale={0.32}]{dam1_10eps}\\\ \end{array}$ Figure 6: Comparison of dam break results between the stochastic MPS with $\alpha=0.003$ (left) and the basic MPS (right) The total energy density with respect to time is plotted in Fig. 7. It is shows that the energy performed by basic MPS has lower state than the stochastic. The oscilation of stochastic energy density is caused by the current algorithm that perform the addition of stochastic velocity only if the total energy on the particular step has lower value than the step before. Since the whole energy system will decrease, the oscilation will occur because of the current stochastic velocity algorithm. Figure 7: Comparison between basic MPS and the stochastic MPS, $\alpha=0.003$, $\alpha=0.001$ and $\alpha=0.0009$ according to total energy density ### 5.2 Standing wave in a rectangular tank Two-dimensional standing waves in a rectangular tank were simulated. This kind of simulation has been studied by Suzuki [5] using Hamiltonian moving particle semi-implicit (HMPS). The results are compared with the analytical solution according to Wu and Taylor [7] based on the water elevation at the center of the tank. In this paper, different approach using surface detection and least square curve fitting is used to determine the amplitude of the wave. #### 5.2.1 Conditions of computation The computation domain was set to be $((x,y)\|0\leq x\leq l,0\leq y\leq l)$, where $l$ was taken to be equal to the wavelength. Here the wavelength $\lambda$ was $1$ m. The depth of water $h$ was $\lambda/3$. The initial configuration of the free surface is given by $y_{0}(x)=A\cos[k(x+l/2)].$ (22) Here $y_{0}$ is the initial surface displacement, $A=0.07\lambda$ is the amplitude and $k$ is the wave number which defined as $2\pi/\lambda$. Periodic boundary conditions were applied, while at the bottom mirror boundary condition was used. The initial velocity and pressure was set to zero for each particle. The distance between particles was set to be $0.02$ m and 834 particles were used in the computation while the simulation is taken until $6$ seconds. The initial configuration of the particles is illustrated in Fig. 8. Figure 8: Initial condition of standing wave in a rectangular tank Surface detection was applied to the solution and the obtained surface was fitted using least square curve fitting. _GNUPLOT_ curve fitting is used. In this case, since the surface seems closed to quadratic function then the data fitted using simple quadratic function $y(x)=ax^{2}+bx+c,$ (23) where constant $a$, $b$ and $c$ are to determined. From the result of the least square curve fitting, the amplitude of the wave is calculated on each time step. From the quadratic function (Eq. 23), the maximum point ($x_{\text{max}},y_{\text{max}}$) is calculated by setting $\frac{dy}{dx}=0$ which results in $\begin{cases}x_{\text{max}}=\frac{-b}{2a}\\\ y_{\text{max}}=ax_{\text{max}}^{2}+bx_{\text{max}}+c.\end{cases}$ (24) To estimated the quality of the fitting, average relative error $\bar{\epsilon}$ is used $\bar{\epsilon}=\frac{1}{N}\sum_{i=0}^{N}\left(\frac{\|y_{\text{eq}}-y_{\text{est}}\|}{y_{\text{eq}}}\right)\times 100\%.$ (25) Here, $N$ denotes the number of data, $y_{\text{eq}}$ denotes the fitted value according to (23), while $y_{\text{est}}$ is the $y$ value of the surface detection result. Figure 9 shows the result of surface detection and error calculation. Figure 9: Example of surface detection and curve fitting result #### 5.2.2 Results In this implementation, the basic MPS and stochastic MPS were simulated. Stochastic MPS used several $\alpha$ values (see (19)). Namely, $\alpha=0.02$, $\alpha=0.03$ and $\alpha=0.04$ were simulated. According to the relative error which was less than $4.5\%$, it can be concluded that the curve fitting worked well. Figure 10: Comparison between basic MPS, stochastic MPS with $\alpha=0.04$ and the analytic solution according to wave height (water elevation) in the center of the tank. The wave height evolution with respect to time is plotted in Fig. 10. It shows that the waves obtained from stochastic MPS survived longer than the basic MPS. The wave from basic MPS decayed rapidly in time, and disappeared already from $t=3.5$ seconds. In comparison with the analytical solution [7], the wave height from stochastic MPS was higher and nearer to the analytical solution. Ideally in qualitative analysis, if the system has perfect mechanical energy conservation, the amplitude of the wave will not decrease. But since the MPS method suffers from the numerical dissipation, the wave will rapidly decay. Using the stochastic velocity, the rapid decay of the wave could be prevented. Figure 11: Comparison between basic MPS and the stochastic MPS using $\alpha=0.04$ according to total energy density The energy density with respect to time is plotted in Fig. 11. It is shown that energy of the basic MPS decayed faster in time compared with the stochastic result (with $\alpha=0.04$). However, the stochastic MPS has a drawback that the particles moved to one direction after the wave disappeared. This is caused by the random movement of the particles plus extra velocity added by the stochastic term. The development of stochastic MPS without this drawback is left for further study. ## 6 Conclusion This paper introduces an improvement of moving particle semi-implicit method through the addition of stochastic velocity. Stochastic velocity is added after the explicit step of basic MPS method. The cases that studied were the dam break problem and standing wave in a rectangular tank. Surface detection and least square curve fitting were used to analyze the waves on the standing wave case. On the dam break problem, it was showed that the wave that performed by the stochastic survived longer than the basic MPS. However, the energy density from the stochastic MPS suffers some oscilation due to the stochastic algorithm. On standing wave case, it was showed that the wave with the stochastic MPS survived longer than the basic MPS. In comparison with the analytical solution [7], the wave height from stochastic MPS was higher and nearer to the analytical solution even it still far from the ideal system in which the mechanical energy is fully conserved. However, on the standing wave case the stochastic MPS has a drawback that the particles moved to one direction after the wave disappeared. This is caused by the random movement of the particles plus extra velocity added by the stochastic term. The development of stochastic MPS without this drawback is left for further study. ## References * [1] Koshizuka S., Oka Y.: _Moving particle semi implicit method for fragmentation of incompressible fluid_. Nuclear Science and Engineering, 123(3):421-434, 1996. * [2] Koshizuka S., Nobe A. and Oka Y. : _Numerical analysis of breaking waves using the moving particle semi-implicit method_. International Journal of Numerical Methods in Fluids, 26: 751-769, 1998. * [3] Nomura et. al: _Numerical Analysis of Droplet Breakup Behaviour using Particle Method_. Journal of Nuclear Science and Technology, 38 (12):1057-1064, 2001. * [4] Khayyer A., Gotoh H. : _Modified Moving Particle Semi-implicit method for the prediction of 2D wave impact pressure_. Coastal Engineering, 56: 419-440, 2009. * [5] Suzuki Y., Koshizuka S., Oka Y. : _Hamiltonian moving-particle semi implicit (HMPS) method for incompressible fluid flows_. Computer methods in applied mechanics and engineering, 196: 2876-2894, 2007. * [6] Zhang et.all, _An improved MPS method for numerical simulations of convective heat transfer problem_. International Journal for Numerical Method in Fluids, 51:31-47, 2005. * [7] Wu G.X., Taylor R. E. : _Finite element analysis of two-dimensional non-linear transient water waves_. Appl. Ocean Res., 16:363-372, 1994. * [8]	arxiv-papers	2013-09-13T17:44:23	2024-09-04T02:49:50.883048	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Christian Fredy Naa, Seiro Omata, Masaki Kazama", "submitter": "Christian Naa", "url": "https://arxiv.org/abs/1309.3524" }
1309.3632	3D Simulation of Dam-break effect on a Solid Wall using Smoothed Particle Hydrodynamic Suprijadia,b, F. Faizalb, C.F. Naaa and A.Trisnawanb aDepartment of Physics, bDepartment of Computational Sciences Faculty of Mathematics and Natural Science, Institut Teknologi Bandung Jl. Ganesha No.10, Bandung 40132, Indonesia email : [email protected] Abstract. Dam is built for water supply, water flow or flooding control and electricity energy storage, but in other hand, dam is one of the most dangerous natural disaster in many countries including in Indonesia. The impact of dam break in neighbour area and is huge and many flooding in remote area, as happen in Dam Situ Gintung in Tangerang (close to Jakarta) in 2009. Smoothed Particle Hydrodynamics (SPH), is one of numerical method based on Lagrangian grid which is applied in astrophysical simulation may be used to solve the simulation on dam break effect. The development of SPH methods become alternative methods to solving Navier Stokes equation, which is main key in fluid dynamic simulation. In this paper, SPH is developed for supporting solid particles in use for 3D dam break effect (3D-DBE) simulation. Solid particle have been treated same as fluid particles with additional calculation for converting gained position became translation and rotation of solid object in a whole body. With this capability, the result of 3D-DBE simulation has been varies and interesting. The goals of this simulation is for analyse fluid and solid particle interaction by using two different scenario. The first scenario relation between height of fluid to a solid wall barrier and second scenario, is to study relation between solid wall and its collapse time by dam break. The results show sliding distance of a solid wall is depend on a fluid height, and relation between fluid heght and wall dimension will be discussed too in this paper. Keywords: smoothed particle hydrodynamic, dam break, solid-fluid interaction ## 1 Introduction Dam is built for water supply, water flow or flooding control and electricity energy storage, but in other hand, dam is one of the most dangerous natural disaster in many countries including in Indonesia. Because of the age, there are some evidence of natural disaster by a dam failure and break were reported for example in Situgintung Tangerang, close to Jakarta was happen in 2009 [1]. The impact of dam break in neighbour area and is huge and many flooding in remote area, the computer calculation to predict effect of the dam broken is important including its vast impact[2]. Smoothed Particle Hydrodynamics (SPH), is one of numerical method based on Lagrangian grid which is applied in astrophysical simulation may be used to solve the simulation on dam break effect. The development of SPH methods become alternative methods to solving Navier-Stokes equation, which is main key in fluid dynamic simulation. As a standard test case in computational fluid dynamic the dam break simulation were often reported in many cases, including using SPH methods [5]. The simulation using SPH became emerge whether using two or three dimension. In other hand, simulation of solid particle is many reported direct numerical simulation [6]. Interaction between fluid and solid particle is still open ended to studying. Including simulation base on SPH method. In this paper, SPH is developed for supporting solid particles in use for 3D dam break effect (3D-DBE) simulation. Solid particle have been treated same as fluid particles with additional calculation for converting gained position became translation and rotation of solid object in a whole body. With this capability, the result of 3D-DBE simulation has been varies and interesting. The goals of this simulation is for analyse fluid and solid particle interaction by using two different scenario. The first scenario relation between height of fluid to a solid wall barrier and second scenario, is to study relation between solid wall and its collapse time by dam break. The results show sliding distance of a solid wall is depend on a fluid height, and relation between fluid height and wall dimension will be discussed too. ## 2 Smoothed Particle Hydrodynamics The governing equation for fluid dynamics is Navier-Stokes equations consist of conservation of momentum, conservation of mass and conservation of energy. Assume the system of dam is in isothermal state, then we can neglect the calculation of fluid’s thermal energy. The momentum equation has been written as follows: $\rho\,\frac{\partial\mathbf{u}}{\partial t}+\rho\,\left(\mathbf{u}\cdot\nabla\,\mathbf{u}\right)=-\nabla p+\mu\nabla^{2}\mathbf{v}+\rho\mathbf{g}$ (1) with $\rho$ is fluid density, t is time, v is fluid velocity, p is pressure, $\mu$ is viscosity, and f is external force. The fluid motion is also governed by the conservation of mass, which stated by $\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{u}\right)=0$ (2) Equations (1) and (2) will be solved by numerical approximation of SPH method. The method simply described by discreet collection of particles or balls for approximating fluid body. Every particle mathematically influence the neighbouring particles via kernel function. $W\left(r,l\right)=\frac{1}{l\sqrt{\pi}}\exp{\left(\frac{-r^{2}}{l^{2}}\right)}$ (3) Here $W$ is kernel function, $x$ is particle distance and $l$ is smoothing factor. The kernel function stated in equation (3) now can be use as weighting function to calculate all physical quantity, gradient and Laplacian attached to the particle, here we calculate the solution of equation (1) and (2) using weighted interpolation concept and solved them numerically. The detail of this method described clearly in [4]. Figure 1: Kernel radius in SPH simulation Figure 1 illustrated the approximation of fluid body by collection of moveable particle interacted along kernel radius $r$. The numerical SPH for momentum and mass conservation int total derivative form respectively are [3]: $\frac{D\mathbf{u}_{a}}{Dt}=\sum_{b}m_{b}\left(\frac{p_{b}}{{\rho_{b}}^{2}}+\frac{p_{a}}{{\rho_{a}}^{2}}+\Pi_{ab}\right)\nabla_{a}W_{ab}+\mathbf{g}$ (4) and $\frac{D{\rho}_{a}}{Dt}=\sum_{b}m_{b}\,\left(\mathbf{u}_{b}-\mathbf{u}_{a}\right)\cdot\nabla_{a}W_{ab}.$ (5) With index $a$ and $b$ denoted particle index, $\Pi_{ab}$ is numerical viscosity described in[3]. Here the kernel function $W_{ab}$ and it’s derivative solved analytically. Equation (4) and (5) can be solved using direct numerical integration such as Euler method. Another well-known methods is Runge-Kuta method, Leap frog method and Verlet method. ## 3 Solid - Fluid Interaction In DBE simulation, one of the important aspect is solid motion exerted by fluid energy. Here we use solid body dynamic to simulate the motion of fluid. Once the total force and total torsion exerted by fluid to surface of solid already known, so do translational and rotational motion of solid can be calculated. Figure 2: Force diagram and solid wall motion Figure 2 show the basic physical scheme of the solid fluid interaction with strong assumption that no flow across the surface of fluid particle. Some important aspect such as floor friction and center mass of solid body is also taking into account. Figure 3: Solid particle motion schematic To simplified the continuum aspect of solid body, solid object will be approximated also by collection rigid particles or balls with certain physical quantity governed by newton equation of translation and rotation. The center of mass of such discreet particle are also easy to calculate. Figure 3 showed the step process of solid particle position update in rotational motion. ## 4 Results and Discussion The scenario of DBE simulation in three dimension described as follow. Initially the fluid position in in cubic shape dam for simplification and without initial velocity of fluid assuming the fluid is at rest. At a certain distance lied solid wall to block flow of the fluid. Another assumption were made that there no crack or fracture in solid wall. Figure 4: Simulation scenario diagram, $h_{2}$ is initial fluid height, $h_{1}$ and $w_{1}$ is height and width of solid Figure 4 shown the description of initial arrangement of solid and fluid particles. As described earlier the arrangement of fluid particle in box shape dam and initially particle at rest. As the time step moves the gravity force work on the fluid particles and they move with the same for as if happened in the dam failure suddenly. Some parameter involved in 3D simulation in Figure 4 is initial fluid height, solid wall dimension. Later in this section will be shown some result in 3D DBE simulation with some variation in simulation parameter. Figure 5: 3D simulation of DBE a $t_{1}=0s$, $t_{2}=0.228s$, $t_{3}=0.543s$ and $t_{4}=1.068s$, $h_{1}=55m$ and $h_{2}=40m$ and $w_{1}=40m$ Figure 5 show general result of DBE simulation wall with height $55m$ and fluid dam with $40m$ in height at time $t_{1}=0s$, $t_{2}=0.228s$, $t_{3}=0.543s$ and $t_{4}=1.068s$. The frame shows the fall process of the wall as the effect of fluid flow. Figure 6: Wall height effect on DBE simulation at $t=0.54s$. Left $h_{1}=55m$, right $h_{1}=60m$ The variation in solid wall height is made to test the stability of moveable wall under the same energy of fluid flow exerted on the wall surface. It is done by setting the fluid height at $40m$. Figure 6 shows the effect of wall height to the fall process under the influence of flow caused by dam break event. Here the frame plot the position of particles at the same time step $0.54s$ and the same initial height of fluid $h_{2}=40m$ but in different wall height. The frames show different response of falling time. Higher wall fall faster than the lower one at other parameter were keep constant. Here we can say that the dimension of the wall influence the stability of the wall object under the same outer forces. If the width of the wall has been increasing the high wall also become relative stable. In another scenario the variation of fluid height were tested to the relatively stable wall object and observe the wall sliding distance. Figure 7: Fluid height effect to the sliding distance on DBE simulation Figure 7 shows sliding distance plot versus initial fluid height. The trend of the data seems to be non linear as the height of fluid increase over 55 m, this is make sense because of volume of the fluid increase by the power of three to the height measure and linearly increase the potential energy of fluid. ## 5 Conclusion We successfully simulate 3D-DBE simulation using SPH methods, and fluid-solid interaction capability was applied to study the dam break effect on a solid wall object, the SPH can be used to simulate a solid particle simulation. Some simplified assumption that no crack or fracture happen in the solid wall object thus no separation of solid body as the impact of the fluid momentum. The simulation results showed that the fluid height and solid wall thickness and height play a significant effect on solid wall translational and rotational motion. For the future work it is also challenging to study the fracture event, crumbling process or separation of solid body by the effect of momentum from the fluid flow and made realistic devastation process from dam failure. This features will increase the computational cost and difficulties in algorithm. ## Acknowledgement A part of this research was funding by Research and Innovation Program of Research Group ITB, grant No:243/I.1.C01/PL/2013. ## References * [1] BBC (2009). Indonesian dam burst toll rises. BBC News Asia-Pacific * [2] D. Munger (2009). Economic Consequences Methodology for Dam Failure Scenarios. Reclamation Managing in the West,Technical Memorandum Number EC-2009-01 T. U.S. Department of the Interior Bureau of Reclamation Technical Service Center Security, Safety and Law Enforcement * [3] J.J. Monaghan (1992). Smoothed Particle Hydrodynamic. Annu. Rev. Astron. Astrophys, 30, 543 - 74. * [4] G.R. Liu and M.B. Liu (2007). Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific Publishing, Singapore * [5] M. Kelager (2006). Lagrangian Fluid Dynamics using Smoothed Particle Hydrodynamics. Master Thesis. Univiersity of Copenhagen. * [6] P.Y. Huang, J. Feng, H.H. Hu and D.D Joseph (1997). Direct simulation of the motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids. J. Fluid Mech., 343, 73 - 94.	arxiv-papers	2013-09-14T07:35:29	2024-09-04T02:49:50.899641	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Suprijadi, Ferry Faizal, Christian Naa, Anggy Trisnawan Putra", "submitter": "Christian Naa", "url": "https://arxiv.org/abs/1309.3632" }
1309.3742	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP-2013-166 LHCb-PAPER-2013-048 November 13, 2013 Measurement of the charge asymmetry in $B^{\pm}\rightarrow\phi K^{\pm}$ and search for $B^{\pm}\rightarrow\phi\pi^{\pm}$ decays The LHCb collaboration†††Authors are listed on the following pages. The $C\\!P$-violating charge asymmetry in $B^{\pm}\rightarrow\phi K^{\pm}$ decays is measured in a sample of $pp$ collisions at 7 TeV centre-of-mass energy, corresponding to an integrated luminosity of 1.0 fb-1 collected by the LHCb experiment. The result is $\mathcal{A}_{CP}(B^{\pm}\rightarrow\phi K^{\pm})=\rm 0.022\pm 0.021\pm 0.009$, where the first uncertainty is statistical and the second systematic. In addition, a search for the $B^{\pm}\rightarrow\phi\pi^{\pm}$ decay mode is performed, using the $B^{\pm}\rightarrow\phi K^{\pm}$ decay rate for normalization. An upper limit on the branching fraction $\mathcal{B}(B^{\pm}\rightarrow\phi\pi^{\pm})<1.5\times 10^{-7}$ is set at 90% confidence level. Submitted to Phys. Lett. B © CERN on behalf of the LHCb collaboration, license CC-BY-3.0. LHCb collaboration R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A. Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben- Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A. Falabella14,e, C. Färber11, C. Farinelli40, S. Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16,e, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,37,f, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Heß60, A. 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Zhong3, A. Zvyagin37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Padova, Padova, Italy 22Sezione INFN di Pisa, Pisa, Italy 23Sezione INFN di Roma Tor Vergata, Roma, Italy 24Sezione INFN di Roma La Sapienza, Roma, Italy 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland 26AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland 27National Center for Nuclear Research (NCBJ), Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 41Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands 42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 43Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 44University of Birmingham, Birmingham, United Kingdom 45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 47Department of Physics, University of Warwick, Coventry, United Kingdom 48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 50School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 52Imperial College London, London, United Kingdom 53School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 54Department of Physics, University of Oxford, Oxford, United Kingdom 55Massachusetts Institute of Technology, Cambridge, MA, United States 56University of Cincinnati, Cincinnati, OH, United States 57University of Maryland, College Park, MD, United States 58Syracuse University, Syracuse, NY, United States 59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11 61Celal Bayar University, Manisa, Turkey, associated to 37 aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oHanoi University of Science, Hanoi, Viet Nam pInstitute of Physics and Technology, Moscow, Russia qUniversità di Padova, Padova, Italy rUniversità di Pisa, Pisa, Italy sScuola Normale Superiore, Pisa, Italy ## 1 Introduction The weak-interaction $B^{\pm}\\!\rightarrow\phi K^{\pm}$ decay is governed by the $b\\!\rightarrow ss\overline{}s$ transition. In the Standard Model (SM), it can only occur through loop diagrams (see Fig. 1), Figure 1: Lowest-order Feynman diagrams of the Standard Model for the decays $B^{+}\\!\rightarrow\phi K^{+}$ (top) and $B^{+}\\!\rightarrow\phi\pi^{+}$ (bottom). The diagrams with an external $\phi$ meson are OZI suppressed. leading to a branching fraction of order $10^{-5}$ [1]. Because the dominant amplitudes have similar weak phases, the $C\\!P$-violating charge asymmetry, defined as $\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})\equiv\frac{{\cal B}(\mbox{$B^{-}\\!\rightarrow\phi K^{-}$})-{\cal B}(\mbox{$B^{+}\\!\rightarrow\phi K^{+}$})}{{\cal B}(\mbox{$B^{-}\\!\rightarrow\phi K^{-}$})+{\cal B}(\mbox{$B^{+}\\!\rightarrow\phi K^{+}$})}\,,$ (1) is predicted to be small in the SM, typically 1–2% with uncertainties of a few percent [2, 3]. A significantly larger value would signal interference with an amplitude not described in the SM. The current experimental world average is $\mathcal{A}_{C\\!P}(\mbox{\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$}})=0.10\pm 0.04$ [1], dominated by a recent measurement from the BaBar collaboration [4]. Large $CP$ violation effects have been seen in some regions of the $B^{\pm}\\!\rightarrow K^{+}K^{-}K^{\pm}$ phase space, but not around the $\phi$ resonance [5]. The $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decay is another flavour-changing neutral current process, driven by the $b\\!\rightarrow ds\overline{}s$ quark- level transition (see Fig. 1). The high suppression, due to the tiny product of the Cabibbo-Kobayashi-Maskawa matrix elements [6, 7] and to the Okubo- Zweig-Iizuka (OZI) rule [8, 9, 10] associated with the creation of the colourless $s\overline{}s$ pair forming the $\phi$ meson, makes this rare loop decay a sensitive probe of the SM. Indeed, even a small non-SM amplitude, e.g. from R-parity violating supersymmetry [11], may dominate over the SM contribution. The current SM prediction for the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ branching fraction suffers from uncertainties originating from the naïve factorization approach, radiative corrections, calculation of the long- distance contribution (e.g. $B\\!\rightarrow KK^{}$ rescattering), and $\omega-\phi$ mixing [12]. The latter is the main source of uncertainty. The physical $\omega$ and $\phi$ meson states do not coincide exactly with the ideal $(\|u\overline{}u\rangle+\|d\overline{}d\rangle)/\sqrt{2}$ and $\|s\overline{}s\rangle$ states, respectively. They appear to be mixtures of these two states characterized by a small mixing angle $\delta_{V}$ [13, 14], which depends on the magnitude of $\mathrm{SU}(3)$ symmetry breaking and can be determined in the framework of chiral perturbation theory. However, more sophisticated treatments based on the full $\rho^{0}-\omega-\phi$ mixing scheme suggest that $\delta_{V}$ is mass dependent, i.e. takes different values at the $\omega$ and $\phi$ masses [15, 16]. In the QCD factorization approach, the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ branching fraction is predicted to be in the range $(5-10)\times 10^{-9}$ [3] if $\omega-\phi$ mixing is neglected, but can be enhanced up to $0.6\times 10^{-7}$ [17, 12] depending on the value of $\delta_{V}$. However, the effect of $\omega-\phi$ mixing has not been observed in a recent search for $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ [18]. Values of the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ branching fraction in excess of $10^{-7}$ would be indicative of non-SM physics. The $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decay mode has not been observed yet. Currently, the most stringent experimental limit is ${\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})<2.4\times 10^{-7}$ at 90% confidence level (CL), obtained by the BaBar collaboration [19]. This Letter presents a measurement of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ charge asymmetry and a search for the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decay mode with the LHCb detector. The results are based on a data sample collected during the 2011 $pp$ run of the Large Hadron Collider at a centre- of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$. The $\phi$ meson is reconstructed in the $K^{+}K^{-}$ final state. We define the $\phi$ signal as any peaking component in the $K^{+}K^{-}$ mass spectrum consistent with the known parameters of the $\phi$ resonance, without attempting a full amplitude analysis of the three-body $K^{+}K^{-}K^{\pm}$ and $K^{+}K^{-}\pi^{\pm}$ final states. In order to suppress several systematic effects, the primary observables measured in this analysis are the difference of $C\\!P$-violating charge asymmetries $\Delta\mathcal{A}_{C\\!P}\equiv\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})-\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$})\,,$ (2) and the branching fraction ratio ${\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})/{\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})$, which are then converted to results on $\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})$ and ${\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})$ using the best known values of $\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$})$ [1, 20] and ${\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})$ [1]. The choice of $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ as reference channel and other features of the analysis follow the approach adopted in inclusive studies of $B^{\pm}\\!\rightarrow K^{+}K^{-}K^{\pm}$ decays with the same data set [5]. The two measurements are performed in a common analysis, i.e. they are based on identical event selections and data descriptions whenever possible. The observables are obtained from two-dimensional maximum likelihood fits to the unbinned $B^{\pm}$ and $\phi$ mass distributions of the reconstructed candidates, using parametric shapes with minimal dependence on simulation. The results of these fits were not examined until the entire analysis procedure was finalized. ## 2 Detector and data set The LHCb detector [21] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system provides a momentum measurement with a relative uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter (IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov detectors [22]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers. The direction of the magnetic field of the spectrometer dipole magnet is reversed regularly. The trigger [23] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. The $B^{\pm}$ candidate decays considered in this analysis must belong to one of two exclusive categories of events, called TOS (triggered on signal) or TIS (triggered independently of signal). A TOS event is triggered at the hardware stage by one of the candidate’s final-state particles being compatible with a transverse energy deposit greater than 3.5$\mathrm{\,Ge\kern-1.00006ptV}$ in the hadron calorimeter. A TIS event does not satisfy the TOS definition and is triggered at the hardware stage by activity in the rest of the event. All candidates must pass a software trigger requiring a two-, three- or four-track secondary vertex with a large scalar sum of the transverse momentum of the tracks and a significant displacement from the primary $pp$ interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with respect to any PV greater than 16, where $\chi^{2}_{\rm IP}$ is defined as the difference in $\chi^{2}$ of a given PV reconstructed with and without the considered track. A multivariate algorithm [24] is used for the identification of secondary vertices consistent with the decay of a $b$ hadron. In the simulation, $pp$ collisions are generated using Pythia 6.4 [25] with a specific LHCb configuration [26]. Decays of hadronic particles are described by EvtGen [27], in which final state radiation is generated using Photos [28]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [29, Agostinelli:2002hh] as described in Ref. [31]. ## 3 Event selection and efficiency The selections of $B^{\pm}\\!\rightarrow\phi K^{\pm}$ and $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidates are identical, except for the particle identification (PID) requirement on the charged hadron combined with the $\phi$ candidate, which is referred to as the bachelor hadron $h^{\pm}$ ($h^{\pm}=K^{\pm}$ or $\pi^{\pm}$). The other requirements are chosen to minimize the relative statistical uncertainty on the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal yield. Only good quality tracks with $\chi^{2}_{\rm IP}>25$ and $\mbox{$p_{\rm T}$}>0.25{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are used in the reconstruction. The $\phi$ meson candidates are reconstructed from two oppositely-charged tracks identified as kaons with the PID requirement ${\rm DLL}_{K\pi}>2$, where ${\rm DLL}_{K\pi}$ is the difference in log-likelihood between the kaon and pion hypotheses, as determined with the ring-imaging Cherenkov detectors in control samples of known particle composition [22]. The $\phi$ candidates are required to have $\mbox{$p_{\rm T}$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, a total momentum, $p$, larger than 10${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an invariant mass, $m_{KK}$, in the range 1.00–1.05${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Bachelor hadrons, reconstructed either as pions if ${\rm DLL}_{K\pi}<-1$ or kaons otherwise, are required to have $p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\mbox{$p_{\rm T}$}>2.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and are combined with $\phi$ candidates to form $B^{\pm}\\!\rightarrow\phi h^{\pm}$ candidates. These $B^{\pm}$ candidates are required to have $\mbox{$p_{\rm T}$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, a three-track vertex $\chi^{2}$ per degree of freedom less than 9, and an invariant mass $m_{KKh}$ in the range 5.0–5.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Furthermore $\cos\theta_{\rm p}$ is required to be greater than 0.9999, where $\theta_{\rm p}$ is the angle between the $B^{\pm}$ momentum vector and the vector joining the $B^{\pm}$ production vertex to the $B^{\pm}$ decay vertex. The production vertex is chosen as the PV for which the $B^{\pm}$ has the smallest $\chi^{2}_{\rm IP}$. Multiple candidates, occurring in 0.2% of the events, are removed by keeping the candidate with the smallest $B^{\pm}$ vertex $\chi^{2}$. The final data sample consists of 6251 $B^{\pm}\\!\rightarrow\phi K^{\pm}$ candidates and 2169 $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidates. The PID performance is determined from a large and high-purity sample of pions from prompt $D^{+}\\!\rightarrow D^{0}(K^{-}\pi^{+})\pi^{+}$ and $D^{-}\\!\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}(K^{+}\pi^{-})\pi^{-}$ decays, as a function of $p$ and $\eta$. After reweighting this calibration sample to the same momentum and pseudorapidity distributions as for the bachelor pion in simulated $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decays, the efficiency of the PID requirement ${\rm DLL}_{K\pi}<-1$ for the bachelor pion is measured to be $\rm 0.846\pm 0.011\,(stat)\pm 0.020\,(syst)$, with a 5% kaon misidentification probability. All other efficiencies, which are slightly different for $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ and $B^{\pm}\\!\rightarrow\phi K^{\pm}$ decays due to their kinematic properties, are determined from simulation. The efficiency ratio $\frac{\epsilon(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})}{\epsilon(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})}=\rm 0.762\pm 0.031\,(stat)\pm 0.036\,(syst)$ (3) is obtained, where the numerator is the total efficiency for a $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decay to be selected as a $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidate and the denominator is the total efficiency for a $B^{\pm}\\!\rightarrow\phi K^{\pm}$ decay to be selected either as a $B^{\pm}\\!\rightarrow\phi K^{\pm}$ candidate or as a $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidate. The statistical uncertainty arises from the size of the calibration and simulation samples, while the systematic uncertainty is the quadratic sum of contributions from the PID ($\pm 0.018$), the trigger ($\pm 0.008$), and other offline kinematic selection requirements ($\pm 0.030)$. ## 4 Fit description The observables of interest, namely the asymmetry between the $B^{-}\\!\rightarrow\phi K^{-}$ and $B^{+}\\!\rightarrow\phi K^{+}$ yields and the ratio between the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ and $B^{\pm}\\!\rightarrow\phi K^{\pm}$ yields, are each determined from a two- dimensional unbinned extended maximum likelihood fit based on probability density functions (PDFs) of the $m_{KKh}$ and $m_{KK}$ masses. In each case, independent subsamples of events, each with either $B^{\pm}\\!\rightarrow\phi K^{\pm}$ candidates or $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidates, are fitted simultaneously. For each subsample, the likelihood is written as ${\cal L}={\textstyle\exp(-\sum_{j}N_{j})}\,\prod_{i}^{N}{\left(\textstyle\sum_{j}N_{j}P_{j}^{i}\right)}\,,$ (4) where $N_{j}$ is the yield of fit component $j$, $P_{j}^{i}$ is the probability of event $i$ for component $j$, and the index $i$ runs over the $N$ events in the subsample. Except for the misidentified components described further below, the probabilities $P_{j}^{i}$ are given by the product of PDFs for the two $K^{+}K^{-}h^{\pm}$ and $K^{+}K^{-}$ invariant masses, evaluated at the values $m_{KKh}^{i}$ and $m_{KK}^{i}$ of event $i$: $P_{j}^{i}=P_{j}^{KKh}(m_{KKh}^{i})P_{j}^{KK}(m_{KK}^{i})\,.$ (5) This assumes that the two mass variables are independent, as supported by data and simulation studies. The correlation between $m_{KKh}$ and $m_{KK}$ is found to be less than $4\%$. The description of the $m_{KKh}$ distributions involves a combination of three contributions: a signal peaking at the $B^{\pm}$ mass, a broad low-mass background with an end-point near $5150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ due to partially-reconstructed $b$-hadron decays such as $B^{0}\\!\rightarrow\phi K^{0}$, and a linear background from random combinations. The peaking signal is modelled with a Crystal Ball function [32] modified such that both the upper and lower tails are power laws. The mean and the width $\sigma_{B}$ of the Crystal Ball function are free in the fit, while the tail parameters are determined from simulation. The partially-reconstructed background is described with an ARGUS function [33] convoluted with a Gaussian resolution function of the same $\sigma_{B}$ as the $B^{\pm}$ signal. The $m_{KK}$ distribution is described with two contributions: a peaking term centred on the $\phi$ mass, described with a relativistic Breit-Wigner function convoluted with a Gaussian resolution function of free width, and a linear term originating from nonresonant, S-wave, or random combinations of two kaons. The above three $m_{KKh}$ contributions and two $m_{KK}$ contributions lead to six components for each subsample: the $B^{\pm}\\!\rightarrow\phi h^{\pm}$ signal, the nonresonant $B^{\pm}\\!\rightarrow K^{+}K^{-}h^{\pm}$ background, the partially-reconstructed $b$-hadron backgrounds with or without a true $\phi$ meson (for example $B\\!\rightarrow\phi h^{\pm}\pi$ or $B\\!\rightarrow K^{+}K^{-}h^{\pm}\pi$), and the combinatorial backgrounds with or without a true $\phi$ meson. The nonresonant $B^{\pm}\\!\rightarrow K^{+}K^{-}h^{\pm}$ components include $b\rightarrow c$ decays, which are found to be negligible from simulation studies. In addition, we consider components for the misidentified $B^{\pm}\\!\rightarrow\phi K^{\pm}$ and $B^{\pm}\\!\rightarrow K^{+}K^{-}K^{\pm}$ decays in the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ sample, while misidentified $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ and $B^{\pm}\\!\rightarrow K^{+}K^{-}\pi^{\pm}$ decays in the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ sample are negligible, and therefore ignored. For these two additional components, the $m_{KK\pi}$ PDF is conditional to the observable $\delta m=m_{KKK}-m_{KK\pi}$, which is the mass difference under the two bachelor hadron mass hypotheses. The probabilities are written as $P_{j}^{i}=P_{\rm misID}^{KK\pi}(m_{KK\pi}^{i}\|\delta m^{i})P_{j}^{KK}(m_{KK}^{i})\,,$ (6) where $P_{j}^{KK}$ is the $m_{KK}$ PDF described above (representing either $\phi$ signal or background) and $\left.P_{\rm misID}^{KK\pi}(m^{i}_{KK\pi}\|\delta m^{i})=P_{\phi K}^{KKK}(m^{i}_{KK\pi}+\delta m^{i})\right\|_{\sigma_{B}\rightarrow\rho\sigma_{B}}\,.$ (7) Here $P_{\phi K}^{KKK}$ is the $m_{KKK}$ PDF of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal, but with an increased $B^{\pm}$ mass resolution to account for the effects of the typically higher momentum of misidentified bachelor kaons. The parameter $\sigma_{B}$ is multiplied here by the central value of a factor $\rho=1.26\pm 0.10$, determined from data as the ratio of the measured $m_{KKK}$ resolutions of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal in the regions $-7<{\rm DLL}_{K\pi}<-1$ and ${\rm DLL}_{K\pi}>-1$. The expression in Eq. 7 is equivalent to $\left.P_{\phi K}^{KKK}(m^{i}_{KKK})\right\|_{\sigma_{B}\rightarrow\rho\sigma_{B}}$, which means that the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ misidentified component in the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ sample would have a $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal distribution if the correct mass was assigned to the bachelor kaon. The advantage of introducing the $\delta m$ observable is to connect the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ shapes in the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ and $B^{\pm}\\!\rightarrow\phi K^{\pm}$ samples, thereby constraining the misidentified $B^{\pm}\\!\rightarrow\phi K^{\pm}$ component in the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ sample using the large signal in the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ sample. This procedure allows to describe the misidentified component with the same parametric shape as the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ signal, and reduces the statistical uncertainty on the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ yield by a factor of two. However, this introduces a bias because the $\delta m$ distribution, which is not accounted for in the likelihood, is not the same for all components [34]. To reduce this bias, the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ sample is divided into four bins of $\delta m$, each with its own eight components. This procedure reduces the bias on the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ signal yield to a negligible level. Other fit parameters that are common to the different subsamples are the $m_{KKh}$ end-point of the partially-reconstructed backgrounds, the peaking $m_{KK}$ PDF parameters for all components containing a $\phi$ meson, and the $m_{KK}$ slope of the nonresonant $B^{\pm}\\!\rightarrow K^{+}K^{-}h^{\pm}$ components. Finally, the ratio of the yield of the misidentified nonresonant $B^{\pm}\\!\rightarrow K^{+}K^{-}K^{\pm}$ background to the yield of misidentified $B^{\pm}\\!\rightarrow\phi K^{\pm}$ background in the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ sample is constrained to the yield ratio of the corresponding correctly-identified components in the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ sample. The fit procedure is validated on simulated data containing the expected proportion of signal and background events. These studies, which take into account the different $\delta m$ distributions and the possible correlation between the fit observables, demonstrate the stability of the fit and show that the fit results are unbiased. ## 5 Measurement of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ charge asymmetry The charge asymmetry of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal is determined from a fit to the $B^{-}\\!\rightarrow\phi K^{-}$ and $B^{+}\\!\rightarrow\phi K^{+}$ candidates in the ${\rm DLL}_{K\pi}>-1$ region. These two samples are each divided into two subsamples according to whether the events were TOS or TIS at the hardware trigger stage. In the fit, each of the six components has therefore four yields. For the signal component, they are expressed as $N^{\pm}_{\rm TOS}=N_{\rm TOS}(1\mp\mathcal{A}_{\rm raw,TOS})/2$ and $N^{\pm}_{\rm TIS}=N_{\rm TIS}(1\mp\mathcal{A}_{\rm raw,TIS})/2$, where $N_{k}$ is the total yield and $\mathcal{A}_{{\rm raw},k}$ is the raw yield asymmetry in subsample $k$ ($k=\rm TOS,TIS$). The fit has a total of 34 free parameters: 10 mass shape parameters, 12 yields and 12 raw asymmetries. \begin{overpic}[width=218.51521pt]{BMass_Asymm_neg.pdf}\put(85.0,73.0){\scriptsize(a)} \end{overpic}\begin{overpic}[width=218.51521pt]{PhiMass_Asymm_neg.pdf} \put(85.0,73.0){\scriptsize(b)} \end{overpic}\begin{overpic}[width=218.51521pt]{BMass_Asymm_pos.pdf}\put(85.0,73.0){\scriptsize(c)} \end{overpic}\begin{overpic}[width=218.51521pt]{PhiMass_Asymm_pos.pdf} \put(85.0,73.0){\scriptsize(d)} \end{overpic} Figure 2: Distributions of the (a) $K^{+}K^{-}K^{-}$ and (b) $K^{+}K^{-}$ masses of the selected $B^{-}\\!\rightarrow\phi K^{-}$ candidates, as well as of the (c) $K^{+}K^{-}K^{+}$ and (d) $K^{+}K^{-}$ masses of the selected $B^{+}\\!\rightarrow\phi K^{+}$ candidates. The solid blue curves represent the result of the simultaneous fit described in the text, with the following components: $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal (dotted red), nonresonant $B^{\pm}\\!\rightarrow K^{+}K^{-}K^{\pm}$ background (dashed red), partially-reconstructed $b$-hadron background with (dotted blue) or without (dashed blue) a true $\phi$ meson, and combinatorial background with (dotted green) or without (dashed green) a true $\phi$ meson. Some of the components are barely visible because the corresponding yields are small. Normalized residuals are displayed below each histogram. Figure 2 shows the projections of the fitting function superimposed on the $m_{KKK}$ and $m_{KK}$ distributions, shown separately for $B^{-}$ and $B^{+}$ candidates, but where TOS and TIS events are summed. The $m_{KKK}$ resolution measured from the fit is $\sigma_{B}=20.4\pm 0.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The fitted raw asymmetries for the signal are shown in the first line of Table 1. They are statistically uncorrelated. Table 1: Raw charge asymmetries for the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ and $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ decays, their difference $\Delta\mathcal{A}_{C\\!P}$, and the fraction of $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal events in each trigger subsample for $k=\rm TOS,TIS$. All uncertainties are statistical only. \| TOS subsample \| TIS subsample ---\|---\|--- $\mathcal{A}_{{\rm raw},k}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})$ \| $+0.027\pm 0.026$ \| $-0.053\pm 0.035$ $\mathcal{A}_{{\rm raw},k}(\mbox{$B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$})$ \| $-0.024\pm 0.008$ \| $-0.008\pm 0.005$ $\Delta\mathcal{A}_{C\\!P}$ \| $+0.052\pm 0.027$ \| $-0.045\pm 0.035$ $N_{k}/(N_{\rm TOS}+N_{\rm TIS})$ \| 66% \| 34% Weighted $\Delta\mathcal{A}_{C\\!P}$ average \| $+0.019\pm 0.021$ Each raw charge asymmetry is related to the $C\\!P$ asymmetry through $\mathcal{A}_{{\rm raw},k}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})=\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})+{\cal A}_{{\rm D},k}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})+{\cal A}_{\rm P}\,,$ (8) where ${\cal A}_{{\rm D},k}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})$ is the detection charge asymmetry for the bachelor $K^{\pm}$ and ${\cal A}_{\rm P}$ is the production asymmetry of $B^{\pm}$ mesons. Equation 8 and the corresponding equation for the $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ reference channel hold because all involved asymmetries are small. Under the assumption that the detection asymmetry is the same for $B^{\pm}\\!\rightarrow\phi K^{\pm}$ and $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$, which is correct in the limit where the bachelor $K^{\pm}$ has the same kinematic properties, the difference in charge asymmetries defined in Eq. 2 can be written as $\Delta\mathcal{A}_{C\\!P}=\mathcal{A}_{{\rm raw},k}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})-\mathcal{A}_{{\rm raw},k}(\mbox{$B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$})$ (9) and should not depend on the trigger category $k$. The raw charge asymmetries of $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ decays have been measured in a previous analysis [5]; they are subtracted from the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ raw asymmetries to obtain two independent measurements of $\Delta\mathcal{A}_{C\\!P}$. Since the two results agree within about two statistical standard deviations, the results are combined. The final $\Delta\mathcal{A}_{C\\!P}$ result is computed as a weighted average, with weights equal to the fractions $N_{k}/(N_{\rm TOS}+N_{\rm TIS})$ of signal events in the two trigger subsamples. All inputs to the calculation are reported in Table 1. The separation between TIS and TOS events is needed because the detection asymmetry ${\cal A}_{{\rm D},k}$ depends on the trigger category $k$ and the fraction of events in the two categories differs between the signal and reference channels. Table 2: Systematic uncertainties on the measurement of $\Delta\mathcal{A}_{C\\!P}$. Source \| Uncertainty ---\|--- Mass shape modelling \| 0.003 Possible S-wave contribution \| 0.002 Trigger \| 0.004 Bachelor kaon kinematic properties \| 0.005 Geometric acceptance \| 0.002 Quadratic sum \| 0.007 Several systematic uncertainties are considered on the weighted $\Delta\mathcal{A}_{C\\!P}$ average, as summarized in Table 2. The contribution due to the mass shape modelling is obtained by repeating the fit (and the calculation of Table 1) with the fixed parameter values of the Crystal Ball and ARGUS functions changed within their uncertainties, as determined from simulation and $B^{\pm}\\!\rightarrow\phi K^{\pm}$ data, respectively, or with an exponential (rather than linear) combinatorial background model. Possible residual effects from S-wave contributions not fully accounted for by the linear component are investigated by comparing the observed angular distribution of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal with the expectation for a peaking structure in the $K^{+}K^{-}$ mass due to a single P-wave state. Other P-wave components are neglected. If these S-wave contributions corresponded to an additional component included in the signal without charge asymmetry, a bias would appear on $\Delta\mathcal{A}_{C\\!P}$, which is taken as a systematic uncertainty. The charge asymmetry in the trigger efficiency for kaons of the TOS subsample does not completely cancel in $\Delta\mathcal{A}_{C\\!P}$, because of the different number of kaons in the two decay modes considered. The difference between the values of $\mathcal{A}_{\rm raw,TOS}(\mbox{$B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$})$ computed with and without a charge-dependent correction for the kaon efficiency determined from calibration data is propagated as a systematic uncertainty on $\Delta\mathcal{A}_{C\\!P}$. Such an effect is absent for the TIS subsample. Another small contribution, due to the TOS events that would still be accepted by the hardware trigger level without considering the particles from the $B^{+}$ candidate decay, has been included in the trigger systematic uncertainty. Due to differences in the kinematic selections of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ and $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ decay modes, the assumption of Eq. 9 cannot be exact, and a further systematic uncertainty is assigned. The fit of the raw charge asymmetries of $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ is repeated with the same kinematic selection on the bachelor kaon as for $B^{\pm}\\!\rightarrow\phi K^{\pm}$, i.e. $p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\mbox{$p_{\rm T}$}>2.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and after reweighting its momentum distribution to that observed in the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ decays. The resulting effect on $\Delta\mathcal{A}_{C\\!P}$ is taken as a systematic uncertainty. Finally, we repeat the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ analysis after requiring the bachelor kaon momentum to point in a fiducial solid angle avoiding detector edge effects, and assign the observed change in $\Delta\mathcal{A}_{C\\!P}$ as a systematic uncertainty due to the geometrical acceptance. The final measurement is $\Delta\mathcal{A}_{C\\!P}=\rm 0.019\pm 0.021(stat)\pm 0.007(syst)\,.$ (10) A recent update of the $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ charge asymmetry measurement by the D0 collaboration [20] has not been included yet in the average of the Particle Data Group (PDG) [1]. Replacing the previous D0 result with the new one yields the world average $\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$})=0.003\pm 0.006$, where the uncertainty is scaled by a factor 1.8 according to the PDG averaging rules. Using this average, we obtain $\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})=\rm 0.022\pm 0.021(stat)\pm 0.009(syst)\,,$ (11) where the uncertainty on the $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ charge asymmetry is incorporated in the systematic uncertainty. ## 6 Search for $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decays \begin{overpic}[width=218.51521pt]{BMassPi_PHIPI_Linear_Pad_cut.pdf}\put(85.0,51.0){\scriptsize(a)} \end{overpic}\begin{overpic}[width=218.51521pt]{PhiMassPi_PHIPI_Linear_Pad_cut.pdf} \put(85.0,51.0){\scriptsize(b)} \end{overpic}\begin{overpic}[width=218.51521pt]{BMassPi_PHIPI_Log.pdf}\put(85.0,73.0){\scriptsize(c)} \end{overpic}\begin{overpic}[width=218.51521pt]{PhiMassPi_PHIPI_Log.pdf} \put(85.0,73.0){\scriptsize(d)} \end{overpic} Figure 3: Distributions of the (a, c) $K^{+}K^{-}\pi^{\pm}$ and (b, d) $K^{+}K^{-}$ masses of the selected $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidates, shown both with linear and logarithmic scales. The solid blue curves represent the result of the simultaneous fit described in the text, with the following components: $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ signal (dotted black), nonresonant $B^{\pm}\\!\rightarrow K^{+}K^{-}\pi^{\pm}$ background (dashed black), $B^{\pm}\\!\rightarrow\phi K^{\pm}$ signal (dotted red), nonresonant $B^{\pm}\\!\rightarrow K^{+}K^{-}K^{\pm}$ background (dashed red), partially-reconstructed $b$-hadron background with (dotted blue) or without (dashed blue) a true $\phi$ meson, and combinatorial background with (dotted green) or without (dashed green) a true $\phi$ meson. Normalized residuals are displayed below the histograms. The search for $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decays is performed using a simultaneous fit to the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ (${\rm DLL}_{K\pi}<-1$) and $B^{\pm}\\!\rightarrow\phi K^{\pm}$ (${\rm DLL}_{K\pi}\geq-1$) candidates, dividing the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidates in four subsamples according to their $\delta m$ values, each with its set of eight yields. The fit has a total of $52$ free parameters: $15$ mass shape parameters, $36$ yields, and the ratio of the total $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ yield to the total $B^{\pm}\\!\rightarrow\phi K^{\pm}$ yield. Figure 3 shows the projections of the fitted function superimposed on the observed mass distributions of the $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidates. The total $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ signal yield is found to be $19\pm 19$, while the total $B^{\pm}\\!\rightarrow\phi K^{\pm}$ yield is $(3486\pm 76)+(280\pm 25)$ summing the samples of $B^{\pm}\\!\rightarrow\phi K^{\pm}$ and $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ candidates. The fitted yield ratio is $\frac{N(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})}{N(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})}=(\rm 5.1\,^{+5.3}_{-5.0}(stat)\pm 2.1(syst))\times 10^{-3}\,,$ (12) where the systematic uncertainty is the quadratic sum of contributions due to the modelling of the mass shapes ($\pm 2.1\times 10^{-3}$), the fit procedure ($\pm 0.2\times 10^{-3}$), and interference effects between the $\phi$ resonance and a $K^{+}K^{-}$ pair in an S-wave state ($\pm 0.4\times 10^{-3}$). The first contribution is obtained by repeating the fit with the parameter values of the Crystal Ball and ARGUS functions changed within their uncertainties, or with an exponential (rather than linear) combinatorial background model. The dominant effect is due to the 8% uncertainty on the ratio $\rho$ of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ mass resolutions in the two ${\rm DLL}_{K\pi}$ regions. Simulation studies show that the fit procedure is unbiased, and the statistical precision of this check is assigned as a systematic uncertainty. The measurement of the branching fraction ratio is obtained as the ratio between Eq. 12 and Eq. 3: $\frac{{\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})}{{\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})}=\rm(6.6\,^{+6.9}_{-6.6}(stat)\pm 2.8(syst))\times 10^{-3}\,.$ (13) Since the result is not significantly different from zero, we also quote upper limits from the integral of the likelihood function of this ratio, considering only the physical (non-negative) region. Including systematic uncertainties we obtain ${\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})/{\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})<0.018\,(0.020)$ at 90% (95%) CL. Using the current world average ${\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})=(8.8\,^{+0.7}_{-0.6})\times 10^{-6}$ [1], we finally obtain $\displaystyle{\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})$ $\displaystyle=$ $\displaystyle(5.8\,^{+6.1}_{-5.8}\pm 2.5)\times 10^{-8}$ (14) $\displaystyle<$ $\displaystyle 1.5\,(1.8)\times 10^{-7}~{}~{}~{}\mbox{at 90\% (95\%) CL}\,.$ (15) ## 7 Conclusions The difference in charge asymmetries between the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ and $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ decay modes is measured in a sample of $pp$ collisions at 7$\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass energy, corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ collected with the LHCb detector. Using the known value of the $B^{\pm}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}$ asymmetry, the $C\\!P$-violating charge asymmetry of $B^{\pm}\\!\rightarrow\phi K^{\pm}$ decays is determined to be $\mathcal{A}_{C\\!P}(\mbox{$B^{\pm}\\!\rightarrow\phi K^{\pm}$})=\rm 0.022\pm 0.021(stat)\pm 0.009(syst)$. This result is almost a factor two more precise than the current world average [1]. It is consistent with both the absence of $C\\!P$ violation and the Standard Model prediction. A search for $B^{\pm}\\!\rightarrow\phi\pi^{\pm}$ decays is also performed. No significant signal is found. Using the known branching fraction of the $B^{\pm}\\!\rightarrow\phi K^{\pm}$ normalization channel, an upper limit of ${\cal B}(\mbox{$B^{\pm}\\!\rightarrow\phi\pi^{\pm}$})<1.5\,(1.8)\times 10^{-7}$ is set at 90% (95%) confidence level. This improves on the previous best upper limit [19], while reaching the upper end of the Standard Model predictions. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. ## References [1] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. D86 (2012) 010001, and 2013 partial update for the 2014 edition * [2] H.-n. Li and S. Mishima, Penguin-dominated $B\rightarrow PV$ decays in NLO perturbative QCD, Phys. Rev. D74 (2006) 094020, arXiv:hep-ph/0608277 * [3] M. Beneke and M. Neubert, QCD factorization for $B\rightarrow PP$ and $B\rightarrow PV$ decays, Nucl. Phys. B675 (2003) 333, arXiv:hep-ph/0308039 * [4] BaBar collaboration, J. P. 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Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C.\n Craik, M. Cruz Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David,\n P.N.Y. David, A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian,\n J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp, M.\n Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F.\n Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, P. Durante, R.\n Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund,\n I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, C. Farinelli, S. Farry,\n D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S.\n Filippov, M. Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O.\n Francisco, M. Frank, C. Frei, M. Frosini, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, P. Gorbounov, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, P. Griffith, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A.\n Hicheur, E. Hicks, D. 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1309.3744	# CaMn2Sb${}_{2}:$ a fully frustrated classical magnetic system I. I. Mazin Code 6393, Naval Research Laboratory, Washington, DC 20375, USA Code 6393, Naval Research Laboratory, Washington, DC 20375, USA (8/9/13) ###### Abstract We show by means of density functional calculations that CaMn2Sb2 is very close to a mean-field critical point known for the classical Heisenberg model on the honeycomb lattice. Three entirely different long range ordered magnetic phases become degenerate at this point: the Neel phase and two different spiral phases. We speculate that the unusual physical properties of this compound, observed in recent experiments, in particular the enigmatic intermediate temperature phase, are due to this proximity. Introduction: CaMn2Sb2 is an interesting implementation of a nearly classical spin system on a honeycomb lattice. Such systems have been attracting the attention of researchers since the seminal paper of Rastelli $et$ $al$ from 1979Rastelli . The basic model involves the nearest neighbor (nn) interaction $J_{1}$ and the second neighbor interaction $J_{2}.$ In the classical limit, the ground state of this $J_{1}-J_{2}$ model, for antiferromagnetic $J_{1,2}$, is the Neel phase where all nearest neighbor bonds are fully antiferromagnetic, if $J_{2}/J_{1}<1/6.$ Interestingly, for $1/6<J_{2}/J_{1}<1/2,$ there are two degenerate solutions, corresponding to two different spiral phases (in one of them the two spins in one unit cell are always antiparallel, and the spiral propagation vector is perpendicular to a nn bond, in the other the two sublattices are rotated by a particular angle with respect to each other, and the spiral vector is parallel to a bond). At $J_{2}/J_{1}>1/2$ the ground state is the stripe phase with alternating FM pairs (Fig. 1). Both critical points, $J_{2}/J_{1}=1/2$ and $J_{2}/J_{1}=1/6$ are triple points: in case there is an additional parameter, for instance the third neighbor exchange $J_{3},$ three phases meet at these points: the Neel phase and the two spiral phases at $J_{2}/J_{1}=1/6,$ and the stripe phase and the two spiral phases at $J_{2}/J_{1}=1/2.$ Whenever a third parameter is added, be it the third neighbor exchangeRastelli $J_{3}$, uniaxial anisotropyRastelli or biquadratic couplingI $K,$ complex phase diagrams arise, with the four phases described above, and additional phases such as zigzag antiferromagnetism. Figure 1: Four antiferromagnetic patterns discussed in the text. Note that the spiral A coincides with the phase N at $\theta=0$ and spiral B with the phase N N at $\phi=\pi$ and $\theta=0$, and with S at $\phi=0$ and $\theta=\pi$. At the same time the two spirals cannot be continuously transformed into each other except in this limiting cases. Figure 2: (color online) Crystal structure of CaMn2Sb2, showing examples of the nearest neighbors ($J_{1}$, red), 2nd neighbors ($J_{2}$, green) and 3rd neighbors ($J_{3}$), blue, inside a MnSb layer, as well as the nearest ($J_{z1}$, yellow) and 2nd ($J_{z2}$, black) interplanar neighbors. Mn atoms are shown by gray spheres, Sb by cyan ones and Ca by green ones. The crystal structure of CaMn2Sb2 is shown in Fig. 2. The Mn sublattice consists of honeycomb layers in which every other atom is shifted perpendicular to the plane. This interesting geometry leads to a rather short Mn-Mn distance, 3.18 Å, with substantial direct overlap between the Mn d orbitals and as a result sizable direct antiferromagnetic exchange. In addition, there are two superexchange paths available. One connects the nearest neghbor Mns, with an Mn-Sb-Mn angle of 70${}^{\circ},$ and the other connects the second neighbors, with an angle of 108${}^{\circ}.$ Mn in this compound has valency 2+, and therefore in the high spin state it would have a magnetic moment of 5 $\mu_{B},$ reduced by hybridization and fluctuations. Experimentally at low temperature the $J_{1}$-driven Neel phase was found, with a magnetic moment of 2.8-3.4 $\mu_{B}$. At $T=85$ K this phase gives rise to another phase of unknown origin, which experimentally resembles weak ferromagnetism, but no long range antiferromagnetic order was detected. Finally, at $T\gtrsim 200$ K the material becomes paramagnetic, exhibiting at $T\gtrsim 300$ K a Curie-Weiss behavior with an effective moment $M_{CW}=1.4$ $\mu_{B}$. The unusual character of the intermediate temperature phase, as well as the very low Curie-Weiss moment, suggest that frustration characteristic of honeycomb magnetic models may play a role. The low- temperature phase shows an activation transport behavior with an activation gap $\sim 40$ meV, while the intermediate temperature phase exhibits a strong increase (up to a factor of 100) of the resistivity.Aronson We have performed first principle calculations of the electronic and magnetic properties of CaMn2Sb2. We found Mn to be in the high spin state and $d({5\uparrow})$ configuration. We also found a small gap consistent with the experiment, and magnetic interactions dominated by the nearest neighbor exchange. Most interestingly, we found that the second neighbor exchange is 4-6 times smaller than the nearest neighbor one, while the third neighbor exchange and the biquadratic exchange are very small, and the magnetic anisotropy is of the easy-plane type. In this regime classical spins on the honeycomb lattice are highly frustrated, with two spiral and two collinear phases nearly degenerate. We speculate that this frustration is the cause of the unusual magnetic phase diagram. Calculations. The calculations in this paper were performed with the Linear Augmented Plane Wave code WIEN2kWIEN2K , using the following crystallography: symmetry group #164, $P\bar{3}m1,$ $a=$4.522 Å, $c=$7.458 Å, $z_{Ca}=0,$ $z_{Mn}=0.3784,$ $z_{Sb}=0.7487,$ and a Generalized Gradient Approximation for the exchange-correlation potential. In agreement with previous calculationsAronson we found that Mn in strongly polarized, and Mn moments are well localized, as demonstrated by the fact that all magnetic configurations converge to about the same magnetic moment, $\sim 4$ $\mu_{B},$ and the calculated exchange energies are much smaller than the magnetization energy. The lowest energy among various collinear states has the Neel state, in which the nn Mn have opposite spins. Figure 3: (color online) Calculated band structure of CaMn2Sb2 in the antiferromagnetic Neel phase. The symbol size is proportional to the Mn character (spin-up in blue, below the Fermi level, and spin-down in red, above the Fermi level. Figure 4: The calculated dielectric function of CaMn2Sb2. The inset shows the corresponding transmission coefficient (the thickness of the slab was chosen to roughly match the amplitude of the transmission coefficient in Ref. Aronson i, and a uniform week absorption was added to account for the experimentally observed in-gap states). The band structure and density of states in this configuration are shown in Fig. 3. One can clearly see that all five $d$ orbitals of Mn are fully polarized, while the reduction of the moment from 5 to 4 $\mu_{B}$ is due to hybridization. There is also an indirect excitation gap (of the spin-flip nature), about 50 meV, consistent with the experimentally measured one. The calculated optical gap (minimal direct transition energy) is about 0.7 meV, but the absorption at this energy starts relatively slow (Fig. 4), owing to the fact that the top of the valence band and the bottom of the conductivity band are formed in the different spin channels (in a ferromagnetic case, these transtions would have been optically forbidden; in an antiferromagnetic one, they are allowed but weak). The main absorption edge is located at 2 eV. Compared to experimentAronson , the calculated transmission drops at a lower frequency. However, the optical experiments reported in Ref. Aronson should be considered as preliminary. One has to wait for more state-of-the-art optical data, preferably for the conductivity, for a full comparison. In order to understand better magnetic interactions in CaMn2Sb${}_{2},$ we have calculated the total energies of 10 different magnetic configurations, and mapped the results onto the model Heisenberg that includes 1st, 2nd and 3rd nearest neighbors inside the MnSb layers ($J_{1},$ $J_{2},$ and $J_{3})$, and 1st and 2nd neighbors between the layers ($J_{z1}$ and $J_{z2}).$ The corresponding distances and values are listed in the Table. Note that the definition of $J$ in this paper is such that the pair interaction is $J\sigma_{i}\sigma_{j},$ where $\sigma=\pm 1.$ Table 1: Calculated exchange parameters and the corresponding Mn-Mn bond lengths. See Fig.2 for graphical examples. \| $J_{1}$ \| $J_{2}$ \| $J_{3}$ \| $J_{z1}$ \| $J_{z2}$ ---\|---\|---\|---\|---\|--- $d$ (Å) \| 3.179 \| 4.522 \| 5.528 \| 6.219 \| 7.458 $J$ (meV) \| 91 \| 22 \| 5 \| 11 \| 3 Since the number of energy differences (9) is larger than the number of fitted exchange constants, one can estimate the accuracy of the fitting from the standard fit error. This is on the order of 7–8 meV. Note that on the mean field level $J_{z}$s simply renormalize the planar interactions, in other words, assuming perfect interplanar ordering, one can map the 3D problem onto 2D with $J_{1eff}=J_{1}+J_{z1}=102$ meV. We have also looked for other possible deviations from the Heisenberg model: the biquadratic interaction, which plays an extraordinary important role in Fe-based superconductors, and single-site anisotropy. The former appears to be zero within computational accuracy, in other words, for all possible angles between the Mn spins in the same unit cell the energy is perfectly cosinusoidal, $E=const+0.5J_{1}\cos\alpha.$ The latter (magnetic anisotropy) is zero in plane (although formally hexagonal symmetry allows for magnetic anisotropy when the field is rotated by 30${}^{o},$ for all practical purposes this effect is negligible), and about 2–3 meV/Fe otherwise, with the direction perpendicular to the plane being the hard axis. Discussion. The fact that essentially any magnetic pattern can be stabilized in the calculations, and that the energy cost of suppressing magnetism entirely (0.8 eV/Fe) is much larger than the exchange constants indicates that CaMn2Sb2 should be considered a local moment system, and overall one expects Mn to be subject to considerable Hubbard correlations. The fact that the nearest-neighbor exchange follows the Heisenberg formula nearly exactly also speaks in favor of classical superexchange and the Hubbard model. Yet DFT calculations reproduce the excitation gap well, in fact, better than dynamic mean field calculationsHaule . The reasons for such an unexpected success of the DFT are unclear st the moment. Given the sharp increase of the resistivity in the other, intermediate temperature phase, one may think that the Neel phase is affected by some cancellation of errors, not operative in the other phases, which leads to an effective increase of the excitation gap in the intermediate phase. In the classical phase diagram of an antiferromagnetic $J_{1},J_{2}$ 2D honeycomb layer there are five planar phases: The Neel (N) phase, where all nearest neighbors order antiferromagnetically, the “stripy” (S) phase where all bond along one direction are ferromagnetic, and forming double stripes order antiferromagnetically, and two spiral phases (A and B) described in the introduction. For $J_{2}/J_{1}<1/6$ the ground state is N, for $1/6<J_{2}/J_{1}<1$ the ground state is degenerate between phases A and B, and for the larger $J_{2}$ the ground state is S. Adding $J_{3}$ creates a complicated phase diagram, with a new “zigzag” phase emerging at large $J_{3},$ which is shown in Fig. 5, where we also for completeness show the phase diagram $J_{1},J_{2},K$ with a biquadratic interaction, not published before. In this phase diagram, the calculated interactions correspond to the point $J_{2}/J_{1eff}\sim 0.21$, $J_{3}/J_{1eff}\sim 0.05,$ but the error in these numbers is close to $\pm 0.08.$ In other words, according to the calculations the material is extremely close to a highly frustrated critical point where three different phases are degenerate, the Neel phase and two qualitatively different spiral phases, $J_{2}/J_{1}=1/6,$ $J_{3}=0.$ Figure 5: Phase diagram of the classical $J_{1}--J_{2}--J_{3}$ Heisenberg model. The phase boundaries are given by the lines $y=(x-\sqrt{x^{2}+2(x-1/2)^{2}})/2$, $y=(1/2+3x-\sqrt{51/12-5x+9x^{2}})/4$, $y=(6x-1)/4$, $y=(2x+1)/4$, $y=0$, $x=1/2$Rastelli . The square shows the region in the phase diagram where, within computational accuracy, the actual system is located. The insert shows the phase diagram for the $J_{1}-J_{2}-K$ model, where K is the biquadratic coupling. The phase boundaries are: $y=(3x-\sqrt{1-2x+9x^{2}})/2$, $y=\sqrt{x-1/4}-x$, $y=(6x-1)/2$, $y=0$I . The thermodynamics of the classical Heisenberg model on the honeycomb lattice has not, to our knowledge, been systematically studied and this subject is beyond the scope of the current paper. It is however likely that the phase boundaries between the zero-temperature phase shift as the entropy is included, so it is tempting to ascribe the transformation at 85 K to a phase transition between the Neel and one of the two spiral phases (more likely the spiral A). Very weak net ferromagnetism is then due to relativistic effects, which can be estimated by the ratio between the magnetic anisotropy energy (2-3 meV) and magnetization energy (800 meV), which is $\sim 3\times 10^{-3},$ to be compared with the experimental ferromagnetic moment of $7\times 10^{-3}$ $\mu_{B}\approx 3\times 10^{-3}$ $M_{Mn}.$ No long range order has been observed between 85 and 210 K in neutron scattering; this suggests that the spirals break after relatively short distance and the emerging spiral chunks are randomly distributed over the three equivalent crystallographic directions. Such a state would be very similar to the “magnetic liquid” state in MnSiReznik , also called a “cholesteric” phase, or a “blue” phase in the original paper. This state is observed in MnSi near the phase transition into a long-range ordered spiral phase, on the high-temperature (high-pressure) side of the phase transition. Neutron scattering in this phase reveals no long range order, but well-defined spirals of considerable length (several hundred Å), propagating in all crystallographically equivalent directions with equal weight. It is still not clear whether the spirals in the “blue phase” of MnSi form domains (possibly dynamic) or are meandering around, periodically switching directions. Both options are open as well in the intermediate temperature phase of CaMn2Sb${}_{2}.$ Another corollary of proximity to the critical point at $J_{2}/J_{1}=1/6$ is that unusual low energy magnetic excitations should be present in the low- temperature Neel phase. More detailed experimental spectroscopic studies are highly desirable. To summarize, we have shown that CaMn2Sb2 is extremely close (within computational accuracy) to a critical point where three entirely different magnetic phases are degenerate on the mean field level. We suggest that unusual properties of this compound, obtained in recent experiments, are related to this unique proximity. The author acknowledges discussions with Girsh Blumberg and Meigan Aronson, and funding from the Office of Naval Research (ONR) through the Naval Research Laboratory’s Basic Research Program, and from the Alexander von Humboldt Foundation. ## References * (1) E. Rastelli, A. Tassi, and L. Reatto, Physica B 97, 1 (1979). * (2) J.B. Fouet, P. Sindzingre, and C. Lhuillier, Eur. Phys. J. B 20, 241 (2001). * (3) I.I. Mazin, unpublished. * (4) J. W. Simonson, G. J. Smith, K. Post, M. Pezzoli, J. J. Kistner-Morris, D. E. McNally, J. E. Hassinger, C. S. Nelson, G. Kotliar, D. N. Basov, and M. C. Aronson, Phys. Rev. B 86, 184430 (2012). * (5) P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J. Luitz, WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (K. Schwarz, Techn. Universit$\ddot{a}$t Wien, Austria, 2001), ISBN 3-9501031-1-2. * (6) K. Haule, unpublished * (7) C. Pfleiderer, D. Reznik, L. Pintschovius, H. v. Löhneysen1, M. Garst and A. Rosch, Nature 427, 227 (2004).	arxiv-papers	2013-09-15T10:23:00	2024-09-04T02:49:50.920300	{ "license": "Public Domain", "authors": "I. I. Mazin", "submitter": "Igor Mazin", "url": "https://arxiv.org/abs/1309.3744" }
1309.3752	# Novel Repair-by-Transfer Codes and Systematic Exact-MBR Codes with Lower Complexities and Smaller Field Sizes Sian-Jheng Lin and Wei-Ho Chung Authors are with the Research Center for Information Technology Innovation, Academia Sinica, Taipei City, Taiwan. (e-mail: [email protected]; [email protected]) ###### Abstract The $(n,k,d)$ regenerating code is a class of $(n,k)$ erasure codes with the capability to recover a lost code fragment from other $d$ existing code fragments. This paper concentrates on the design of exact regenerating codes at Minimum Bandwidth Regenerating (MBR) points. For $d=n-1$, a class of $(n,k,d=n-1)$ Exact-MBR codes, termed as repair-by-transfer codes, have been developed in prior work to avoid arithmetic operations in node repairing process. The first result of this paper presents a new class of repair-by- transfer codes via congruent transformations. As compared with the prior work, the advantages of the proposed codes include: i). The minimum of the finite field size is significantly reduced from $n\choose 2$ to $n$. ii). The encoding complexity is decreased from $n^{4}$ to $n^{3}$. As shown in simulations, the proposed repair-by-transfer codes have lower computational overhead when $n$ is greater than a specific constant. The second result of this paper presents a new form of coding matrix for product-matrix Exact-MBR codes. The proposed coding matrix includes a number of advantages: i). The minimum of the finite field size is reduced from $n-k+d$ to $n$. ii). The fast Reed-Solomon erasure coding algorithms can be applied on the Exact-MBR codes to reduce the time complexities. ###### Index Terms: Distributed storage, maximum-distance-separable (MDS) codes, partial downloading, Reed-Solomon codes, repair-by-transfer. ## 1 Introduction In a distributed storage system, the source data (message) is dispersed across nodes in the network, and a data collector (DC) can retrieve the whole source data by accessing a subset of the nodes. To tradeoff between the storage efficiency and the system reliability, the erasure codes, such as maximum- distance-separable (MDS) codes [1], random linear codes [2] or fountain codes [3, 4], are usually adopted as the base of data format in distributed storage systems [5, 6, 27]. For an unstable network, the nodes may frequently join and depart. When a node departs or crashes, the system manager will place a replacement node in the distributed storage network to replace the functionality of the failed node. Suppose the replacement node does not store any information about the data (code fragments) stored in the prior failed node. To reconstruct the data, the replacement node broadcasts a request to a subset of other helper nodes, and those helper nodes reply with the requisite information to the replacement node. If the distributed storage systems is based on conventional Reed-Solomon (RS) codes, an intuitive method is to reconstruct the entire source data in the replacement node, and then extract the desired code fragment from the source data. By such method, the total amount of downloaded symbols is not less than the size of whole source data. However, as the size of data stored in a single node is much smaller than the entire source data, it is possible to design a new class of storage codes to reduce the amount of downloaded symbols in node-repairing process. The new class of storage codes, termed as regenerating codes, is introduced by the pioneer paper [7]. ### 1.1 Coding system description In this paper, the regenerating code over $GF(q)$ is associated with a set of parameters $\\{n,k,d,\alpha,\beta,B\\}$ elaborated in the following. The value $B$ is the number of source symbols over $GF(q)$ to be encoded. The $n$ is the number of produced code fragments, which will be respectively stored in $n$ network nodes. The $\alpha$ is the number of symbols of a code fragment. In data reconstruction process, the DC individually downloads $\alpha$ symbols from each of a subset of $k$ nodes to reconstruct the message. In the node- repairing process, the replacement node individually downloads $\beta$ symbols from each of a subset of $d$ integrity nodes to rebuild the code fragment. Those parameters $\\{n,k,d\\}$ follows the inequality $k\leq d\leq n-1.$ The theoretical bound of storage-bandwidth trade-off have been given by [8] based on the cut-set bound of network coding: $B\leq\sum_{i=0}^{k-1}min\\{\alpha,(d-i)\beta\\}.$ (1) By the theoretical bound (1), two extreme points on the storage-bandwidth trade-off have been adequately investigated in prior works. The first extreme point, termed as minimum storage regeneration (MSR) point, is firstly to minimize the $\alpha$ and then minimize the $\beta$. The parameter configuration is $\begin{array}[]{l}\alpha=B/k;\\\ \beta=B/\left(k(d-k+1)\right).\end{array}$ (2) The second extreme point, termed as the minimum bandwidth regenerating (MBR) point, is firstly to minimize the $\beta$, and then minimize the $\alpha$. The parameter configuration is $\begin{array}[]{l}\beta=2B/\left(k(2d-k+1)\right);\\\ \alpha=d\beta.\end{array}$ (3) By the so-called data striping technique [9], the regenerating codes at $\beta=1$ can be used to construct the regenerating codes for any $\beta$. Thus, here in after, we focus on the design of regenerating codes at the $\beta=1$ MBR points, and the corresponding parameter configuration is $\alpha=d,\beta=1\textrm{, and }B={k+1\choose 2}+k(d-k).$ (4) In the node-regenerating process, if the restored fragment is always the same with the fragment in the prior failed node, this property is called the exact regeneration. This is in contrast to the functional regeneration without imposing restrictions on the content of the stored fragment. Practically, the exact regeneration is a good property to simplify the hardware and software designs for distributed storage systems. However, the non-existence of exact regeneration codes at the interior points on the storage-bandwidth trade-off curve have been proved [10]. In this paper, the abbreviations ”Exact-MSR” and ”Exact-MBR” respectively indicate the regenerating codes at MSR and MBR points with the exact regeneration property. ### 1.2 Definitions of terminologies #### 1.2.1 Systematic regenerating codes The [9] defines the systematic regenerating code as a class of regenerating code whose $B$ message symbols appear on a certain set of $k$ systematic code fragments. The nodes storing those systematic fragments are termed as the systematic nodes. A major work of this paper is to construct the systematic regenerating codes at MBR points. Systematic codes are useful in data reconstruction: If the DC can download those systematic code fragments, the DC can directly obtain the corresponding pieces of source data without any computational cost. This is a good property for practical systems. #### 1.2.2 Repair-by-transfer codes In the node-repairing process, the replacement node broadcasts a request to a subset of helper nodes, and each helper node returns certain number of responding symbols to the replacement node. In general, each helper node should compute the responding symbols via a function of the fragment stored in the node. The repair-by-transfer codes are a class of distributed storage codes where each helper node simply needs to pass a portion of the stored fragment without any arithmetic operations. The repair-by-transfer codes are particularly beneficial to the unstable network environment with frequent occurrence of the node regenerations. A repair-by-transfer code at $(n,k,d=n-1)$ Exact-MBR case is proposed by Shah et al. [10], and the non- existence of other cases $d<n-1$ is shown in [19]. The details [10] are introduced in Section 3. Furthermore, the generalized form of [10] is presented in [20, 21]. A system implementation for $k=n-1$ and $k=n-2$ is demonstrated by Hu et al. [26]. A objective of this paper is to construct the $(n,k,d=n-1)$ Repair-by-transfer codes with smaller finite fields and lower computational costs. By assigning $d=n-1$ to the (4), the parameters for $(n,k,d=n-1)$ repair-by-transfer codes are $\alpha=d=n-1,\beta=1\textrm{, and }B=(n-1)k-{k\choose 2}.$ (5) #### 1.2.3 Partial downloading scheme By the MBR data reconstruction process in [9], the DC should download the whole data stored in the set of connected nodes. To reduce the total amount of downloaded symbols, Gong and Wang [18] present a data decoding algorithm, termed as partial downloading scheme, on the non-systematic Exact-MBR codes [9]. By the partial downloading scheme, the DC can download a partial portion of code fragment from each connected node. The partial downloading scheme is useful to mitigate the network congestion. Thus, the partial downloading schemes are also developed on the proposed repair-by-transfer codes and Exact- MBR codes. ### 1.3 Previous works The exact regenerating codes at MSR and MBR points have been proposed in recent years. For Exact-MSR codes, the [13] discovers the code constructions at $(n=4,k=2,d=3)$ and ($n=5,k=3,d=4)$ via computer searching. The [14] presents the Exact-MSR codes for $d=n-1\geq 2k-1$ based on interference alignment technique. The non-existence of Exact-MSR code for $d<2k-3$ with $\beta=1$ is shown in [14]. The [15, 16] have shown the existence of exact-MSR codes for all $(n,k,d)$, while the size of message approaches infinity. By interference alignment technique, the [17] describes the Exact-MSR codes for the following cases: i) $k/n\leq 1/2,d\geq 2k-1$; and ii) $k\leq 3$. Rashmi et al. [9] present an construction for $(n,k,d\geq 2k-2)$ Exact-MSR codes via a product matrix framework. In Exact-MBR codes, the [10] presents the $(n=d+1,k,d)$ Exact-MBR codes with no arithmetic operations in node regeneration process, and the [9] presents the constructions for all feasible $(n,k,d)$ Exact-MBR codes. Furthermore, the cooperative repair codes [11, 12] are the generalized version of regenerating codes to address multiple node failures. ### 1.4 Results and organizations of the paper In this paper, we developed two classes of Exact-MBR codes. The first result is the repair-by-transfer code at $(n,k,d=n-1)$ Exact-MBR points via the congruences of skew-symmetric matrices. The systematic version and the partial downloading scheme are also proposed. The second result is the systematic version of Exact-MBR code for all feasible values of $(n,k,d)$ based on the framework defined by [9]. We design a new encoding matrix for systematic Exact-MBR code, and the partial downloading scheme are also proposed. To emphasize the contributions of the paper, Section 5 shows the comparisons of the proposed codes with the previous works. Notations and conventions are declared as follows. Throughout this paper, the operations and symbols are drawn from the field $GF(q)$. For a vector $x$, the underlined notation as $\underline{x}$ represents a row vector, and the over- lined notation as $\overline{x}$ represents a column vector. The $x[i]$ denotes the $i$-th element of the vector $x$. For a matrix $X$, the $X[i,j]$ denotes the entry at $i$-th row and $j$-th column. For a matrix (vector) $X$, the superscript ${}^{\prime}t^{\prime}$ on a matrix (vector) $X^{t}$ denotes the transpose of this $X$. The $I_{k}$ represents a $k\times k$ identity matrix. The rest of this paper is organized as follows. Section 2 reviews the previous works, such as repair-by-transfer codes and Exact-MBR codes. Section 3 presents the new class of repair-by-transfer codes. Section 4 presents the proposed systematic Exact-MBR codes based on partially systematic Reed-Solomon (PSRS) codes. Another construction approach is placed in Appendix. The comparisons and discussions are placed in Section 5. Section 6 concludes this paper. ## 2 Previous works This section reviews a number of related works, such as repair-by transfer codes [10], Exact-MBR codes [9], and partial downloading scheme [18]. ### 2.1 Repair-by-transfer codes [10] Figure 1: Graphical representation of the $(n=5,k=3,d=4)$ repair-by-transfer code proposed by [10]. This subsection briefly introduces the $(n,k,d=n-1)$ repair-by-transfer codes [10] by a simple example $(n=5,k=3,d=4)$ shown in Figure 1. In beginning, the $B=9$ message symbols are encoded with $(N,K)=\left({n\choose 2},B\right)$ MDS codes, to generate ${n\choose 2}$ code packets. Then each code packet is stored in two distinct nodes. The assignment rule can be visualized with a complete graph of $n$ vertices. As shown in Figure 1, each vertex is recognized as an individual node, and each edge corresponds to a distinct code packet. Each node (vertex) stores the $n-1$ code packets linked to this node. The node regeneration is very simple. If one node fails, the lost $n-1$ code packets in this node can be directly downloaded from each of other $n-1$ nodes. To reconstruct the data, a DC download the code packets from $k$ nodes. It can be shown that the DC accesses a total of $B$ distinct code packets, so the message symbols can be reconstructed via the $\left({n\choose 2},B\right)$ MDS decoding. The [10] suggests that the doubly extended RS codes can be chosen as the $(N,K)$ MDS coding technique, and the minimal field size is ${n\choose 2}\leq N+1$. ### 2.2 Exact-MBR codes [9] and partial downloading scheme [18] This section reviews the Exact-MBR codes [9] at (4) through product-matrix framework. In code constructions, the $B$ message symbols are formed as a $d\times d$ message matrix $M$, which is then multiplied by an $n\times d$ encoding matrix $\Psi$, resulting in an $n\times d$ $C=\Psi M.$ (6) code matrix. Let $\underline{c}_{i}^{t}$ denote the $i$-th row of $C$, for $1\leq i\leq n$. The $\underline{c}_{i}^{t}$ is computed through $\underline{c}_{i}^{t}=\underline{\psi}_{i}^{t}M,$ (7) where the $\underline{\psi}_{i}^{t}$ denotes the $i$-th row of $\Psi$. Each $\underline{c}_{i}^{t}$ is then stored in a network node with index $i$. The message matrix $M$ is expressed as $M=\begin{bmatrix}S&T\\\ T^{t}&\mathbf{0}\end{bmatrix},$ (8) where the $\mathbf{0}$ denotes a $(d-k)\times(d-k)$ zero matrix, the $T$ is a $k\times(d-k)$ matrix filled with $k(d-k)$ distinct message symbols, and the $S$ is a $k\times k$ symmetric matrix determined by ${k+1\choose 2}$ message symbols. The upper triangular part of $S$ is filled with the message symbols, and other entries assign the corresponding values such that the symmetry holds. Then, the encoding matrix $\Psi=\begin{bmatrix}\Phi&\Delta\end{bmatrix}$ (9) is the concatenation of a $n\times k$ matrix $\Phi$ with a $n\times(d-k)$ matrix $\Delta$. The coding matrix is chosen in such a way that: i) Any $d$ rows of $\Psi$ are linearly independent; ii) Any $k$ rows of $\Phi$ are linearly independent. For the non-systematic case, a feasible form of $\Psi$ is a Vandermonde matrix [9]. #### 2.2.1 Node-repairing process Suppose the node $f$ fails, and a replacement node is placed in the network to replace the functionality of the failure node. To reconstruct the code fragment (7) in the failure node, the replacement node connects to a subset of $d$ helper nodes $\\{h_{1},h_{2},\ldots,h_{d}\\}$. Then each helper node $h_{j}$ computes the scalar value $\upsilon_{h_{j}}=\underline{c}_{h_{j}}^{t}\underline{\psi}_{f},$ (10) and passes this value on to the replacement node. Thus, the replacement node gather $d$ downloaded symbols expressed as a $d$-element column vector $\Upsilon_{\mathrm{repair}}=[\upsilon_{h_{1}},\upsilon_{h_{2}},\ldots,\upsilon_{h_{d}}]^{t}$. By definition, the $\Upsilon_{\mathrm{repair}}$ possesses the equality $\Upsilon_{\mathrm{repair}}=C_{\mathrm{repair}}\underline{\psi}_{f}=\Psi_{\mathrm{repair}}M\underline{\psi}_{f},$ (11) where the $C_{\mathrm{repair}}$ is a $d\times\alpha$ matrix consisting of $d$ rows $\\{\underline{c}_{h_{1}}^{t},\underline{c}_{h_{2}}^{t},\ldots,\underline{c}_{h_{d}}^{t}\\}$ taken from the $C$, and the $\Psi_{\mathrm{repair}}$ is a $d\times d$ matrix consisting of $d$ corresponding encoding rows $\\{\underline{\psi}_{h_{1}}^{t}\underline{\psi}_{h_{2}}^{t},\ldots,\underline{\psi}_{h_{d}}^{t}\\}$. As the $\Psi_{\mathrm{repair}}$ is invertible by the first condition of the MBR encoding matrix, the decoding formula is formulated as $\Psi_{\mathrm{repair}}^{-1}\times\Upsilon_{\mathrm{repair}}=M\underline{\psi}_{f}=\underline{c}_{f},$ (12) which is the transpose of the desired fragment $\underline{c}_{f}^{t}$. #### 2.2.2 Data reconstruction process To reconstruct the message, the DC connects to $k$ active nodes $\\{i_{1},i_{2},\ldots,i_{k}\\}$ and then downloads $\\{\underline{c}_{i_{1}}^{t},\underline{c}_{i_{2}}^{t},\ldots,\underline{c}_{i_{k}}^{t}\\}$ from those connected nodes. The $k$ rows $\\{\underline{c}_{i_{1}}^{t},\ldots,\underline{c}_{i_{k}}^{t}\\}$ are formulated as a $k\times\alpha$ matrix $C_{\mathrm{DC}}$ following the order $[g_{1},\ldots,g_{k}]$. That is, each $\underline{c}_{i_{j}}^{t}$ is placed at the $g_{j}$-th row of the matrix $C_{\mathrm{DC}}$. In many cases, the sequence $[g_{1},\ldots,g_{k}]$ can be defined as a monotonically increasing sequence $g_{i}=i$, $1\leq i\leq k$. However, the proposed partial decoding scheme, addressed in Sec. 4.3, requires that systematic codeword fragments should be placed at a specific row of $C_{\mathrm{DC}}$. Based on above definitions, the DC accesses $k$ vectors expressed as $C_{\mathrm{DC}}=\Psi_{\mathrm{DC}}M,$ (13) where the $\Psi_{\mathrm{DC}}$ denotes a $k\times d$ matrix consisting of $k$ corresponding encoding rows $\\{\underline{\psi}_{i_{1}}^{t},\underline{\psi}_{i_{2}}^{t},\ldots,\underline{\psi}_{i_{k}}^{t}\\}$. By definition (9), the $k\times d$ matrix $\Psi_{\mathrm{DC}}$ can be represented as the concatenation of two sub-matrices, given by $\Psi_{\mathrm{DC}}=\begin{bmatrix}\Phi_{\mathrm{DC}}&\Delta_{\mathrm{DC}}\end{bmatrix},$ (14) where the $k\times k$ matrix $\Phi_{\mathrm{DC}}$ and the $k\times(d-k)$ matrix $\Delta_{\mathrm{DC}}$ are drawn from the sub-matrices of $\Phi$ and $\Delta$. Then the (13) can be rewritten as $C_{\mathrm{DC}}=\begin{bmatrix}\Phi_{\mathrm{DC}}S+\Delta_{\mathrm{DC}}T^{t}&\Phi_{\mathrm{DC}}T\end{bmatrix}.$ (15) The $C_{\mathrm{DC}}$ is split into two parts $C_{\mathrm{DC}}=[\begin{array}[]{cc}C_{\mathrm{DC}}^{\Phi}&C_{\mathrm{DC}}^{\Delta}\end{array}]$, where the $k$-column part $C_{\mathrm{DC}}^{\Phi}$ corresponds to $\Phi_{\mathrm{DC}}S+\Delta_{\mathrm{DC}}T^{t}$, and the $(d-k)$-column part $C_{\mathrm{DC}}^{\Delta}$ corresponds to $\Phi_{\mathrm{DC}}T$. Then the (15) is reformulated as $C_{\mathrm{DC}}^{\Phi}=\Phi_{\mathrm{DC}}S+\Delta_{\mathrm{DC}}T^{t};$ (16) $C_{\mathrm{DC}}^{\Delta}=\Phi_{\mathrm{DC}}T.$ (17) As the $\Phi_{\mathrm{DC}}$ is non-singular by the second condition of the encoding matrix, the DC can compute the matrix $T=\Phi_{\mathrm{DC}}^{-1}C_{\mathrm{DC}}^{\Delta}$, and subsequently, the $S=\Phi_{\mathrm{DC}}^{-1}(C_{\mathrm{DC}}^{\Phi}-\Delta_{\mathrm{DC}}T^{t})$. #### 2.2.3 Partial downloading scheme Chen and Wang [18] indicate that the above data reconstruction process involves a certain amount of redundancy. In the data reconstruction process, the DC completely downloads $k$ vectors $\\{\underline{c}_{i_{j}}^{t}\|j=1,\ldots,k\\}$ with length $d$ for each $\underline{c}_{i_{j}}^{t}$, to be used to reconstruct the $B={k+1\choose 2}+k(d-k)$ message symbols. As $(kd-B)={k\choose 2}\geq 0$, this process potentially downloads ${k\choose 2}$ redundant symbols. To avoid the wasted transmission resource, the [18] develops a partial downloading scheme on the Exact-MBR code. By the scheme, the DC can only download the $C_{\mathrm{DC}}^{\Delta}$ and the upper triangular part of $C_{\mathrm{DC}}^{\Phi}$. Totally, the DC exactly download $B$ symbols. In data reconstruction process, the sub-matrix $T$ can be solved by the equality (17). Let $D_{\mathrm{DC}}=C_{\mathrm{DC}}^{\Phi}-\Delta_{\mathrm{DC}}T^{t}$ (18) denote the solvable part in (16). Thus, the (16) is rewritten as $\Phi_{\mathrm{DC}}S=D_{\mathrm{DC}}.$ (19) In the scheme [18], the DC only downloads the upper triangular part of $C_{\mathrm{DC}}^{\Phi}$, so the upper triangular part of $D_{\mathrm{DC}}$ is also accessible. The main idea of solving (19) is to utilize the symmetry of $S$. The process can be divided into $k$ stages, and each stage solves a column of $S$ in the backward order. While the $d$-th column of $S$ have been solved, the $d$-th row of $S$ is also obtained by symmetry of $S$. The obtained $d$-th row of $S$ will be utilized in the later decoding stages. By such recursive decoding process, a symmetric matrix $S$ can be completely solved. ## 3 Repair-by-transfer codes This section proposes a new class of $(n,k,d=n-1)$ repair-by-transfer codes at (5). Upon describing the code constructions, two basic entities, termed as the message matrix $\hat{M}$ and the encoding matrix $\hat{\Phi}$, are defined as follows. The $\hat{M}$ is a $n\times n$ matrix constructed from two sub- matrices $\hat{S}$ and $\hat{T}$. The $\hat{S}$ is a $k\times k$ skew- symmetric matrix determined by ${k\choose 2}$ message symbols. The skew- symmetric matrix is defined as a square matrix $A$ satisfying $A=-A^{t}$. For each entry $A[i,j]$ in the skew-symmetric matrix, the equality holds $A[i,j]=-A[j,i]$. Note that the diagonal entries of skew-symmetric matrix $A$ are filled with zeros $A[i,i]=0$. By the above definition, the strictly upper triangular part of $\hat{S}$ (excluding the diagonal entries) is filled with ${k\choose 2}$ message symbols, and the lower triangular part $\hat{S}$ is filled with the corresponding values such that the skew symmetric condition holds. The remaining $B-{k\choose 2}=k(n-k)$ message symbols are formed as the second matrix $\hat{T}$ with $k\times(n-k)$. The $n\times n$ message matrix $\hat{M}$ is defined as $\hat{M}=\begin{bmatrix}\hat{S}&\hat{T}\\\ -\hat{T}^{t}&\mathbf{0}\end{bmatrix},$ (20) where the $\mathbf{0}$ denotes a $(n-k)\times(n-k)$ zero matrix. Notably, the $\hat{M}$ is also a skew-symmetric matrix. For the encoding matrix, this matrix is defined as a $n\times n$ square matrix of the form $\hat{\Psi}=\begin{bmatrix}\hat{\Phi}&\hat{\Delta}\end{bmatrix},$ (21) where the size of the matrix $\hat{\Phi}$ is $n\times k$, and the size of matrix $\hat{\Delta}$ is $n\times(n-k)$. The $\hat{\Psi}$ is chosen in such a way that i) Any $k$ rows of $\hat{\Phi}$ are linearly independent; ii) The matrix $\hat{\Psi}$ is non-singular. The above conditions can be met by choosing $\hat{\Phi}$ to be a $n\times k$ Vandermonde matrix, and the $\hat{\Delta}$ is defined as $\hat{\Delta}=\begin{bmatrix}\mathbf{0}\\\ I_{n-k}\end{bmatrix},$ (22) where the $\mathbf{0}$ is a $k\times(n-k)$ zero matrix, and the $I_{n-k}$ is a $(n-k)\times(n-k)$ identity matrix. By above definitions, the feasible range of $n$ is $n\leq q$ over $GF(q)$. Furthermore, the $\hat{\Phi}$ can adopt the extended Vandermonde matrix, which is the encoding matrix of the $(q+1,k)$ doubly extended RS code, as the form. Then the $n$ can be extended to $n=q+1$. By above matrices, the construction of repair-by-transform code is formulated as a congruence $\hat{C}=\hat{\Psi}\hat{M}\hat{\Psi}^{t}.$ There is a useful theorem used in the code constructions: The $n\times n$ matrix $\hat{C}$ congruent to a skew-symmetric matrix $\hat{M}$ is also skew- symmetric. Next, we modify the $\hat{C}$ to obtain a symmetric one $\check{C}$. Each entry in strictly lower triangular part of $\hat{C}$ is replaced with its negation value, resulting in a symmetric matrix $\check{C}$. Equivalently, for each row $\hat{c}_{j}^{t}$ in $\hat{C}$, a modified row $\check{c}_{j}^{t}$ in $\check{C}$ is obtained by assigning each entry to $\check{c}_{j}^{t}[i]=\begin{cases}\hat{c}_{j}^{t}[i]&\textrm{if }i\geq j;\\\ -\hat{c}_{j}^{t}[i]&\textrm{otherwise.}\end{cases}$ (23) The output $\check{C}$ is the generated codewords. The $n$ rows of $\check{C}$ are then respectively stored in $n$ distinct nodes. For $1\leq i\leq n$, the $n$-element row $\check{c}_{i}^{t}$ is stored in an individual network node indexed as $i$. As the diagonal entries $\\{\check{c}_{i}^{t}[i]=0\\}_{i=1}^{n}$ are always zeros, those zero symbols do not require storage space. Thus, each node takes $n-1$ units of memory space to store a row of $\check{C}$, and the parameter configuration (5) holds $\alpha=n-1$. _Example 1:_ We give an example for $(n=5,k=3)$ repair-by-transfer codes over $GF(4)$. By (5), other parameters are set as $d=\alpha=4$, $\beta=1$, and $B=9$. By the definition of message matrix (20), the matrix $\hat{M}$ is filled with 5 message symbols $\\{u_{i}\\}_{i=1}^{9}$ as follows: $\hat{M}=\begin{bmatrix}0&u_{1}&u_{2}&u_{3}&u_{4}\\\ -u_{1}&0&u_{5}&u_{6}&u_{7}\\\ -u_{2}&-u_{5}&0&u_{8}&u_{9}\\\ -u_{3}&-u_{6}&-u_{8}&0&0\\\ -u_{4}&-u_{7}&-u_{9}&0&0\end{bmatrix}.$ As $-u_{i}=u_{i}$ over the field of characteristic two, the $\hat{M}$ is also a symmetric matrix. For the encoding matrix, the matrix $\hat{\Phi}$ is chosen as the $5\times 3$ extended Vandermonde matrix given by $\hat{\Phi}=\begin{bmatrix}1&0&0\\\ 1&1&1\\\ 1&\omega&\omega^{2}\\\ 1&\omega^{2}&\omega^{4}\\\ 0&0&1\end{bmatrix},$ where the $\omega$ denotes the primitive element of $GF(4)$. By the $\hat{\Delta}$ defined in (22), the encoding matrix is expressed as $\hat{\Psi}=\begin{bmatrix}1&0&0&0&0\\\ 1&1&1&0&0\\\ 1&\omega&\omega^{2}&0&0\\\ 1&\omega^{2}&\omega^{4}&1&0\\\ 0&0&1&0&1\end{bmatrix}.$ As the $\hat{M}$ is skew-symmetric, the congruence $\hat{C}=\hat{\Psi}\hat{M}\hat{\Psi}^{t}$ is also skew-symmetric, expressed as $\hat{C}=\begin{bmatrix}0&c_{1}&c_{2}&c_{3}&c_{4}\\\ -c_{1}&0&c_{5}&c_{6}&c_{7}\\\ -c_{2}&-c_{5}&0&c_{8}&c_{9}\\\ -c_{3}&-c_{6}&-c_{8}&0&c_{10}\\\ -c_{4}&-c_{7}&-c_{9}&-c_{10}&0\end{bmatrix}.$ (24) Then each entry of strictly lower triangular part of $\hat{C}$ is replaced with its additive inverse value, resulting in $\check{C}=\begin{bmatrix}0&c_{1}&c_{2}&c_{3}&c_{4}\\\ c_{1}&0&c_{5}&c_{6}&c_{7}\\\ c_{2}&c_{5}&0&c_{8}&c_{9}\\\ c_{3}&c_{6}&c_{8}&0&c_{10}\\\ c_{4}&c_{7}&c_{9}&c_{10}&0\end{bmatrix}.$ (25) Notably, as the $\hat{C}$ is over the field of characteristic two, the $\check{C}=\hat{C}$ can be directly obtained without any arithmetic operations. ### 3.1 Node-repairing process The node-repairing process utilizes the symmetry of $\check{C}$. Suppose the node $h_{0}$ fails, and the failure node stores the vector $\check{c}_{h_{0}}^{t}$ at the $h_{0}$-th row of $\check{C}$. By the symmetry of $\check{C}$, the $h_{0}$-th row of $\check{C}$ is equivalent to the $h_{0}$-th column of $\check{C}$, whose entries (excluding the entry at main diagonal) are respectively stored in $n-1$ non-failure nodes. Thus, the replacement node can directly download the elements at the $h_{0}$-th column $\check{C}$ from other $n-1$ nodes. Let the $\check{c}_{j}^{t}[i]$ denote the $i$-th element of the row $\check{c}_{j}^{t}$. The formulation is given by $\check{c}_{h_{0}}^{t}[i]=\begin{cases}0&\textrm{if }i=h_{0};\\\ \check{c}_{i}^{t}[h_{0}]&\textrm{Otherwise.}\end{cases}$ (26) Consequently, this node-repairing process does not involve any arithmetic operations at the helper nodes and the replacement node, as illustrated in Example 1. In (25), if any one row of $\check{C}$ is erased, this row can be regenerated through the aid of corresponding column in $\check{C}$. ### 3.2 Data reconstruction process with full downloading In data reconstruction, the DC accesses the $k$ rows $\\{\check{c}_{i_{1}}^{t},\check{c}_{i_{2}}^{t},\ldots,\check{c}_{i_{k}}^{t}\\}$, which are respectively downloaded from $k$ connected nodes $\\{i_{1},i_{2},\ldots,i_{k}\\}$. To begin with, each row $\check{c}_{i_{j}}^{t}$, $1\leq j\leq k$, is restored to the original vector $\hat{c}_{i_{j}}^{t}$ via the inversion of formula (23). The restored results $\\{\hat{c}_{i_{1}}^{t},\ldots,\hat{c}_{i_{k}}^{t}\\}$ are formed as a $k\times n$ matrix $\hat{C}_{\mathrm{DC}}$ which is a sub-matrix of $\hat{C}$. By construction, the $\hat{C}_{\mathrm{DC}}$ possesses the equality $\hat{C}_{\mathrm{DC}}=\hat{\Psi}_{\mathrm{DC}}\hat{M}\hat{\Psi}^{t},$ (27) where the $k\times(n-1)$ matrix $\hat{\Psi}_{\mathrm{DC}}=\begin{bmatrix}\hat{\Phi}_{\mathrm{DC}}&\hat{\Delta}_{\mathrm{DC}}\end{bmatrix}=\left[\begin{array}[]{c}\psi_{i_{1}}^{t}\\\ \vdots\\\ \psi_{i_{k}}^{t}\end{array}\right]$ consists of the $k$ encoding rows of $\hat{C}_{\mathrm{DC}}$. As the $\hat{\Psi}$ is non-singular by the second condition of $\hat{\Psi}$, the $\hat{C}_{\mathrm{DC}}$ in (27) is then post-multiplied by its inversion $(\hat{\Psi}^{t})^{-1}$, to obtain a $k\times n$ matrix $\hat{D}_{\mathrm{DC}}=\hat{C}_{\mathrm{DC}}(\hat{\Psi}^{t})^{-1}=\hat{\Psi}_{\mathrm{DC}}\hat{M}.$ (28) The term $\hat{\Psi}_{\mathrm{DC}}\hat{M}$ in (28) is then decomposed as two parts: $\hat{\Psi}_{\mathrm{DC}}\hat{M}=\begin{bmatrix}\hat{\Phi}_{\mathrm{DC}}\hat{S}-\hat{\Delta}_{\mathrm{DC}}\hat{T}^{t}&\hat{\Phi}_{\mathrm{DC}}\hat{T}\end{bmatrix}.$ To elaborate the process, the $\hat{D}_{\mathrm{DC}}$ is split into two parts $\hat{D}_{\mathrm{DC}}=[\begin{array}[]{cc}\hat{D}_{\mathrm{DC}}^{\Phi}&\hat{D}_{\mathrm{DC}}^{\Delta}\end{array}]$, where the left part $\hat{D}_{\mathrm{DC}}^{\Phi}$ has $k$ columns and the right part $\hat{D}_{\mathrm{DC}}^{\Delta}$ has $(n-k)$ columns, so $\hat{D}_{\mathrm{DC}}^{\Phi}=\hat{\Phi}_{\mathrm{DC}}\hat{S}-\hat{\Delta}_{\mathrm{DC}}\hat{T}^{t};$ (29) $\hat{D}_{\mathrm{DC}}^{\Delta}=\hat{\Phi}_{\mathrm{DC}}\hat{T}.$ (30) By the first definition of $\hat{\Psi}$, the $\hat{\Phi}_{\mathrm{DC}}$ is non-singular. Thus, the DC can compute the matrix $\hat{T}=\hat{\Phi}_{\mathrm{DC}}^{-1}\hat{D}_{\mathrm{DC}}^{\Delta}$; and subsequently, the $\hat{S}=\hat{\Phi}_{\mathrm{DC}}^{-1}(\hat{D}_{\mathrm{DC}}^{\Phi}+\hat{\Delta}_{\mathrm{DC}}\hat{T}^{t})$. ### 3.3 Systematic version of repair-by-transfer codes To construct a systematic version of repair-by-transfer codes, a message- symbol remapping procedure is employed to determine the entries of $\hat{M}$. Without loss of generality, we declare that the source data are embedded in the first $k$ rows of $\hat{C}$. To reduce the computational cost, the $\hat{\Phi}$, which is the sub-matrix of $\hat{\Psi}$, is defined as the encoding matrix of $(n,k)$ systematic RS codes. The matrix contains two parts expressed as $\hat{\Phi}=\begin{bmatrix}I_{k}\\\ \check{\Phi}\end{bmatrix},$ (31) where the first $k$ rows of $\hat{\Phi}$ is an identity matrix $I_{k}$. By the $\hat{\Delta}$ defined in (22), the encoding matrix $\hat{\Psi}$ is thus formulated as $\hat{\Psi}=\begin{bmatrix}I_{k}&\mathbf{0}\\\ \check{\Phi}&I_{n-k}\end{bmatrix}.$ By the above encoding matrix, the encoding formula $\hat{C}=\hat{\Psi}\hat{M}\hat{\Psi}^{t}$ can be rewritten as $\hat{C}=\begin{bmatrix}S&S\check{\Phi}^{t}+T\\\ \check{\Phi}S-T^{t}&\check{\Phi}S\check{\Phi}^{t}+\check{\Phi}T-T^{t}\check{\Phi}^{t}\end{bmatrix}.$ To achieve the systematic condition, the first $k$ rows of $\hat{C}$, expressed as $\begin{bmatrix}S&S\check{\Phi}^{t}+T\end{bmatrix}$, are defined as the source data. Let $U=[\begin{array}[]{cc}U_{\mathrm{L}}&U_{\mathrm{R}}\end{array}]$ denote a $k\times n$ matrix consisting of $B$ source symbols. The $U_{\mathrm{L}}$ is a $k\times k$ skew-symmetric matrix whose strictly upper-triangular part is filled with ${k\choose 2}$ source symbols, and other entries are filled with the corresponding values to satisfy the skew-symmetry condition. The $U_{\mathrm{R}}$ is a $k\times(n-k)$ matrix filled with $k\times(n-k)$ source symbols. The systematic condition gives two equations $S=U_{\mathrm{L}};S\check{\Phi}^{t}+T=U_{\mathrm{R}}.$ By above two equations, the $\hat{C}$ can be rewritten as $\hat{C}=\begin{bmatrix}U_{\mathrm{L}}&U_{\mathrm{R}}\\\ -U_{\mathrm{R}}^{t}&V\end{bmatrix},$ where the $V$ is a $(n-k)\times(n-k)$ matrix defined as $V=\check{\Phi}U_{\mathrm{R}}-U_{\mathrm{R}}^{t}\check{\Phi}^{t}-\check{\Phi}U_{\mathrm{L}}\check{\Phi}^{t}$. As other three parts of $\hat{C}_{\mathrm{DC}}$, namely $U_{\mathrm{L}}$ and $\pm U_{\mathrm{R}}$, are available without the arithmetic computations, the matrix $V$ is the remaining unknown objective to be computed. It is noted that the matrix $V$ is a skew-symmetric matrix, so is the $\hat{C}$. The computation of $T$ involves the matrix product $\check{\Phi}U_{\mathrm{R}}$ and the congruence $\check{\Phi}U_{\mathrm{L}}\check{\Phi}^{t}$, and the term $U_{\mathrm{R}}^{t}\check{\Phi}^{t}$ can be directly obtained via transposing the result $\check{\Phi}U_{\mathrm{R}}$. As the $\check{\Phi}$ identifies the encoding matrix of parity part in the $(n,k)$ systematic RS codes, the product $\check{\Phi}U_{\mathrm{R}}$ denotes the parity parts of RS codes for each column of $U_{\mathrm{R}}$. For the congruence $\check{\Phi}U_{\mathrm{L}}\check{\Phi}^{t}$, the $(n,k)$ systematic RS encoding is applied on each column of $U_{\mathrm{L}}$ to obtain the parity part $\check{\Phi}U_{\mathrm{L}}$. Then the $(n,k)$ systematic RS encoding is applied on each row of $\check{\Phi}U_{\mathrm{L}}$, resulting in the $\check{\Phi}U_{\mathrm{L}}\check{\Phi}^{t}$ at the parity part. By above steps, the product $\check{\Phi}U_{\mathrm{R}}$ requires $O(k(n-k)^{2})$ operations, and the transformation $\check{\Phi}U_{\mathrm{L}}\check{\Phi}^{t}$ requires $O(2k^{2}(n-k))$ operations. ### 3.4 Partial downloading scheme For the data reconstruction in Sec. 3.2, we suppose that the DC completely downloads the $k$ vectors $\\{\check{c}_{i_{j}}^{t}\|j=1,\ldots,k\\}$, and the length of each vector $\check{c}_{i_{j}}^{t}$ is $n-1$. Thus, the total number of downloaded symbols is $(n-1)k$, which is much larger than the size of message $B=(n-1)k-{k\choose 2}$. By utilizing the symmetry of $\check{C}$, the DC can exactly download $B$ symbols. For any two distinct codeword vectors $\check{c}_{i_{j}}^{t}$ and $\check{c}_{i_{l}}^{t}$ in $\check{C}$, we have $\check{c}_{i_{j}}^{t}[i_{l}]=\check{c}_{i_{l}}^{t}[i_{j}]$ by the symmetric property, so the DC can download this symbol only from either the node $i_{j}$ or the node $i_{l}$. Based on this observation, the $k$ connected nodes can avoid the total of ${k\choose 2}$ symbols to be transmitted. An simple transmission strategy is that, the first node $i_{1}$ transmits the whole $n-1$ symbols $\check{c}_{i_{1}}^{t}$ to the DC. Then the second node $i_{2}$ can only transmit $n-2$ symbols of $\check{c}_{i_{2}}^{t}$ to the DC, as the symbol $\check{c}_{i_{2}}^{t}[i_{1}]=\check{c}_{i_{1}}^{t}[i_{2}]$ does not need to be transmitted. Inductively, the connected node $i_{j}$ can only transmit $n-j$ symbols of $\check{c}_{i_{j}}^{t}$ to the DC, for $j=1,\ldots,n$. The above policy is simple, but the data throughputs for each node is imbalanced. Thus, an alternative transmission policy is presented in the following. It is noted that each node can save $(k-1)/2$ symbols of data transmission on average, and this value is achieved for odd $k$ by the proposed transmission policy. For even $k$, as the value $(k-1)/2$ is not an integer, the proposed transmission policy can save $k/2-1$ symbols in each odd-index node, and $k/2$ symbols in each even-index node. Given any two connected nodes with indices $i_{j}$, $i_{l}$ and $1\leq i,l\leq k$, we define a decision criterion as $D(j,l)=\begin{cases}min\\{j,l\\}&\textrm{if }j+l\textrm{ is even};\\\ max\\{j,l\\}&\textrm{otherwise.}\end{cases}$ (32) As any two distinct nodes $i_{j}$ and $i_{l}$ simultaneously store a common symbol, the $D(j,l)\in\\{j,l\\}$ returns the index of the chosen node to avoid the transmission of this common symbol. Hence the DC downloads this element from another un-chosen node. Two examples are given in Figure 2 tabulating the exhaustive outputs of $D(i,j)$ for $k=5$ and $6$. In the case $k=5$, each node omits two symbols in transmission. In the case $k=6$, the nodes $\\{g_{1},g_{3},g_{5}\\}$ omit two symbols in transmission, and the nodes $\\{g_{2},g_{4},g_{6}\\}$ omit three symbols in transmission. The valid of decision criterion (32) is explained as follows. Given a node indexed by $X$, we consider the output of $D(X,y)$ for $y=1,\ldots,k$. If the (32) outputs $X=D(X,y)$ for a specific $y$, the node $X$ can omit the transmission of a symbol, and DC will download this symbol from another node $y$. To satisfy the equality $X=D(X,y)$, the range of $y$ are drawn from $y\in\\{\ldots,X-1-2i,\ldots,X-1,X+2,\ldots,X+2i,\ldots\\}$ and $1\leq y\leq k$. Thus, there are about $k/2$ distinct symbols of $y$, and the condition for bandwidth balance holds. \| 5 \| 4 \| 3 \| 2 ---\|---\|---\|---\|--- 1 \| 1 \| 4 \| 1 \| 2 2 \| 5 \| 2 \| 3 \| 3 \| 3 \| 4 \| \| 4 \| 5 \| \| \| (a) \| 6 \| 5 \| 4 \| 3 \| 2 ---\|---\|---\|---\|---\|--- 1 \| 6 \| 1 \| 4 \| 1 \| 2 2 \| 2 \| 5 \| 2 \| 3 \| 3 \| 6 \| 3 \| 4 \| \| 4 \| 4 \| 5 \| \| \| 5 \| 6 \| \| \| \| (b) Figure 2: Two examples of the outputs of decision criterion $D(j,l)$. (a) $k=5$. (b) $k=6$. ## 4 Systematic Exact-MBR coding algorithm Based on the framework of $(n,k,d)$ Exact-MBR codes [9] in Sec. 2.2, this section presents a systematic form of encoding matrix $\Psi$, where the feasible range of $n$ are $n\leq q$ over $GF(q)$. Then the partial downloading scheme is developed on the proposed Exact-MBR codes. Upon describing the proposed encoding matrix, the encoding (6) can be divided into $\alpha$ individual columns given by $\overline{c}_{i}=\Psi\overline{m}_{i},$ (33) where the $\overline{m}_{i}$ indicates the $i$-th column of $M$, and the result $\overline{c}_{i}$ is the $i$-th column in $C$. The (33) can be rewritten as $\overline{c}_{i}=\Psi\overline{m}_{i}=\begin{bmatrix}\Phi&\Delta\end{bmatrix}\left[\begin{array}[]{c}\overline{m}_{i}^{a}\\\ \overline{m}_{i}^{b}\end{array}\right]=\Phi\overline{m}_{i}^{a}+\Delta\overline{m}_{i}^{b},$ (34) where the $\overline{m}_{i}^{a}$ denotes the $k$-element vector located in the upper part of the $\overline{m}_{i}$, and the $\overline{m}_{i}^{b}$ denotes the remaining $(d-k)$-elements located in the lower part of the $\overline{m}_{i}$. By the first condition of Exact-MBR encoding matrix, the $\overline{m}_{i}$ can be reconstructed from arbitrary $d$ elements in $\overline{c}_{i}$. By the second condition, if the term $\Delta\overline{m}_{i}^{b}$ is given, the $\overline{m}_{i}^{a}$ can be reconstructed from arbitrary $k$ elements in $\overline{c}_{i}$. Under above observations, Section 4.1 presents a class of modified version of Reed-Solomon codes, termed as partially systematic Reed- Solomon (PSRS) codes, to satisfy those conditions. Section 4.2 shows that the encoding matrix of the systematic Exact-MBR codes. Section 4.3 presents the partial downloading scheme. ### 4.1 Partially systematic Reed-Solomon codes We define the partially systematic Reed-Solomon (PSRS) code associated with three parameters $(n,k,d)$ where $k\leq d<n$. The $n$ is the codeword length, the $d$ is the message length, and the $k$ is the length of systematic part. The input is expressed as a $d$-element vector $\underline{c}=\begin{bmatrix}\underline{a}&\underline{b}\end{bmatrix}$, where the sub-vector $\underline{a}=[a_{1}...a_{k}]$ denotes the $k$ systematic symbols, and the sub-vector $\underline{b}=[b_{1}...b_{d-k}]$ denotes the remaining $d-k$ non-systematic symbols. By definition, the systematic part $\underline{a}$ is embedded in the first $k$ elements of the generated codeword. This subsection presents the constructions of $(n,k,d)$ PSRS codes via the polynomial evaluation approach. Let the $G(x)$ denote the coding polynomial constructed from the message $\underline{c}$. The degree of $G(x)$ is $deg(C(x))<d$. The codeword symbols are the evaluations of $C(x)$ at $n$ distinct points: $\\{C(x_{1}),C(x_{2}),\ldots,C(x_{n})\\}.$ (35) As the code is over $GF(q)$, the code suffices for $n\leq q$. By the partially systematic condition, the first $k$ codeword symbols are equivalent to the systematic message symbols. Thus, $C(x_{i})=a_{i},\forall i=1,2,\ldots,k.$ (36) In the following, the $C(x)$ is properly defined to satisfy the partial systematic condition. The $C(x)$ is defined as the sum of two polynomials $C(x)=\Phi(x)+\Delta(x),$ (37) where the polynomial $\Phi(x)$ is constructed from $\underline{a}$, and the $\Delta(x)$ is constructed from $\underline{b}$. The $\Phi(x)$, and $deg(\Phi(x))<k$, is defined as $\Phi(x)=\sum_{i=1}^{k}a_{i}\prod_{j\neq i}\frac{x-x_{j}}{x_{i}-x_{j}}.$ (38) This follows the form of Lagrange polynomial. Thus, the $\Phi(x)$ possesses the systematic property: $\Phi(x_{i})=a_{i},\forall i=1,2,\ldots,k.$ The polynomial $\Delta(x)$ is defined as the multiplication of two polynomials: $\Delta(x)=\Gamma(x)B(x).$ (39) The polynomial $\Gamma(x)$ has $k$ roots located in the evaluation points of systematic part: $\Gamma(x)=\prod_{i=1}^{k}(x-x_{i}).$ (40) The $B(x)$ is constructed from the $(d-k)$-element vector $\underline{b}$. The $B(x)$ can be chosen as the systematic or non-systematic form. For example, a non-systematic form with geometric progression is expressed as $B(x)=\sum_{i=1}^{d-k}b_{i}x^{i-1}.$ (41) By the above definitions, it can be shown that the partial systematic condition (36) holds: $\displaystyle C(x_{i})$ $\displaystyle=\Phi(x_{i})+\Delta(x_{i})$ (42) $\displaystyle=\Phi(x_{i})+0\times B(x_{i})=a_{i},\forall i=1,2,\ldots,k.$ In summary, the encoding algorithm includes four major steps listed as follows: i). Compute the coefficients of $\Phi(x)$. ii). Compute the product $\Delta(x)=\Gamma(x)B(x)$, where the coefficients of $\Gamma(x)$ can be computed in advance. iii). Compute the summation $C(x)=\Phi(x)+\Delta(x)$. iv). Evaluate the values $\\{C(x_{1}),\ldots,C(x_{n})\\}$ to obtain the codeword symbols. If the encoding algorithm is implemented in the native way, the computational complexities of the four steps are $O(k^{2})$, $O(k(d-k))$, $O(k)$, and $O(dn)$, respectively. To reduce the complexity complexity, we observe that the fast Fourier transforms (FFT) can be utilized to reduce the computational cost in steps (i), (ii) and (iv). The conceptual ideas are addressed below. In step (i), the (38) can be calculated via fast Lagrange interpolation [22] with complexity $O(k\log^{2}k)$. Alternatively, the fast Reed-Solomon encoding algorithms can also be used in (38). If the code is operated on Fermat field $GF(q+1),q\in\\{2,4,16,65536\\}$, the (38) can be calculated via inverse fast Fourier transform with complexity $O(k\log k)$ (see [24] and [25]). If the code is operated on finite field with characteristic two $GF(q),q\in\\{2,4,8,\ldots\\}$, the [23] proposed an coding algorithm with complexity $O(q\log^{2}q)$. The step (ii) is a polynomial multiplication. By using FFT, the complexity can be reduced to $O(d\log d)$. In step (iv), the polynomial evaluations can be computed with FFT, and the complexity is $O(n\log n)$. It is noted that the PSRS codes can also be implemented with generator polynomials. The details are placed in appendix. #### 4.1.1 Full erasure decoding from $d$ codeword symbols The message vector $\underline{c}$ can be reconstructed from arbitrary $d$ out of $n$ codeword symbols $\\{y_{i}=C(z_{i})\|1\leq i\leq d\\}$. By the subset of codeword symbols, the $C(x)$ is constructed via Lagrange interpolation: $C(x)=\sum_{i=1}^{d}y_{i}\prod_{j\neq i}\frac{x-z_{j}}{z_{i}-z_{j}}.$ (43) The $C(x)$ is then divided by $\Gamma(x)$ to obtain a quotient $B(x)$ and a remainder $\Phi(x)$. The $k$ evaluations $a_{i}=\Phi(x_{i})$, $1\leq i\leq k$, are the systematic part $\underline{a}$, and the coefficients of $B(x)$ are the non-systematic part $\underline{b}$. #### 4.1.2 Partial erasure decoding from $k$ codeword symbols Suppose the non-systematic part $\underline{b}$ is given. In this case, we shows that the systematic part $\underline{a}$ can be reconstructed from arbitrary $k$ out of $n$ codeword symbols $\\{y_{i}=C(z_{i})\|1\leq i\leq k\\}$. By the given $\underline{b}$, the polynomial $\Delta(x)$ can be constructed. Then the $k$ evaluation values of $\Phi(x)$ are calculated via $\Phi(z_{i})=C(z_{i})-\Delta(z_{i}),\forall i=1...k.$ (44) By the $k$ evaluation values of $\Phi(x)$, the $\Phi(x)$ can be interpolated via Lagrange polynomial, and the $\underline{a}$ is the $k$ evaluations $a_{i}=\Phi(x_{i})$. ### 4.2 Encoding matrix of proposed Exact-MBR codes As the $(n,k,d)$ PSRS codes satisfy the conditions of Exact-MBR codes, the encoding matrix of $(n,k,d)$ PSRS codes can be chosen as the $\Psi$. For the systematic part $\underline{a}$, the coding polynomial $\Phi(x)$ formulates a generator matrix corresponding to the component $\Phi$ in encoding matrix $\Psi$. By the definition of $\Phi(x)$, the entries of matrix $\Phi$ are $\Phi[l,i]=\prod_{j=1;j\neq i}^{k}\frac{x_{l}-x_{j}}{x_{i}-x_{j}},\textrm{for }i=1,\ldots,k.$ (45) Consequently, the first $k$ rows of $\Phi$ is a $k\times k$ identity matrix $I_{k}$. For the non-systematic part $\underline{b}$, the coding polynomial $\Delta(x)$ formulates a generator matrix corresponding to the component $\Delta$ in encoding matrix $\Psi$. By the definition of $\Delta(x)$, the entries of matrix $\Delta$ are $\Delta[l,i]=x_{l}^{(i-1)}\Gamma(x_{l})\textrm{, for }i=1,\ldots,d-k.$ (46) As $\Gamma(x_{l})=0$ for $1\leq l\leq k$, the first $k$ rows of $\Delta$ are entirely filled with zeros. Then the encoding matrix $\Psi$ is obtained by combining the $\Phi$ and $\Delta$. Thus, the first $k$ rows of $\Psi$ are in the form $\begin{bmatrix}I_{k}&\mathbf{0}\end{bmatrix}$, so that the corresponding first $k$ rows of the code matrix $C$ are expressed as $[\begin{array}[]{cc}S&T\end{array}]$. Hence, the proposed Exact-MBR code is systematic. As stated previously, the proposed $\Psi$ satisfies the two conditions of Exact-MBR encoding matrix, which enables the node-repairing algorithm and data reconstruction algorithm addressed in Sec. 2.2. _Example 2:_ We give an example for $(n=6,k=3,d=4)$ Exact-MBR codes over $GF(7)$. By (4), other parameters are set as $\alpha=4$, $\beta=1$, and $B=9$. By the definition of message matrix (8), the matrices $M$ is filled with 9 message symbols $\\{u_{i}\\}_{i=1}^{9}$. The $S$, $T$ and $M$ are given by $\displaystyle S$ $\displaystyle=\begin{bmatrix}u_{1}&u_{2}&u_{3}\\\ u_{2}&u_{5}&u_{6}\\\ u_{3}&u_{6}&u_{8}\end{bmatrix},T=\begin{bmatrix}u_{4}\\\ u_{7}\\\ u_{9}\end{bmatrix};$ $\displaystyle M$ $\displaystyle=\begin{bmatrix}u_{1}&u_{2}&u_{3}&u_{4}\\\ u_{2}&u_{5}&u_{6}&u_{7}\\\ u_{3}&u_{6}&u_{8}&u_{9}\\\ u_{4}&u_{7}&u_{9}&0\end{bmatrix}.$ The coding polynomial $C(x)$ of $(n=6,k=3,d=4)$ PSRS code is chosen as $\displaystyle\Phi(x)=$ $\displaystyle a_{1}\times\frac{(x-2)(x-3)}{2}+a_{2}\times\frac{(x-1)(x-3)}{6}$ $\displaystyle+a_{3}\times\frac{(x-1)(x-2)}{2};$ $\displaystyle\Delta(x)=$ $\displaystyle(x-1)(x-2)(x-3)b_{1}.$ By above definitions, the corresponding matrices $\Phi$ and $\Delta$ are as follows: $\Phi=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ 1&4&3\\\ 3&6&6\\\ 6&6&3\end{bmatrix};\Delta=\begin{bmatrix}0\\\ 0\\\ 0\\\ 6\\\ 3\\\ 4\end{bmatrix}.$ The encoding matrix $\Psi=\begin{bmatrix}\Phi&\Delta\end{bmatrix}$ is the combination of $\Phi$ and $\Delta$. ### 4.3 Partial downloading scheme This subsection presents the partial downloading scheme on the proposed systematic Exact-MBR codes. Similar to the [18], the proposed scheme only downloads the entire $C_{\mathrm{DC}}^{\Delta}$ and the lower (or upper, alternatively) triangular part of $C_{\mathrm{DC}}^{\Phi}$. Precisely, each connected node $i_{j}$ passes a portion of the code fragment $c_{i_{j}}^{t}$ in the lower/upper triangular part of $C_{\mathrm{DC}}$. By (17), the $T$ can be successfully solved. Then the lower/upper triangular part of $D_{\mathrm{DC}}$ can be computed via (18). The two cases are respectively considered as follows. #### 4.3.1 Data collector downloads the lower triangular part of $C_{\mathrm{DC}}^{\Phi}$ In this case, the DC can access the lower triangular part of $D_{\mathrm{DC}}$. The computational structure can be divided into $k$ stages, and the $l$-th stage solves the $l$-th column $\overline{s}_{l}$ of $S$. In the first stage, as the first column of $D_{\mathrm{DC}}$ are fully located in the lower triangular part of $D_{\mathrm{DC}}$, the first column $\overline{s}_{1}$ of $S$ can be solved successfully. By the symmetry of $S$, the first row of $S$ is also obtained $\underline{s}_{1}=\overline{s}_{1}^{t}$. Let $\underline{i}_{l}^{t}$ denote a row vector with one at the $l$-th position and zeros elsewhere. By the definition of proposed encoding matrix, the obtained $\underline{s}_{1}$ is at the first row (systematic part) of $\Phi$. Thus, we have the equation $\underline{i}_{1}^{t}S=\underline{s}_{1}$ which will be utilized in the upcoming decoding stages. In the $l$-th stage, $1\leq l\leq k$, the DC can access the $\\{\underline{d}_{i_{l}}^{t}[l],\ldots,\underline{d}_{i_{k}}^{t}[l]\\}$ in the $l$-th column of lower triangular part of $D_{\mathrm{DC}}$, and the corresponding encoding rows are $\\{\underline{\phi}_{i_{l}}^{t},\ldots,\underline{\phi}_{i_{k}}^{t}\\}$. In the previous stages, we obtain $l-1$ equations: $\underline{i}_{j}^{t}\overline{s}_{l}=\underline{s}_{j}[l],\forall j=1,\ldots,l-1.$ It is noted that the $\\{\underline{i}_{j}^{t}\|1\leq j\leq l-1\\}$ are the first $l-1$ rows of $\Phi$. The above equations are combined to obtain $\begin{bmatrix}\underline{i}_{1}^{t}\\\ \vdots\\\ \underline{i}_{l-1}^{t}\\\ \underline{\phi}_{i_{l}}^{t}\\\ \vdots\\\ \underline{\phi}_{i_{k}}^{t}\end{bmatrix}\overline{s}_{l}=\begin{bmatrix}\underline{s}_{1}[l]\\\ \vdots\\\ \underline{s}_{l-1}[l]\\\ \underline{d}_{i_{l}}^{t}[l]\\\ \vdots\\\ \underline{d}_{i_{k}}^{t}[l]\end{bmatrix}.$ (47) Let the $D_{1}^{l}$ denote the matrix at the left-hand-side of (47). To solve the $\overline{s}_{l}$ successfully, the $D_{1}^{l}$ should be non-singular. Then we have $\underline{s}_{l}=\overline{s}_{l}^{t}$, and the $\underline{i}_{l}^{t}S=\underline{s}_{l}$ can be utilized in the upcoming decoding stages. The non-singularity of $D_{1}^{l}$ is discussed below. In the $D_{1}^{l}$, the set $\underline{i}^{t}=\\{\underline{i}_{1}^{t},\ldots,\underline{i}_{l-1}^{t}\\}$ are the first $l-1$ rows of $\Phi$, and the set $\underline{\phi}^{t}=\\{\underline{\phi}_{i_{l}}^{t},\ldots,\underline{\phi}_{i_{k}}^{t}\\}$ are $k-l$ rows in $\Phi$. As any $k$ rows of $\Phi$ are non-singular, the $D_{1}^{l}$ is also non-singular, as long as the two sets are mutually exclusive $\underline{i}^{t}\cap\underline{\phi}^{t}=\emptyset$. To satisfy this condition, the order of fragments $[g_{1},\ldots,g_{k}]$ in $C_{\mathrm{DC}}$ should follow a special condition: For the systematic fragment $\underline{c}_{l}^{t}$, $1\leq l\leq k$, downloaded from the node $i_{j}$, the $\underline{c}_{l}^{t}$ is placed at the $g_{j}$-th row of $C_{\mathrm{DC}}$, where $g_{j}\leq l$. _Example 3:_ By following the codes given by Example 1, we assume that the DC connects to nodes 1, 2, and 4 respectively corresponding to encoding rows $\begin{bmatrix}1&0&0&0\end{bmatrix}$, $\begin{bmatrix}0&1&0&0\end{bmatrix}$ and $\begin{bmatrix}1&4&3&6\end{bmatrix}$. The three rows of $C_{\mathrm{DC}}$ are arranged as $C_{\mathrm{DC}}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 1&4&3&6\end{bmatrix}M.$ The DC downloads the whole $C_{\mathrm{DC}}^{\Delta}$ and the lower triangular part of $C_{\mathrm{DC}}^{\Phi}$. The $C_{\mathrm{DC}}^{\Delta}$ possesses the equation given by $C_{\mathrm{DC}}^{\Delta}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 1&4&3\end{bmatrix}T.$ By the equation, the $T$ can be solved to obtain $\\{\tilde{u}_{4},\tilde{u}_{7},\tilde{u}_{9}\\}$, where the tilde symbol $\tilde{\bullet}$ indicates the solved terms. By the solved $T$, the DC calculates the lower triangular part of $D_{\mathrm{DC}}$ via $D_{\mathrm{DC}}=C_{\mathrm{DC}}^{\Phi}-\begin{bmatrix}0\\\ 0\\\ 6\end{bmatrix}\begin{bmatrix}\tilde{u}_{4}&\tilde{u}_{7}&\tilde{u}_{9}\end{bmatrix}.$ Let $D[i,j]$ denote the entry of $D_{\mathrm{DC}}$ at the $i$-th row and $j$-th column. The accessible part of $D_{\mathrm{DC}}$ is $\begin{bmatrix}D[1,1]&-&-\\\ D[2,1]&D[2,2]&-\\\ D[3,1]&D[3,2]&D[3,3]\end{bmatrix}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 1&4&3\end{bmatrix}S,$ where the notation ”$-$” indicates the inaccessible entries. Firstly, by the first column of $D_{\mathrm{DC}}$, the first column of $S$ is solved. The solved symbols possesses the equality: $\begin{bmatrix}\tilde{u}_{1}&\tilde{u}_{2}&\tilde{u}_{3}\end{bmatrix}=\begin{bmatrix}1&0&0\end{bmatrix}S.$ (48) Secondly, to decode the second column of $S$, we have $\begin{bmatrix}\tilde{u}_{2}\\\ D[2,2]\\\ D[3,2]\end{bmatrix}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 1&4&3\end{bmatrix}\begin{bmatrix}u_{2}\\\ u_{5}\\\ u_{6}\end{bmatrix}.$ Then the symbols $\\{u_{5},u_{6}\\}$ are solved. The solved symbols possess the equality: $\begin{bmatrix}\tilde{u}_{2}&\tilde{u}_{5}&\tilde{u}_{6}\end{bmatrix}=\begin{bmatrix}0&1&0\end{bmatrix}S.$ (49) By the third column of $D_{\mathrm{DC}}$ and the (48)(49), we have $\begin{bmatrix}\tilde{u}_{3}\\\ \tilde{u}_{6}\\\ D[3,3]\end{bmatrix}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 1&4&3\end{bmatrix}\begin{bmatrix}u_{3}\\\ u_{6}\\\ u_{8}\end{bmatrix}.$ Then the symbol $u_{8}$ is solved successfully. #### 4.3.2 Data collector downloads the upper triangular part of $C_{\mathrm{DC}}^{\Phi}$ In this case, the DC accesses the upper triangular part of $D_{\mathrm{DC}}$ defined in (19). The steps are very similar to the above decoding scheme. The decoding structure can be expressed as $k$ stages, and each stage extracts a column of $S$ in backward order. That is, the $l$-th stage extracts the $(k+1-l)$-th column $\overline{s}_{k+1-l}$ of $S$. In the $l$-th stage, the DC can access the $\\{\underline{d}_{i_{1}}^{t}[l],\ldots,\underline{d}_{i_{k+1-l}}^{t}[l]\\}$ taken from the $(k+1-l)$-th column of $D_{\mathrm{DC}}$ in upper triangular part, and the corresponding encoding rows are $\\{\underline{\phi}_{i_{1}}^{t},\ldots,\underline{\phi}_{i_{k+1-l}}^{t}\\}$. Furthermore, we also have $l-1$ equations by the previous stages: $\underline{i}_{j}^{t}\overline{s}_{l}=\underline{s}_{j}[l],\forall j=k+2-l,\ldots,k.$ Those equations are combined to obtain $\begin{bmatrix}\underline{\phi}_{i_{1}}^{t}\\\ \vdots\\\ \underline{\phi}_{i_{k+1-l}}^{t}\\\ \underline{i}_{k+2-l}^{t}\\\ \vdots\\\ \underline{i}_{k}^{t}\end{bmatrix}\overline{s}_{k+1-l}=\begin{bmatrix}\underline{d}_{i_{1}}^{t}[l]\\\ \vdots\\\ \underline{d}_{i_{k+1-l}}^{t}[l]\\\ \underline{s}_{k+2-l}[l]\\\ \vdots\\\ \underline{s}_{k}[l]\end{bmatrix}.$ (50) Let the $D_{2}^{l}$ denote the left-hand-side matrix in (50). To decode the $\overline{s}_{k+1-l}$, the $D_{2}^{l}$ should be non-singular, and this condition induces that $\\{\underline{\phi}_{i_{1}}^{t},\ldots,\underline{\phi}_{i_{k+1-l}}^{t}\\}\cap\\{\underline{i}_{k+2-l}^{t},\ldots,\underline{i}_{k}^{t}\\}=\emptyset$, for $1\leq l\leq k$. By the above condition, the systematic fragment $\underline{c}_{l}^{t}$ downloaded from the node $i_{j}$ is placed at the $g_{j}$-th row of $C_{\mathrm{DC}}$, where $1\leq l\leq g_{j}\leq k$. Then the $\overline{s}_{k+1-l}$ can be solved successfully, and the formula $\underline{i}_{k+1-l}^{t}S=\overline{s}_{k+1-l}^{t}$ is utilized in the upcoming decoding stages. #### 4.3.3 The time-sharing policy to balance the bandwidth requirements on each connected node In the above two partial downloading schemes, both partial downloading schemes have the disadvantage that the transmission amounts for $k$ connected nodes are excessively unbalanced. To overcome this drawback, we can iteratively switch the two partial downloading schemes during the whole transmission rounds. Specifically, if a node $i_{j}$ transmits the elements of a code fragment in the lower triangular of $C_{\mathrm{DC}}^{\Phi}$ at this transmission round, this node will transmit the elements of next code fragment in the upper triangular of $C_{\mathrm{DC}}^{\Phi}$ at the next transmission round. By this time-sharing policy, each node transmits $d-(k-1)/2$ symbols in each transmission round on average. As stated in Sections 4.3.1 and 4.3.2, the two partial downloading schemes respectively give two different conditions on the order $[g_{1},\ldots,g_{k}]$ of the downloaded fragments in $C_{\mathrm{DC}}$. Since the time-sharing policy iteratively applies two partial downloading schemes, the two conditions should be satisfied simultaneously. The intersection of two conditions is that, the systematic fragment $\underline{c}_{l}^{t}$ downloaded from the node $i_{j}$ is placed at the $g_{j}$-th row of $C_{\mathrm{DC}}$, where $1\leq g_{j}=l\leq k$. ## 5 Comparisons and discussions In this section, we compare the proposed codes with prior works. The results are briefly summarized in Tables I and II. ### 5.1 Comparisons for Repair-by-transfer codes TABLE I: Comparisons for repair-by-transfer codes over $GF(q)$. \| Down. policy \| Range of $n$ \| Enc. comp. ---\|---\|---\|--- Shah et al. [10] \| - \| ${n\choose 2}\leq q+1$ \| $O(n^{4})$ Ours (Section 3) \| Partial \| $n\leq q+1$ \| $O(n^{3})$ Figure 3: The simulations of [10] and the proposed repair-by-transfer codes. This subsection compares the proposed repair-by-transfer codes with the [10] introduced in Sec. 2.1. As shown in Sec. 2.1, the field size of is at least ${n\choose 2}\leq N+1$. For the proposed repair-by-transfer codes, Section 3.3 states that the feasible range of $n$ can be extended up to $n\leq N+1$ via the extended Vandermonde matrix. Hence we conclude that the size of finite field is significantly reduced. Another issue is the computational complexities. We compare the complexities of both codes over the same finite field $GF(q)$. For the [10], it is evident that the $(N,K)=\left({n\choose 2},B\right)$ MDS code dominates the whole computational overhead. By employing the $\left({n\choose 2},B\right)$ systematic RS code, the encoding complexity is given by $O(({n\choose 2}-B)B)$. For the proposed scheme, the systematic version Sec. 3.3 computes the matrix $V$, whose computational cost is dominated by two terms $\check{\Phi}U_{\mathrm{R}}$ and the $\check{\Phi}U_{\mathrm{L}}\check{\Phi}^{t}$. As stated in Sec. 3.3, both terms take a total of $O(k(n-k)^{2})+O(2k^{2}(n-k))=O(k(n^{2}-k^{2}))$ operations. To magnify the difference between both codes further, we consider the case $k=cn$ with a constant $c$. In this case, the big-O representation of both codes are simplified into $O(({n\choose 2}-B)B)=O(n^{4})$ , and $O(k(n^{2}-k^{2}))=O(n^{3})$, respectively. Thus, the proposed code reduces one order of magnitude in big-O complexity representation. The real simulations of two codes are shown in Fig. 3. Both codes are written in JAVA, and the programs are running on Intel i7-950, 4GB RAM, Windows 8. We test the case $k=n/2$ at $n=\\{8,12,\ldots,64\\}$ over $GF(2^{16})$. In the simulation, the source data are generated by a random number generator. The Y-axis represents the logarithm of the encoding time of the $B$ input symbols on average. As shown in Fig. 3, the performance of the proposed codes is better than the [10] if the $n$ is larger than a specific value. For the small value of $n$, we conjecture that the structure of [10] is more simple, and the proposed algorithm contains a number of redundant arithmetic operations in the the congruence $V$, so that the [10] is better. ### 5.2 Comparisons for systematic Exact-MBR codes TABLE II: Comparisons for Exact-MBR codes over $GF(q)$. \| Syst. \| Down. policy \| Range of $n$ \| Enc. complexity ---\|---\|---\|---\|--- Rashmi et al. [9] \| N \| Full \| $n\leq q$ \| $O(nd^{2})$ Y \| Full \| $n\leq(q+k-d)$ or $n\leq q$ \| $O(nd^{2})$ Gong and Wang[18] \| N \| Partial \| $n\leq q$ \| $O(nd^{2})$ Ours (Section 4) \| Y \| Partial \| $n\leq q$ \| $O(nd^{2})$ or $O(n\log n)$ In the following, we compare the proposed systematic Exact-MBR codes with the [9], in terms of the range of $n$ and the encoding complexity. For the range of $n$, the [9] presents two distinct forms for the encoding matrix, so the $n$ has two distinct upper bounds. The first form is expressed as $\Psi=\left[\begin{array}[]{cc}I_{k}&\mathbf{0}\\\ \tilde{\Phi}&\tilde{\Delta}\end{array}\right],$ (51) where $I_{k}$ denotes a $k\times k$ identity matrix, $\mathbf{0}$ is a $k\times(d-k)$ zero matrix. The $[\begin{array}[]{cc}\tilde{\Phi}&\tilde{\Delta}\end{array}]$ is a $(n-k)\times d$ Cauchy matrix, where the sizes of $\tilde{\Phi}$ and $\tilde{\Delta}$ are $(n-k)\times k$ and $(n-k)\times(d-k)$, respectively. As stated by [9], the (51) meets the two conditions of Exact-MBR encoding matrix. By definition, a $(n-k)\times d$ Cauchy matrix requires $n-k+d$ distinct symbols. As the $GF(q)$ contains a total of $q$ distinct symbols, the feasible range of $n$ is $n-k+d\leq q\Rightarrow n\leq q+k-d.$ As addressed in Sec. 4.1, the range of $n$ for the PSRS codes is $n\leq q$, so is the proposed Exact-MBR code. Due to $k\leq d$, the proposed codes have larger range of $n$. In the second form of encoding matrix [9], the range of $n$ is also $n\leq q$. However, the second form is not explicit and the matrix generation requires an additional matrix inversion and multiplication step. An explicit form can facilitate the further development on the codes. For example, the partial decoding algorithm proposed in Sec. 4.3 is based on the observations on the form of encoding matrix. If the encoding matrix is not explicit, the partial decoding algorithm may become more difficult to be designed. Furthermore, by Appendix, the proposed $(n,k)$ PSRS codes can be implemented by generator polynomials. The size of generator polynomial is $(n-k)$, which is lower than the size of encoding matrix $(n-k)\times k$ in parity part. Thus, the generator polynomial approach is more common in usage. The encoding complexities of those codes are discussed below. Suppose those three codes are implemented with native matrix product approach. As the sizes of encoding matrix and message matrix are $n\times d$ and $d\times d$ for the three codes, the encoding complexity is $O(nd^{2})$. Furthermore, Section 4.1 indicates that the PSRS codes can be implemented with fast Fourier transforms. By FFTs, the encoding complexity can be reduced to $O(n\log n)$. ### 5.3 Comparisons for partial downloading schemes on Exact-MBR codes The partial downloading scheme is useful to reduce the requisite throughput to reconstruct data. This subsection highlights the differences between [18] and ours. First, the proposed scheme requires that the systematic fragments should be placed at a specific row of $C_{\mathrm{DC}}$. On the other hand, the [18] do not require this condition as the [18] is developed on non-systematic codes. Second, in our survey, this is the first work of considering the throughput balance on the connected nodes. ### 5.4 Simulations for systematic Exact-MBR codes Figure 4: The simulations of systematic Exact-MBR codes with native approach and fast approach. As stated in Section 4.1, the PSRS codes can be implemented with FFT. By employing the fast algorithm of PSRS code, we expected that the encoding time of Exact-MBR codes can be reduced. Based on this motivation, we implement the native and fast approaches of Exact-MBR codes, and the simulation results are shown in Figure 4. Both codes are written in JAVA, and the programs are running on Intel i7-950, 4GB RAM, Windows 8. We test the case $k=3/8\times n$, $d=n/2$, at $n\in\\{32,64,128,256,512\\}$ over Fermat field $GF(2^{16}+1)$. As shown in Figure 4, the fast approach works better for larger $n$. Otherwise, the native approach is suggested. ## 6 Conclusions The contributions of this paper can be organized in two parts. First, a new class of repair-by-transfer codes are proposed at $d=n-1$ MBR points. As compared with prior works, the proposed repair-by-transfer code demands smaller finite field and lower big-O complexity. The partial downloading scheme is also developed on the proposed repair-by-transfer codes to avoid the unnecessary symbol transmissions. The simulation shows that the proposed repair-by-transfer codes require fewer arithmetic operations than the prior work when $n$ is larger than a specific value. Second, for all feasible parameters $(n,k,d)$, we present an encoding matrix for systematic Exact-MBR codes via the partially systematic Reed-Solomon codes. To minimize the number of transmitted symbols in data reconstruction process, the partial downloading scheme is designed on the proposed Exact-MBR codes. However, the transmission amount for those connected nodes are excessively unbalanced. Thus, a time- sharing scheme is presented to balance the bandwidths requirements on those connected nodes. The proposed Exact-MBR codes can be implemented via fast Fourier transforms. As shown in the simulations, the fast approach has better encoding performance for large $n$. ## Appendix A Partially systematic Reed-Solomon codes by generator polynomial The appendix presents another approach of $(n,k,d)$ PSRS codes by generator polynomials. In this approach, the messages and codewords are formulated as polynomials. Thus, the message $\underline{a}$ and $\underline{b}$ are $a(x)=a_{0}+a_{1}x+\ldots+a_{k-1}x^{k-1};$ $b(x)=b_{0}+b_{1}x+\ldots+b_{d-k-1}x^{d-k-1}.$ The codeword polynomial is defined as $c(x)=c_{0}(x)+c_{1}(x),$ where the $c_{0}(x)$ is the codeword generated from $a(x)$ via $(n,k)$ systematic RS code, and the $c_{1}(x)$ is the codeword generated from $b(x)$ via $(n-k,d-k)$ RS code. Precisely, for the construction of $c_{0}(x)$, the generator polynomial of $(n,k)$ systematic RS code is defined as $g_{0}(x)=(x-1)(x-\alpha)\ldots(x-\alpha^{n-k-1}).$ Then the parity polynomial $r_{0}(x)$ is calculated through polynomial division $r_{0}(x)=x^{n-k}a(x)\textrm{ (mod }g_{0}(x)).$ The codeword $c_{0}(x)$ is expressed as the concatenation of $a(x)$ and $r_{0}(x)$: $c_{0}(x)=x^{n-k}a(x)-r_{0}(x).$ (52) For the construction of $c_{1}(x)$, the generator polynomial of $(n-k,d-k)$ RS code is defined as $g_{1}(x)=(x-1)(x-\alpha)\ldots(x-\alpha^{n-d-1}).$ The $c_{1}(x)$ can be formed as the systematic or non-systematic version. For the systematic case, the codeword polynomial is defined as $r_{1}(x)=x^{n-d}b(x)\textrm{ (mod }g_{1}(x));$ $c_{1}(x)=x^{n-d}b(x)-r_{1}(x).$ (53) The polynomial $a(x)$ is embedded in the $c(x)$ between $x^{n-k}$ and $x^{n-1}$, as the degree of $c_{1}(x)$ is less than $n-k$. Thus, the partially systematic condition holds. By generator polynomial, the length of this $(n,k,d)$ coding algorithm gets up to $n\leq q-1$ over $GF(q)$. The decoding algorithms are explained in the following. ### A.1 Full erasure decoding from $d$ codeword symbols The $a(x)$ and $b(x)$ can be reconstructed by arbitrary $d$ out of $n$ coefficients of the $c(x)$. As $g_{0}(x)$ and $g_{1}(x)$ are respectively the factors of $c_{0}(x)$ and $c_{1}(x)$, the $\gcd(g_{0}(x),g_{1}(x))=g_{0}(x)$ is also the factor of the $c(x)$. Therefore the $(n,k,d)$ PSRS code is isomorphic to the $(n,d)$ RS code with the generator polynomial $g_{0}(x)$. Thus, the $c(x)$ can be reconstructed from arbitrary $d$ out of $n$ coefficients via Forney algorithm. Forney algorithm is a method to compute the erasures of BCH codes at known error locations. When the $c(x)$ is completely reconstructed, the $a(x)$ is located in the systematic part of $c(x)$. Then the $c_{0}(x)$ can be computed from $a(x)$, and subsequently the $c_{1}(x)=c(x)-c_{0}(x)$. Thus, the $b(x)$ is decoded from $c_{1}(x)$. ### A.2 Partial erasure decoding from $k$ codeword symbols Given the $b(x)$, the message $a(x)$ can be reconstructed by arbitrary $k$ out of $n$ coefficients in $c(x)$. By (53), the $c_{1}(x)$ is calculated from $b(x)$. Since we have $k$ coefficients in $c(x)$, the corresponding $k$ coefficients in $c_{0}(x)=c(x)-c_{1}(x)$ can also be calculated. As the $c_{0}(x)$ is the codeword of $(n,k)$ systematic RS code, the $c_{0}(x)$ can be completely recovered via Forney algorithm. Then the message $a(x)$ is obtained from $c_{0}(x)$. ## References * [1] I. S. Reed and G. Solomon, ”Polynomial Codes over Certain Finite Fields”, _Journal of the Society for Industrial and Applied Mathematics_ , vol. 8, no. 2, pp. 300–304, 1960. * [2] Y. Lin, B. Liang, and B. Li, ”Priority Random Linear Codes in Distributed Storage Systems,” _IEEE Transactions on Parallel and Distributed Systems_ , vol. 20, no. 11, pp. 1653–1667, 2009. * [3] M. 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Hu, ”Exact Minimum-Repair-Bandwidth Cooperative Regenerating Codes for Distributed Storage Systems,” _in Proc. 2011 IEEE International Symposium on Information Theory Proceedings (ISIT 2011)_ , pp. 1442-1446, 2011. * [13] D. Cullina, A. G. Dimakis, and T. Ho, ”Searching for Minimum Storage Regenerating Codes,” _in Proc. 47th Annual Allerton Conference on Communication, Control_ , and Computing, Urbana-Champaign, 2009. * [14] N. B. Shah, K. V. Rashmi, P. V. Kumar, and K. Ramchandran, ”Interference alignment in regenerating codes for distributed storage: necessity and code constructions,” _IEEE Transactions on Information Theory_ , vol. 58, no. 4, pp. 2134-2158, 2012. * [15] V. R. Cadambe, S. A. Jafar, and H. Maleki, ”Distributed data storage with minimum storage regenerating codes - exact and functional repair are asymptotically equally efficient,” _in Proc. 2010 Wireless Network Coding (WINC) Workshop_ , 2010. * [16] C. Suh and K. Ramchandran, ”On the existence of optimal exact-repair MDS codes for distributed storage,” _technical report_ , 2010. * [17] C. Suh and K. Ramchandran, ”Exact regeneration codes for distributed storage repair using interference alignment,” _in Proc. 2010 IEEE International Symposium on Information Theory (ISIT)_ , pp. 161-165, 2010. * [18] Chen Gong and Xiaodong Wang, ”On partial downloading for wireless distributed storage networks,” _IEEE Transactions on Signal Processing_ , vol. 60, no. 6, pp. 3278-3288, 2012. * [19] N. B. Shah, ”Characterising exact repair-by-transfer for MBR,” _technical report_ , 2012. * [20] S. El Rouayheb and K. Ramchandran, ”Fractional Repetition Codes for Repair in Distributed Storage Systems,” _in Proceedings of Annual Allerton Conference on Communication, Control, and Computing_ , 2010. * [21] S. Pawar, N. Noorshams, and S. Y. El Rouayheb, and K. Ramchandran, ”DRESS codes for the storage cloud: Simple randomized constructions,” _in Proc. 2011 IEEE International Symposium on Information Theory Proceedings (ISIT 2011)_ , pp. 2338-2342, 2011. * [22] D. Bini and V. Y. Pan, ”Polynomial and matrix computations fundamental algorithms vol. 1,” Birkhäuser Boston, 1994. * [23] F. Didier, ”Efficient erasure decoding of Reed-Solomon codes,” Computing Research Repository - CORR, vol. abs/0901.1886, 2009. * [24] S. J. Lin and W. H. Chung, ”An Efficient (n, k) Information Dispersal Algorithm for High Code Rate System over Fermat Fields,” IEEE Communications Letters, vol. 16, no. 12, pp. 2036-2039, 2012. * [25] S. J. Lin and W. H. Chung, ”An Efficient (n, k) Information Dispersal Algorithm based on Fermat Number Transforms,” to appear in IEEE Transactions on Information Forensics and Security, doi: 10.1109/TIFS.2013.2270892. * [26] Y. Hu, C. M. Yu, Y. K. Li, P. P. C. Lee, and J. C. S. Lui, ”NCFS: On the Practicality and Extensibility of a Network-Coding-Based Distributed File System,” Proceedings of the 2011 International Symposium on Network Coding (NETCOD), Beijing, China, July 2011. * [27] O. Khan, R. Burns, J. S. Plank, W. Pierce and C. Huang, ”Rethinking Erasure Codes for Cloud File Systems: Minimizing I/O for Recovery and Degraded Reads,” FAST 2012: 10th USENIX Conference on File and Storage Technologies, San Jose, CA, Feb. 2012. \| Sian-Jheng Lin was born in Taichung, Taiwan, in 1981. He received the B.S., M.S., and Ph.D. degrees in computer science from National Chiao Tung University, in 2004, 2006, and 2010, respectively. He is currently a postdoctoral fellow with the Research Center for Information Technology Innovation, Academia Sinica. His recent research interests include data hiding and error control coding. ---\|--- \| Wei-Ho Chung was born in Kaohsiung, Taiwan, in 1978. He received the B.Sc. and M.Sc. degrees in Electrical Engineering from National Taiwan University, Taipei City, Taiwan, in 2000 and 2002 respectively. From 2005 to 2009, he was with the Electrical Engineering Department at University of California, Los Angeles, where he obtained his Ph.D. degree. From 2000 to 2002, he worked on routing protocols in the mobile ad hoc networks in the M.Sc. program in National Taiwan University. From 2002 to 2005, he was a system engineer at ChungHwa Telecommunications Company, where he worked on data networks. In 2008, he was an research intern working on CDMA systems in Qualcomm, Inc. From 2007 to 2009, he was a Teaching Assistant at UCLA. From June to December 2009, Dr. Chung had been working as a research associate in San Diego, California, on wireless communications for multimedia communications and unequal error protection for video transmission. His research interests include communications, signal processing, and networks. Dr. Chung received the Taiwan Merit Scholarship from 2005 to 2009, and the Best Paper Award in IEEE WCNC 2012. Dr. Chung has been an assistant research fellow in Research Center for Information Technology Innovation in Academia Sinica, Taiwan, since January 2010. ---\|---	arxiv-papers	2013-09-15T11:54:39	2024-09-04T02:49:50.927875	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sian-Jheng Lin and Wei-Ho Chung", "submitter": "Sian-Jheng Lin", "url": "https://arxiv.org/abs/1309.3752" }
1309.3838	# Linear isometries between real JB∗-triples and C∗-algebras Maria Apazoglou Queen Mary, University of London, London, United Kingdom [email protected] and Antonio M. Peralta Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias 18071, Granada, Spain [email protected] ###### Abstract. Let $T:A\to B$ be a (not necessarily surjective) linear isometry between two real JB∗-triples. Then for each $a\in A$ there exists a tripotent $u_{a}$ in the bidual, $B^{\prime\prime},$ of $B$ such that 1. $(a)$ $\\{u_{a},T(\\{f,g,h\\}),u_{a}\\}=\\{u_{a},\\{T(f),T(g),T(h)\\},u_{a}\\}$, for all $f,g,h$ in the real JB∗-subtriple, $A_{a},$ generated by $a$; 2. $(b)$ The mapping $\\{u_{a},T(\cdot),u_{a}\\}:A_{a}\rightarrow B^{\prime\prime}$ is a linear isometry. Furthermore, when $B$ is a real C∗-algebra, the projection $p=p_{a}=u_{a}^{}u_{a}$ satisfies that $T(\cdot)p:A_{a}\rightarrow B^{\prime\prime}$ is an isometric triple homomorphism. When $A$ and $B$ are real C∗-algebras and $A$ is abelian of real type, then there exists a partial isometry $u\in B^{\prime\prime}$ such that the mapping $T(\cdot)u^{}u:A\rightarrow B^{\prime\prime}$ is an isometric triple homomorphism. These results generalise, to the real setting, some previous contributions due to C.-H. Chu and N.-C. Wong, and C.-H. Chu and M. Mackey in 2004 and 2005. We give an example of a non-surjective real linear isometry which cannot be complexified to a complex isometry, showing that the results in the real setting can not be derived by a mere complexification argument. Second author partially supported by the Spanish Ministry of Science and Innovation, D.G.I. project no. MTM2011-23843, and Junta de Andalucía grants FQM0199 and FQM3737. Part of the results of this paper are contained in the first author’s PhD thesis [1] presented to Queen Mary, University of London, funded by EPSRC. The first author wishes to thank Professor Cho-Ho Chu for his valuable guidance and support. ## 1\. Introduction Despite many recent applications of real C∗-algebras in operator theory [8, 13], in JB∗-triples and infinite dimensional geometry [7, 8, 9, 17, 18, 22, 36, 37], the theory of real C∗-algebras has not been fully developed. In this paper, we study the linear geometry of real C∗-algebras and in particular, the not necessarily surjective linear isometries between real C∗-algebras, thereby extending the isometry results in [10], [11] and [25] for complex C∗-algebras. A celebrated result of Kadison [25] states that a surjective linear isometry $T:A\rightarrow B$ between complex C∗-algebras is necessarily a Jordan triple isomorphism, that is, $T(ab^{}c+cb^{}a)=T(a)T(b)^{}T(c)+T(c)T(b)^{}T(a)\qquad(a,b,c\in A).$ This result need not be true when the hypothesis of $T$ being surjective is not assumed. Nevertheless, it has been shown by C.-H. Chu and N.-C. Wong [9] that even if $T$ is non-surjective, for each $a\in A$, there is a largest projection $p_{a}\in B^{}$, such that $T(\cdot)p_{a}:A_{a}\rightarrow B^{}$ is an isometric Jordan triple homomorphism, where $A_{a}$ denotes the JB∗-subtriple of $A$ generated by $a$. In [10], C.-H. Chu and M. Mackey have further extended this result to the case in which $A$ and $B$ are JB∗-triples, showing, among other things, that, in this case, for each $a\in A$ there exists a tripotent $u_{a}\in B^{*}$ such that $\\{u_{a},T(\\{f,g,h\\}),u_{a}\\}=\\{u_{a},\\{T(f),T(g),T(h)\\},u_{a}\\}\ \ (f,g,h\in A_{a})$ and the mapping $\\{u_{a},T(\cdot),u_{a}\\}$ is an isometry. Our objective is not only to extend these results to real C∗-algebras but also to clarify the relationship between the projection $p_{a}$ and the tripotent $u_{a}$ which was not given in [10]. Indeed, we show that both results above are valid for real C∗-algebras and moreover, we have $p_{a}=u_{a}^{}u_{a}$. Prior to this result, we establish that when $A$ is a real C∗-algebra which admits a complex character of real type, then for every not necessarily surjective linear isometry $T$ from $A$ to another C∗-algebra $B$, there exists a minimal projection $p$ in $B^{\prime\prime}$ satisfying: 1. $(a)$ $T\\{a,b,c\\}p=\\{T(a),T(b),T(c)\\}p$ and $pT(a)^{}T(b)=T(a)^{}T(b)p,$ for all $a,b,c$ in $A$; 2. $(b)$ $T(\cdot)p:A\to B^{\prime\prime}$ is a non-zero triple homomorphism (see Proposition 3.1 and Theorem 3.4). When $A$ does not admit complex characters of real type, that is, when $A$ is of complex type we can find an example of a linear surjection $T$ from $A$ into another real C∗-algebra such that $T(\cdot)p=0$ whenever $T(\cdot)p$ is a triple homomorphism, $T\\{a,b,c\\}p=\\{T(a),T(b),T(c)\\}p$ and $pT(a)^{}T(b)=T(a)^{}T(b)p,$ for all $a,b,c$ in $A$ (compare Example 3.2). We culminate the paper showing that every non-surjective linear isometry between real JB∗-triples $T:A\to B$, although need not be a triple homomorphism, locally reduces to a triple homomorphism, that is, for each $a\in A$, there exists a tripotent $u\in B^{\prime\prime}$ such that 1. $(a)$ $\\{u,T(\\{f,g,h\\}),u\\}=\\{u,\\{T(f),T(g),T(h)\\},u\\}$, for all $f,g,h$ in the real JB∗-subtriple generated by $a$; 2. $(b)$ The mapping $\\{u,T(\cdot),u\\}:A_{a}\rightarrow B^{\prime\prime}$ is a linear isometry. Furthermore, when $B$ is a real C∗-algebra, the projection $p=u^{}u$ satisfies that $T(\cdot)p:A_{a}\rightarrow B^{\prime\prime}$ is an isometric triple homomorphism (compare Theorem 3.9). Our results cannot be achieved by simple complexification since, as shown later, the complexification of a non-surjective real isometry need not be, in general, an isometry. We develop some new techniques to accomplish our task. ## 2\. Preliminaries: Jordan structures in real C∗-algebras Let $A$ be a real Banach-algebra. Then $A$ is called a _real C ∗-algebra_ if $\\|a^{}a\\|=\\|a\\|^{2}$ and $1+a^{}a$ is invertible for all $a\in A$. If $A$ is non-unital, then we require that $1+a^{}a$ is invertible in the unit extension $A\oplus\mathbb{R}1$. Equivalently, a real Banach -algebra $A$ is a real C∗-algebra if, and only if, it is isometrically -isomorphic to a norm- closed real -algebra of bounded operators on a real Hilbert space (cf. [29, Corollary 5.2.11]). Trivially, every (complex) C∗-algebra is a real C∗-algebra when scalar multiplication is restricted to the real field. Let $A$ be a real C∗-algebra. We denote by $A^{\prime}$ the dual space of $A$. The complexification $A_{c}=A\oplus iA$ is a C∗-algebra in a suitable norm such that $A$ identifies with a real closed -subalgebra of $A_{c}$. The dual space of a complex C∗-algebra $B$ will be denoted by $B^{}$. Throughout the paper, given a real or complex C∗-algebra $A$, $A_{sa}$ will stand for the set of all self-adjoint elements in $A$. We shall now survey some Jordan structures associated with real and complex C∗-algebras. A (complex) _JB ∗-triple_ is a complex Banach space $A$ equipped with a triple product $\\{\cdot,\cdot,\cdot\\}:A\times A\times A\rightarrow A$ which is linear and symmetric in the outer variables, conjugate linear in the middle one and satisfies the following conditions: 1. (1) (Jordan identity) for $a,b,x,y,z\in A$, (2.1) $\\{a,b,\\{x,y,z\\}\\}=\\{\\{a,b,x\\},y,z\\}-\\{x,\\{b,a,y\\},z\\}+\\{x,y,\\{a,b,z\\}\\};$ 2. (2) $D(a,a):A\rightarrow A$ is an hermitian linear operator with non-negative spectrum, where $D(a,b)(x)=\\{a,b,x\\}$ with $a,b,x\in A$; 3. (3) $\\|\\{x,x,x\\}\\|=\\|x\\|^{3}$ for all $x\in A$. A complex subspace $B$ of $A$ is called a _subtriple_ if it is closed with respect to the triple product, that is, $x,y,z\in B$ implies $\\{x,y,z\\}\in B$. A real subtriple of $A$ is a real subspace of $A$ which is closed with respect to the triple product. We define a _real JB ∗-triple_ to be a closed real subtriple of a JB∗-triple. When restricted to real scalar multiplication, every (complex) JB∗-triple is also a real JB∗-triple. It has been shown in [22] that a real JB∗-triple can be complexified to become a JB∗-triple. Furthermore, every real JB∗-triple $B$ is a real form of a complex JB∗-triple, that is, there exist a (complex) JB∗-triple $B_{c}$ and a conjugate linear isometry $\tau:B_{c}\rightarrow B_{c}$ of period 2 such that $B=\\{b\in B_{c}\>:\>\tau(b)=b\\}$. We can actually identify $B_{c}$ with the complexification of $B$. A real C∗-algebra $A$ is a real JB∗-triple in the triple product: (2.2) $\\{a,b,c\\}=\frac{1}{2}(ab^{}c+cb^{}a)\qquad(a,b,c\in A).$ We shall always assume the above triple product in a real or complex C∗-algebra. An element $e$ in a complex or real JB∗-triple $A$ is called _tripotent_ if $\\{e,e,e\\}=e$. Let $A$ be a real JB∗-triple. For an element $a\in A$, we define the operators $D(a):A\rightarrow A$ and $Q(a):A\rightarrow A$ by $D(a)(x)=\\{a,a,x\\}\;\textrm{and}\;Q(a)(x)=\\{a,x,a\\}\qquad(x\in A).$ The Peirce projections associated with a tripotent $e\in A$ are given by $\displaystyle P_{2}(e)$ $\displaystyle=Q(e)^{2}$ $\displaystyle P_{1}(e)$ $\displaystyle=2(D(e)-Q(e)^{2})$ $\displaystyle P_{0}(e)$ $\displaystyle=I-2D(e)+Q(e)^{2}.$ The space $A_{j}(e)=P_{j}(e)(A)$ is called the Peirce-$j$-space associated to the tripotent $e$. Thus, each tripotent $e$ in a real or complex JB∗-tripe, $A$, defines a decomposition of $A$ in terms of Peirce subspaces $A=A_{0}(e)\oplus A_{1}(e)\oplus A_{2}(e).$ Another decomposition can be given in terms of the eigensubspaces of $Q(e)$, $A=A^{1}(e)\oplus A^{-1}(e)\oplus A^{0}(e)$ where $A^{k}(e):=\\{x\in A:Q(e)(x):=\\{e,x,e\\}=kx\\}$ is a real Banach subspace of $A$. Furthermore, the following identities and rules hold: $\displaystyle A_{2}(e)=A^{1}(e)\oplus A^{-1}(e),$ $\displaystyle A_{1}(e)\oplus A_{0}(e)=A^{0}(e)$ $\displaystyle\\{A^{i}(e),A^{j}(e),A^{k}(e)\\}\subseteq A^{ijk}(e),\hbox{ whenever }ijk\neq 0.$ A real JB∗-triple which is also a dual Banach space is called a real (respectively, complex) _JBW ∗-triple_. Every real or complex JBW∗-triple has a (unique) predual (cf. [3], [17, Corollary 3.2], [21] and [31]) and its triple product is separately weak∗ continuous (see [31]). Given a tripotent $e$ in a real or complex JB∗-triple $A$, the Peirce space $A_{2}(e)$ is a real JB∗-algebra and $A^{1}(e)$ is a JB-algebra with product and involution given by $a\circ_{e}b:=\\{a,e,b\\}$ and $a^{\sharp_{e}}:=\\{e,a,e\\}$, respectively. It can be easily seen that, for each $a\in A$, the expression (2.3) $P_{2}(e)\left\\{a,a,e\right\\}=\left\\{P_{2}(e)(a),P_{2}(e)(a),e\right\\}+\left\\{P_{1}(e)(a),P_{1}(e)(a),e\right\\}$ $=P^{1}(e)\left\\{a,a,e\right\\}$ defines a positive element in the JB-algebra $A^{1}(e)$ (see, for example the arguments given in [5, Lemma 2.1], which remain valid in the real setting). Two elements $a,b$ in a real or complex JB∗-triple $A$ are said to be _orthogonal_ (written $a\perp b$) if $L(a,b)=0$. By Lemma 1 in [6] (whose proof is also valid for real JB∗-triples), two elements $a,b$ in $A$ are orthogonal if, and only if, $\left\\{a,a,b\right\\}=0$ (equivalently, $\left\\{b,b,a\right\\}=0$). It follows from the Peirce rules that, for each tripotent $e$ in $A$, the sets $A_{0}(e)$ and $A_{2}(e)$ are orthogonal. In a real or complex JB∗-triple, $A$, a partial order relation, $\leq$, can be defined in the set of all tripotents elements in $A$ given by $e\leq f$ if, and only if, $f-e$ is a tripotent in $A$ with $f-e\perp e$. A non-zero tripotent $e$ is called _minimal_ whenever $A^{1}(e)=\mathbb{R}e$ (since in the complex case $A^{-1}(e)=iA^{1}(e)$, this definition is equivalent to say $A_{2}(e)=\mathbb{C}e$). However, in the real setting the dimensions of $A^{1}(e)$ and $A^{-1}(e)$ are not correlated as there exist examples of real Cartan factors $A$ containing a minimal tripotent $e$ satisfying dim$(A^{-1}(e))=\infty$. Another illustrating example is $A=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$ and $e=1$, then $e$ is minimal and dim$(A^{-1}(e))=0,1$ and $3,$ respectively. In general, given a minimal tripotent $e$ in a real JB∗-triple $A$, $\\{e,A,e\\}=A_{2}(e)$ need not coincide with $\mathbb{R}e=A^{1}(e)$. In [37] a tripotent $e$ is called minimal when it is minimal with respect to the partial order relation $\leq$, that is, $e$ is minimal iff $e\geq f\neq 0$ implies $e=f$. To avoid confusion, let us call these kind of tripotents _order minimal tripotents_. Clearly every minimal tripotent is order-minimal, however, the reciprocal statement is not, in general, true. For example, the unit element $1$ in $C[0,1]$ (real or complex valued) is order minimal because there exists no tripotent $e\in C[0,1]$ satisfying $1>e$. When $A$ is a real JBW∗-triple minimal and order minimal tripotents in $A$ coincide (cf. Proposition 2.2 and Remark 2.3, [37]). We recall that a minimal tripotent $e$ in a real JB∗-triple $E$ is called _reduced_ whenever $E_{2}(e)=\mathbb{R}e$ (equivalently, $E^{-1}(e)=0$). The real JB∗-triple $E$ is said to be _reduced_ when every minimal tripotent $e\in E$ is reduced (cf. [30, 11.9]). A real W∗-algebra is a real C∗-algebra $A$ having a predual, $A_{{}_{{}^{\prime}}}$, in which case, $A_{{}_{{}^{\prime}}}$ is unique up to linear isometric isomorphism and the product of $A$ is separately weak∗ continuous (cf. [24]). We note that the second dual $A^{\prime\prime}$ of a real C∗-algebra $A$ is a real W∗-algebra. Moreover, let $A_{c}$ and $A_{c}^{}$ denote the complexifications of $A$ and its bidual, respectively, then $A_{c}^{}=A^{\prime\prime}\oplus iA^{\prime\prime}$. Every W∗-algebra is a real JB∗-triple with respect to the product given in $(\ref{eq triple product in C-algebras})$. An element $p$ in a real C∗-algebra $A$ is called a _projection_ if $p=p^{}=p^{2}$. A projection $p$ in $A$ is _minimal_ whenever it is a minimal tripotent in $A$, that is, $A^{1}(p)=\\{x\in A:px^{}p=x\\}=\mathbb{R}p$. It is well known that, when $A$ is regarded as a real JB∗-triple, tripotents in $A$ correspond to _partial isometries_ in $A$ (i.e. elements $u\in A$ with $u^{}u$ being a projection in $A$). The next lemma should be known for experts but we were not able to find an explicit reference. ###### Lemma 2.1. Let $u$ be a tripotent in a real C∗-algebra $A$ regarded as a real JB∗-triple. Then $u$ is minimal if, and only if, $p=u^{}u$ (respectively, $q=uu^{}$) is a minimal projections in $A$. ###### Proof. The equivalence follows from the following identities: $A^{1}(p)=pA_{sa}p,$ $uA^{1}(p)=A^{1}(u)=\\{x\in A:ux^{}u=x\\}.$ ∎ Let $A$ be a real W∗-algebra. Two projections $p$ and $q$ in $A$ are called _orthogonal_ if $pq=0$. A projection $p$ is _majorised_ by $q$, i.e. $p\leq q$ in the partial order of the cone $A_{+}$, if and only if $p(1-q)=0$. When $A$ is regarded as a JB∗-triple this partial order agrees with the partial order defined in the set of tripotents (i.e. partial isometries) of $A$. ###### Lemma 2.2. ([29],Proposition 4.3.4) Let $A\subset B(H)$ be a real W∗-algebra. Then the supremum $\bigvee_{i\in I}p_{i}$ of a family $\\{p_{i}\\}_{i\in I}$ of projections in $A$ exists with respect to the ordering induced by the cone $A_{+}$. Further, we have $\bigvee_{i\in I}p_{i}:H\rightarrow\overline{[\cup_{i\in I}p_{i}H]}.$ is the projection onto the closed linear span $\overline{[\cup_{i\in I}p_{i}H]}$ of $\cup_{i\in I}p_{i}H$. ###### Lemma 2.3. Let $\\{p_{\phi}\\}_{\phi\in Q}$ be a family of projections in a real W∗-algebra $W$. Let $p=\bigvee_{\phi\in Q}p_{\phi}$. If $ap_{\phi}=0$ for some $a\in W$ and all $\phi\in Q$, then $ap=0$. ###### Proof. By hypothesis, $a\in W$ annihilates on every element in the linear span of $\bigcup_{i\in I}p_{i}(H)$, and by continuity $ap(\xi)=0$, for every $\xi\in H$. ∎ If $B$ is a real JB∗-triple, then its second dual, $B^{\prime\prime}$, is also a real JB∗-triple. Moreover, $B$ is the real form of a complex JB∗-triple $B_{c}$, that is, there is a conjugate linear isometry $\tau:B_{c}\rightarrow B_{c}$ of period 2 such that $B=\\{b\in B_{c}\>:\>\tau(b)=b\\}$. Further, the bidual map $\sigma=\tau^{}:B_{c}^{}\rightarrow B_{c}^{}$ is a conjugate linear isometry of period 2 and $B^{\prime\prime}=\\{b\in B_{c}^{}\>:\>\sigma(b)=b\\}$. The “dual” map $\widetilde{\tau}:B_{c}{{}^{}}\rightarrow B_{c}^{}$ defined by $\widetilde{\tau}(\phi)(b)=\overline{\phi(\tau(b))}\qquad(\phi\in B_{c}^{},\,b\in B_{c})$ is a conjugate linear isometry of period 2 and the mapping $(B_{c}^{})^{\widetilde{\tau}}\to B^{\prime}$ $\varphi\mapsto\varphi\|_{B}$ is a surjective linear isometry. We observe that if $u$ is a tripotent in $B_{c}^{}$, then $\sigma(u)$ is also a tripotent in $B_{c}^{}$. In fact, letting $U(B_{c}^{})$ be the set of all tripotents in $B_{c}^{}$, the set $U(B^{\prime\prime})$ of tripotents in $B^{\prime\prime}$ is $U(B_{c}^{})\cap B^{\prime\prime}=\\{u\in U(B_{c}^{})\>:\>\sigma(u)=u\\}$ [17]. For a functional $\phi\in B_{c}^{}$, there exists a unique tripotent $u_{\phi}\in B_{c}^{}$, called the _support tripotent_ of $\phi$, such that $\phi=\phi\circ P_{2}(u_{\phi})$ and $\phi\|_{P_{2}(u_{\phi})(B_{c}^{})}$ is a faithful normal positive functional on the JBW∗-algebra $P_{2}(u_{\phi})(B_{c}^{})$ (cf. [19]). We note that if $u_{\phi}\in B_{c}^{}$ is the support tripotent of $\phi\in B_{c}^{}$, then $\sigma(u_{\phi})$ is the support tripotent of $\sigma_{}(\phi)$. On the other hand, if $\phi$ is in $B^{\prime}$, its complex extension $\phi_{c}\in B_{c}^{}$ has support tripotent $u_{\phi_{c}}\in B_{c}^{}$ such that $\sigma(u_{\phi_{c}})=u_{\phi_{c}}$ since $\sigma^{\sharp}(\phi_{c})=\phi_{c}$. Hence $u_{\phi_{c}}\in U(B^{\prime\prime})$ and we call it the _support tripotent_ of $\phi$ in $B^{\prime\prime}$, denoted by $u_{\phi}$ (cf. [36, Lemma 2.2]). Finally, we note that $\phi$ is an extreme point of the closed unit ball of $A_{}$ if and only if its support tripotent $u_{\phi}$ is a minimal tripotent in $A$ [37, Corollary 2.1]. Moreover, in this case $\phi=\phi P^{1}(u_{\phi})$ (and hence $\phi=\phi Q(u_{\phi})$) ([37, Lemma 2.7]). Therefore $\phi(x)u_{\phi}=\frac{1}{2}\left(Q(u_{\phi})^{2}(x)+Q(u_{\phi})(x)\right)=P^{1}(u_{\phi})(x),$ for every $x\in A$. Based on these observations and the complex results in [2, Proposition 1.2] we have the following lemma. ###### Lemma 2.4. ([36, Lemma 2.4]) Let $B^{\prime\prime}$ be the second dual of a real C∗-algebra $B$ and let $\phi\in B^{\prime}$ with $\\|\phi\\|=1$. Let $u,v\in B^{\prime\prime}$ such that $\phi(u)=\\|u\\|=1=\phi(v)=\\|v\\|$ then 1. (1) $\phi\\{x,y,v\\}=\phi\\{x,y,u\\}=\phi\\{y,x,u\\}$; 2. (2) $\phi\\{x,x,u\\}\geq 0$; 3. (3) $\|\phi\\{x,y,u\\}\|^{2}\leq\phi\\{x,x,u\\}\phi\\{y,y,u\\}$ (Cauchy-Schwarz inequality), for all $x,y\in B^{\prime\prime}$. ###### Lemma 2.5. Let $B^{\prime\prime}$ be the second dual of a real C∗-algebra $B$ and let $\phi\in B^{\prime}$ be an extreme point in the closed unit ball $B^{\prime}_{1}$. Let $N_{\phi}=\\{b\in B^{\prime\prime}\>:\>\phi\\{b,b,u\\}=0\\}$ where $u$ is the support tripotent of $\phi$. Then $N_{\phi}=P_{0}(u)(B^{\prime\prime})$. ###### Proof. By $(\ref{equation positive element})$ and the faithfulness of $\phi\|_{(B^{\prime\prime})^{1}(u)}$, we have $\\{b,b,u\\}=0$, that is, $b\perp u$, and hence $b\in P_{0}(u)(B^{\prime\prime})$. ∎ A (real) linear functional $\varphi$ on a real C∗-algebra $A$ is said to be positive when it is hermitian and maps positive elements in $A$ into $\mathbb{R}^{+}_{0}$ (i.e. $\varphi(a)\geq 0$, for every positive element $a$ in $A$, and $\varphi(b^{})=\varphi(b)$, for every $b$ in $A$). Following standard notation, given a real C∗-algebra $A$, the quasi-state space of $A$, $\mathcal{Q}(A),$ is defined as the set of all positive $\varphi\in A^{\prime}$ satisfying $\\|\varphi\\|\leq 1$ (see, for example [29, §5.2]). It is known that $\mathcal{Q}(A)$ is a weak∗ closed convex subset of $(A^{\prime})_{1}$. We shall require later some results on the facial structure of $\mathcal{Q}(A)$. Having in mind that $A_{sa}$ is a JC-algebra with respect to the canonical Jordan product, the following Lemma was proved in [32, Theorem 5.2]. Finally, let $A$ be a real or complex C∗-algebra or a JB-algebra. Following standard notation (see [34, §3.11] and [14]), we recall that a projection $p$ in the bidual, $A^{}$, of $A$ is said to be open if $A^{}_{2}(p)\cap A$ is weak∗-dense in $A^{}_{2}(p)$. The projection $p$ is called closed when $1-p$ is open. ###### Lemma 2.6. [32, Theorem 5.2] Let $\mathcal{Q}(A)$ be the quasi-state space of a real C∗-algebra $A$. For every norm-closed face $F$ of $\mathcal{Q}(A)$ there exists a (unique) closed projection $p\in A^{\prime\prime}$ satisfying that $F=\\{\varphi\in\mathcal{Q}(A):\varphi(1-p)=0\\}.\hfill\Box$ ## 3\. Isometries between real C∗-algebras Our goal in this section is to show that any linear isometry $T:A\rightarrow B$, surjective or not, between real C∗-algebras $A$ and $B$ reduces _locally_ to an (isometric) Jordan triple isomorphism by a projection $p=u^{}u\in B^{\prime\prime}$ for some partial isometry $u\in B^{\prime\prime}$. In our first result we shall study contractive linear projections between real C∗-algebras. In the complex setting, C.-H. Chu and N.-C. Wong proved in [11, Proposition 2.2] that for every linear contraction $T$ between (complex) C∗-algebras $A$ and $B$, there exists a largest projection $p\in B^{*}$ such that 1. $(a)$ $T\\{a,a,a\\}p=\\{T(a),T(a),T(a)\\}p$; 2. $(b)$ $pT(a)^{}T(a)=T(a)^{}T(a)p$ for every $a\in A$. A standard complex polarisation argument implies 1. $(a)$ $T\\{a,b,c\\}p=\\{T(a),T(b),T(c)\\}p$ and $pT(a)^{}T(b)=T(a)^{}T(b)p$ for all $a,b,c$ in $A$; 2. $(b)$ $T(\cdot)p:A\to B^{}$ is a triple homomorphism. In the real setting, the lacking of a standard polarisation identity forces us to make a slight modification of the argument given by Chu and Wong in [11, Proposition 2.2], an sketch of the proof is included here for completeness. ###### Proposition 3.1. Let $T:A\to B$ be a linear contraction between real C∗\- algebras. Then there is a largest projection $p$ in $B^{\prime\prime}$ such that 1. $(a)$ $T\\{a,b,c\\}p=\\{T(a),T(b),T(c)\\}p$ and $pT(a)^{}T(b)=T(a)^{}T(b)p$ for all $a,b,c$ in $A$; 2. $(b)$ $T(\cdot)p:A\to B^{\prime\prime}$ is a triple homomorphism. Furthermore, the projection $p$ is closed. ###### Proof. Let us define $F_{1}:=\bigcap_{a,b\in A_{1}}\Big{\\{}\varphi\in\mathcal{Q}(B):\\|T(Q(a)b)-Q(T(a))(T(b))\\|_{\varphi}=0\Big{\\}}$ $=\bigcap_{a,b\in A_{1}}\Big{\\{}\varphi\in\mathcal{Q}(B):{\varphi}\Big{(}(\Theta_{a,b})(\Theta_{a,b})^{}+(\Theta_{a,b})^{}(\Theta_{a,b})\Big{)}=0\Big{\\}},$ where, for each $a,b\in A,$ $\Theta_{a,b}=T(Q(a)b)-Q(T(a))(T(b)).$ Clearly, $F_{1}$ is a weak∗ closed face of $\mathcal{Q}(B)$ containing zero. Given $a,b\in A_{1},$ we define a continuous affine mapping $\Phi_{a,b}:\mathcal{Q}(B)\to\mathcal{Q}(B),$ $\varphi\mapsto\Phi_{a,b}(\varphi)(\cdot):=\varphi(T(b)^{}T(a)\cdot T(a)^{}T(b)).$ For each natural $n\geq 2$, $F_{n}:=\bigcap_{a,b\in A_{1}}\Big{(}F_{n-1}\bigcap\Phi_{a,b}^{-1}(F_{n-1})\Big{)}$ is a weak∗ closed face of $\mathcal{Q}(B)$ containing zero and $F_{n}\subseteq F_{n-1}.$ Therefore $F=\bigcap_{n=1}^{\infty}F_{n}$ is a weak∗ closed face of $\mathcal{Q}(B)$ containing zero, and hence there exists a closed projection $p\in B^{\prime\prime}$ satisfying that $F=\\{\varphi\in\mathcal{Q}(B):\varphi(1-p)=0\\}$ (compare Lemma 2.6 or [32, Theorem 5.2]). In particular, for each $a,b\in A$ and $\varphi\in F$ we have $0=\Phi_{a,b}(\varphi)(1-p)=\varphi(T(b)^{}T(a)(1-p)T(a)^{}T(b))$ $=\varphi\Big{(}((1-p)T(a)^{}T(b))^{}((1-p)T(a)^{}T(b))\Big{)}.$ Since the support tripotent of $\varphi$, $s(\varphi)$, is a projection with $s(\varphi)\leq p\leq 1$, we have: $\left\\|(1-p)T(a)^{}T(b)\right\\|_{\varphi}^{2}=\varphi\left\\{(1-p)T(a)^{}T(b),(1-p)T(a)^{}T(b),p\right\\}$ $=\varphi P_{2}(p)\left\\{(1-p)T(a)^{}T(b),(1-p)T(a)^{}T(b),p\right\\}$ $=\varphi P^{1}(p)\left\\{(1-p)T(a)^{}T(b),(1-p)T(a)^{}T(b),p\right\\}$ $=\frac{1}{2}\varphi\Big{(}((1-p)T(a)^{}T(b))((1-p)T(a)^{}T(b))^{}p\Big{)}$ $+\frac{1}{2}\varphi\Big{(}p((1-p)T(a)^{}T(b))^{}((1-p)T(a)^{}T(b))\Big{)}$ $=\frac{1}{2}\varphi\Big{(}((1-p)T(a)^{}T(b))^{}((1-p)T(a)^{}T(b))\Big{)}=0$ for every $a,b\in A$ and $\varphi\in F$. Observing that $P_{2}(p)\left\\{(1-p)T(a)^{}T(b),(1-p)T(a)^{}T(b),p\right\\}$ $=P^{1}(p)\left\\{(1-p)T(a)^{}T(b),(1-p)T(a)^{}T(b),p\right\\}$ is a positive element in $(B^{\prime\prime})^{1}(p)$ (see $(\ref{equation positive element})$) and $F=\\{\varphi\in B^{\prime}:\varphi(p)=\varphi(1)=\\|\varphi\\|\leq 1\\}$ is the normal quasi- state of the JBW-algebra $(B^{\prime\prime})^{1}(p)$, we have $P_{2}(p)\left\\{(1-p)T(a)^{}T(b),(1-p)T(a)^{}T(b),p\right\\}=0$ for every $a,b\in A$. This implies that $p\Big{(}(1-p)T(a)^{}T(b)T(b)^{}T(a)(1-p)+pT(b)^{}T(a)(1-p)T(a)^{}T(b)\Big{)}^{}p=0,$ or equivalently, $\Big{(}(1-p)T(a)^{}T(b)p\Big{)}^{}\Big{(}(1-p)T(a)^{}T(b)p\Big{)}=pT(b)^{}T(a)(1-p)T(a)^{}T(b)p=0,$ and hence (3.1) ${(1-p)T(a)^{}T(b)}p=0$ for every $a,b\in A$, which proves the second statement in $(a)$. By definition of $F$, for each $a,b\in A$ and $\varphi\in F$, we have ${\varphi}\Big{(}(\Theta_{a,b})(\Theta_{a,b})^{}+(\Theta_{a,b})^{}(\Theta_{a,b})\Big{)}=0,$ which gives $0={\varphi}P_{2}(p)\Big{(}(\Theta_{a,b})^{}(\Theta_{a,b})\Big{)}={\varphi}P^{1}(p)\Big{(}(\Theta_{a,b})^{}(\Theta_{a,b})\Big{)}.$ Recalling that $F$ coincides with the normal quasi-state of the JBW-algebra $(B^{\prime\prime})^{1}(p)$ and $P^{1}(p)\Big{(}(\Theta_{a,b})^{}(\Theta_{a,b})\Big{)}=p(\Theta_{a,b})^{}(\Theta_{a,b})p=(\Theta_{a,b}\ p)^{}(\Theta_{a,b}\ p)$ is a positive element in $(B^{\prime\prime})^{1}(p)$, we deduce that $\Theta_{a,b}\ p=0$, and accordingly $T(\left\\{a,b,a\right\\})p=\left\\{T(a),T(b),T(a)\right\\}p.$ On the other hand, $\left\\{T(a)p,T(b)p,T(a)p\right\\}=T(a)pT(b)^{}T(a)p$ $=(\hbox{by }(\ref{eq 1 in prop linear contraction}))=T(a)T(b)^{}T(a)p=\left\\{T(a),T(b),T(a)\right\\}p,$ and we establish $(b)$. Finally, when $q$ is a another projection in $B^{\prime\prime}$ satisfying $(a)$ and $(b)$, it follows straightforwardly that $F(q):=\\{\varphi\in\mathcal{Q}(B):\varphi(q)=\\|\varphi\\|\leq 1\\}$ is a norm- closed face contained in $F_{1}$ and hence $F(q)\subseteq F$. It is known that, under these conditions, $q\leq p$ (see, for example [32, Theorem 2.2]). ∎ The projection $p\in B^{\prime\prime}$ given by Proposition 3.1 is called the _structure projection_ of $T$ and is denoted by $p_{T}$. Unfortunately, in some cases, the structure projection $p_{T}$ satisfies that $T(.)p_{T}$ reduces to zero, even under the hypothesis of $T$ being an isometry. In the complex setting, Chu and Wong proved in [11, Therorem 3.10] that $T(.)p_{T}$ is an isometry whenever $T:A\to B$ is an (complex) linear isometry between C∗-algebras with $A$ abelian. We shall see next that the same statement is not valid in the setting of real C∗-algebras. ###### Example 3.2. Let $K$ be a compact Hausdorff space and let $C(K)_{\mathbb{R}}$ be the real abelian C∗-algebra of complex continuous functions on $K$ regarded as a real Banach space. Let $C(K,M_{2}(\mathbb{R}))$ be the real C∗-algebra of continuous $M_{2}(\mathbb{R})$-valued functions on $K$, where $M_{2}(\mathbb{R})$ denotes the real C∗-algebra of $2\times 2$ real matrices. Define $T:C(K)_{\mathbb{R}}\rightarrow C(K,M_{2}(\mathbb{R}))$ by $T(f)=\left(\begin{array}[]{ccc}\textrm{Re}(f)&\textrm{Im}(f)\\\ 0&0\\\ \end{array}\right)\;\;\;\;(f\in C(K)_{\mathbb{R}}).$ Then $T$ is a real linear isometry. We have $T(f)^{3}=T(f^{3})$ for all $f\in C(K)_{\mathbb{R}}$, but in general $T(\\{f,g,h\\})\neq\\{T(f),T(g),T(h)\\}$. We shall now show that the structure projection of $T$ satisfies $p_{T}\leq\left(\begin{array}[]{cc}0&0\\\ 0&1\\\ \end{array}\right)$, and in this case, $T(.)p_{T}=0$ (cf. [1]). Let $p$ be a projection satisfying the thesis of Proposition 3.1. Then $\displaystyle\\{T(f),T(g),T(f)\\}$ $\displaystyle=\left(\begin{array}[]{ccc}\textrm{Re}(f)&\textrm{Im}(f)\\\ 0&0\\\ \end{array}\right)\left(\begin{array}[]{ccc}\textrm{Re}(g)&0\\\ \textrm{Im}(g)&0\\\ \end{array}\right)\left(\begin{array}[]{ccc}\textrm{Re}(f)&\textrm{Im}(f)\\\ 0&0\\\ \end{array}\right)$ $\displaystyle=\left(\begin{array}[]{ccc}\textrm{Re}^{2}(f)\textrm{Re}(g)&\textrm{Im}^{2}(f)\textrm{Im}(g)\\\ +\textrm{Re}(f)\textrm{Im}(f)\textrm{Im}(g)&+\textrm{Re}(f)\textrm{Im}(f)\textrm{Re}(g)\\\ 0&0\\\ \end{array}\right)$ and $T(\\{f,g,f\\})=\left(\begin{array}[]{ccc}\textrm{Re}^{2}(f)\textrm{Re}(g)&\textrm{Im}^{2}(f)\textrm{Im}(g)\\\ -\textrm{Im}^{2}(f)\textrm{Re}(g)&-\textrm{Re}^{2}(f)\textrm{Im}(g)\\\ +2\textrm{Re}(f)\textrm{Im}(f)\textrm{Im}(g)&+2\textrm{Re}(f)\textrm{Im}(f)\textrm{Re}(g)\\\ 0&0\\\ \end{array}\right).$ By assumptions: $\displaystyle\left(\begin{array}[]{ccc}\textrm{Re}^{2}(f)\textrm{Re}(g)&\textrm{Im}^{2}(f)\textrm{Im}(g)\\\ +\textrm{Re}(f)\textrm{Im}(f)\textrm{Im}(g)&+\textrm{Re}(f)\textrm{Im}(f)\textrm{Re}(g)\\\ 0&0\\\ \end{array}\right)p$ $\displaystyle=\left(\begin{array}[]{ccc}\textrm{Re}^{2}(f)\textrm{Re}(g)&\textrm{Im}^{2}(f)\textrm{Im}(g)\\\ -\textrm{Im}^{2}(f)\textrm{Re}(g)&-\textrm{Re}^{2}(f)\textrm{Im}(g)\\\ +2\textrm{Re}(f)\textrm{Im}(f)\textrm{Im}(g)&+2\textrm{Re}(f)\textrm{Im}(f)\textrm{Re}(g)\\\ 0&0\\\ \end{array}\right)p$ which simplifies to (3.2) $\left(\begin{array}[]{ccc}\textrm{Re}(f)\textrm{Im}(f)\textrm{Im}(g)&\textrm{Re}(f)\textrm{Im}(f)\textrm{Re}(g)\\\ -\textrm{Im}^{2}(f)\textrm{Re}(g)&-\textrm{Re}^{2}(f)\textrm{Im}(g)\\\ 0&0\\\ \end{array}\right)p=0.$ Let $f,g\in C(K)_{\mathbb{R}}$ be the constant functions $g(x)=1$ and $f(x)=\imath$. Substituting $f$ and $g$ into (3.2) we obtain $\left(\begin{array}[]{ccc}-1&0\\\ 0&0\\\ \end{array}\right)p(x)=0\qquad(x\in K),$ which gives the desired statement. According to the result established by Chu and Wong in [11, Therorem 3.10], the mapping $T$ cannot be complexified to an isometry on the complexification $C(K)_{\mathbb{R}}\oplus\imath C(K)_{\mathbb{R}}$. Here is a direct argument: let $T_{c}$ denote the complexification of $T$. $\displaystyle T_{c}(g+if)$ $\displaystyle=T(g)+iT(f)$ $\displaystyle=\left(\begin{array}[]{ccc}\textrm{Re}(g)+i\textrm{Re}(f)&\textrm{Im}(g)+i\textrm{Im}(f)\\\ 0&0\\\ \end{array}\right)\in C(K,M_{2}(\mathbb{C})).$ Let $g(x)=1$ and $f(x)=\imath$ be the constant functions considered before. Then $\\|g+if\\|=2$ while $\\|T_{c}(g+if)\\|=\left\\|\left(\begin{array}[]{ccc}1&i\\\ 0&0\\\ \end{array}\right)\right\\|_{M_{2}(\mathbb{R})_{c}}=\sqrt{2}.$ Therefore $T_{c}:C(K)_{\mathbb{R}}\bigoplus\imath C(K)_{\mathbb{R}}\rightarrow C(K,M_{2}(\mathbb{C}))$ is not an isometry. We shall see later, that the obstacle in the above example relies on the “complex nature” of $C(K)_{\mathbb{R}}$ and the real linearity of $T$. Let $A$ be a real C∗-algebra, a _“complex” character_ of $A$ is a real linear homomorphism $\rho:A\to\mathbb{C}$. Since every complex character $\rho:A\to\mathbb{C}$ admits a complex linear extension to a character $\widetilde{\rho}:A_{c}\to\mathbb{C}$, we may assume, via Gelfand theory, that $\rho$ is a real linear ∗-homomorphism. Every non-zero complex character of $A$ is unital whenever $A$ has a unit element, or can be extended to a unital complex character of the unitisation of $A$ otherwise. Following standard notation, $\Omega(A)$ will denote the space of all complex characters of $A$. We write $\Omega(A)_{\mathbb{F}}:=\\{\rho\in\Omega(A):\rho(A)=\mathbb{F}\\},\hbox{ where }\mathbb{F}=\mathbb{R}\hbox{ or }\mathbb{C}.$ The elements in $\Omega(A)_{\mathbb{R}}$ (resp., in $\Omega(A)_{\mathbb{C}}$) are called _complex characters of real type_ (resp., _complex characters of complex type_). When $A$ is abelian, we shall say that $A$ is of _real type_ if $\Omega(A)_{\mathbb{C}}=\emptyset$, and of _complex type_ if $\Omega(A)_{\mathbb{C}}=\emptyset$. An abelian real C∗-algebra $A$ is of real type (resp., of complex type) if and only if all non-zero complex characters of $A$ are onto $\mathbb{R}$ (resp., onto $\mathbb{C}$) (compare [29, Theorem 2.7.7]). In the more general setting of real commutative J∗-algebras, real and complex types where studied by L.J. Bunce and C.-H. Chu in [4, §3]. When $A$ is an abelian complex C∗-algebra, then the real C∗-algebra underlying $A$, $A_{\mathbb{R}}$, is a real C∗-algebra of complex type. Given a locally compact Hausdorff space $X$, the real C∗-algebra $C_{0}(X,\mathbb{R})$ is of real type. Furthermore, it follows from [4, Lemma 3.4] that every abelian real C∗-algebra of real type if of the form $C_{0}(X,\mathbb{R})$, for some locally compact Hausdorff space $X$. Our next result shows that the main result in [11] can be also proved when the domain is an abelian real C∗-algebra of real type. Henceforth, the closed unit ball of a Banach space $E$ will be denoted by $E_{1}$ and the set of extreme points of a convex set $S$ will be denoted by $\partial S$. ###### Proposition 3.3. Let $T:X\to Y$ be a (not necessarily surjective) linear isometry between Banach spaces and let $E=T(X)$. For each $\psi\in\partial E^{\prime}_{1},$ the set $\mathcal{Q}_{\psi}=\left\\{\phi\in\partial Y^{\prime}_{1}:\phi\|_{E}=\psi\right\\}$ is non-empty, that is, there exists $\phi\in\partial Y^{\prime}_{1}$ satisfying $\psi=\phi\|_{E}$ and $T^{\prime}(\phi)\in\partial X^{\prime}_{1}$. ###### Proof. Let $T^{\prime}:E^{\prime}\rightarrow X^{\prime}$ be the dual map of the surjective isometry $T:X\rightarrow E$. Then $T^{\prime}$ is a linear surjective isometry. Let $\psi$ be an extreme point of the unit ball, $E_{1}^{\prime}$, of $E^{\prime}$. By the Hahn-Banach theorem, the set $\mathcal{F}_{\psi}=\left\\{\phi\in Y^{\prime}_{1}:\phi\|_{E}=\psi\right\\}$ is a non-empty weak∗ closed face of $Y^{\prime}_{1}$, thus, by the Krein- Milman theorem, there exists $\phi\in\mathcal{F}_{\psi}\bigcap\partial Y^{\prime}_{1}=\mathcal{Q}_{\psi}$. By an abuse of notation, we also denote by $T^{\prime}:E^{\prime}\to X^{\prime}$ the dual map of $T:X\to Y$ since confusion is unlikely. One sees readily that $T^{\prime}(\phi)=T^{\prime}(\psi)$. Since $\psi=\phi\|_{E}$ is an extreme point in $E_{1}^{\prime}$, $T^{\prime}(\psi)$ is an extreme point in $X^{\prime}_{1}$ because $T^{\prime}:E^{\prime}\rightarrow X^{\prime}$ is a surjective linear isometry. ∎ We show below that if a real C∗-algebra $A$ admits a complex character of real type (i.e. $\Omega(A)_{\mathbb{R}}\neq\emptyset$) then the mapping $T(\cdot)p_{T}$ is non-zero for every linear isometry from $A$ into another C∗-algebra. Our next result is in fact an appropriate real version of [10, Proposition 2] and [11, Proposition 4.3]. ###### Theorem 3.4. Let $T:A\to B$ be a (not necessarily surjective) linear isometry between real C∗-algebras. Suppose $\rho\in\Omega_{\mathbb{R}}(A)$, then there exists $\phi\in\partial B^{\prime}_{1}$ and a minimal partial isometry $u_{\phi}\in B^{\prime\prime}$ such that $T^{\prime}(\phi)=\rho$, $u_{\phi}$ is the support partial isometry of $\phi$, ${T^{\prime\prime}(x)}=\rho(x){u_{\phi}}+P_{0}(u_{\phi})(T^{\prime\prime}(x)),$ $\\{u_{\phi},u_{\phi},\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}\\}=\\{u_{\phi},u_{\phi},T^{\prime\prime}(\\{x,y,z\\})\\},$ and $\\{u_{\phi},\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\},u_{\phi}\\}=\\{u_{\phi},T^{\prime\prime}(\\{x,y,z\\}),u_{\phi}\\},$ for every $x,y,z\in A^{\prime\prime}.$ Moreover, the minimal projection $p_{\phi}=u_{\phi}^{}u_{\phi}$ satisfies statements $(a)$ and $(b)$ in Proposition 3.1 and the mapping $T(\cdot)p_{\phi}:A\to B^{\prime\prime}$ is a non-zero triple homomorphism. In particular, $T(\cdot)p_{T}$ is non-zero. ###### Proof. Let $\rho\in\Omega(A)_{\mathbb{R}}$, where $A$ is a (not necessarily abelian) real C∗-algebra. The map $\rho^{\prime\prime}:A^{\prime\prime}\to\mathbb{R}$ is a surjective norm-one weak∗ continuous -homomorphism. The set $K=\ker(\rho^{\prime\prime})$ is a weak∗ closed ideal of $A^{\prime\prime}$, thus there exists a minimal and reduced central projection $p\in A^{\prime\prime}$ such that $A^{\prime\prime}=K\oplus^{\infty}\mathbb{R}p$ and $K=A^{\prime\prime}(1-p).$ In particular, $\rho\in\partial A^{\prime}_{1}.$ If $T:A\to B$ is a linear isometry, taking $E=T(A),$ $T^{\prime}:E^{\prime}\to A^{\prime}$ is a surjective linear isometry, then there exists $\psi\in\partial E^{\prime}_{1}$ such that $T^{\prime}(\psi)=\rho$. Proposition 3.3 assures the existence of $\phi\in\partial B^{\prime}_{1}$ such that $T^{\prime}(\phi)=T^{\prime}(\psi)=\rho$. We observe that $(A^{\prime\prime})^{-1}(p)=\\{0\\},$ $P^{1}(p)(x)=xp$ and $T^{\prime\prime}:A^{\prime\prime}\to B^{\prime\prime}$ is a weak* continuous linear isometry. By the remarks following Lemma 2.3 (cf. [37, Corollary 2.1]), the minimal tripotents of $B^{\prime\prime}$ are the support tripotents of the extreme points of $B^{\prime}_{1}$. Let $u_{\phi}\in B^{\prime\prime}$ be the support tripotent of $\phi$. Then $\phi(.)=\phi Q(u_{\phi})(.)$ and $P^{1}(u_{\phi})(.)=\phi(.)u_{\phi}$ (cf. [37, Lemma 2.7]). Thus, $\phi T(x)=T^{\prime}(\phi)(x)=\rho(x)$ implies (3.3) $P^{1}(u_{\phi})T(x)=\phi(T(x))u_{\phi}=\rho(x)u_{\phi},$ for every $x\in A$. Since $T^{\prime\prime}:A^{\prime\prime}\to B^{\prime\prime}$ and $\rho\in A^{\prime}$ are weak* continuous and $A_{1}$ is weak* dense in $A^{\prime\prime}_{1}$, we have (3.4) $P^{1}(u_{\phi})T^{\prime\prime}(x)=\phi(T^{\prime\prime}(x))u_{\phi}=\rho(x)u_{\phi},$ for every $x\in A^{\prime\prime}$. In particular, $P^{1}(u_{\phi})T^{\prime\prime}(\\{x,y,z\\})=\rho\left\\{x,y,z\right\\}u_{\phi}=\rho(x)\rho(y)\rho(z)\\{u_{\phi},u_{\phi},u_{\phi}\\}$ $=\\{P^{1}(u_{\phi})T^{\prime\prime}(x),P^{1}(u_{\phi})T^{\prime\prime}(y),P^{1}(u_{\phi})T^{\prime\prime}(z)\\},$ for every $x,y,z\in A^{\prime\prime}$, that is, $P^{1}(u_{\phi})T^{\prime\prime}:A^{\prime\prime}\to B^{\prime\prime}$ is a non-zero triple homomorphism. We shall show that $\\{u_{\phi},u_{\phi},T(a)\\}=u_{\phi}$ for every $a\in A^{\prime\prime}$ satisfying $\\|a\\|=1$ and $P^{1}(p)(a)=p$. By $(\ref{correct eq 3.1b})$ and the Cauchy-Schwarz inequality (cf. Lemma 2.4), we have $\displaystyle 1$ $\displaystyle=\\|P^{1}(p)(a)\\|=\|\phi\circ T^{\prime\prime}(a)\|^{2}=\|\phi(\\{u_{\phi},T^{\prime\prime}(a),u_{\phi}\\})\|^{2}$ $\displaystyle\leq\phi(\\{u_{\phi},u_{\phi},u_{\phi}\\})\phi(\\{T^{\prime\prime}(a),T^{\prime\prime}(a),u_{\phi}\\})$ $\displaystyle\leq\\|T^{\prime\prime}(a)\\|^{2}=\\|a\\|^{2}=1$ which implies $\phi(\\{T^{\prime\prime}(a),T^{\prime\prime}(a),u_{\phi}\\})=1.$ Let $N_{\phi}=\\{b\in B^{\prime\prime}\;:\;\phi(\\{b,b,u_{\phi}\\})=0\\}$. By Lemma 2.5, (3.5) $N_{\phi}=P_{0}(u_{\phi})(B^{\prime\prime}).$ Now we claim that $T^{\prime\prime}(a)-u_{\phi}\in N_{\phi}$, for any $a$ as above. Indeed, taking into account that, by Lemma 2.4, $\phi(\\{T^{\prime\prime}(a),u_{\phi},u_{\phi}\\})=\phi(\\{u_{\phi},T^{\prime\prime}(a),u_{\phi}\\})$, we have $\displaystyle\phi(\\{T^{\prime\prime}(a)-u_{\phi},T^{\prime\prime}(a)-u_{\phi},u_{\phi}\\})$ $\displaystyle=\phi(\\{T^{\prime\prime}(a),T^{\prime\prime}(a),u_{\phi}\\})-2\phi(\\{u_{\phi},T^{\prime\prime}(a),u_{\phi}\\})+\phi(u_{\phi})$ $\displaystyle=\phi(\\{T^{\prime\prime}(a),T^{\prime\prime}(a),u_{\phi}\\})-2\phi(T^{\prime\prime}(a))+\phi(u_{\phi})$ $\displaystyle=1-2\rho(a)+1=0.$ Therefore, by (3.5), $T^{\prime\prime}(a)-u_{\phi}\in P_{0}(u_{\phi})(B^{\prime\prime})$ and then (3.6) $\\{u_{\phi},u_{\phi},T^{\prime\prime}(a)\\}=u_{\phi},$ for every $a$ as above. In the next step we prove that $\phi(\\{T^{\prime\prime}(b),T^{\prime\prime}(b),u_{\phi}\\})=0$ whenever $b\in A^{\prime\prime}$ satisfies $P^{1}(p)(b)=0$. Without loss of generality we can take $\\|b\\|=1$. Since $A^{\prime\prime}=A^{\prime\prime}(1-p)\oplus^{\infty}\mathbb{R}p$, $p$ and $b$ are orthogonal in $A^{\prime\prime}$, $\\|p+b\\|=1$ and $P^{1}(p)(b+p)=p$. We deduce, from the above arguments, that $T^{\prime\prime}(b+p)+N_{\phi}=u_{\phi}+N_{\phi}=T^{\prime\prime}(p)+N_{\phi},$ which implies $T^{\prime\prime}(b)\in N_{\phi}$ and $\phi(\\{T^{\prime\prime}(b),T^{\prime\prime}(b),u_{\phi}\\})=0$. Now, let $c\in A^{\prime\prime}$ with $\\|c\\|=1$. Then $P^{1}(p)(c-\rho(c)p)=0$ and, by the arguments of the previous paragraph, we have $T^{\prime\prime}(c-\rho(c)p)\in N_{\phi}$. That is, $\\{u_{\phi},u_{\phi},T^{\prime\prime}(c-\rho(c)p)\\}=0$ and hence, by $(\ref{eq new theorem 1})$, (3.7) $\\{u_{\phi},u_{\phi},T^{\prime\prime}(c)\\}=\rho(c)\\{u_{\phi},u_{\phi},T^{\prime\prime}(p)\\}=\rho(c)u_{\phi}.$ Left multiply by $u_{\phi}u_{\phi}^{}$, right multiply by $u_{\phi}^{}u_{\phi}$ and subtracting, we have $u_{\phi}u_{\phi}^{}T^{\prime\prime}(c)=T^{\prime\prime}(c)u_{\phi}^{}u_{\phi}=u_{\phi}u_{\phi}^{}T^{\prime\prime}(c)u_{\phi}^{}u_{\phi}$, for every $c\in A^{\prime\prime}$. Therefore (3.8) ${T^{\prime\prime}(c)}=\rho(c){u_{\phi}}+P_{0}(u_{\phi})(T^{\prime\prime}(c)),$ for every $c\in A^{\prime\prime}$. It follows that $T^{\prime\prime}(\\{x,y,z\\})=\rho\\{x,y,z\\}\ u_{\phi}+P_{0}(u_{\phi})(T^{\prime\prime}(\\{x,y,z\\})),$ and by Peirce rules $\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}=$ $=\rho(x)\rho(y)\rho(z)u_{\phi}+\\{{P_{0}(u_{\phi})(T^{\prime\prime}(x))},{P_{0}(u_{\phi})(T^{\prime\prime}(y))},{P_{0}(u_{\phi})(T^{\prime\prime}(z))}\\}$ $=\rho\\{x,y,z\\}u_{\phi}+\\{{P_{0}(u_{\phi})(T^{\prime\prime}(x))},{P_{0}(u_{\phi})(T^{\prime\prime}(y))},{P_{0}(u_{\phi})(T^{\prime\prime}(z))}\\},$ which assures that (3.9) $\\{u_{\phi},u_{\phi},\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}\\}=\\{u_{\phi},u_{\phi},T^{\prime\prime}(\\{x,y,z\\})\\},$ and (3.10) $\\{u_{\phi},\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\},u_{\phi}\\}=\\{u_{\phi},T^{\prime\prime}(\\{x,y,z\\}),u_{\phi}\\},$ for every $x,y,z\in A^{\prime\prime}.$ Let $p_{\phi}=u_{\phi}^{}u_{\phi}$ and $q_{\phi}=u_{\phi}u_{\phi}^{}$ be the initial and final minimal projections of the minimal tripotent $u_{\phi}$ (see Lemma 2.1). We then have $(\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}-T^{\prime\prime}(\\{x,y,z\\}))p_{\phi}+$ $q_{\phi}(\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}-T^{\prime\prime}(\\{x,y,z\\}))=0,$ and $u_{\phi}(\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}-T^{\prime\prime}(\\{x,y,z\\}))^{}u_{\phi}=0,$ and consequently: $q_{\phi}(\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}-T^{\prime\prime}(\\{x,y,z\\}))p_{\phi}=0,$ and (3.11) $(\\{T^{\prime\prime}(x),T^{\prime\prime}(y),T^{\prime\prime}(z)\\}-T^{\prime\prime}(\\{x,y,z\\}))p_{\phi}=0,$ witnessing that $T^{\prime\prime}(\cdot)p_{\phi}:A\to B^{\prime\prime}$ is a non-zero triple homomorphism (observe that $T^{\prime\prime}(\cdot)p_{\phi}=\rho(\cdot)u_{\phi}\neq 0$). Proposition 3.1 implies that $p\leq p_{T}$ and $T(\cdot)p_{T}\neq 0$. ∎ Suppose $A$ is an abelian real C∗-algebra of complex type, that is, $\Omega(A)_{\mathbb{R}}=\emptyset$. In general, $A$ need not be C∗-isomorphic to $C_{0}(X)_{\mathbb{R}}$ (compare, for example, [4, Remark 3.8]). However, by [4, Theorem 3.7], $A^{\prime\prime}$ is isometric and C∗-isomorphic to $C(\Omega)_{\mathbb{R}}$, for some compact hyperstonean space $\Omega$. Let $J_{A}:A\hookrightarrow A^{\prime\prime}$ denote the canonical inclusion of $A$ into $A^{\prime\prime}$, then $J_{A}$ is an isometric C∗-embedding. Let $T:A^{\prime\prime}=C(\Omega)_{\mathbb{R}}\rightarrow C(\Omega,M_{2}(\mathbb{R}))$ be the isometry given in Example 3.2. That is $T(f)=\left(\begin{array}[]{ccc}\textrm{Re}(f)&\textrm{Im}(f)\\\ 0&0\\\ \end{array}\right)\;\;\;\;(f\in C(\Omega)_{\mathbb{R}}).$ The mapping $S=TJ_{A}:A\to C(\Omega,M_{2}(\mathbb{R}))$ is an isometry satisfying $S(.)p_{S}=0$. When $T$ is an isometry from an abelian real C∗-algebra of real type to another real C∗-algebra, we can prove that $T(.)p_{T}$ is an isometry. ###### Theorem 3.5. Let $B$ be a real C∗-algebra and $T:C_{0}(X,\mathbb{R})\rightarrow B$ be a (not necessarily surjective) linear isometry. Let $p_{T}\in B^{\prime\prime}$ be the structure projection of $T$. Then $T(.)p_{T}:C_{0}(X,\mathbb{R})\rightarrow B^{\prime\prime}$ is an isometric triple homomorphism. ###### Proof. Let us denote $A=C_{0}(X,\mathbb{R})$ and $E=T(A)$ and let $T^{\prime}:E^{\prime}\rightarrow A^{\prime}$ be the dual map of the surjective isometry $T:A\rightarrow E$. The real C∗-algebra $A$ is of real type and $\Omega(A)_{\mathbb{R}}=\\{\delta_{x}:x\in X\\}$. By Proposition 3.3, the set $\mathcal{Q}=\left\\{\phi\in\partial B^{\prime}_{1}:\phi\|_{E}\in\Omega(A)_{\mathbb{R}}\right\\}$ is non-empty. By Theorem 3.4, for each $x\in X$, there exists $\phi_{x}\in\mathcal{Q}$ and a minimal partial isometry $u_{x}\in B^{\prime\prime}$ such that $T^{\prime}(\phi_{x})=\delta_{x}$, $u_{x}$ is the support partial isometry of $\phi_{x}$, (3.12) ${T(f)}=\delta_{x}(f){u_{x}}+P_{0}(u_{x})(T(f)),$ $(\\{T(f),T(g),T(h)\\}-T(\\{f,g,h\\}))p_{x}=0,$ and $p_{x}T(f)^{}T(g)=T(f)^{}T(g)p_{x},$ for every $f,g,h\in A$, where $p_{x}=u_{x}^{}u_{x}$ is a minimal projection in $B^{\prime\prime}$. Let $p=\bigvee_{x\in X}p_{x}$ be the lattice supremum in $B^{\prime\prime}$. By Lemma 2.3 and (3.12), we obtain (3.13) $\\{T(f),T(g),T(h)\\}p=T(\\{f,g,h\\})p\hbox{ and }pT(f)^{}T(g)=T(f)^{}T(g)p,$ for every $f,g,h\in A$. Thus, $T(\cdot)p:A\to B^{\prime\prime}$ is a triple homomorphism. Proposition 3.1 implies $p\leq p_{T}.$ We claim that $T(\cdot)p$ is an isometry. Indeed, for each $f\in A$, we can pick $x_{0}\in X$ with $\\|f\\|=\|f(x_{0})\|$. Let $\phi_{x_{0}}\in\mathcal{Q}$ be such that $T^{\prime}(\phi_{x_{0}})=\delta_{x_{0}}$. In this case, $\\|f\\|=\\|T(f)\\|\geq\\|T(f)p\\|\geq\\|T(f)pp_{x_{0}}\\|=\\|T(f)p_{x_{0}}\\|$ $=\\|T(f)u_{x_{0}}^{}u_{x_{0}}\\|\geq\\|u_{x_{0}}u_{x_{0}}^{}T(f)u_{x_{0}}^{}u_{x_{0}}\\|$ $=\\|\\{u_{x_{0}},\\{u_{x_{0}},T(f),u_{x_{0}}\\},u_{x_{0}}\\}\\|=\\|f(x_{x_{0}})u_{x_{0}}\\|=\\|f\\|,$ which implies $\\|T(f)p\\|=\\|f\\|$ for all $f\in A$. Finally, since $p\leq p_{T}$ we have $\\|f\\|=\\|T(f)p\\|=\\|T(f)p_{T}p\\|\leq\\|T(f)p_{T}\\|\leq\\|T(f)\\|=\\|f\\|,$ which completes the proof. ∎ Let $E$ be a real or complex JBW∗-triple, two elements $\phi,\psi\in E_{{}_{{}^{\prime}}}$ are said to be _orthogonal_ (written $\phi\perp\psi$) if their support tripotents are orthogonal, that is, $u_{\phi}\perp u_{\psi}$. C.M. Edwards and G. Rüttimann proved in [16, Theorem 5.4] that in the setting of complex JBW∗-triples, $\phi$ and $\psi$ are orthogonal if, and only if, $\\|\phi\pm\psi\\|=\\|\phi\\|+\\|\psi\\|$. Suppose $E$ is a real JBW∗-triple whose complexification, $E_{c}$, is a (complex) JBW∗-triple. We have already commented that there is a conjugate linear isometry $\tau:E_{c}\rightarrow E_{c}$ of period 2 such that $E=E_{c}^{\tau}=\\{b\in E_{c}\>:\>\tau(b)=b\\}$. The “dual” map $\widetilde{\tau}:E_{c}{{}^{}}\rightarrow E_{c}^{}$ defined by $\widetilde{\tau}(\phi)(b)=\overline{\phi(\tau(b))}\qquad(\phi\in E_{c}^{},\,b\in E_{c})$ is a conjugate linear isometry of period 2 and the mapping $(E_{c}^{})^{\widetilde{\tau}}\to E^{\prime}$ $\varphi\mapsto\varphi\|_{E}$ is a surjective linear isometry. Moreover, by [31, Proposition 2.3], $\tau$ is weak continuous and hence $\widetilde{\tau}((E_{c})_{})=(E_{c})_{}$ and the restricted mapping $((E_{c})_{})^{\widetilde{\tau}}\to E_{{}_{{}^{\prime}}}$ $\varphi\mapsto\varphi\|_{E}$ is a surjective linear isometry. Take two elements $\phi,\psi\in E_{{}_{{}^{\prime}}}$, then $\phi\perp\psi$ (in $E_{{}_{{}^{\prime}}}$) if, and only if, $\phi\perp\psi$ as elements in $E_{}$, therefore $\\|\phi\pm\psi\\|=\\|\phi\\|+\\|\psi\\|$ as elements in $E_{}$, which is equivalent to $\\|\phi\pm\psi\\|=\\|\phi\\|+\\|\psi\\|$ as elements in $E_{{}_{{}^{\prime}}}$. Let $\mathcal{U}(E)$ and $\mathcal{U}(E_{c})$ denote the sets of all tripotents in $E$ and $E_{c}$, respectively. Let $\mathcal{U}(E_{c})^{\sim}$ denote the union of the set $\mathcal{U}(E_{c})$ and a one point set $\\{\omega\\}$ such that $e\leq\omega$ for every $e\in\mathcal{U}(E_{c})$. It is known that $\mathcal{U}(E_{c})^{\sim}$ is a complete lattice and $\mathcal{U}(E)\cup\\{\omega\\}=\\{e\in\mathcal{U}(E_{c}):\tau(e)=e\\}\cup\\{\omega\\}$ is a sub-complete lattice of $\mathcal{U}(E_{c})^{\sim}$. The supremum of a family $(e_{i})\subset\mathcal{U}(E_{c})$ (resp., in $\mathcal{U}(E)$) need not exist, in general, in $\mathcal{U}(E_{c})$ (resp., in $\mathcal{U}(E)$). However, for every family $(e_{i})$ of mutually orthogonal elements in $\mathcal{U}(E_{c})$ (resp., in $\mathcal{U}(E)$), the supremum $\bigvee_{i}e_{i}$ exists in $\mathcal{U}(E_{c})$ or in (resp., in $\mathcal{U}(E)$) (cf. [16, Theorem 5.1] and [17]). Moreover, suppose that an element $a\in E$ satisfies $a\perp e_{i}$ for every $i$, then $a\perp\bigvee_{i}e_{i}$. The next lemma subsumes some of the above results. ###### Lemma 3.6. Let $E$ be a real JBW∗-triple. The following statements hold: 1. $(a)$ Let $\phi$ and $\psi$ be two elements in $E_{{}_{{}^{\prime}}}$, then $\phi\perp\psi$ if, and only if, $\\|\phi\pm\psi\\|=\\|\phi\\|+\\|\psi\\|$. 2. $(b)$ Let $(e_{i})$ be a family of mutually orthogonal elements in $\mathcal{U}(E)$. Then the supremum $\bigvee_{i}e_{i}$ exists in $\mathcal{U}(E)$. Moreover, suppose $a\in E$ satisfies $a\perp e_{i}$, for every $i$, then $\bigvee_{i}e_{i}\perp a$. $\hfill\Box$ The next result reveals the connection between the real versions of Theorem 1 in [10] and Theorem 3.10 in [11]. ###### Theorem 3.7. Let $B$ be a real JB∗-triple and $T:C_{0}(X,\mathbb{R})\rightarrow B$ be a not necessarily surjective linear isometry. Then there exists a partial isometry $u\in B^{\prime\prime}$ such that ${T(f)}=\delta_{x}(f){u_{x}}+P_{0}(u_{x})(T(f)),$ $\\{u,T(\\{f,g,h\\}),u\\}=\\{u,\\{T(f),T(g),T(h)\\},u\\},$ for every $f,g,h\in C_{0}(X,\mathbb{R})$, and $\\{u,T(\cdot),u\\}:C(X,\mathbb{R})\rightarrow B^{\prime\prime}$ is an isometry. ###### Proof. Keeping in mind the notation in the proof of Theorem 3.5, we write $A=C_{0}(X,\mathbb{R})$, $\Omega(A)_{\mathbb{R}}=\\{\delta_{x}:x\in X\\}$ and $\mathcal{Q}=\left\\{\phi\in\partial B^{\prime}_{1}:\phi\|_{E}\in\Omega(A)_{\mathbb{R}}\right\\}.$ By Theorem 3.4, whose proof is valid when $B$ is a real JB∗-triple, for each $x\in X$, there exists $\phi_{x}\in\mathcal{Q}$ and a minimal partial isometry $u_{x}\in B^{\prime\prime}$ such that $T^{\prime}(\phi_{x})=\delta_{x}$, $u_{x}$ is the support partial isometry of $\phi_{x}$, (3.14) ${T(f)}=\delta_{x}(f){u_{x}}+P_{0}(u_{x})(T(f)),$ and hence $T(\\{f,g,h\\})-\\{T(f),T(g),T(h)\\}\in B^{\prime\prime}_{0}(u_{x})$ for every $f,g,h\in A$ and $x\in X$. We shall prove now that $u_{x}\perp u_{y}$ whenever $x\neq y$ in $X$. Indeed, for $x\neq y$ we have $\delta_{x}\perp\delta_{y}$, and thus $2=\\|\delta_{x}\pm\delta_{y}\\|=\\|T^{\prime}(\phi_{x}\pm\phi_{y})\\|\leq\\|\phi_{x}\pm\phi_{y}\\|\leq\\|\phi_{x}\\|+\\|\phi_{y}\\|=2.$ It follows from Lemma 3.6$(a)$ and the comments preceding it, that $\phi_{x}\perp\phi_{y}$, or equivalently, $u_{x}\perp u_{y}$. Therefore $(u_{x})_{x\in X}$ is a family of mutually orthogonal minimal tripotents in $B^{\prime\prime}$. The supremum $u=\bigvee_{x}u_{x}$ exists and defines a tripotent in $B^{\prime\prime}$ (cf. Lemma 3.6$(b)$). Since, by $(\ref{4.3.13c})$, for each $f,g,$ and $h$ in $A$ and $x\in X$, we have $T(\\{f,g,h\\})-\\{T(f),T(g),T(h)\\}\perp u_{x}$, we deduce from Lemma 3.6$(b)$, that $T(\\{f,g,h\\})-\\{T(f),T(g),T(h)\\}\perp u,$ for every $f,g,h\in A$. In order to prove the last statement, observe that for each $f\in A$ we can take $x_{0}\in X$ with $\\|f\\|=\|f(x_{0})\|$. Let $\phi_{x_{0}}\in\partial B^{\prime}_{1}$ such that $T^{\prime}(\phi_{x_{0}})=\delta_{x_{0}}$. Then, $\displaystyle\\|f\\|=\\|T(f)\\|$ $\displaystyle\geq\\|\\{u,T(f),u\\}\\|\geq\\|\\{u_{0},\\{u,T(f),u\\},u_{0}\\}\\|$ $\displaystyle\geq\\|Q(u_{x_{0}})Q(u)(T(f))\\|=\\|Q(u_{x_{0}})^{2}(T(f))\\|$ $\displaystyle=\\|\\{u_{0},\\{u_{0},T(f),u_{0}\\},u_{0}\\}\\|=\\|f(x_{x_{0}})u_{x_{0}}\\|=\\|f\\|,$ which proves $\\|\\{u,T(f),u\\}\\|=\\|f\\|\qquad(f\in A).$ ∎ ###### Remark 3.8. When in the above Theorem 3.7, $B$ is a real C∗-algebra, the projection $p=u^{}u$ coincides with the one considered in the proof of Theorem 3.5. Using the previous results, we can now show that if $T$ is an isometry between two real C∗-algebras $A$ and $B$, then $T$ is a local triple homomorphism via a tripotent in $B^{\prime\prime}$. That does not however imply that $T$ is a triple homomorphism on the whole algebra $A$ (compare Example 3.2). The following theorem is an extension of the results in [10, 11]. We recall that given an element $a$ in a real JB∗-triple $A$, the real JB∗-subtriple, $A_{a},$ generated by $a$ is linearly isometric to the real JB∗-triple $C_{0}(X,\mathbb{R})$ of real-valued continuous functions on $X$ vanishing at infinity (see [7] and [27]). ###### Theorem 3.9. Let $T:A\rightarrow B$ be a not necessarily surjective linear isometry between two real JB∗-triples. Then, for each $a\in A$, there exists a tripotent $u\in B^{\prime\prime}$ such that 1. $(a)$ $\\{u,T(\\{f,g,h\\}),u\\}=\\{u,\\{T(f),T(g),T(h)\\},u\\}$, for all $f,g,h$ in the real JB∗-subtriple generated by $a$; 2. $(b)$ The mapping $\\{u,T(\cdot),u\\}:A_{a}\rightarrow B^{\prime\prime}$ is a linear isometry. Furthermore, when $B$ is a real C∗-algebra, the projection $p=u^{}u$ satisfies that $T(\cdot)p:A_{a}\rightarrow B^{\prime\prime}$ is an isometric triple homomorphism. ###### Proof. Since $A_{a}$ is linearly isometric to the JB∗-triple $C_{0}(X,\mathbb{R})$ of all real-valued continuous functions on $X$ vanishing at infinity (cf. [27]), the result follows immediately from Theorem 3.5, Theorem 3.7 and Remark 3.8. ∎ Acknowledgements: The useful comments and suggestions made by the Referee in his/her report were fundamental to improve the presentation and quality of the paper. We are grateful to the referee for his/her thorough suggestions. ## References [1] M. Apazoglou, _Linear maps on real C ∗-algebras and related structures_, _Ph.D. Thesis, University of London_ , 2010. * [2] T. Barton, Y. Friedman, Grothendieck’s inequality for JB∗-triples and applications, _J. London Math. Soc. (2)_ 36, 513-523 (1987). * [3] T.J. Barton, R.M. Timoney, Weak∗-continuity of Jordan triple products and its applications, _Math. Scand._ 59, 177-191 (1986). * [4] L.J. Bunce, C.-H. Chu, Real contractive projections on commutative C∗-algebras, _Math. 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Kaup, Hermitian Jordan triple systems and the automorphisms of bounded symmetric domains, _Non-Associative Algebra and Applications_ , Kluver Academic Publishers, 204-214, (1994). * [28] W. Kaup, On real Cartan factors, _Manuscripta Math._ 92, 191-222 (1997). * [29] B.R. Li, _Real Operator Algebras_ , World Scientific Publishing (Singapore), 2003\. * [30] O. Loos, _Bounded symmetric domains and Jordan pairs_ , Math. Lectures, University of California, Irvine, 1977. * [31] J. Martínez Moreno, A.M. Peralta, Separate weak-continuity of the triple product in dual real JB∗-triples, _Math. Z._ 234, 635-646 (2000). [32] M.P. Neal, Inner ideals and facial structure of the quasi-state space of a JB-algebra, _J. Funct. Anal._ 173 no. 2, 284-307 (2000). * [33] A.L.T. Paterson, A.M. Sinclair, Characterisation of isometries between C∗-algebras, _J. London Math. Soc._ 5, 755-761 (1972). * [34] G.K. Pedersen, _C-algebras and their automorphism groups,_ London Mathematical Society Monographs, 14. Academic Press, Inc., London-New York, 1979. [35] A. Rodríguez Palacios, Banach space characterizations of unitaries: A survey, _J. Math. Anal. Appl._ 369 168-178 (2010). * [36] A.M. Peralta, _Little Grothendieck’s theorem for real JB ∗-triples_, _Math. Z._ 237, 531-545 (2001). * [37] A.M. Peralta, L.L. Stacho, Atomic decomposition of real JBW∗-triples, _Quart J. Math._ 52, 79-87 (2001). * [38] J.D.M. Wright, M. Youngson, On isometries of Jordan algebras, _J. London Math. Soc._ 17, 339-344 (1978).	arxiv-papers	2013-09-16T07:27:52	2024-09-04T02:49:50.941845	{ "license": "Public Domain", "authors": "Maria Apazoglou and Antonio M. Peralta", "submitter": "Antonio M. Peralta", "url": "https://arxiv.org/abs/1309.3838" }
1309.3839	# Orthogonal forms and orthogonality preservers on real function algebras Jorge J. Garcés [email protected] Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. and Antonio M. Peralta [email protected] Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain. ###### Abstract. We initiate the study of orthogonal forms on a real C∗-algebra. Motivated by previous contributions, due to Ylinen, Jajte, Paszkiewicz and Goldstein, we prove that for every continuous orthogonal form $V$ on a commutative real C∗-algebra, $A$, there exist functionals $\varphi_{1}$ and $\varphi_{2}$ in $A^{}$ satisfying $V(x,y)=\varphi_{1}(xy)+\varphi_{2}(xy^{}),$ for every $x,y$ in $A$. We describe the general form of a (not-necessarily continuous) orthogonality preserving linear map between unital commutative real C∗-algebras. As a consequence, we show that every orthogonality preserving linear bijection between unital commutative real C∗-algebras is continuous. ###### Key words and phrases: Orthogonal form, real C∗-algebra, orthogonality preservers, disjointness preserver, separating map ###### 2010 Mathematics Subject Classification: Primary 46H40; 4J10, Secondary 47B33; 46L40; 46E15; 47B48. Authors partially supported by the Spanish Ministry of Economy and Competitiveness, D.G.I. project no. MTM2011-23843, and Junta de Andalucía grants FQM0199 and FQM3737. ## 1\. Introduction and preliminaries Elements $a$ and $b$ in a real or complex C∗-algebra, $A$, are said to be _orthogonal_ (denoted by $a\perp b$) if $ab^{}=b^{}a=0$. A bounded bilinear form $V:A\times A\to\mathbb{K}$ is called _orthogonal_ (resp., _orthogonal on self-adjoint elements_) whenever $V(a,b^{})=0$ for every $a\perp b$ in $A$ (resp., in the self-adjoint part of $A$). All the forms considered in this paper are assumed to be continuous. Motivated by the seminal contributions by K Ylinen [51] and R. Jajte and A. Paszkiewicz [29], S. Goldstein proved that every orthogonal form $V$ on a (complex) C∗-algebra, $A,$ is of the form $V(x,y)=\phi(xy)+\psi(xy)\ \ (x,y\in A),$ where $\phi$ and $\psi$ are two functionals in $A^{}$ (cf. [21, Theorem 1.10]). A simplified proof of Goldstein’s theorem was published by U. Haagerup and N.J. Laustsen in [24]. This characterisation has emerged as a very useful tool in the study of bounded linear operators between C∗-algebras which are orthogonality or disjointness preserving (see, for example, [11, 12]). The first aim of this paper is to study orthogonal forms on the wider class of real C∗-algebras. Little or nothing is known about the structure of an orthogonal form $V$ on a real C∗-algebra. At first look, one is tempted to consider the canonical complex bilinear extension of $V$ to a form on the complexification, $A_{\mathbb{C}}=A\oplus iA$, of $A$ and, when the latter is orthogonal, to apply Goldstein’s theorem. However, the complex bilinear extension of $V$ to $A_{\mathbb{C}}\times A_{\mathbb{C}}$, need not be, in general, orthogonal (see Example 2.7). The study of orthogonal forms on real C∗-algebras requires a completely independent strategy; surprisingly the resulting forms will enjoy a different structure to that established by S. Goldstein in the complex setting. In section 2 we establish some structure results for orthogonal forms on a general real C∗-algebra, showing, among other properties, that every orthogonal form on a real C∗-algebra extends to an orthogonal form on its multiplier algebra (see Proposition 1.3). It is also proved that, for each orthogonal and symmetric form $V$ on a real C∗-algebra, $A,$ there exists a functional $\phi\in A^{}$ satisfying $V(a,b)=\phi(ab+ba),$ for every $a,b\in A$ with $a=a^{}$, $b^{}=b$ (cf. Proposition 1.5). In the real setting, the skew-symmetric part of a real C∗-algebra, $A,$ is not determined by the self- adjoint part of $A,$ so the information about the behavior of $V$ on the rest of $A$ is very limited. Section 3 contains one of the main results of the paper: the characterisation of all orthogonal forms on a commutative real C∗-algebra. Concretely, we prove that a form $V$ on a commutative real C∗-algebra $A$ is orthogonal if, and only if, there exist functionals $\varphi_{1}$ and $\varphi_{2}$ in $A^{}$ satisfying $V(x,y)=\varphi_{1}(xy)+\varphi_{2}(xy^{}),$ for every $x,y\in A$ (see Theorem 2.4). Among the consequences, it follows that the complex bilinear extension of $V$ to the complexification of $A$ is orthogonal if, and only if, we can take $\varphi_{2}=0$ in the above representation. We recall that a mapping $T:A\to B$ between real or complex C∗-algebras is said to be _orthogonality or disjointness preserving_ (also called _separating_) whenever $a\perp b$ in $A$ implies $T(a)\perp T(b)$ in $B.$ The mapping $T$ is _bi-orthogonality preserving_ whenever the equivalence $a\perp b\Leftrightarrow T(a)\perp T(b)$ holds for all $a,b$ in $A$. As noticed in [13], every bi-orthogonality preserving linear surjection, $T:A\to B$ between two C∗-algebras is injective. The study of orthogonality preserving operators between C∗-algebras started with the work of W. Arendt [1] in the setting of unital abelian C∗-algebras. Subsequent contributions by K. Jarosz [30] extended the study to the setting of orthogonality preserving (not necessarily bounded) linear mappings between abelian C∗-algebras. The first study on orthogonality preserving symmetric (bounded) linear operators between general (complex) C∗-algebras is originally due to M. Wolff (cf. [49]). Orthogonality preserving bounded linear maps between C∗-algebras, JB∗-algebras and JB∗-triples were completely described in [11] and [12]. The pioneer works of E. Beckenstein, L. Narici, and A.R. Todd in [8] and [9] (see also [7]) were applied by K. Jarosz to prove that every orthogonality preserving linear bijection between $C(K)$-spaces is (automatically) continuous (see [30]). More recently, M. Burgos and the authors of this note proved in [13] that every bi-orthogonality preserving linear surjection between two von Neumann algebras (or between two compact C∗-algebras) is automatically continuous (compare [40], [41] for recent additional generalisations). The main goal of section 4 is to describe the orthogonality preserving linear mappings between unital commutative real C∗-algebras (see Theorem 3.2). As a consequence, we shall prove that every orthogonality preserving linear bijection between unital commutative real C∗-algebras is automatically continuous. We shall exhibit some examples illustrating that the results in the real setting are completely independent from those established for complex C∗-algebras. We further give a characterisation of those linear mappings between real forms of $C(K)$-spaces which are bi-orthogonality preserving. ### 1.1. Preliminary results Let us now introduce some basic facts and definitions required later. A _real C ∗-algebra_ is a real Banach -algebra $A$ which satisfies the standard C∗-identity, $\\|a^{}a\\|=\\|a\\|^{2}$, and which also has the property that $1+a^{\ast}a$ is invertible in the unitization of $A$ for every $a\in A$. It is known that a real Banach -algebra, $A,$ is a real C∗-algebra if, and only if, it is isometrically -isomorphic to a norm-closed real -subalgebra of bounded operators on a real Hilbert space (cf. [39, Corollary 5.2.11]). Clearly, every (complex) C∗-algebra is a real C∗-algebra when scalar multiplication is restricted to the real field. If $A$ is a real C∗-algebra whose algebraic complexification is denoted by ${B}=A\oplus iA,$ then there exists a C∗-norm on ${B}$ extending the norm of $A$. It is further known that there exists an involutive conjugate-linear ∗-automorphism $\tau$ on ${B}$ such that $A={B}^{\tau}:=\\{x\in B:\tau(x)=x\\}$ (compare [39, Proposition 5.1.3] or [47, Lemma 4.1.13], and [22, Corollary 15.4]). The dual space of a real or complex C∗-algebra $A$ will be denoted by $A^{}$. Let $\widetilde{\tau}:B^{}\rightarrow B^{}$ denote the map defined by $\widetilde{\tau}(\phi)(b)=\overline{\phi(\tau(b))}\qquad(\phi\in B^{},\,b\in B).$ Then $\widetilde{\tau}$ is a conjugate-linear isometry of period 2 and the mapping $(B^{})^{\widetilde{\tau}}\to A^{}$ $\varphi\mapsto\varphi\|_{A}$ is a surjective linear isometry. We shall identify $(B^{})^{\widetilde{\tau}}$ and $A^{}$ without making any explicit mention. When $A$ is a real or complex C∗-algebra, then $A_{sa}$ and $A_{skew}$ will stand for the set of all self-adjoint and skew-symmetric elements in $A$, respectively. We shall make use of standard notation in C∗-algebra theory. Given Banach spaces $X$ and $Y$, $L(X,Y)$ will denote the space of all bounded linear mappings from $X$ to $Y$. We shall write $L(X)$ for the space $L(X,X)$. Throughout the paper the word “operator” (respectively, multilinear or sesquilinear operator) will always mean bounded linear mapping (respectively bounded multilinear or sesquilinear mapping). The dual space of a Banach space $X$ is always denoted by $X^{}$. Let us recall that a series $\sum_{n}x_{n}$ in a Banach space is called _weakly unconditionally Cauchy (w.u.C.)_ if there exists $C>0$ such that for any finite subset $F\subset{\mathbb{N}}$ and $\varepsilon_{n}=\pm 1$ we have $\displaystyle\left\\|\sum_{n\in F}\varepsilon_{n}x_{n}\right\\|\leq C$. A (linear) operator $T:X\longrightarrow Y$ is _unconditionally converging_ if for every w.u.C. series $\sum_{n}x_{n}$ in $X,$ the series $\sum_{n}T(x_{n})$ is unconditionally convergent in $Y$, that is, every subseries of $\sum_{n}T(x_{n})$ is norm converging. It is known that $T:X\to Y$ is unconditionally converging if, and only if, for every w.u.C. series $\sum_{n}x_{n}$ in $X,$ we have $\\|T(x_{n})\\|\to 0$ (compare, for example, [44, page 1257]) Let us also recall that a Banach space $X$ is said to have _Pełczyński’s_ _property_ (V) if, for every Banach space $Y$, every unconditionally converging operator $T:X\to Y$ is weakly compact. The proof of the following elementary lemma is left to the reader. ###### Lemma 1.1. Let $X$ be a complex Banach space, $\tau:X\to X$ a conjugate-linear period-2 isometry. Then the real Banach space $X^{\tau}:=\\{x\in X:\tau(x)=x\\}$ satisfies property (V) whenever $X$ does. $\hfill\Box$ We shall require, for later use, some results on extensions of multilinear operators. Let $X_{1},\dots,X_{n}$, and $X$ be Banach spaces, $T:X_{1}\times\cdots\times X_{n}\to X$ a (continuous) $n$-linear operator, and $\pi:\\{1,\dots,n\\}\to\\{1,\dots,n\\}$ a permutation. It is known that there exists a unique $n$-linear extension $AB(T)_{\pi}:X_{1}^{}\times\cdots\times X_{n}^{}\to X^{}$ such that for every $z_{i}\in X_{i}^{}$ and every net $(x_{\alpha_{i}}^{i})\in X_{i}$ ($1\leq i\leq n$), converging to $z_{i}$ in the weak topology we have $AB(T)_{\pi}(z_{1},\ldots,z_{n})=\mbox{weak-}\lim_{\alpha_{\pi(1)}}\cdots\mbox{weak-}\lim_{\alpha_{\pi(n)}}T(x_{\alpha_{1}}^{1},\ldots,x_{\alpha_{n}}^{n}).$ Moreover, $AB(T)_{\pi}$ is bounded and has the same norm as $T$. The extensions $AB(T)_{\pi}$ coincide with those considered by Arens in [2, 3] and by Aron and Berner for polynomials in [4]. The $n$-linear operators $AB(T)_{\pi}$ are usually called the Arens or _Aron-Berner extensions_ of $T$. Under some additional hypothesis, the Arens extension of a multilinear operator also is separately weak∗ continuous. Indeed, if every operator from $X_{i}$ to $X_{j}^{}$ is weakly compact ($i\not=j)$ the Arens extensions of $T$ defined above do not depend on the chosen permutation $\pi$ and they are all separately weak∗ continuous (see [5], and Theorem 1 in [10]). In particular, the above requirements always hold when every $X_{i}$ satisfies Pelczynski’s property $(V)$ (in such case $X_{i}^{}$ contains no copies of $c_{0}$, therefore every operator from $X_{i}$ to $X_{j}^{}$ is unconditionally converging, and hence weakly compact by property $(V)$, see [43]). When all the Arens extensions of $T$ coincide, the symbol $AB(T)=T^{}$ will denote any of them. We should note at this point that every C∗-algebra satisfies property $(V)$ (cf. Corollary 6 in [46]). Since every real C∗-algebra is, in particular, a real form of a (complex) C∗-algebra, it follows from Lemma 1.1 that every real C∗-algebra satisifes property $(V)$. We therefore have: ###### Lemma 1.2. Let $A_{1},\ldots,A_{k}$ be real C∗-algebras and let $T$ be a multilinear continuous operator from $A_{1}\times\ldots\times A_{k}$ to a real Banach space $X$. Then $T$ admits a unique Arens extension $T^{}:A_{1}^{}\times\ldots\times A_{k}^{}\to X^{}$ which is separately weak∗ continuous.$\hfill\Box$ Given a real or complex C∗-algebra, $A$, the _multiplier algebra_ of $A$, $M(A)$, is the set of all elements $x\in A^{}$ such that, for each element $a\in A$, $xa$ and $ax$ both lie in $A$. We notice that $M(A)$ is a C∗-algebra and contains the unit element of $A^{}$. It should be recalled here that $A=M(A)$ whenever $A$ is unital. ###### Proposition 1.3. Let $A$ be a real C∗-algebra. Suppose that $V:A\times A\to\mathbb{R}$ is an orthogonal bounded bilinear form. Then the continuous bilinear form $\tilde{V}:M(A)\times M(A)\to\mathbb{R},\ \ \tilde{V}(a,b):=V^{}(a,b)$ is orthogonal. ###### Proof. Let $a$ and $b$ be two orthogonal elements in $M(A)$. Let $a^{[\frac{1}{3}]}$ (resp., $b^{[\frac{1}{3}]}$) denote the unique element $z$ in $M(A)$ satisfying $zz^{}z=a$ (resp., $zz^{}z=b$). We notice that $a^{[\frac{1}{3}]}$ and $b^{[\frac{1}{3}]}$ are orthogonal, so, for each pair $x,y$ in $A$, $a^{[\frac{1}{3}]}xa^{[\frac{1}{3}]}$ and $b^{[\frac{1}{3}]}yb^{[\frac{1}{3}]}$ are orthogonal elements in $A$. Since $V$ is orthogonal, we have $V(a^{[\frac{1}{3}]}xa^{[\frac{1}{3}]},(b^{[\frac{1}{3}]})^{}y(b^{[\frac{1}{3}]})^{})=0$ for every $x,y\in A$. Goldstine’s theorem (cf. Theorem V.4.2.5 in [18]) guarantees that the closed unit ball of $A$ is weak-dense in the closed unit ball of $A^{}$. Therefore we can pick two bounded nets $(x_{\lambda})$ and $(y_{\mu})$ in $A$, converging in the weak∗ topology of $A^{}$ to $(a^{[\frac{1}{3}]})^{}$ and $b^{[\frac{1}{3}]}$, respectively. We have already mentioned that $V^{}:A^{}\times A^{}\to\mathbb{R}$ is separately weak∗ continuous. Since $0=V(a^{[\frac{1}{3}]}x_{\lambda}a^{[\frac{1}{3}]},(b^{[\frac{1}{3}]})^{}y_{\mu}(b^{[\frac{1}{3}]})^{}),$ for every $\lambda$ and $\mu$, taking limits, first in $\lambda$ and subsequently in $\mu$, we deduce that $V^{}(a^{[\frac{1}{3}]}(a^{[\frac{1}{3}]})^{}a^{[\frac{1}{3}]},(b^{[\frac{1}{3}]})^{}b^{[\frac{1}{3}]}(b^{[\frac{1}{3}]})^{}))=\tilde{V}(a,b^{})=0,$ which shows that $\tilde{V}$ is orthogonal. ∎ Since the multiplier algebra of a real or complex C∗-algebra always has a unit element, Proposition 1.3 allows us to restrict our study on orthogonal bilinear forms on a real C∗-algebra $A$ to the case in which $A$ is unital. A _real von Neumann algebra_ is a real C∗-algebra which is also a dual Banach space (cf. [28] or [39, §6.1]). Clearly, the self adjoint part of a real von Neumann algebra is a JW-algebra in the terminology employed in [25], so every self-adjoint element in a real von Neumann algebra $W$ can be approximated in norm by a finite real linear combination of mutually orthogonal projections in $W$ (cf. [25, Proposition 4.2.3]). We shall explore now the validity in the real setting of some of the results established by S. Goldstein in [21]. ###### Lemma 1.4. Let $A$ be a real von Neumann algebra with unit $1$. Suppose that $V:A\times A\to\mathbb{R}$ is a bounded bilinear form. The following are equivalent: 1. $(a)$ $V$ is orthogonal on $A_{sa}$; 2. $(b)$ $V(p,q)=0$, whenever $p$ and $q$ are two orthogonal projections in $A$; 3. $(c)$ $V(a,b)=V(ab,1)$ for every $a,b\in A_{sa}$ with $ab=ba$. If any of the above statements holds and $V$ is symmetric, then defining $\phi_{1}(x):=V(x,1)$ ($x\in A$), we have $V(a,b)=\phi_{1}(\frac{ab+ba}{2})$, for every $a,b\in A_{sa}$. ###### Proof. Applying the existence of spectral resolutions for self-adjoint elements in a real von Neumann algebra, the argument given by S. Goldstein in [21, Proposition 1.2] remains valid to prove the equivalence of $(a)$, $(b)$ and $(c)$. Suppose now that $V$ is symmetric. Let $a=\sum_{j=1}^{m}\lambda_{j}p_{j}$ be an algebraic element in $A_{sa}$, where the $\lambda_{j}$’s belong to $\mathbb{R}$ and $p_{1},\ldots,p_{m}$ are mutually orthogonal projections in $A$. Since $V$ is orthogonal, for every projection $p\in A$, we have $V(p,1)=V(p,1-p)+V(p,p)=V(p,p).$ Thus, $V(a,a)=\sum_{j=1}^{m}\lambda_{j}^{2}V(p_{j},p_{j})=\sum_{j=1}^{m}\lambda_{j}^{2}V(p_{j},1)=V\left(\sum_{j=1}^{m}\lambda_{j}^{2}p_{j},1\right)=V(a^{2},1).$ The (norm) density of algebraic elements in $A_{sa}$ and the continuity of $V$ imply that $V(a,a)=V(a^{2},1),$ for every $a\in A_{sa}$. Finally, applying that $V$ is symmetric we have $V(a^{2},1)+V(b^{2},1)+V(ab+ba,1)=V((a+b)^{2},1)$ $=V(a+b,a+b)=V(a,a)+V(b,b)+2V(a,b),$ for every $a,b\in A_{sa}$, and hence $V(a,b)=V(\frac{ab+ba}{2},1),$ for all $a,b\in A_{sa}$. ∎ The above result holds for every monotone $\sigma$-complete unital real C∗-algebra $A$ (that is, each upper bounded, monotone increasing sequence of selfadjoint elements of $A$ has a least upper bound). Surprisingly, the final conclusion of the above Lemma can be established for unital real C∗-algebras with independent basic techniques. ###### Proposition 1.5. Let $A$ be a unital real C∗-algebra with unit $1$. Suppose that $V:A\times A\to\mathbb{R}$ is an orthogonal, symmetric, bounded, bilinear form. Then defining $\phi_{1}(x):=V(x,1)$ ($x\in A$), we have $V(a,b)=\phi_{1}(\frac{ab+ba}{2})$, for every $a,b\in A_{sa}$. ###### Proof. Let $a$ be a selfadjoint element in $A$. The real C∗-subalgebra, $C$, of $A$ generated by $1$ and $a$ is isometrically isomorphic to the space $C(K,\mathbb{R})$ of all real-valued continuous functions on a compact Hausdorff space $K$. The restriction of $V$ to $C\times C$ is orthogonal, therefore the mapping $x\mapsto V(x,x)$ is a $2$-homogeneous orthogonally additive polynomial on $C$. The main result in [45] implies the existence of a functional $\varphi_{a}\in C^{}$ such that $V(x,x)=\varphi_{a}(x^{2})$, for every $x\in C$. It is clear that $\varphi_{a}(x)=V(x,1)$ for every $x\in C$. In particular $V(a,a)=\varphi_{a}(a^{2})=V(a^{2},1).$ The argument given at the end of the proof of Lemma 1.4 gives the desired statement. ∎ The above proposition shows that we can control the form of a symmetric orthogonal form on the self adjoint part of a (unital) real C∗-algebra. The form on the skew-symmetric part remains out of control for the moment. ## 2\. Orthogonal forms on abelian real C∗-algebras Throughout this section, $A$ will denote a unital, abelian, real C∗-algebra whose complexification will be denoted by $B$. It is clear that $B$ is a unital, abelian C∗-algebra. It is known that there exists a period-2 conjugate-linear ∗-automorphism $\tau:B\to B$ such that $A=B^{\tau}:=\\{x\in B:\tau(x)=x\\}$ (cf. [47, 4.1.13] and [22, 15.4] or [39, §5.2]). By the commutative Gelfand theory, there exists a compact Hausdorff space $K$ such that $B$ is C∗-isomorphic to the C∗-algebra $C(K)$ of all complex valued continuous functions on $K$. The Banach-Stone Theorem implies the existence of a homeomorphism $\sigma:K\to K$ such that $\sigma^{2}(t)=t$, and $\tau(a)(t)=\overline{a(\sigma(t))},$ for all $t\in K$, $a\in C(K)$. Real function algebras of the form $C(K)^{\tau}$ have been studied by its own right and are interesting in some other settings (cf. [37]). Henceforth, the symbol $\mathfrak{B}$ will stand for the $\sigma$-algebra of all Borel subsets of $K$, $S(K)$ will denote the space of $\mathfrak{B}$-simple scalar functions defined on $K$, while the _Borel algebra over $K$_, $B(K)$, is defined as the completion of $S(K)$ under the supremum norm. It is known that $B=C(K)\subset B(K)\subset C(K)^{}.$ The mapping $\tau^{}:C(K)^{}\to C(K)^{}$ is a period-2 conjugate-linear ∗-automorphism on $B^{}=C(K)^{}.$ It is easy to see that $\tau^{}(B(K))=B(K),$ and hence $\tau^{}\|_{B(K)}:B(K)\to B(K)$ defines a period-2 conjugate-linear ∗-automorphism on $B(K)$. By an abuse of notation, the symbol $\tau$ will denote $\tau$, $\tau^{}$ and $\tau^{}\|_{B(K)}$ indistinctly. It is clear that, for each Borel set $B\in\mathfrak{B},$ $\tau(\chi_{{}_{B}})=\chi_{{}_{\sigma(B)}}$. Let $a$ be an element in $B(K)$. For each $\varepsilon>0,$ there exist complex numbers $\lambda_{1},\ldots,\lambda_{r}$ and disjoint Borel sets $B_{1},\ldots,B_{r}$ such that $\displaystyle{\left\\|a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right\\|<{\varepsilon}.}$ When $a\in A$ is $\tau$-symmetric (i.e. $\tau(a)=a$) then, since $a=\frac{1}{2}(a+\tau(a)),$ we have $\left\\|a-\frac{1}{2}\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}+\overline{\lambda_{k}}\chi_{{}_{\sigma(B_{k})}}\right\\|\leq\frac{1}{2}\left\\|a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right\\|+\frac{1}{2}\left\\|a-\sum_{k=1}^{r}\overline{\lambda_{k}}\chi_{{}_{\sigma(B_{k})}}\right\\|$ $\leq\frac{1}{2}\left\\|a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right\\|+\frac{1}{2}\left\\|\tau\left(a-\sum_{k=1}^{r}\lambda_{k}\chi_{{}_{B_{k}}}\right)\right\\|<\varepsilon.$ Consequently, every element in $B(K)^{\tau}$ can be approximated in norm by finite linear combinations of the form $\sum_{k}\alpha_{k}\chi_{{}_{B_{k}}}+\overline{\alpha_{k}}\chi_{{}_{\sigma(B_{k})}},$ where $\alpha_{1},\ldots,\alpha_{n}$ are complex numbers and $B_{1},\ldots,B_{n}$ are mutually disjoint Borel sets. Having in mind that, for each Borel set $B\in\mathfrak{B}$ and each $\alpha\in\mathbb{C},$ $\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)^{}=\overline{\alpha}\chi_{{}_{B}}+{\alpha}\chi_{{}_{\sigma(B)}},$ we have $\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)+\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)^{}=2\Re\hbox{e}(\alpha)\left(2\chi_{{}_{\sigma(B)\cap B}}+\chi_{{}_{\sigma(B)\backslash B}}+\chi_{{}_{B\backslash\sigma(B)}}\right)$ $=2\Re\hbox{e}(\alpha)\left(2\chi_{{}_{\sigma(B)\cap B}}+\chi_{{}_{(\sigma(B)\backslash B)\cup\sigma(\sigma(B)\backslash B)}}\right),$ and $\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)-\left(\alpha\chi_{{}_{B}}+\overline{\alpha}\chi_{{}_{\sigma(B)}}\right)^{}=2i\Im\hbox{m}(\alpha)\left(\chi_{{}_{B\backslash\sigma(B)}}-\chi_{{}_{\sigma(B)\backslash B}}\right).$ Suppose now that $a\in B(K)^{\tau}$ is -symmetric (i.e. $a^{}=a$). It follows from the above that $a$ can be approximated in norm by linear combinations of the form $\displaystyle{\sum_{k=1}^{r}\alpha_{k}\chi_{{}_{E_{k}}},}$ where $\alpha_{k}\in\mathbb{R}$ and $E_{1},\ldots,E_{r}$ are mutually disjoint Borel subsets of $K$ with $\sigma(E_{i})=E_{i}$. Let $b$ be an element in $B(K)^{\tau}$ satisfying $b^{}=-b$. Similar arguments to those given for -symmetric elements, allow us to show that $b$ can be approximated in norm by finite linear combinations of the form $\displaystyle{\sum_{k=1}^{r}i\ \alpha_{k}(\chi_{{}_{E_{k}}}-\chi_{{}_{\sigma(E_{k})}}),}$ where $\alpha_{k}\in\mathbb{R}$ and $E_{1},\ldots,E_{r}$ are mutually disjoint Borel subsets of $K$ with $\sigma(E_{i})\cap E_{i}=\emptyset$. ###### Lemma 2.1. Let $A$ be a unital, abelian, real C∗-algebra whose complexification is denoted by $B=C(K)$, for a suitable compact Hausdorff space $K$. Let $\tau:B\to B$ be a period-2 conjugate-linear ∗-automorphism satisfying $A=B^{\tau}$ and $\tau(a)(t)=\overline{a(\sigma(t))},$ for all $t\in K$, $a\in C(K)$, where $\sigma:K\to K$ is a period-2 homeomorphism. Then the set $N=\\{t\in K:\sigma(t)\neq t\\}$ is an open subset of $K$, $F=\\{t\in K:\sigma(t)=t\\}$ is a closed subset of $K$ and there exists an open subset $\mathcal{O}\subset K$ maximal with respect to the property $\mathcal{O}\cap\sigma(\mathcal{O})=\emptyset.$ ###### Proof. That $F$ is closed follows easily from the continuity of $\sigma,$ and consequently, $N=K/F$ is open. Let $\mathcal{F}$ be the family of all open subsets $O\subseteq K$ such that $O\cap\sigma(O)=\emptyset$ ordered by inclusion. Let $S=\\{O_{\lambda}\\}_{\lambda}$ be a totally ordered subset of $\mathcal{F}.$ We shall see that $O=\bigcup_{\lambda}O_{\lambda}$ is an open set which also lies in $\mathcal{F},$ that is, $O\cap\sigma(O)=\emptyset.$ Let us suppose, on the contrary, that there exists $t\in O\cap\sigma(O)\neq\emptyset.$ Then there exist $\lambda,\beta$ such that $t\in O_{\lambda}$ and $t\in\sigma(O_{\beta}).$ Since $S$ is totally ordered, $O_{\lambda}\subseteq O_{\beta}$ or $O_{\beta}\subseteq O_{\lambda}.$ We shall assume that $O_{\lambda}\subseteq O_{\beta}.$ Then $\sigma(O_{\lambda})\subseteq\sigma(O_{\beta})$ and $t$ lies in $O_{\beta}\cap\sigma(O_{\beta})=\emptyset,$ which is a contradiction. Finally, Zorn’s Lemma gives the existence of a maximal element $\mathcal{O}$ in $\mathcal{F}.$ ∎ It should be noticed here that, in Lemma 2.1, $\mathcal{O}\cup\sigma(\mathcal{O})=N$, an equality which follows from the maximality of $\mathcal{O}$. Our next lemma analyses the “spectral resolution” of a -skew-symmetric element in $B(K)^{\tau}.$ ###### Lemma 2.2. In the notation of Lemma 2.1, let $B(A)=B(K)^{\tau}$, let $a\in B(K)^{\tau}_{sa},$ and let $b$ be an element in $B(A)_{skew}$. Then the following statements hold: 1. $a)$ $b\|F=0$; 2. $b)$ For each $\varepsilon>0$, there exist mutually disjoint Borel sets $B_{1},\ldots,B_{m}\subset\mathcal{O}$ and real numbers $\lambda_{1},\ldots,\lambda_{m}$ satisfying $\displaystyle{\left\\|b-\sum_{j=1}^{m}i\ \lambda_{j}(\chi_{{}_{B_{j}}}-\chi_{{}_{\sigma(B_{j})}})\right\\|<\varepsilon;}$ 3. $c)$ For each $\varepsilon>0$, there exist mutually disjoint Borel sets $C_{1},\ldots,C_{m}\subset K$ and real numbers $\mu_{1},\ldots,\mu_{m}$ satisfying $\sigma(C_{j})=C_{j},$ and $\displaystyle{\left\\|a-\sum_{j=1}^{m}\mu_{j}\chi_{{}_{C_{j}}}\right\\|<\varepsilon.}$ ###### Proof. $a)$ Since $b^{}=-b$, we have $Re(b(t))=0,\forall t\in K.$ Now, let $t\in F$, applying $\sigma(t)=t$ and $\tau(b)=b$ we get $\overline{b(t)}=\overline{b(\sigma(t))}=b(t),$ and hence $\Im\hbox{m}(b(t))=0.$ Statements $b)$ and $c)$ follow from the comments prior to Lemma 2.1 and the maximality of $\mathcal{O}$ in that Lemma. ∎ It is clear that in a commutative real (or complex) C∗-algebra, $A,$ two elements $a,b$ are orthogonal if and only if they have zero-product, that is, $ab=0.$ Therefore, $V(a,b^{})=0=V(a,b)$ whenever $V:A\times A\to{\mathbb{R}}$ is an orthogonal bilinear form on an abelian real C∗-algebra and $a,b$ are two orthogonal elements in $A$. We shall make use of this property without an explicit mention. We shall keep the notation of Lemma 2.1 throughout the section. Henceforth, for each $C\subseteq\mathcal{O}$ we shall write $u_{{}_{C}}=i\ (\chi_{{}_{C}}-\chi_{{}_{\sigma(C)}}).$ The symbol $u_{{}_{0}}$ will stand for the element $u_{{}_{\mathcal{O}}}.$ It is easy to check $1=\chi_{{}_{F}}+u_{{}_{0}}u_{{}_{0}}^{},$ where $1$ is the unit element in $B(K)^{\tau}.$ By Lemma 2.2 $a)$, for each $b\in B(K)^{\tau}_{skew}$ we have $b\perp\chi_{{}_{F}},$ and so $b=bu_{{}_{0}}u_{{}_{0}}^{}.$ ###### Proposition 2.3. Let $K$ be a compact Hausdorff space, $\tau$ a period-2 conjugate-linear isometric ∗-homomorphism on $C(K)$, $A=C(K)^{\tau}$, and $V:A\times A\to\mathbb{R}$ be an orthogonal bounded bilinear form whose Arens extension is denoted by $V^{}:A^{}\times A^{*}\to\mathbb{R}$. Let $\sigma:K\to K$ be a period-2 homeomorphism satisfying $\tau(a)(t)=\overline{a(\sigma(t))},$ for all $t\in K$, $a\in C(K)$. Then the following assertions hold for all Borel subsets $D,B,C$ of $K$ with $\sigma(B)\cap B=\sigma(C)\cap C=\emptyset$ and $\sigma(D)=D$: 1. $a)$ $V(\chi_{{}_{D}},u_{{}_{B}})=V(u_{{}_{B}},\chi_{D})=0,$ whenever $D\cap B=\emptyset$; 2. $b)$ $V(u_{{}_{B}},u_{{}_{C}})=0,$ whenever $B\cap C=\emptyset;$ 3. $c)$ $V((u_{{}_{0}}u_{{}_{0}}^{}-u_{{}_{C}}u^{}_{{}_{C}})u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{C}},(u_{{}_{0}}u_{{}_{0}}^{}-u_{{}_{C}}u^{}_{{}_{C}})u_{{}_{B}})=0.$ ###### Proof. By an abuse of notation, we write $V$ for $V$ and $V^{}$. Let $K_{1},K_{2}$ be compact subsets of $K$ such that $K_{1},K_{2}$ and $\sigma(K_{2})$ are mutually disjoint. By regularity and Urysohn’s Lemma there exist nets $(f_{\lambda})_{{}_{\lambda}}$, $(g_{\gamma})_{{}_{\gamma}}$ in $C(K)^{+}$ such that $\chi_{{}_{K_{1}}}\leq f_{\lambda}\leq\chi_{{}_{K\backslash(K_{2}\cup\sigma(K_{2}))}},$ $\chi_{{}_{K_{2}}}\leq g_{\gamma}\leq\chi_{{}_{K\backslash(K_{1}\cup\sigma(K_{1})\cup\sigma(K_{2}))}},$ $(f_{\lambda})_{{}_{\lambda}}$ (respectively, $(g_{\gamma})_{{}_{\gamma}}$) converges to $\chi_{{}_{K_{1}}}$ (resp., to $\chi_{{}_{K_{2}}}$) in the weak∗ topology of $C(K)^{}$. The nets $\widetilde{f}_{\lambda}=\frac{1}{2}(f_{\lambda}+\tau(f_{\lambda}))$ and $\widetilde{g}_{\gamma}=i(g_{\gamma}-\tau(g_{\gamma}))$ lie in $C(K)^{\tau}$ and converge in the weak∗ topology of $C(K)^{}$ to $\frac{1}{2}(\chi_{{}_{K_{1}}}+\chi_{{}_{\sigma(K_{1})}})$ and $u_{{}_{K_{2}}},$ respectively. It is also clear that $f_{\lambda}\perp g_{\gamma},$ $\tau(f_{\lambda})\perp g_{\gamma},$ and hence $\widetilde{f}_{\lambda}\perp\widetilde{g}_{\gamma},$ for every $\lambda,\gamma.$ By the separate weak∗ continuity of $V^{}\equiv V$ we have (1) $V\left(\frac{1}{2}(\chi_{{}_{K_{1}}}+\chi_{{}_{\sigma(K_{1})}}),u_{{}_{K_{2}}}\right)=w^{}-\lim_{\lambda}\left(w^{}-\lim_{\gamma}V\left(\widetilde{f_{\lambda}},\widetilde{g}_{\gamma}\right)\right)=0,$ and $V\left(u_{{}_{K_{2}}},\frac{1}{2}(\chi_{{}_{K_{1}}}+\chi_{{}_{\sigma(K_{1})}})\right)=0.$ We can similarly prove that (2) $V\left(u_{{}_{K_{1}}},u_{{}_{K_{2}}}\right)=0,$ whenever $K_{1}$ and $K_{2}$ are two compact subsets of $K$ such that $K_{1},K_{2},\sigma(K_{1})$ and $\sigma(K_{2})$ are pairwise disjoint. $a)$ Let now $D,B$ be two disjoint Borel subsets of $K$ such that $\sigma(D)=D$ and $B\subseteq\mathcal{O}.$ By inner regularity there exist nets of the form $(\chi_{{}_{K_{\lambda}^{{}^{D}}}})_{\lambda}$ and $(\chi_{{}_{K_{\gamma}^{{}^{B}}}})_{\gamma}$ such that $(\chi_{{}_{K_{\lambda}^{{}^{D}}}})_{\lambda}$ and $(\chi_{{}_{K_{\gamma}^{{}^{B}}}})_{\gamma}$ converge in the weak∗ topology of $C(K)^{}$ to $\chi_{{}_{D}}$ and $\chi_{{}_{B}}$, respectively, where each $K_{\lambda}^{{}^{D}}\subseteq D$ and each $K_{\gamma}^{{}^{B}}\subseteq B$ is compact subset of $K$. By the assumptions made on $D$ and $B$ we have that $K_{\lambda}^{{}^{D}}\cap K_{\gamma}^{{}^{B}}=K_{\lambda}^{{}^{D}}\cap\sigma(K_{\gamma}^{{}^{B}})=\emptyset$ and $K_{\gamma}^{{}^{B}}\subseteq\mathcal{O}$ for all $\lambda$ and $\gamma.$ By (1) and the separate weak∗ continuity of $V$ we have (3) $V(\chi_{{}_{D}},u_{{}_{B}})=w^{}-\lim_{\lambda}\left(w^{}-\lim_{\gamma}V\left(\frac{\chi_{{}_{K_{\lambda}^{{}^{D}}}}+\chi_{{}_{\sigma(K_{\lambda}^{{}^{D}})}}}{2},u_{{}_{K_{\gamma}^{{}^{B}}}}\right)\right)=0,$ and (4) $V(u_{{}_{B}},\chi_{{}_{{}_{D}}})=0.$ A similar argument, but replacing (1) with (2), applies to prove $b)$. To prove the last statement, we observe that $(u_{{}_{0}}u_{{}_{0}}^{}-u_{c}u_{c}^{})u_{{}_{B}}=(\chi_{{}_{\mathcal{O}}}+\chi_{{}_{\sigma(\mathcal{O}})}-\chi_{{}_{C}}-\chi_{{}_{\sigma(C)}})u_{{}_{B}}=(\chi_{{}_{\mathcal{O}\setminus C}}+\chi_{{}_{\sigma(\mathcal{O}\setminus C)}})u_{{}_{B}}=u_{{}_{(\mathcal{O}\setminus C)\cap B}},$ and hence the statement $c)$ follows from $b)$. ∎ We can now establish the description of all orthogonal forms on a commutative real C∗-algebra. ###### Theorem 2.4. Let $V:A\times A\to\mathbb{R}$ be a continuous orthogonal form on a commutative real C∗-algebra, then there exist $\varphi_{1},$ and $\varphi_{2}$ in $A^{}$ satisfying $V(x,y)=\varphi_{1}(xy)+\varphi_{2}(xy^{}),$ for every $x,y\in A.$ ###### Proof. We may assume, without loss of generality, that $A$ is unital (compare Proposition 1.3). Let $B$ denote the complexification of $A$. In this case $B$ identifies with $C(K)$ for a suitable compact Hausdorff space $K$ and $A=C(K)^{\tau}$, where $\tau$ is a conjugate-linear period-2 -homomorphism on $C(K)$. We shall follow the notation employed in the rest of this section. The form $V:A\times A\to\mathbb{R}$ extends to a continuous form $V^{}:A^{}\times A^{}\to\mathbb{R}$ which is separately weak∗ continuous (cf. Lemma 1.2). The restriction $V^{}\|_{B(K)^{\tau}\times B(K)^{\tau}}:B(K)^{\tau}\times B(K)^{\tau}\to\mathbb{R}$ also is a continuous extension of $V$. We shall prove the statement for $V^{}\|_{B(K)^{\tau}\times B(K)^{\tau}}.$ Henceforth, the symbol $V$ will stand for $V$, $V^{}$ and $V^{*}\|_{B(K)^{\tau}\times B(K)^{\tau}}$ indistinctly. Let us first take two self-adjoint elements $a_{1},a_{2}$ in $B(K)^{\tau}$. By Proposition 1.5, (5) $V(a_{1},a_{2})=V(a_{1}a_{2},1).$ To deal with the skew-symmetric part, let $D,B,C$ be Borel subsets of $K$ with, $D=\sigma(D)$ and $B,C\subseteq\mathcal{O}.$ From Proposition 2.3 $a)$, we have (6) $V(\chi_{{}_{D}},u_{{}_{B}})=V(\chi_{{}_{D}},u_{{}_{B}}(1-\chi_{{}_{D}}+\chi_{{}_{D}}))=V(\chi_{{}_{D}},u_{{}_{B\cap(K\backslash D)}})+V(\chi_{{}_{D}},u_{{}_{B}}\chi_{{}_{D}})$ $=V(\chi_{{}_{D}}-1+1,u_{{}_{B}}\chi_{{}_{D}})=V(-\chi_{{}_{(K\backslash D)}}+1,u_{{}_{(B\cap D)}})=V(1,u_{{}_{B}}\chi_{{}_{D}}).$ Similarly, (7) $V(u_{{}_{B}},\chi_{{}_{D}})=V(u_{{}_{B}}\chi_{{}_{D}},1).$ Now, Proposition 2.3 $b)$ and $c)$, repeatedly applied give: $V(u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{B}}(\chi_{{}_{F}}+u_{{}_{0}}u_{{}_{0}}^{}),u_{{}_{C}})=V(u_{{}_{B}}u_{{}_{0}}u_{{}_{0}}^{},u_{{}_{C}})$ $=V(u_{{}_{B}}(u_{{}_{0}}u_{{}_{0}}^{}+u_{{}_{C}}u^{}_{{}_{C}}-u_{{}_{C}}u^{}_{{}_{C}}),u_{{}_{C}})=V(u_{{}_{B}}u_{{}_{C}}u^{}_{{}_{C}},u_{{}_{C}})$ $=V(u_{{}_{B}}u_{{}_{C}}u^{}_{{}_{C}},u_{{}_{C}}-u_{{}_{0}}+u_{{}_{0}})=V(u_{{}_{(B\cap C)}},-u_{{}_{(\mathcal{O}\backslash C)}}+u_{{}_{0}})=V(u_{{}_{(B\cap C)}},u_{{}_{0}})$ $=V(u_{{}_{B}}u_{{}_{C}}(u^{}_{{}_{C}}-u_{{}_{0}}^{}+u_{{}_{0}}^{}),u_{{}_{0}})=V(u_{{}_{B}}u_{{}_{C}}u_{{}_{0}}^{},u_{{}_{0}}).$ Thus, we have (8) $V(u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{B}}u_{{}_{C}}u_{{}_{0}}^{},u_{{}_{0}}),$ and similarly (9) $V(u_{{}_{B}},u_{{}_{C}})=V(u_{{}_{0}},u_{{}_{B}}u_{{}_{C}}u_{{}_{0}}^{}).$ Let $\displaystyle{a_{l}=\sum_{j=1}^{m_{l}}\mu_{l,j}\chi_{{}_{D^{l}_{j}}},}$ $\displaystyle{b_{l}=\sum_{k=1}^{p_{l}}\lambda_{l,k}u_{{}_{B^{l}_{k}}}}$ ($l\in\\{1,2\\}$) be two simple elements in $B(K)^{\tau}_{sa}$ and $B(K)^{\tau}_{skew}$, respectively, where $\lambda_{l,k},\mu_{l,j}\in\mathbb{R},$ for each $l\in\\{1,2\\},$ $\\{D^{l}_{1},\ldots,D^{l}_{m_{l}}\\}$ and $\\{B^{l}_{1},\ldots,B^{l}_{p_{l}}\\}$ are families of mutually disjoint Borel subsets of $K$ with $\sigma(D^{l}_{j})=D^{l}_{j}$ and $B^{l}_{i}\subseteq\mathcal{O}.$ By $(\ref{eq theorem -1})$, $(\ref{eq sym against anti})$, $(\ref{eq anti against sym})$, and $(\ref{eq anti anti 1}),$ we have $V(a_{1}+b_{1},a_{2}+b_{2})=V(a_{1}a_{2},1)+\sum_{j=1}^{m_{1}}\sum_{k=1}^{p_{2}}\mu_{1,j}\lambda_{2,k}V\left(\chi_{{}_{D^{1}_{j}}},u_{{}_{B^{2}_{k}}}\right)$ $+\sum_{k=1}^{p_{1}}\sum_{j=1}^{m_{2}}\mu_{2,j}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}},\chi_{{}_{D^{2}_{j}}}\right)+\sum_{k=1}^{p_{1}}\sum_{k=1}^{p_{2}}\lambda_{2,k}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}},u_{{}_{B^{2}_{k}}}\right)$ $=V(a_{1}a_{2},1)+\sum_{j=1}^{m_{1}}\sum_{k=1}^{p_{2}}\mu_{1,j}\lambda_{2,k}V\left(1,\chi_{{}_{D^{1}_{j}}}u_{{}_{B^{2}_{k}}}\right)$ $+\sum_{k=1}^{p_{1}}\sum_{j=1}^{m_{2}}\mu_{2,j}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}}\chi_{{}_{D^{2}_{j}}},1\right)+\sum_{k=1}^{p_{1}}\sum_{k=1}^{p_{2}}\lambda_{2,k}\lambda_{1,k}V\left(u_{{}_{B^{1}_{k}}}u_{{}_{B^{2}_{k}}}u_{{}_{0}}^{},u_{{}_{0}}\right)$ $=V(a_{1}a_{2},1)+V\left(1,a_{1}b_{2}\right)+V\left(b_{1}a_{2},1\right)+V\left(b_{1}b_{2}u_{{}_{0}}^{},u_{{}_{0}}\right)$ $=\psi_{1}(a_{1}a_{2})+\psi_{2}\left(a_{1}b_{2}\right)+\psi_{1}\left(b_{1}a_{2}\right)+\psi_{4}\left(b_{1}b_{2}\right),$ where $\psi_{1},\psi_{2},$ and $\psi_{4}$ are the functionals in $A^{}$ defined by $\psi_{1}(x)=V(x,1),$ $\psi_{2}(x)=V(1,x),$ and $\psi_{4}(x)=V(xu_{{}_{0}}^{},u_{{}_{0}}),$ respectively. Since, by Proposition 2.2, simple elements of the above form are norm-dense in $B(K)^{\tau}_{sa}$ and $B(K)^{\tau}_{skew}$, respectively, and $V$ is continuous, we deduce that $V(a_{1}+b_{1},a_{2}+b_{2})=\psi_{1}(a_{1}a_{2})+\psi_{2}\left(a_{1}b_{2}\right)+\psi_{1}\left(b_{1}a_{2}\right)+\psi_{4}\left(b_{1}b_{2}\right),$ for every $a_{1},a_{2}\in B(K)^{\tau}_{sa}$, $b_{1},b_{2}\in B(K)^{\tau}_{skew}.$ Now, taking $\phi_{1}=\frac{1}{4}(2\psi_{1}+\psi_{2}+\psi_{4}),$ $\phi_{2}=\frac{1}{4}(2\psi_{1}-\psi_{2}-\psi_{4}),$ $\phi_{3}=\frac{1}{4}(\psi_{2}-\psi_{4}),$ and $\phi_{4}=\frac{1}{4}(\psi_{4}-\psi_{2}),$ we get $V(a_{1}+b_{1},a_{2}+b_{2})=\phi_{1}((a_{1}+b_{1})(a_{2}+b_{2}))+\phi_{2}\left((a_{1}+b_{1})(a_{2}+b_{2})^{}\right)$ $+\phi_{3}\left((a_{1}+b_{1})^{}(a_{2}+b_{2})\right)+\phi_{4}\left((a_{1}+b_{1})^{}(a_{2}+b_{2})^{}\right),$ for every $a_{1},a_{2}\in B(K)^{\tau}_{sa}$, $b_{1},b_{2}\in B(K)^{\tau}_{skew}.$ Finally, defining $\varphi_{1}(x)=\phi_{1}(x)+\phi_{4}(x^{})$ and $\varphi_{2}(x)=\phi_{2}(x)+\phi_{3}(x^{})$ $(x\in A)$, we get the desired statement. ∎ ###### Remark 2.5. The functionals $\varphi_{1}$ and $\varphi_{2}$ appearing in Theorem 2.4 need not be unique. For example, let $(\varphi_{1},\varphi_{2})$ and $(\phi_{1},\phi_{2})$ be two couples of elements in the dual of a commutative real C∗-algebra $A$. It is not hard to check that $\varphi_{1}(xy)+\varphi_{2}(xy^{})=\phi_{1}(xy)+\phi_{2}(xy^{}),$ for every $x,y\in A$ if, and only if, $\varphi_{1}+\varphi_{2}=\phi_{1}+\phi_{2}$, $(\varphi_{1}-\varphi_{2})(z)=(\phi_{1}-\phi_{2})(z)$ and $(\varphi_{1}-\varphi_{2})(zw)=(\phi_{1}-\phi_{2})(zw),$ for every $z,w\in A_{skew}.$ These conditions are not enough to guarantee that $\phi_{i}=\varphi_{i}$. Take, for example, $A=\mathbb{R}\oplus^{\infty}\mathbb{C}_{\mathbb{R}},$ $\phi_{1}(a,b)=a+\Re\hbox{e}(b)+\Im\hbox{m}(b),$ $\phi_{2}(a,b)=0,$ $\varphi_{1}(a,b)=\frac{a}{2}+\Re\hbox{e}(b)+\Im\hbox{m}(b),$ and $\varphi_{2}(a,b)=\frac{a}{2}.$ ###### Corollary 2.6. Let $V:A\times A\to\mathbb{R}$ be a continuous orthogonal form on a commutative real C∗-algebra, then its (unique) Arens extension $V^{}:A^{}\times A^{*}\to\mathbb{R}$ is an orthogonal form.$\hfill\Box$ Clearly, the statement of the above Theorem 2.4 doesn’t hold for bilinear forms on a commutative (complex) C∗-algebra. The real version established in this paper is completely independent to the result proved by K. Ylinen for commutative complex C∗-algebras in [51] and [21]. It seems natural to ask whether the real result follows from the complex one by a mere argument of complexification. Our next example shows that the (canonical) extension of an orthogonal form on a commutative real C∗-algebra need not be an orthogonal form on the complexification. ###### Example 2.7. Let $K=\\{t_{1},t_{2}\\}.$ We define $\sigma:K\rightarrow K$ by $\sigma(t_{1})=t_{2}.$ Let $A=C(K)^{\tau}$ be the real C∗-algebra whose complexification is $C(K)$ and let $V:A\times A\to\mathbb{R},$ be the orthogonal form defined by $V(x,y)=\phi_{{}_{t_{1}}}(xy^{})=\Re\hbox{e}(x(t_{1})\overline{y(t_{1})})=\Re\hbox{e}(x(t_{1})y(t_{2})),$ where $\phi_{t_{1}}=\Re\hbox{e}(\delta_{{}_{t_{1}}}).$ In this case, the canonical complex bilinear extension $\widetilde{V}:C(K)\times C(K)\to\mathbb{C}$ is given by $\widetilde{V}(x,y)=\phi_{{}_{t_{1}}}(x\tau(y)^{})=x(t_{1})y(t_{2})$ $(x,y\in C(K)).$ It is clear that $\chi_{{}_{t_{1}}}\perp\chi_{{}_{t_{2}}}$ in $C(K),$ however $\widetilde{V}(\chi_{{}_{t_{1}}},\chi_{{}_{t_{2}}})=1\neq 0,$ which implies that $\widetilde{V}$ is not orthogonal. The (complex) bilinear extension of an orthogonal form $V$ on a real C∗-algebra to its complexification is orthogonal precisely when $V$ satisfies the generic form of an orthogonal form on a (complex) C∗-algebra given by the main result in [21]. ###### Corollary 2.8. Let $V:A\times A\to\mathbb{R}$ be a continuous orthogonal form on a commutative real C∗-algebra, let $B$ denote the complexification of $A$ and let $\widetilde{V}:B\times B\to\mathbb{R}$ be the (complex) bilinear extension of $V$. Then the form $\widetilde{V}$ is orthogonal if, and only if, $V$ writes in the form $V(x,y)=\varphi_{1}(xy)$ $(x,y\in A),$ where $\varphi_{1}$ is a functional in $A^{}$. ###### Proof. Let $\tau$ be the period-2 ∗-automorphism on $B$ satisfying that $B^{\tau}=B$ and let $\widetilde{\tau}:B^{}\to B^{}$ be the involution defined by $\widetilde{\tau}(\phi)(b)=\overline{\phi(\tau(b))}$. Suppose $\widetilde{V}$ is orthogonal. By the main result in [21] (see also [51]), there exists $\phi\in B^{}$ satisfying $\widetilde{V}(x,y)=\phi(xy),$ for every $x,y\in B$. Since $\widetilde{V}$ is an extension of $V$, we get $V(a,b)=\Re\hbox{e}\phi(ab)=\phi(ab),$ for every $a,b\in A$. In particular, $\phi(a)\in\mathbb{R}$, for every $a\in A$ and hence $\widetilde{\tau}(\phi)=\phi$ lies in $(B^{})^{\widetilde{\tau}}\equiv A^{}$. Let us assume that $V$ writes in the form $V(x,y)=\varphi_{1}(xy)$ $(x,y\in A)$, where $\varphi_{1}$ is a functional in $A^{}$. The functional $\varphi_{1}$ can be regarded as an element in $B^{}$ satisfying $\widetilde{\tau}(\varphi_{1})=\varphi_{1}.$ It is easy to check that $\widetilde{V}(x,y)=\varphi_{1}(xy),$ for every $x,y\in B$. ∎ ## 3\. Orthogonality preservers between commutative real C∗-algebras Throughout this section, $A_{1}=C(K_{1})^{\tau_{1}}$ and $A_{2}=C(K_{2})^{\tau_{2}}$ will denote two unital commutative real C∗-algebras, $K_{1}$ and $K_{2}$ will be two compact Hausdorff spaces and $\tau_{i}$ will denote a conjugate-linear period-2 ∗-automorphism on $C(K_{i})$ given by $\tau_{i}(f)(t)=\overline{f(\sigma_{i}(t))}$ ($t\in K_{i}$, $f\in C(K_{i})$), where $\sigma_{i}:K_{i}\to K_{i}$ is a period-2 homeomorphism. We shall write $B_{1}=C(K_{1})$ and $B_{2}=C(K_{2})$ for the corresponding complexifications of $A_{1}$ and $A_{2}$, respectively. By the Banach-Stone theorem, every surjective isometry $T:C(K_{1})\to C(K_{2})$ is a composition operator, that is, there exist a unitary element $u$ in $C(K_{2})$ and a homeomorphism $\sigma:K_{2}\to K_{1}$ such that $T(f)(t)=(uC_{\sigma})(f)(t):=u(t)\ f(\sigma(t))$ ($t\in K_{2}$, $f\in C(K_{1})$). This result led to the study of the so-called Banach-Stone theorems in different classes of Banach spaces containing $C(K)$-spaces, in which their algebraic and geometric properties are mutually determined. That is the case of general C∗-algebras (R. Kadison [31] and Paterson and Sinclair [42]), JB- and JB∗-algebras (Wright and M. Youngson [50] and Isidro and A. Rodríguez [27]), JB∗-triples (Kaup [33] and Dang, Friedman and Russo [15]), real C∗-algebras (Grzesiak [23], Kulkarni and Arundhathi [36], Kulkarni and Limaye [37] and Chu, Dang, Russo and Ventura [14]) and real JB∗-triples (Isidro, Kaup and Rodrguez [26], Kaup [34] and Fernndez-Polo, Martnez and Peralta [19]). In what concerns us, we highlight that any surjective linear isometry $T:C(K_{1})^{\tau_{1}}\to C(K_{2})^{\tau_{2}}$ is a composition operator given by a homeomorphism $\phi:K_{2}\to K_{1}$ which satisfies $\sigma_{1}\circ\phi=\phi\circ\sigma_{2}$ (cf. [23] or [36] or [37, Corollary 5.2.4]). The class of orthogonality preserving (continuous) operators between $C(K)$-spaces is strictly bigger than the class of surjective isometries. Actually, a bounded linear operator $T:C(K_{1})\to C(K_{2})$ is orthogonality preserving (equivalently, disjointness preserving) if, and only if, there exist $u$ in $C(K_{2})$ and a mapping $\varphi:K_{2}\to K_{1}$ which is continuous on $\\{t\in K_{2}:u(t)\neq 0\\}$ such that $T(f)(t)=(uC_{\varphi})(f)(t)=u(t)\ f(\varphi(t))$ (compare [1, Example 2.2.1]). Developing ideas given by E. Beckenstein, L. Narici, and A.R. Todd in [8] and [9] (see also [7]), K. Jarosz showed, in [30], that the above hypothesis of $T$ being continuous can be, in some sense, relaxed. More concretely, for every orthogonality preserving linear mapping $T:C(K_{1})\to C(K_{2})$, there exists a disjoint decomposition $K_{2}=S_{1}\cup S_{2}\cup S_{3}$ (with $S_{2}$ open, $S_{3}$ closed), and a continuous mapping $\varphi$ from $S_{1}\cup S_{2}$ into $K_{1}$ such that $T(f)(s)=\chi(s)f(\varphi(s))$ for all $s\in S_{1}$ (where $\chi$ is a continuous, bounded, non-vanishing, scalar-valued function on $S_{1}$), $T(f)(s)=0$ for all $s\in S_{3}$, $\varphi(S_{2})$ is finite and, for each $s\in S_{2}$, the mapping $f\mapsto T(f)(s)$ is not continuous. As a consequence, every orthogonality preserving linear bijection between $C(K)$-spaces is (automatically) continuous. More recently, M. Burgos and the authors of this note prove, in [13], that every bi-orthogonality preserving linear surjection between two von Neumann algebras (or between two compact C∗-algebras) is automatically continuous (compare [40], [41] for recent additional generalisations). The main goal of this section is to describe the orthogonality preserving linear mappings between $C(K)^{\tau}$-spaces. Among the consequences, we establish that every orthogonality preserving linear bijection between unital commutative real C∗-algebras is automatically continuous. We shall provide an example of an orthogonality preserving linear bijection between $C(K)^{\tau}$-spaces which is not bi-orthogonality preserving and give a characterisation of bi-orthogonality preserving linear maps. We shall borrow and adapt some of the ideas developed in those previously mentioned papers (cf. [8, 9] and [30]). In order to have a good balance between completeness and conciseness, we just give some sketch of the refinements needed in our setting. In any case, the results presented here are independent innovations and extensions of those proved by Beckenstein, Narici, and Todd and Jarosz for $C(K)$-spaces. Let $T:C(K_{1})^{\tau_{1}}\to C(K_{2})^{\tau_{2}}$ be an orthogonality preserving linear mapping. Keeping in mind the notation in the previous section, we write $L_{i}:=\mathcal{O}_{i}\cup F_{i},$ where $\mathcal{O}_{i}$ and $F_{i}$ are the subsets of $K_{i}$ given by Lemma 2.1. The map sending each $f$ in $C(Ki)^{\tau_{i}}$ to its restriction to $L_{i}$ is a C∗-isomorphism (and hence a surjective linear isometry) from $C(Ki)^{\tau_{i}}$ onto the real C∗-algebra $C_{r}(L_{i})$ of all continuous functions $f:L_{i}\to\mathbb{C}$ taking real values on $F_{i}$. Thus, studying orthogonality preserving linear maps between $C(K)^{\tau}$ spaces is equivalent to study orthogonality preserving linear mappings between the corresponding $C_{r}(L)$-spaces. Henceforth, we consider an orthogonality preserving (not necessarily continuous) linear map $T:C_{r}(L_{1})\to C_{r}(L_{2})$, where $L_{1}$ and $L_{2}$ are two compact Hausdorff spaces and each $F_{i}$ is a closed subset of $L_{i}$. Let us consider the sets $Z_{1}=\\{s\in L_{2}:{\delta}_{s}T\hbox{ is a non-zero bounded real-linear mapping}\\},$ $Z_{3}=\\{s\in L_{2}:{\delta}_{s}T=0\\},\hbox{ and }Z_{2}=L_{2}\backslash(Z_{1}\cup Z_{3}).$ It is easy to see that $Z_{3}$ is closed. Following a very usual technique (see, for example, [8, 9, 30, 16] and [17]), we can define a continuous support map $\varphi:Z_{1}\cup Z_{2}\to L_{1}$. More concretely, for each $s\in Z_{1}\cup Z_{2}$, we write $\hbox{supp}(\delta_{s}T)$ for the set of all $t\in L_{1}$ such that for each open set $U\subseteq L_{1}$ with $t\in U$ there exists $f\in C_{r}(L_{1})$ with $\hbox{coz}(f)\subseteq U$ and $\delta_{s}(T(f))\neq 0$. Actually, following a standard argument, it can be shown that, for each $s\in Z_{1}\cup Z_{2},$ $\hbox{supp}(\delta_{s}T)$ is non-empty and reduces exactly to one point $\varphi(s)\in L_{1}$, and the assignment $s\mapsto\varphi(s)$ defines a continuous map from $Z_{1}\cup Z_{2}$ to $L_{1}$. Furthermore, the value of $T(f)$ at every $s\in Z_{1}$ depends strictly on the value $f(\varphi(s))$. More precisely, for each $s\in Z_{1}$ with $\varphi(s)\notin F_{1}$, the value $T(g)(s)$ is the same for every function $g\in C_{r}(L_{1})$ with $g\equiv i$ on a neighborhood of $\varphi(s).$ Thus, defining $T(i)(s):=0$ for every $s\in Z_{3}\cup Z_{2}$ and for every $s\in Z_{1}$ with $\varphi(s)\in F_{1}$, and $T(i)(s):=T(g)(s)$ for every $s\in Z_{1}\cup Z_{2}$ with $\varphi(s)\notin F_{1}$, where $g$ is any element in $C_{r}(L_{1})$ with $g\equiv i$ on a neighborhood of $\varphi(s),$ we get a (well-defined) mapping $T(i):L_{2}\to\mathbb{C}.$ It should be noticed that “$T(i)$” is just a symbol to denoted the above mapping and not an element in the image of $T$. In this setting, the identity $T(f)(s)=T(1)(s)\ \Re\hbox{e}f(\varphi(s))+T(i)(s)\ \Im\hbox{m}f(\varphi(s)),$ holds for every $s\in Z_{1}.$ Clearly, $T(1)(s),T(i)(s)\in\mathbb{R}$, for every $s\in F_{2}$ and $\left\|T(1)(s)\right\|+\left\|T(i)(s)\right\|\neq 0$, for every $s\in Z_{1}$. The following property also follows from the definition of $\varphi$ by standard arguments: Under the above conditions, let $s$ be an element in $Z_{1}\cup Z_{2},$ then (10) $\delta_{s}T(f)=0\hbox{ for every }f\in C_{r}(L_{1})\hbox{ with }\varphi(s)\notin\overline{\hbox{coz}(f)}.$ ###### Lemma 3.1. The mapping $T(i)$ is bounded on the set $\varphi^{-1}(\mathcal{O}_{1})$. Furthermore, the inequality $\left\|T(f)(s)\right\|\leq\\|T(1)\\|+\sup_{\widetilde{s}\in\varphi^{-1}(\mathcal{O}_{1})}\|T(i)(\widetilde{s})\|$ holds for all $s\in Z_{1}$ and all $f\in C_{r}(L_{1})$ with $\|\Re\hbox{e}(f)\|,\|\Im\hbox{m}(f)\|\leq 1.$ ###### Proof. Arguing by contradiction, we suppose that, for each natural $n$, there exists $s_{n}\in\varphi^{-1}(\mathcal{O}_{1})$ such that $\left\|T(i)(s_{n})\right\|>n^{3}$. The elements $s_{n}^{\prime}s$ can be chosen so that $\varphi(s_{n})\neq\varphi(s_{m})$ for $n\neq m$, and consequently we can find a sequence of pairwise disjoint open subsets $(U_{n})$ of $\mathcal{O}_{1}$ with $\varphi(s_{n})\in U_{n}$. It is easily seen that we can define a function $\displaystyle g=\sum_{n=1}^{\infty}i\ g_{n}\in C_{r}(L_{1})$ with coz$(g_{n})\subset U_{n}$, $0\leq g_{n}\leq\frac{1}{n^{2}}$, and $g_{n}\equiv\frac{1}{n^{2}}$ on a neighborhood of $s_{n}$, for all $n$. By the form of $g$, and since $T$ is orthogonality preserving, we have $\|T(g)(s_{n})\|=n^{2}\|T(i)(s_{n})\|>n$ for all $n$, which is absurd. ∎ We can easily show now that $Z_{2}$ is an open subset of $L_{2}$. With this aim, we consider an element $s_{0}$ in $Z_{2}$. We can pick a function $f\in C_{r}(L_{1})$ such that $\\|f\\|\leq 1$ and $\|T(f)(s_{0})\|>1+\\|T(1)\\|+\sup_{\widetilde{s}\in\varphi^{-1}(\mathcal{O}_{1})}\|T(i)(\widetilde{s})\|.$ Since $\displaystyle\left\|T(f)(s)\right\|\leq\\|T(1)\\|+\sup_{\widetilde{s}\in\varphi^{-1}(\mathcal{O}_{1})}\|T(i)(\widetilde{s})\|<\|T(f)(s_{0})\|-1$, for every $s\in Z_{1}\cup Z_{3},$ we conclude that there exists an open neighborhood of $s_{0}$ contained in $Z_{2}$. The next theorem resumes the above discussion. ###### Theorem 3.2. In the notation above, let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an orthogonality preserving linear mapping. Then $L_{2}$ decomposes as the union of three mutually disjoint subsets $Z_{1},Z_{2},$ and $Z_{3}$, where $Z_{2}$ is open and $Z_{3}$ is closed, there exist a continuous support map $\varphi:Z_{1}\cup Z_{2}\to L_{1}$, and a bounded mapping $T(i):L_{2}\to\mathbb{C}$ which is continuous on $\varphi^{-1}(\mathcal{O}_{1})$ satisfying: $T(i)(s)\in\mathbb{R}\ (\forall s\in F_{2}),\ T(i)(s)=0,\ (\forall s\in Z_{3}\cup Z_{2}\hbox{ and }\forall s\in Z_{1}\hbox{ with }\varphi(s)\in F_{1}),$ (11) $\left\|T(1)(s)\right\|+\left\|T(i)(s)\right\|\neq 0,\ (\forall s\in Z_{1}),$ (12) $T(f)(s)=T(1)(s)\ \Re\hbox{e}f(\varphi(s))+T(i)(s)\ \Im\hbox{m}f(\varphi(s)),\hbox{ {\rm(}$\forall s\in Z_{1},f\in C_{r}(L_{1}){\rm)},$}$ $T(f)(s)=0,\hbox{ {\rm(}$\forall s\in Z_{3},f\in C_{r}(L_{1})${\rm),}}$ and for each $s\in L_{2}$, the mapping $C_{r}(L_{1})\to\mathbb{C}$, $f\mapsto T(f(s)),$ is unbounded if, and only if, $s\in Z_{2}$. Furthermore, the set $\varphi(Z_{2})$ is finite. ###### Proof. Everything has been substantiated except perhaps the statement concerning the set $\varphi(Z_{2})$. Arguing by contradiction, we assume the existence of a sequence $(s_{n})$ in $Z_{2}$ such that $\varphi(s_{n})\neq\varphi(s_{m})$ for every $n\neq m$. Find a sequence $(U_{n})$ of mutually disjoint open subsets of $L_{1}$ satisfying $\varphi(s_{n})\in U_{n}$ and a sequence $(f_{n})\subseteq C_{r}(L_{1})$ such that $\\|f_{n}\\|\leq\frac{1}{n}$, $\hbox{coz}(f_{n})\subseteq U_{n}$ and $\|\delta_{s_{n}}T(f_{n})\|>n$, for every $n\in\mathbb{N}$. The element $\displaystyle f=\sum_{n=1}^{\infty}f_{n}$ lies in $C_{r}(L_{1}),$ and for each natural $n_{0}$, $\displaystyle{f_{n_{0}}\perp\sum_{n=1,n\neq n_{0}}^{\infty}f_{n}}.$ Thus, $\|\delta_{s_{n_{0}}}T(f)\|\geq\|\delta_{s_{n_{0}}}T(f_{n_{0}})\|>n_{0},$ which is impossible. ∎ ###### Remark 3.3. The mapping $T(i):L_{2}\to\mathbb{C}$ has been defined to satisfy $T(i)(s)=0,$ for all $s\in Z_{3}\cup Z_{2}$ and for all $s\in Z_{1}$ with $\varphi(s)\in F_{1}.$ It should be noticed here that the value $T(i)(s)$ is uniquely determined only when $s\in Z_{1}$ and $\varphi(s)\notin F_{1}$. There are some other choices for the values of $T(i)(s)$ at $s\in Z_{3}\cup Z_{2}$ and at $s\in Z_{1}$ with $\varphi(s)\in F_{1}$ under which conditions $(\ref{eq 1 thm 32})$ and $(\ref{eq 2 thm 32})$ are satisfied. ###### Remark 3.4. We shall now explore some of the consequences derived from Theorem 3.2. Let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an orthogonality preserving linear mapping. 1. $(a)$ The set $Z_{3}$ is empty whenever $T$ is surjective. 2. $(b)$ $Z_{3}=\emptyset$ implies that $Z_{1}=L_{2}\backslash Z_{2}$ is a compact subset of $L_{2}.$ 3. $(c)$ $\varphi(Z_{2})$ is a finite set of non-isolated points in $L_{1}$. Indeed, if $\varphi(s_{0})=t_{0}$ is isolated for some $s_{0}\in Z_{2}$, then we can find an open set $U\subseteq L_{1}$ such that $U\cap K_{1}=\\{t_{0}\\}$. Therefore, for each $f\in C_{r}(L_{1})$ with $f(t_{0})=0$ we have $\delta_{s_{0}}T(f)=0$. Pick an arbitrary $h\in C_{r}(L_{1})$. Clearly, $\chi_{{}_{t_{0}}}\in C_{r}(L_{1})$, while $i\chi_{{}_{t_{0}}}$ lies in $C_{r}(L_{1})$ if, and only if, $t_{0}\notin F_{1}$. Therefore, $h_{0}=\Re\hbox{e}(h(t_{0}))\chi_{{}_{t_{0}}}+\Im\hbox{m}(h(t_{0}))\ i\chi_{{}_{t_{0}}}$ lies in $C_{r}(L_{1})$ and $(h-h_{0})(t_{0})=0$. Assume first that $t_{0}\notin F_{1}$. Denoting $\lambda_{0}=\delta_{s_{0}}T(\chi_{{}_{t_{0}}})$ and $\mu_{{}_{0}}=\delta_{s_{0}}T(i\chi_{{}_{t_{0}}}),$ we have $\delta_{s_{0}}T(h)=\delta_{s_{0}}T(h_{0})=\lambda_{0}\Re\hbox{e}(h(t_{0}))+\mu_{{}_{0}}\Im\hbox{m}(h(t_{0}))$ $=\frac{\lambda_{0}-i\mu_{{}_{0}}}{2}\delta_{t_{0}}(h)+\frac{\lambda_{0}+i\mu_{{}_{0}}}{2}\overline{\delta_{t_{0}}}(h).$ This shows that $\delta_{s_{0}}T=\frac{\lambda_{0}-i\mu_{{}_{0}}}{2}\delta_{t_{0}}+\frac{\lambda_{0}+i\mu_{{}_{0}}}{2}\overline{\delta_{t_{0}}}$ is a continuous mapping from $C_{r}(L_{1})$ to $\mathbb{C}$, which is impossible. When $t_{0}\in F_{1}$ we have $\delta_{s_{0}}T=\lambda_{0}\delta_{t_{0}}$ is a continuous mapping from $C_{r}(L_{1})$ to $\mathbb{R}$, which is also impossible. 4. $(d)$ $T$ surjective implies $\varphi(Z_{1}\cap\mathcal{O}_{2})\subseteq\mathcal{O}_{1}$. Suppose, on the contrary that there exists $s_{0}\in Z_{1}\cap\mathcal{O}_{2}$ with $\varphi(s_{0})\in F_{1}$. By $(\ref{eq 2 thm 32})$, $T(f)(s_{0})=T(1)(s_{0})\Re\hbox{e}f(\varphi(s_{0})),$ for every $f\in C_{r}(L_{1})$. It follows from the surjectivity of $T$, together with the condition $s_{0}\in\mathcal{O}_{2},$ that for every complex number $\omega$ there exists a real $\lambda$ satisfying $\omega=T(1)(s_{0})\lambda,$ which is impossible. 5. $(e)$ Suppose $T$ is surjective and fix $s_{0}\in Z_{1}\cap\mathcal{O}_{2}$. The mapping $\delta_{s_{0}}T$ is a bounded real-linear mapping from $C_{r}(L_{1})$ onto $\mathbb{C}$. On the other hand, by $(\ref{eq 2 thm 32})$, $\delta_{s_{0}}T(f)=T(1)(s_{0})\Re\hbox{e}f(\varphi(s_{0}))+T(i)(s_{0})\Im\hbox{m}f(\varphi(s_{0})),\ (\forall f\in C_{r}(L_{1})).$ Thus, $T$ being surjective implies that the space $\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}$ is linearly spanned by the elements $T(1)(s_{0})$ and $T(i)(s_{0})$. Therefore, for each $s_{0}\in Z_{1}\cap\mathcal{O}_{2},$ the set $\\{T(1)(s_{0}),T(i)(s_{0})\\}$ is a basis of $\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}.$ Consequently, when $T$ is surjective and $s_{0}\in Z_{1}\cap\mathcal{O}_{2},$ the condition $T(f)(s_{0})=0$ implies $f(\varphi(s_{0}))=0.$ For any other $s_{1}\in Z_{1}\cap\mathcal{O}_{2}$ with $\varphi(s_{0})=\varphi(s_{1})$, we have: $T(f)(s_{0})=0\Rightarrow f(\varphi(s_{0}))=0\Rightarrow T(f)(s_{1})=0.$ The fact that $C_{r}(L_{2})$ separates points implies that $s_{1}=s_{0}$. Thus, $\varphi$ is injective on $Z_{1}\cap\mathcal{O}_{2}$. We can now state the main result of this section which affirms that every orthogonality preserving linear bijection between unital commutative real C∗-algebras is (automatically) continuous. ###### Theorem 3.5. Every orthogonality preserving linear bijection between unital commutative (real) C∗-algebras is (automatically) continuous. ###### Proof. Since $T$ is surjective, $Z_{3}=\emptyset$, and hence $Z_{1}=L_{2}\backslash Z_{2}$ is a compact subset of $L_{2}.$ It is also clear that $\varphi(L_{2})$ is compact. We claim that $\varphi(L_{2})=L_{1}.$ Otherwise, there would exist a non-zero function $f\in C_{r}(L_{1})$ with $\overline{\hbox{coz}(f)}\subseteq L_{1}\backslash\varphi(L_{2})$. Thus, by $(\ref{eq varphi notin cozero})$, $T(f)=0,$ contradicting the injectivity of $T$. By Remark 3.4$(c)$, $\varphi(Z_{1})=\overline{\varphi(Z_{1})}=\varphi(L_{2})=\varphi(Z_{1})\cup\varphi(Z_{2})=L_{1}.$ We next see that $Z_{2}=\emptyset.$ Otherwise we can take $g\in C_{r}(L_{2})$ with $\emptyset\neq coz(g)\subset Z_{2}.$ Let $h=T^{-1}(g).$ Obviously $Th(s)=0$ whenever $s\in Z_{1}.$ We claim that $h(t)=0,$ for every $t\in\varphi(Z_{1})\setminus\varphi(Z_{2}).$ Let us fix $t\in\varphi(Z_{1})\setminus\varphi(Z_{2}).$ Since $\varphi(Z_{2})$ is a finite set there are disjoint open sets $U_{1},U_{2}$ such that $t\in U_{1},\varphi(Z_{2})\subset U_{2}.$ Let $f\in C(L_{1},{\mathbb{R}})$ be such that $f(t)\neq 0$ and $\overline{coz(f)}\subset U_{1}.$ We see that $T(fh)=0.$ Indeed, let $s\in L_{2}=Z_{1}\cup Z_{2}.$ If $s$ lies in $Z_{1},$ then the maps $fh$ and $f(\varphi(s))h$ lie in $C_{r}(L_{1})$ and coincide at $\varphi(s).$ Since $T$ is linear over ${\mathbb{R}}$ and $f$ takes real values, we deduce, by $(\ref{eq 2 thm 32})$, that $T(fh)(s)=f(\varphi(s))Th(s)=0.$ If $s\in Z_{2}$ then, since $\varphi(s)\notin\overline{coz(fh)}$, then $\delta_{s}T(fh)=T(fh)(s)=0$. We have shown that $T(fh)=0$. Thus, since $T$ is injective, $fh=0$ and therefore $h(t)=0.$ We have therefore proved that $coz(h)\subset\varphi(Z_{2})$ which is a finite set. This means that $h$ must be a finite linear combination of characteristic function on points of $\varphi(Z_{2})$ and these points must be isolated which is impossible, since by $c)$ in Remark 3.4 no point in $\varphi(Z_{2})$ can be isolated. We have proved that $Z_{2}=\emptyset.$ Now the fact that $T$ is continuous follows easily. ∎ The above theorem is the first step toward extending, to the real setting, those results proved in [30], [6], [13], [40], [38] and [48] for (complex) C∗-algebras. Orthogonality preserving linear bijections enjoy an interesting additional property. ###### Proposition 3.6. In the notation of this section, let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an orthogonality preserving linear bijection. Then $T^{-1}$ preserves invertible elements, that is, $T^{-1}(g)$ is invertible whenever $g$ is an invertible element in $C_{r}(L_{2})$. ###### Proof. Take an invertible element $g\in C_{r}(L_{2})$. Let $f$ be the unique element in $C_{r}(L_{1})$ satisfying $T(f)=g$. Theorem 3.2 implies that $0\neq g(s)=T(f)(s)=T(1)(s)\ \Re\hbox{e}f({\varphi(s)})+T(i)(s)\ \Im\hbox{m}f({\varphi(s)}),$ for every $s\in Z_{1}$. This assures that $f({\varphi(s)})\neq 0,$ for every $s\in Z_{1},$ and since $\varphi(Z_{1})=L_{1},$ $f=T^{-1}(g)$ must be invertible in $C_{r}(L_{1})$. ∎ In the setting of complex Banach algebras, it follows from the Gleason-Kahane- Żelazko theorem that a linear transformation $\phi$ from a unital, commutative, complex Banach algebra $A$ into $\mathbb{C}$ satisfying $\phi(1)=1$ and $\phi(a)\neq 0$ for every invertible element $a$ in $A$ is multiplicative, that is, $\phi(ab)=\phi(a)\phi(b)$ (see [20, 32]). Although, the Gleason-Kahane-Żelazko theorem fails for real Banach algebras, S.H. Kulkarni found in [35] the following reformulation: a linear map $\phi$ from a real unital Banach algebra ${A}$ into the complex numbers is multiplicative if $\varphi(1)=1$ and $\phi(a)^{2}+\phi(b)^{2}\neq 0$ for every $a,b\in{A}$ with $ab=ba$ and $a^{2}+b^{2}$ invertible. It is not clear that statement $(b)$ in the above proposition can be improved to get the hypothesis of Kulkarni’s theorem. The structure of orthogonality preserving linear mappings between $C_{r}(L)$-spaces described in Theorem 3.2 invites us to affirm that they are not necessarily multiplicative. ### 3.1. Bi-orthogonality preservers As a consequence of the description of orthogonality preserving linear maps given in [30], it can be shown that an orthogonality preserving linear bijection between (complex) $C(K)$-spaces is bi-orthogonality preserving. It is natural to ask wether every orthogonality preserving linear bijection between commutative (unital) real C∗-algebras is bi-orthogonality preserving. This is known to be true in two cases: first, between spaces $C_{\mathbb{R}}(K)$ of real (and also complex) valued functions on a compact Hausdorff space $K$, as it is well-known; second, between spaces of the type $C_{\mathbb{R}}(K;\mathbb{R}^{n})$ (compare [16, Section 3]). Spaces like those we are dealing with in this paper need not satisfy this property, that is, there exists an orthogonality preserving linear bijection $T:C_{r}(L_{1})\to C_{r}(L_{2})$ which is not bi-orthogonality preserving (and even $L_{1}$ and $L_{2}$ are not homeomorphic either). ###### Example 3.7. Let $L_{1}=\\{t_{1},t_{2},t_{3}\\}$ $L_{2}=\\{s_{1},s_{2},s_{3},s_{4}\\}$ with $\mathcal{O}_{1}=\\{t_{1},t_{3}\\}$, $\mathcal{O}_{2}=\\{s_{1}\\}$, $F_{1}=\\{t_{2}\\}$ and $F_{2}=\\{s_{2},s_{3},s_{4}\\}$. Define $\varphi:L_{2}\to L_{1}$ by $\varphi(s_{i})=t_{i},$ for $i=1,2,$ and $\varphi(s_{i})=t_{3},$ for $i=3,4$. It is easy to check that $T(f)(s_{i})=f(\varphi(s_{i}))$ if $i=1,2$, and $T(f)(s_{3})=\Re\hbox{e}f(t_{3}),$ $T(f)(s_{4})=\Im\hbox{m}f(t_{3})$ is an orthogonality preserving linear bijection, but $T^{-1}$ is not orthogonality preserving. In the above example, $\varphi^{-1}(\mathcal{O}_{1})\cap F_{2}$ is non-empty. Our next result shows that a topological condition on $F_{2}$ assures that an orthogonality preserving linear bijection between unital commutative real C∗-algebras is bi-orthogonality preserving. ###### Proposition 3.8. In the notation of this section, let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be an orthogonality preserving linear bijection (not assumed to be bounded). The following statements hold: 1. $(a)$ If $T$ is bi-orthogonality preserving then $\varphi:L_{2}\to L_{1}$ is a (surjective) homeomorphism, $\varphi(F_{2})=F_{1},$ and $\varphi(\mathcal{O}_{2})=\mathcal{O}_{1}$. In particular, $\varphi^{-1}(\mathcal{O}_{1})\cap F_{2}=\emptyset$. 2. $(b)$ If $F_{2}$ has empty interior then $T$ is biorthogonality preserving. ###### Proof. $(a)$ If $T$ is bi-orthogonality preserving, it can be easily seen that $\varphi:L_{2}\to L_{1}$ is a homeomorphism, and for each $s\in L_{2}$, supp$(\delta_{\varphi(s)}T^{-1})=\\{s\\}$. By Remark 3.4$(d)$, applied to $T$ and $T^{-1}$, we have $\varphi(F_{2})=F_{1}$ and $\varphi(\mathcal{O}_{2})=\mathcal{O}_{1}$. Then $\varphi^{-1}(\mathcal{O}_{1})\cap F_{2}=\emptyset.$ So, $a)$ is clear. $(b).$ Let us assume that $F_{2}$ has empty interior. Arguing by contradiction we suppose that $T^{-1}$ is not orthogonality preserving. Then there exist $f_{1},f_{2}\in C_{r}(L_{1})$ with $f_{1}f_{2}\neq 0$, but $T(f_{1})\perp T(f_{2})$. Thus $U:=\hbox{coz}(f_{1})\cap\hbox{coz}(f_{2})$ is a non-empty open subset of $L_{1}.$ Keeping again the notation of Theorem 3.2 for $T$, we recall that, by Theorem 3.5 and Remark 3.4, $Z_{3}=\emptyset,$ $Z_{2}=\emptyset$, $\varphi(L_{2})=L_{1}$, $\varphi(\mathcal{O}_{2})\subset\mathcal{O}_{1},$ $\varphi\|_{\mathcal{O}_{2}}$ is injective, and for each $s\in\mathcal{O}_{2}$, and $\\{T(1)(s),T(i)(s)\\}$ is a basis of $\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}.$ By the form of $T$, there are no points of $\varphi(\mathcal{O}_{2})$ in $U=\hbox{coz}(f_{1})\cap\hbox{coz}(f_{2})$ (because for each $s\in\mathcal{O}_{2},$ $T(f)(s)\neq 0$ when $f(\varphi(s))\neq 0$). Now, let $k$ be a non-zero element in $C(L_{1},\mathbb{R})$, with coz$(k)\subseteq\hbox{coz}(f_{1})\cap\hbox{coz}(f_{2})$. By Theorem 3.2 $(\ref{eq 2 thm 32})$, it is clear that $\varphi(\hbox{coz}(T(k)))\subseteq\hbox{coz}(k)$, and hence, since $\varphi(\mathcal{O}_{2})\subseteq\mathcal{O}_{1}$, $\hbox{coz}(T(k))$ is a non-empty subset of $F_{2}$, against our hypotheses. ∎ As we have already seen, an orthogonality preserving linear bijection between $C_{r}(L)$-spaces needs not to be biorthogonality preserving. Example 3.7 also shows that, unlike in the complex case, the existence of an orthogonality preserving linear bijection between $C_{r}(L)$-spaces does not guarantee that the corresponding compacts spaces are homeomorphic. We next provide a characterisation of those (linear) mappings which are bi-orthogonality preserving. As a consequence, we shall see that if there exists a bi- orthogonality preserving linear map $T:C_{r}(L_{1})\to C_{r}(L_{2})$ then $L_{1}$ and $L_{2}$ are homeomorphic. ###### Theorem 3.9. Let $T:C_{r}(L_{1})\to C_{r}(L_{2})$ be a mapping. The following statements are equivalent: 1. $(a)$ $T$ is a bi-orthogonality preserving linear surjection; 2. $(b)$ There exists a (surjective) homeomorphism $\varphi:L_{2}\to L_{1}$ with $\varphi(\mathcal{O}_{2})=\mathcal{O}_{1},$ a function $a_{1}=\gamma_{1}+i\gamma_{2}$ in $C_{r}(L_{2})$ with $a_{1}(s)\neq 0$ for all $s\in L_{2},$ and a function $a_{2}=\eta_{1}+i\eta_{2}:L_{2}\to{\mathbb{C}}$ continuous on $\mathcal{O}_{2}$ with the property that $0<\inf_{s\in\mathcal{O}_{2}}\left\|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\ \gamma_{2}(s)&\eta_{2}(s)\\\ \end{array}\right)\right\|\leq\sup_{s\in\mathcal{O}_{2}}\left\|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\ \gamma_{2}(s)&\eta_{2}(s)\\\ \end{array}\right)\right\|<+\infty,$ such that $T(f)(s)=a_{1}(s)\ \Re\hbox{e}f({\varphi(s)})+a_{2}(s)\ \Im\hbox{m}f({\varphi(s)})$ for all $s\in L_{2}$ and $f\in C_{r}(L_{1}).$ ###### Proof. $(a)\Rightarrow(b)$. Since every bi-orthogonality preserving linear mapping is injective, we can assume that $T:C_{r}(L_{1})\to C_{r}(L_{2})$ is a bi- orthogonality preserving linear bijection. We keep the notation given in Theorem 3.2. We have already shown that $Z_{3}=\emptyset,$ $Z_{2}=\emptyset$, $\varphi:L_{2}\to L_{1}$ is a surjective homeomorphism, $\varphi(\mathcal{O}_{2})=\mathcal{O}_{1},$ and for each $s\in\mathcal{O}_{2}$, $\\{T(1)(s),T(i)(s)\\}$ is a basis of $\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R}$ (compare Theorem 3.5, Remark 3.4 and Proposition 3.8). Taking $a_{1}=T(1)=\gamma_{1}+i\gamma_{2}$ and $a_{2}=T(i)=\eta_{1}+i\eta_{2}$ we only have to show that $0<\inf_{s\in\mathcal{O}_{2}}\left\|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\ \gamma_{2}(s)&\eta_{2}(s)\\\ \end{array}\right)\right\|\leq\sup_{s\in\mathcal{O}_{2}}\left\|\det\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\ \gamma_{2}(s)&\eta_{2}(s)\\\ \end{array}\right)\right\|<+\infty.$ Let us denote $M_{s}=\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\ i\gamma_{2}(s)&i\eta_{2}(s)\\\ \end{array}\right).$ Clearly $\det(M_{s})\neq 0$, for every $s\in\mathcal{O}_{2}$ and $T(f)(s)=M_{s}\cdot\left(\begin{array}[]{c}\Re\hbox{e}f(\varphi(s))\\\ \Im\hbox{m}f(\varphi(s))\\\ \end{array}\right),$ for every $f\in C_{r}(L_{1}),s\in L_{2}$. By the boundedness of $T(1):L_{2}\to\mathbb{C}$ and $T(i)\|_{\mathcal{O}_{2}}:\mathcal{O}_{2}\to\mathbb{C}$ (see Lemma 3.1) there exists $M>0$ such that $\left\|det(M_{s})\right\|\leq M$ for all $s\in\mathcal{O}_{2}.$ Applying the above arguments to the mapping $T^{-1}$ we find a surjective homeomorphism $\psi=\varphi^{-1}:L_{1}\to L_{2}$, a mapping $T^{-1}(i):L_{1}\to L_{2}$ and $m>0,$ such that $\psi(\mathcal{O}_{1})=\mathcal{O}_{2}$, for each $t\in\mathcal{O}_{1}$, $\\{T^{-1}(1)(t),T^{-1}(i)(t)\\}$ is a basis of $\mathbb{C}_{\mathbb{R}}=\mathbb{R}\times\mathbb{R},$ $T^{-1}(g)(t)=N_{t}\cdot\left(\begin{array}[]{c}\Re\hbox{e}g(\psi(t))\\\ \Im\hbox{m}g(\psi(t))\\\ \end{array}\right)$ $(g\in C_{r}(L_{2}),t\in L_{1})$, $\left\|\det(N_{t})\right\|\leq m,$ for all $t\in\mathcal{O}_{1},$ where $N_{t}=\left(\begin{array}[]{cc}\Re\hbox{e}T^{-1}(1)(t)&\Re\hbox{e}T^{-1}(i)(t)\\\ i\Im\hbox{m}T^{-1}(1)(t)&i\Im\hbox{m}T^{-1}(i)(t)\\\ \end{array}\right).$ It can be easily seen that, for each $s\in\mathcal{O}_{2},$ $N_{\varphi(s)}=M_{s}^{-1},$ which shows that $\left\|det(M_{s})\right\|\geq\frac{1}{m}$, for all $s\in\mathcal{O}_{2}.$ $(b)\Rightarrow(a)$. Let $T:C_{r}(L_{1}):\to C_{r}(L_{2})$ be a mapping satisfying the hypothesis in $(b).$ Clearly, $T$ is linear, and since $\varphi(F_{2})=F_{1},$ $Tf(s)\in{\mathbb{R}}$ for all $s\in F_{2}$ and $f\in C_{r}(L_{1})$ (that is, $T(f)\in C_{r}(L_{2})$). We can easily check that, under these hypothesis, $T$ is injective and preserves orthogonality. We shall now prove that $T$ is surjective. Indeed, for each $s\in\mathcal{O}_{2}$ $T(f)(s)=\left(\begin{array}[]{c}\Re\hbox{e}g(s)\\\ \Im\hbox{m}g(s)\\\ \end{array}\right)=\left(\begin{array}[]{cc}\gamma_{1}(s)&\eta_{1}(s)\\\ i\gamma_{2}(s)&i\eta_{2}(s)\\\ \end{array}\right)\cdot\left(\begin{array}[]{c}\Re\hbox{e}f({\varphi(s)})\\\ \Im\hbox{m}f({\varphi(s)})\\\ \end{array}\right)$ $=M_{s}\cdot\left(\begin{array}[]{c}\Re\hbox{e}f({\varphi(s)})\\\ \Im\hbox{m}f({\varphi(s)})\\\ \end{array}\right),$ thus, $\left(\begin{array}[]{c}\Re\hbox{e}f({\varphi(s)})\\\ \Im\hbox{m}f({\varphi(s)})\\\ \end{array}\right)=M_{s}^{-1}\cdot\left(\begin{array}[]{c}\Re\hbox{e}g(s)\\\ \Im\hbox{m}g(s)\\\ \end{array}\right).$ We define $b_{1}(t):L_{1}\to{\mathbb{C}}$ and $b_{2}:\mathcal{O}_{1}\to{\mathbb{C}}$ by $b_{1}(t)=\widetilde{\gamma}_{1}(t)+i\widetilde{\gamma}_{2}(t)$ and $b_{2}=\widetilde{\eta}_{1}(t)+i\widetilde{\eta}_{2}(t)$ ($t\in\mathcal{O}_{1}$), where $M_{\varphi^{-1}(t)}^{-1}=\left(\begin{array}[]{cc}\widetilde{\gamma}_{1}(t)&\widetilde{\eta}_{1}(t)\\\ i\widetilde{\gamma}_{2}(t)&i\widetilde{\eta}_{2}(t)\\\ \end{array}\right),$ and $b_{1}(t)=\frac{1}{\gamma_{1}(\varphi^{-1}(t))},$ for every $t\in F_{1}$. Then $S:C_{r}(L_{2})\to C_{r}(L_{1}),$ defined by $S(g)(t)=b_{1}(t)\ \Re\hbox{e}g(\varphi^{-1}(t))+b_{2}(t)\ \Im\hbox{m}g(\varphi^{-1}(t)),$ is linear, preserves orthogonality and it is easy to check that $S=T^{-1}.$ It follows that $T$ is bi-orthogonality preserving. ∎ Let $T$ be a bi-orthogonality preserving linear mapping with associated homeomorphism $\varphi:L_{2}\to L_{1}$. Clearly, the operator $S:C_{r}(L_{1})\to C_{r}(L_{2})$, $S(f)(s):=f(\varphi(s))$ is a ∗-isomorphism. Having in mind that a linear mapping $T:A\to B$ between real C∗-algebras is a ∗-isomorphism if, and only if, the complex linear extension $\widetilde{T}:A\oplus iA\to B\oplus iB$, $\widetilde{T}(a+ib)=T(a)+iT(b)$ is a ∗-isomorphism, we get the following corollary. ###### Corollary 3.10. 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A I Math._ 1, no. 2, 355-385 (1975).	arxiv-papers	2013-09-16T07:32:31	2024-09-04T02:49:50.953116	{ "license": "Public Domain", "authors": "Jorge J. Garc\\'es and Antonio M. Peralta", "submitter": "Antonio M. Peralta", "url": "https://arxiv.org/abs/1309.3839" }
1309.4074	Three-Dimensional Smoothed Particle Hydrodynamics Method for Simulating Free Surface Flows Rizal Dwi Prayogoa,b, Christian Fredy Naaa aFaculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132 Indonesia, E-mail: [email protected], [email protected] bGraduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa 920-1192 Japan, Abstract. In this paper, we applied an improved Smoothing Particle Hydrodynamics (SPH) method by using gradient kernel renormalization in three- dimensional cases. The purpose of gradient kernel renormalization is to improve the accuracy of numerical simulation by improving gradient kernel approximation. This method is implemented for simulating free surface flows, in particular dam break case with rigid ball structures and the propagation of waves towards a slope in a rectangular tank. Keywords: Smoothed particle hydrodynamics, free surface flows, gradient kernel renormalization ## 1 Introduction Computational Fluid Dynamics (CFD) using Smoothed Particle Hydrodynamics (SPH) has a wide range of applications to solve problem in engineering and science. SPH is a mesh-free Lagrangian method and well suited to the simulation of complex and free surface flows. The SPH method was originally used to model astrophysical problems by Lucy [7] and Gingold and Monaghan [8]. Three- dimensional SPH method has been studied numerically by Monaghan [6] in a field of astrophysical fluid dynamics processes. We obtain the SPH equations from the continuum equations of fluid dynamics by interpolating from set of points which may be disordered. This interpolation is based on the theory of integral interpolants using interpolation kernels which approximate the delta function. The interpolants being analytic functions can be differentiated without using grids. Monaghan [4] studied the application of the particle method SPH to free surface problems in two-dimensional cases. In this paper, we applied an improved SPH method by using gradient kernel renormalization in three- dimensional cases. The purpose of gradient kernel renormalization is to improve the accuracy of the simulations [1]. In the following, first the general concept of SPH method is given. The improved SPH method using gradient kernel renormalization is introduced and described in detail. This improved SPH method is implemented for simulating free surface flows, in particular dam break case with rigid ball structures and the propagation of waves towards a slope in a rectangular tank. ## 2 Smoothed particle hydrodynamics The SPH equations are described in detail by Liu and Liu [2]. In this paper, we consider the application of three-dimensional SPH to free surface problems. The SPH method represents continuous fluid using a set of particles. Each particle $i$ has physical quantities, such as mass $m_{i}$, position $\mathbf{r}_{i}$, velocity $\mathbf{v}_{i}$, density $\rho_{i}$, and pressure $P_{i}$. Each particle in the SPH method is associated with a support domain. The SPH approximation, which consists of the particle approximation and the kernel approximation, is performed within the current support domain. The value of a function defining a physical quantity can be approximated by its values at a number of neighboring particles. The SPH method uses the concept of integral representation of a field function $f(x)$ by the following identity $<f(\mathbf{r})>=\int_{\Omega}f(\mathbf{r^{\prime}})W(\|\mathbf{r}-\mathbf{r^{\prime}}\|,h)d\mathbf{r^{\prime}},$ (1) where $\mathbf{r}$ and $\mathbf{r}^{\prime}$ are the position vectors, $W$ is the smoothing function or kernel function, and $h$ is the smoothing length defining the influence radius of $W$. Figure 1: The influence radius of $W$. In SPH approximation, there are various kernel functions. Since it is affects the accuracy and stability of numerical results, the choice of kernel function $W$ is important to consider. The integral representation should satisfies several conditons. That is the normalization condition $\int_{\Omega}W(\|\mathbf{r}-\mathbf{r^{\prime}}\|,h)d\Omega=1,$ delta function property $\lim_{h\to 0}W(\|\mathbf{r}-\mathbf{r^{\prime}}\|,h)=\delta(\|\mathbf{r}-\mathbf{r^{\prime}}\|),$ moreover, often the compact support condition is required $W(\|\mathbf{r}-\mathbf{r^{\prime}}\|,h)=0\text{ outside of support domain.}$ We converted the continuous integral representation (1) into discretized forms as a summation over all the particles in the support domain. This process is also commonly known as particle approximation in the SPH literature [2]. Writing the particle approximation as follows $<f(\mathbf{r})>=\sum_{j=1}^{N}\dfrac{m_{j}}{\rho_{j}}f(\mathbf{r_{j}})W(\|\mathbf{r}-\mathbf{r_{j}}\|,h),$ where $m_{j}$ and $\rho_{j}$ are the mass and density of the particle $j$, respectively, and $j=1,2,\ldots,N$, where $N$ is the total number of neighboring particles in the influence domain $\Omega$. In this paper, we use cubic spline kernel as follows [5] $W(q,h)=\dfrac{1}{\pi h^{3}}\begin{cases}1-\frac{3}{2}q^{2}+\frac{3}{4}q^{3},&\text{$0\leq q<1$}\\\ \frac{1}{4}(2-q)^{3},&\text{$1\leq q<2$}\\\ 0,&\text{otherwise,}\end{cases}$ where $q=\dfrac{\|\mathbf{r_{i}}-\mathbf{r_{j}}\|}{h}$ is the relative distance of particle $i$ and $j$. ## 3 Numerical model ### 3.1 The continuity equation The continuity equation is based on the conservation of mass. We write the continuity equation in the form $\dfrac{D\rho}{Dt}=-\rho\nabla\mathbf{v},$ (2) where $\mathbf{v}$ and $\rho$ are velocity and density, respectively. Writing (2) in SPH discretization form as in [3], we obtain $\dfrac{D\rho_{i}}{Dt}=\rho_{i}\sum_{j}\dfrac{m_{j}}{\rho_{j}}(\mathbf{v_{i}}-\mathbf{v_{j}})\nabla_{r_{i}}W(\|\mathbf{r_{i}}-\mathbf{r_{j}}\|,h),$ (3) where $\rho_{k}$ and $\mathbf{v}_{k}$ are density and velocity of particle $k$ (evaluated at $k=i$ or $k=j$), respectively, $m_{j}$ is mass of particle $j$ and $\nabla_{r_{i}}W(\|\mathbf{r_{i}}-\mathbf{r_{j}}\|,h)=\dfrac{\mathbf{r_{i}}-\mathbf{r_{j}}}{\|\mathbf{r_{i}}-\mathbf{r_{j}}\|}\dfrac{\partial W}{\partial r}.$ ### 3.2 The momentum equation The momentum equation is based on the conservation of momentum which is given by $\dfrac{D\mathbf{v}}{Dt}=-\dfrac{1}{\rho}\nabla P+\mathbf{F},$ (4) where $\mathbf{v},\rho$ and $P$ are velocity, density, and pressure, respectively. Here, $F$ is external force, in this case gravitational acceleration. Writing (4) in SPH discretization form as in [3], we get $\dfrac{D\mathbf{v_{i}}}{Dt}=-\sum_{j}m_{j}\Big{(}\dfrac{P_{i}+P_{j}}{\rho_{i}\rho_{j}}+\Pi_{ij}\Big{)}\nabla_{r_{i}}W(\|\mathbf{r_{i}}-\mathbf{r_{j}}\|,h)+\mathbf{F},$ (5) where $P_{k}$ is pressure of particle $k$ (evaluated at $k=i$ or $k=j$). In SPH, there are various formulations for viscosity. In the momentum equation, the introduction of a viscous term is necessary not only to consider viscid fluids and no slip boundary conditions, but also to provide the stability to the system and to prevent inter-particle penetration. The artificial viscosity term $\Pi_{ij}$ is added to pressure terms within the momentum equation (5). The artificial viscosity $\Pi_{ij}$ has the form [6] $\Pi_{ij}=\begin{cases}\dfrac{-\alpha c\mu_{ij}+\beta\mu_{ij}^{2}}{(\rho_{i}+\rho_{j})/2},&\text{$(\mathbf{v_{i}-\mathbf{v_{j}}}).(\mathbf{r_{i}-\mathbf{r_{j}}})<0$}\\\ 0,&\text{$(\mathbf{v_{i}-\mathbf{v_{j}}}).(\mathbf{r_{i}-\mathbf{r_{j}}})>0$}\end{cases}$ where $\displaystyle\mu_{ij}=\dfrac{h(\mathbf{r_{i}-\mathbf{r_{j}}}).(\mathbf{v_{i}-\mathbf{v_{j}}})}{\|\mathbf{r_{i}}-\mathbf{r_{j}}\|^{2}+\eta^{2}}.$ In these expressions, $c$ is the speed of sound, $\eta=0.001$, $\alpha$ and $\beta$ represent shear and bulk viscosity, respectively. For the problems described here, we choose $\alpha=0.03$ and $\beta=0$. ### 3.3 The equation of state The equation of state is used to relate density to pressure. In this paper, the Tait’s equation of state has the form $p=\dfrac{\rho_{0}c^{2}}{\gamma}\Big{[}\Big{(}\dfrac{\rho}{\rho_{0}}\Big{)}^{\gamma}-1\Big{]},$ where $c,\rho_{0}$, and $\gamma$ are the speed of sound, density reference, and the polytropic constant, respectively. Note that $\gamma=7$ is usually used for water simulations. The speed of sound $c$ is approximately $\sqrt{100gH}$ and it is chosen in respect of a low Mach number ($Ma<0.1$) to ensure low compressibility effects [6]. ## 4 Improvement of the SPH method In this paper, we applied an improvement to the standard SPH method by using renormalization. This technique is to improve the accuracy of the method [1]. ### 4.1 Gradient kernel renormalization The velocity gradient in (3) can be approached by using $\displaystyle\nabla\mathbf{v}=\nabla\mathbf{v}-\mathbf{v}\nabla 1.$ We can generalize this approach for any field $f$ by using $\nabla f=\nabla f-f\nabla 1$ (6) and transforming (6) into its continuous convoluted form we have $\langle\nabla f(\mathbf{r})\rangle=\int_{\Omega}f(\mathbf{r^{\prime}})\nabla Wd\mathbf{r^{\prime}}-f(\mathbf{r})\int_{\Omega}\nabla Wd\mathbf{r^{\prime}}.$ We recall the second order Taylor expansion $\displaystyle\int_{\Omega}f(\mathbf{r^{\prime}})\nabla Wd\mathbf{r^{\prime}}$ $\displaystyle=f(\mathbf{r})\int_{\Omega}\nabla Wd\mathbf{r^{\prime}}+\dfrac{\partial f(\mathbf{r})}{\partial\mathbf{r_{1}}}\underbrace{\int_{\Omega}(\mathbf{r_{1}^{\prime}}-\mathbf{r_{1}})\nabla Wd\mathbf{r^{\prime}}}_{A}+$ $\displaystyle\dfrac{\partial f(\mathbf{r})}{\partial\mathbf{r_{2}}}\underbrace{\int_{\Omega}(\mathbf{r_{2}^{\prime}}-\mathbf{r_{2}})\nabla Wd\mathbf{r^{\prime}}}_{B}+\dfrac{\partial f(\mathbf{r})}{\partial\mathbf{r_{3}}}\underbrace{\int_{\Omega}(\mathbf{r_{3}^{\prime}}-\mathbf{r_{3}})\nabla Wd\mathbf{r^{\prime}}}_{C}+O(h^{2}).$ In order to ensure gradient interpolations of linear fields, it is necessary to ensure that the discrete approximation of $A$, $B$, and $C$ are $\displaystyle A=\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix}\quad B=\begin{pmatrix}0\\\ 1\\\ 0\end{pmatrix}\quad C=\begin{pmatrix}0\\\ 0\\\ 1\end{pmatrix}.$ By the renormalization procedure [1], we modify $\nabla W$ as follows $\displaystyle\sum_{j}\dfrac{m_{j}}{\rho_{j}}(\mathbf{r_{j}}-\mathbf{r})\footnotesize{\bigotimes}L(\mathbf{r})\nabla W(\|\mathbf{r}-\mathbf{r}_{j}\|)=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},$ where $L(\mathbf{r})$ is a $(d,d)$ correction matrix, $d$ is the dimension of the case. In this paper, we consider three-dimensional cases ($d=3$) and calculate $L(\mathbf{x})$ to increase the accuracy of gradient kernel approximation. The continuity equation is discretized by the following manner $\dfrac{D\rho_{i}}{Dt}=\rho_{i}\sum_{j}\dfrac{m_{j}}{\rho_{j}}(\mathbf{v_{i}}-\mathbf{v_{j}})L(\mathbf{r_{i}})\nabla_{r_{i}}W(\|\mathbf{r_{i}}-\mathbf{r_{j}}\|,h).$ This discretized form ensures exact interpolations for both constant and linear fields. Note that we can discretize the conservation of momentum by the following manner $\dfrac{D\mathbf{v_{i}}}{Dt}=-\sum_{j}m_{j}\Big{(}\dfrac{P_{i}+P_{j}}{\rho_{i}\rho_{j}}+\Pi_{ij}\Big{)}L(\mathbf{r_{i}})\nabla_{r_{i}}W(\|\mathbf{r_{i}}-\mathbf{r_{j}}\|,h)+\mathbf{F}.$ ### 4.2 Numerical time integration with renormalization As the other explicit hydrodynamic methods, different numerical time integrations can be applied in SPH simulation, such as Leap-Frog, predictor- corrector, Runge-Kutta, and Beeman schemes. The advantages of the Leap-Frog algorithm are its low memory usage on storage and its computational efficiency. We applied it in this paper with its improvement by using gradient kernel renormalization. Therefore, $\displaystyle\mathbf{r_{i}^{}}=\mathbf{r_{i}^{n}}+\dfrac{dt}{2}\mathbf{v_{i}^{n}},$ $\displaystyle p_{i}^{n}=\dfrac{\rho_{0}c_{0}^{2}}{\gamma}\Big{[}\Big{(}\dfrac{\rho_{i}^{n}}{\rho_{0}}\Big{)}^{\gamma}-1\Big{]},$ , $\displaystyle\mathbf{v}_{i}^{n+1/2}=\mathbf{v}_{i}^{n-1/2}-dt\sum_{j}m_{j}\Big{(}\dfrac{p_{i}^{n}+p_{j}^{n}}{\rho_{i}^{n}\rho_{j}^{n}}+\Pi_{ij}\Big{)}L(r_{i}^{n+1/2})\nabla_{r_{i}}W(\mathbf{r}_{ij}^{n+1/2},h)+\mathbf{F}_{i}^{n+1/2},$ $\displaystyle\rho_{i}^{n+1/2}=\rho_{i}^{n-1/2}+dt\rho_{i}\sum_{j}\dfrac{m_{j}}{\rho_{j}}\big{(}\mathbf{v_{i}^{n+1/2}}-\mathbf{v_{j}^{n+1/2}}\big{)}L(r_{i}^{n+1/2})\nabla_{r_{i}}W(\mathbf{r}_{ij}^{n+1/2},h),$ $\displaystyle\mathbf{r}_{i}^{n+1}=\mathbf{r}_{i}^{n+1/2}+\dfrac{dt}{2}\mathbf{v}_{i}^{n+1/2}.$ ## 5 Implementation In the following section, the results of numerical simulations for improved SPH method are given. This method was implemented in dam-break problem with rigid ball structures and water waves generated by oblique piston type wave- maker. ### 5.1 Dam-break and structure In this implementation, we consider a rectangular tank with three-dimensional problem, in particular on interaction between waves and structures. Here we examine the impact of a single wave with rigid ball structures over the slope by means of a three-dimensional SPH method. A rectangular tank contains fixed structures and we used 10075 particles for this simulation. The geometry is shown in Fig. 2. Figure 2: Sketch of dam-break problem considered, side view (upper panel) and top view (lower panel). Fig. 3 shows the motion of a single wave which moves through rigid ball structures in a rectangular tank. The frame at $T=0.0$ s shows the initial configuration. In the next frame at $T=0.11175$ s, the wave generated by the dam break and the initial layer of water on the bottom collides with the front of the rigid ball structures and at $T=0.1425$ s the wave wraps around the rigid ball structures. At $T=0.1725$ s, the waves collide from both sides of the rigid ball structures then continue moving toward the right vertical wall. The wave reflects after colliding with the opposite wall of the tank at $T=0.3255$ s. The last movement, at $T=0.47625$ s, the reflected wave hits the back of rigid ball structures. Figure 3: ($T=0.0$ s) initial configuration; ($T=0.11175$ s) wave hitting the rigid ball structures; ($T=0.1425$ s) wave wrapping around the rigid ball structures; ($T=0.1725$ s) waves colliding after passing the rigid ball structures; ($T=0.3255$ s) wave colliding with the opposite wall of the tank; ($T=0.47625$ s) reflected wave hitting the back of the rigid ball structures. ### 5.2 Oblique piston type wave-maker In this implementation, we consider a rectangular tank with three-dimensional problem, in particular the propagation of waves towards a slope. This simulation involves a wave-maker in the form of an oscillating oblique piston on the left-hand side and used 80682 particles. The geometry is shown in Fig. 4. Figure 4: Sketch of oblique piston type wave-maker, side view (upper panel) and top view (lower panel). Figure 5: ($T=0.0$ s) initial configuration; ($T=0.2865$ s) the wave is generated by the oscillating oblique piston type wave-maker; at ($T=0.823125$ s), ($T=0.917625$ s), and ($T=0.9915$ s) the water wave propagates onto the slope; ($T=1.18575$ s) the water wave reaches shallow water area and hits the right wall. Fig. 5 shows the simulation of free surface flows. The water waves generated by oscillating piston type wave-maker were simulated. In Fig. 5 the waves are shown propagating onto the slope. The frame at $T=0.0$ s shows the initial configuration with water lying on the slope. In the next frame at $T=0.2865$ s, the wave is generated by the oscillating oblique piston type wave-maker. At $T=0.823125$ s, $T=0.917625$ s, and $T=0.9915$ s the water wave propagates onto the slope. Finally, at $T=1.18575$ s the water wave reaches shallow water area and hits the right wall. ## 6 Summary This paper presents the application of an improved SPH method by using gradient kernel renormalization for simulating free surface flows. We have implemented this method on three-dimensional cases, in particular on the interaction between waves and structures; and the propagation of waves towards a slope for waves generated by oblique piston type wave-maker. The three- dimensional case of the model has been shown to produce three-dimensional phenomenon, i.e., the collision of a single wave with rigid ball structures and its passing around the obstacle. In summary, an improved SPH model based on renormalization can be successfully used to simulate three-dimensional wave problems. ## References [1] G. Oger, M. Doring, B. Alessandrini, P. Ferrant (2007). An improved SPH method: Towards higher order convergence. Journal of Computational Physics., 225, 1472-1492. * [2] G. R. Liu and M. B. Liu (2003). Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Publishing Co. Pte. Ltd, Singapore. * [3] J. J. Monaghan (1988). An Introduction to SPH. Computer Physics Communications., 48, 1, 89-96. * [4] J. J. Monaghan (1994). Simulating Free Surface Flows with SPH. Journal of Computational Physics., 110, 399 - 406. * [5] J. J. Monaghan and J. C. Lattanzio (1985). A Refined Method for Astrophysical Problems. Astron. Astrophys., 149, 399 - 406. * [6] J. J. Monaghan (1992). Smoothed Particle Hydrodynamics. Annu. Rev. Astron. Astrophys., 30, 543 - 574. * [7] L. B. Lucy (1977). A numerical approach to the testing of the fission hypothesis. Astron. J., 82, 1013 - 1024. * [8] R. A. Gingold and J. J. Monaghan (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astr. Soc., 181, 375 - 389.	arxiv-papers	2013-09-16T19:18:48	2024-09-04T02:49:50.972062	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rizal Dwi Prayogo, Christian Fredy Naa", "submitter": "Christian Naa", "url": "https://arxiv.org/abs/1309.4074" }
1309.4146	# Systematic Uncertainties in the Analysis of the Reactor Neutrino Anomaly A.C. Hayes1, J.L. Friar1, G.T. Garvey1, Gerard Jungman1, G. Jonkmans2 1Los Alamos National Laboratory, Los Alamos, NM, USA 87545 2AECL, Chalk River Laboratories, Chalk River, Ontario, Canada, K0J 1J0 ###### Abstract We examine uncertainties in the analysis of the reactor neutrino anomaly, wherein it is suggested that only about 94% of the emitted antineutrino flux was detected in short baseline experiments. We find that the form of the corrections that lead to the anomaly are very uncertain for the 30% of the flux that arises from forbidden decays. This uncertainty was estimated in four ways, is larger than the size of the anomaly, and is unlikely to be reduced without accurate direct measurements of the antineutrino flux. Given the present lack of detailed knowledge of the structure of the forbidden transitions, it is not possible to convert the measured aggregate fission beta spectra to antineutrino spectra to the accuracy needed to infer an anomaly. Neutrino physics conclusions based on the original anomaly need to be revisited, as do oscillation analyses that assumed that the antineutrino flux is known to better than approximately $4\%$. The term “reactor neutrino anomaly” first appeared in a publication by G. Mention et al. mention , where it generally referred to the $3\sigma$ deficit of neutrinos detected in short-baseline reactor neutrino experiments relative to the number predicted. The predicted number of detected neutrinos has evolved upward over time, largely as a consequence of a predicted increase in the energy of the neutrino flux and an increased $\bar{\nu}_{e}+p\rightarrow n+e^{+}$ cross section associated with smaller values for the neutron lifetime. This cross section is used to infer the neutrino flux in a presumably well-characterized detector. The changes in the predicted neutrino flux are mostly associated with improved knowledge of the beta decays of the isotopes created in fission reactors. Such an anomaly would potentially be extremely significant, if a shortfall in the detected neutrino flux could be ascribed to $\bar{\nu}_{e}$ oscillation into a light sterile neutrino with a mass of about 1 eV. There is an extensive recent literature dealing with the reactor anomaly, starting with a seminal paper by Mueller et al. mueller that reexamined the reactor antineutrino flux. The latter publication sought to improve the earlier flux estimates based on the ILL on-line measurements schreck ; schreck2 ; schreck3 of the integral beta spectrum of the fission products. An antineutrino spectrum can be inferred from a beta spectrum provided one knows the linear combination of operators involved in the decay, the end-point energy, and the nuclear charge. The fission beta spectra involve about 6000 beta transitions, of which about 1500 are forbidden endf-7 . Clearly some assumptions are required in order to infer the fission antineutrino flux. The improvements mention ; mueller on the earlier analyses of ILL integral measurements led to an increased energy of the antineutrino flux, which was subsequently verified in an independent analysis huber . The present contribution examines the consequences of the forbidden transitions known to be present (at the 30% level) in the beta decay of fission products. We analyze the antineutrino flux, using a first-principles derivation of the finite size (FS) and weak magnetism (WM) corrections that were the main focus of the analyses in Refs. mueller ; huber . In addition, we examine the shape factors needed to describe the forbidden transitions. We find that the forbidden transitions introduce a large uncertainty in the predicted antineutrino flux irrespective of whether the antineutrino spectra were deduced using nuclear databases or by inverting measured aggregate fission beta spectra. As detailed below, this finding results from the fact that the corrections are nuclear-operator dependent and that an undetermined combination of matrix elements contributes to non-unique forbidden transitions. The beta-decay spectrum $S$ for a single transition in nucleus $(Z,A)$ with end-point energy $E_{0}=E_{e}+E_{\nu}$ is $S(E_{e},Z,A)=S_{0}(E_{e})F(E_{e},Z,A)C(E_{e})(1+\delta(E_{e},Z,A))\,,$ (1) where $S_{0}=G_{F}^{2}\,p_{e}E_{e}(E_{0}-E_{e})^{2}/2\pi^{3}$, $E_{e}(p_{e})$ is the electron total energy (momentum), $F(E_{e},Z,A)$ is the Fermi function needed to account for the Coulomb interaction of the outgoing electron with the charge of the daughter nucleus, and $C(E_{e})$ is a shape factor schopper for forbidden transitions due to additional lepton momentum terms. For allowed transitions $C(E)=1$. The term $\delta(E_{e},Z,A)$ represents fractional corrections to the spectrum that were the central focus of the original anomaly studies. The primary corrections to beta decay are radiative, finite size, and weak magnetism, or $\delta(E_{e},Z,A)=\delta_{\mathrm{rad}}+\delta_{\mathrm{FS}}+\delta_{\mathrm{WM}}$. Before discussing the details of the corrections $C(E_{e})$ and $\delta(E_{e})$, we briefly summarize the treatments used in earlier work. The radiative corrections as derived by Sirlin sirlin were included in the description of the beta spectra (though not in the antineutrino spectra) in the original analyses of Schreckenbach _et al._ schreck ; schreck2 ; schreck3 . In the later ILL work schreck2 ; schreck3 an approximation for the FS and WM corrections was included by first deducing the antineutrino spectrum from the measured beta spectra without these corrections, and then applying a linear correction to the deduced antineutrino spectrum of the form, $\delta_{\mathrm{FS}}+\delta_{\mathrm{WM}}=0.0065(E_{\nu}-{\rm 4\;MeV})$. In that work no corrections were made for the shape factors $C(E_{e})$. In the analyses of Refs. mueller ; huber ; fallot an approximation (derived by Vogel vogel-2 ) for the FS and WM corrections was applied on a transition-by- transition basis. In Refs. mueller ; fallot the shape factor appropriate for unique forbidden transitions was used for all forbidden transitions. In Ref. huber it was argued that these shape factors only play a small role in inferring antineutrino spectra from measured beta spectra because for vanishing electron mass, $m_{e}$, they are symmetric under $E_{e}\leftrightarrow E_{\nu}$. In the present work, we derived ab initio analytic expressions for the FS and WM corrections for allowed GT transitions, as well as WM and shape factors for first-forbidden GT operators. We used the radiative corrections derived by Sirlin sirlin . We now turn to the form of the corrections. The attractive Coulomb interaction increases the electron density near the nuclear surface and increases the beta-decay rate, while the finite nuclear size decreases the electron density and decreases the rate (relative to the point-nucleus Fermi function). Using first-order perturbation theory in $Z\alpha$, we find that the finite-size correction to the Fermi function, $\delta_{\mathrm{FS}}$, for allowed GT transitions is $\delta_{\mathrm{FS}}=-\frac{3}{2}\frac{Z\alpha}{\hbar c}\left<r\right>_{(2)}\left(E_{e}-\frac{E_{\nu}}{27}+\frac{m_{e}^{2}c^{4}}{3E_{e}}\right)\\\ \;\;.$ (2) The quantity $\langle r\rangle_{(2)}=\int d^{3}r\,\rho_{W}(r)\int d^{3}s\,\rho_{\rm ch}(s)\mid\vec{r}-\vec{s}\mid$ is the first moment of the convoluted nuclear weak and charge densities (called a Zemach moment zemach ). We assume uniform distributions of radius $R$ for the weak and charge densities, for which $\left<r\right>_{(2)}=\frac{36}{35}R$ friar1 . The FS corrections do depend on the beta-decay operator, but in this work we always use Eq. (2) and $R=1.2\;A^{1/3}$ fm. The WM correction arises from the interference of the magnetic moment distribution of the vector current, $\vec{J}_{V}=\vec{\nabla}\times\vec{\mu}$, with the spin distribution $\vec{\Sigma}$ of the axial current. We derived the WM corrections for allowed and first-forbidden operators. There are four possible operators in the case of first-forbidden GT transitions, and all have well-defined WM corrections, as listed in Table 1. Table 1: The shape factors and leading-order weak magnetism corrections to allowed and first-forbidden Gamow-Teller beta decays are shown in the top panel. The shape factors for allowed and first-forbidden Fermi beta decays are shown in the bottom panel. All agree with Ref. milliner for $Z=0$. The entries for $\vec{J}_{V}$ and $\rho_{A}$ are discussed in siegert . The weak magnetism correction for $\vec{J}_{V}$ involves the unknown overlap of very different $1^{-}$ matrix elements and is therefore not listed. The nucleon isovector magnetic moment is $\mu_{v}=4.7$, $M_{N}$ is the nucleon mass, $g_{A}$ is the axial vector coupling constant, and $\beta=p_{e}/E_{e}$. No meson currents were used in the magnetic moment operator, and a truncated orbital current led to the factor of “1/2” in $\delta_{\mathrm{WM}}$. Classification \| $\;\Delta J^{\pi}\;$ \| Operator \| Shape Factor $C(E_{e})$ \| Fractional Weak Magnetism Correction $\delta_{\mathrm{WM}}(E_{e})$ ---\|---\|---\|---\|--- Allowed GT \| $1^{+}$ \| $\Sigma\equiv\sigma\tau$ \| 1 \| $\frac{2}{3}\left[\frac{\mu_{v}-1/2}{M_{N}g_{A}}\right](E_{e}\beta^{2}-E_{\nu})$ Non-unique 1st Forbidden GT \| $0^{-}$ \| $\left[\Sigma,r\right]^{0-}$ \| $p_{e}^{2}+E_{\nu}^{2}+2\beta^{2}E_{\nu}E_{e}$ \| 0 Non-unique 1st Forbidden $\rho_{A}$ \| $0^{-}$ \| $\left[\Sigma,r\right]^{0-}$ \| $\lambda\,E_{0}^{2}$ \| 0 Non-unique 1st Forbidden GT \| $1^{-}$ \| $\left[\Sigma,r\right]^{1-}$ \| $p_{e}^{2}+E_{\nu}^{2}-\frac{4}{3}\beta^{2}E_{\nu}E_{e}$ \| $\;\;\;\;\left[\frac{\mu_{v}-1/2}{M_{N}g_{A}}\right]\left[\frac{(p_{e}^{2}+E_{\nu}^{2})(\beta^{2}E_{e}-E\nu)+2\beta^{2}E_{e}E_{\nu}(E_{\nu}-E_{e})/3}{(p_{e}^{2}+E_{\nu}^{2}-4\beta^{2}E_{\nu}E_{e}/3)}\right]$ Unique 1st Forbidden GT \| $2^{-}$ \| $\left[\Sigma,r\right]^{2-}$ \| $p_{e}^{2}+E_{\nu}^{2}$ \| $\frac{3}{5}\left[\frac{\mu_{v}-1/2}{M_{N}g_{A}}\right]\left[\frac{(p_{e}^{2}+E_{\nu}^{2})(\beta^{2}E_{e}-E\nu)+2\beta^{2}E_{e}E_{\nu}(E_{\nu}-E_{e})/3}{(p_{e}^{2}+E_{\nu}^{2})}\right]$ Allowed F \| $0^{+}$ \| $\tau$ \| 1 \| 0 Non-unique 1st Forbidden F \| $1^{-}$ \| $r\tau$ \| $p_{e}^{2}+E_{\nu}^{2}+\frac{2}{3}\beta^{2}E_{\nu}E_{e}$ \| 0 Non-unique 1st Forbidden $\vec{J}_{V}$ \| $1^{-}$ \| $r\tau$ \| $E_{0}^{2}$ \| - Our FS and WM corrections for allowed GT transitions are identical to those derived by Holstein holstein , and differ from the forms used in mueller ; huber ; fallot ; vogel-2 . The first-forbidden shape factors, $C(E_{e})$, which depend on the operator in question, were derived and are displayed in Table 1. To determine the implications of the corrections we take advantage of the fact that the ENDF/B-VII.1 decay library endf-7 contains branching ratios and end-point energies for over 90% of the transitions making up the full spectra. Thus, the library allows a representative estimate of the effect of the corrections, provided we know what assumptions to make about the forbidden transitions. We first fit the Schreckenbachschreck2 electron spectrum for 235U before examining the effect of the corrections in an analysis based on the summation of the known beta branches of the fission products. We include the corrections both with and without a treatment of the forbidden transitions. We fit the beta spectrum with 40 fictitious equally spaced end-point energies using non- negative least-squares fitting nnls . The charge Z associated with each end- point is assigned according to the method given in schreck2 . There is no unique physical prescription for beta-decay operator assignments to the fictitious end-points. For this reason we examine four prescriptions: (1) all transitions are assumed to be allowed; (2) all end-point energies can be associated with either an allowed or forbidden transition; (3) 30% of the branches are selected to be forbidden at equal energy intervals; (4) 30% of the branches are selected to be forbidden with a bias towards higher energies. In addition, we examine fits in which the operator determining the forbidden decays was taken to be $[\Sigma,r]^{0-},[\Sigma,r]^{1-}$, $[\Sigma,r]^{2-}$ or a combination of these. As a natural consequence of the non-negative least- squares procedure, the fit results in a significant fraction of the end-points having zero amplitude. The subset of end-points with non-zero amplitudes varies depending on the operator assignment. We find excellent fits to the electron spectrum in all cases. However, different treatments of the forbidden transitions can lead to antineutrino spectra that differ both in shape and magnitude at about the 4% level. Two examples are shown in Fig.1, where we present the fits obtained when the WM and FS corrections are included. In one case all transitions are assumed to be allowed, while in the second case the best fit results from about 25% forbidden decays. For the assumption of all allowed transitions, we see a systematic increase of about 2.2% in the number of antineutrinos relative to Schreckenbach, while including forbidden transitions leads to no increase relative to Schreckenbach. Other prescriptions for the forbidden transitions lead to changes relative to the Schreckenbach antineutrino spectrum ranging from 0-4%, including changes in the shape of the antineutrino spectrum. These examples help to clarify the difficulty in inferring with high certainty the antineutrino spectrum from a measured electron spectrum when information on the forbidden transitions is not available. Figure 1: The fit to the electron spectrum for 235U (left) for two different assumptions on how to treat forbidden transitions, and the ratio of the corresponding antineutrino spectra to that of Schreckenbach (right). The electron data are those of schreck2 . The electron spectra are fit assuming (a) all allowed GT branches, or (b) up to 30% forbidden GT transitions. In both cases the WM and FS corrections are included. When folded over the neutrino detection cross section, the case for all allowed (25% forbidden) transitions results in a 2.2% (0.06%) increase in the number of detectable antineutrinos. In calculating the aggregate fission spectra from the database library an analogous uncertainty arises because detailed structure information does not exist for the majority of the roughly 1500 forbidden transitions. In addition, several of the transitions are forbidden at second or higher order, for which analytic corrections are not available. Thus, there is no clear prescription for applying the corrections to this component of the spectra, and it is more beneficial to examine the effect of different approximations in order to estimate the uncertainty involved. In all approximations we treat unique forbidden transitions as unique first-forbidden GT transitions, and treat non- unique forbidden transitions in one of the following ways: (1) as allowed GT; (2) as unique first-forbidden GT with the operator $\left[\Sigma,r\right]^{2-}$; (3) with the operator $\left[\Sigma,r\right]^{0-}$; (4) with the operator $\left[\Sigma,r\right]^{1-}$. None of the these treatments is correct, but they provide estimates for changes in the spectra induced by forbidden transitions. The aggregate fission beta spectrum under equilibrium reactor burning conditions for a given actinide is determined by the beta spectra $S(E_{e},Z_{i},A_{i})$ of the individual unstable fission fragments weighted by their cumulative fission yields, $Y_{F_{i}}$ tal : $N_{\beta}(E_{e})=\sum_{F_{i}}Y_{F_{i}}S(E_{e},Z_{i},A_{i}).$ (3) The beta spectrum $S$ for each fragment $(Z_{i},A_{i})$ summed over all decay branches must be normalized to unity: $\int S(E,Z,A)\,dE=1$. Thus, Eq. (3) is a statement that under equilibrium burning conditions the beta-decay rates are determined by the fission rate nonequilibrium . If the antineutrino spectrum is inferred from a measured aggregate beta spectrum, Eq. (3) must be replaced by a sum over a set of end-point energies $\\{E_{0_{i}}\\}$, weighted by a fitted set of coefficients $\\{a_{i}\\}$: $N_{\beta}=\sum_{i}a_{i}S(E_{e},E_{0_{i}})$. There is no unique method for determining the uncertainty in the antineutrino spectrum introduced by the forbidden transitions. Another possibility (in addition to the fitting exercise above) is to consider changes in the bi- variant function $k(E_{e},E_{\nu})$, where $k(E_{e},E_{\nu})=N_{\nu}(E_{\nu})/N_{\beta}(E_{e})$. If $k(E_{e},E_{\nu})$ only changes by some small percentage for some path in the $(E_{e},E_{\nu})$ plane as we change our treatment of forbidden transitions, then there exists a prescription for inferring the antineutrino spectrum to that accuracy. We calculated the function $k(E_{e},E_{\nu})$ for each of our four assumptions (above). We found no path in the $(E_{e},E_{\nu})$ plane that left $k(E_{e},E_{\nu})$ unchanged by as little as 5% as our assumptions for the forbidden transitions changed. Fig. (2) depicts the result for the path $E_{\nu}=K_{\beta}\equiv E_{e}-m_{e}c^{2}$ borovoi . Similar or larger differences were found for all other paths. The very non-smooth and non-linear shape of $k$ arises from the shape factors $C(E_{e})$. Figure 2: The ratio of the function $k(E_{e},E_{\nu})$ for 235U using ENDFB/VII.1 relative to using $E_{\nu}=K_{\beta}$ borovoi . A third prescription for estimating the uncertainties is to examine the rate of change in the antineutrino spectrum relative to the rate of change in the beta spectrum, using the fact that the beta spectrum is fit to amplitudes ${a_{i}}$ on a fixed grid of end-point energies ${E_{0_{i}}}$. We calculated $T=\sum_{i}[\partial N_{\nu}(E_{\nu})/\partial a_{i}]/[\partial N_{\beta}(E_{e})/\partial a_{i}]$, and examined the changes in $T$ as the assumptions for the forbidden transitions were varied, and again found no path in $(E_{e},E_{\nu})$ over which the changes were $<$ 5%. Figure 3: Different treatments of the forbidden GT transitions contributing to the antineutrino spectrum summed over all actinides in the fission burn in mid-cycle kopeikin of a typical reactor. The left panel shows the ratio of these antineutrino spectra relative to that using the assumptions of Ref. schreck2 . The right panel shows the spectra weighted by the detection cross section, where the additional curve in black uses the assumptions of Ref. schreck2 . The spectra are strongly distorted by the forbidden operators, being lower below the peak and in some cases more than 20% larger above the peak than Ref. schreck2 . The corresponding change in the number of detectable antineutrinos relative to schreck2 is -0.75%, 5.8% and 1.85% for the $0^{-},1^{-}$, and $2^{-}$ forbidden operators, respectively. Our final method considers the ratio of the actual antineutrino spectra themselves in Fig. (3), although this method does not take into account the corresponding changes in the beta spectra. We carried out identical analyses of the role of the corrections and the associated uncertainties for the other actinides 239,241Pu and 238U, and found very similar results. Fig. (3) shows the change in the total antineutrino spectrum at a representative time kopeikin in mid-cycle in the reactor burn history. The antineutrino spectra differ significantly, depending on our treatment of the forbidden transitions. The cross-section-weighted spectra are quite distorted, being lower than the Schreckenbach-Vogel schreck2 ; vogel-1 ; explain spectra up to the peak, and higher or lower above the peak depending on the operator. The actual spectrum is unlikely to be as distorted as in Fig. (3) because no single operator dominates the forbidden transitions. The original ILL analysis schreck2 assumed that the $Z$ of the daughter fragments satisfies $Z_{i}=49.5-0.7E_{0_{i}}-0.09E_{0_{i}}^{2}$ for $Z_{i}\geq 34$. We find that taking the $Z$ of the fission fragments directly from ENDF/B-VII.1 instead would increase the antineutrino spectrum by less than 1% for $E_{\nu}\leq$ 7 MeV, and less than 1.5% for $E_{\nu}>$ 7 MeV. In summary we find that the component of the aggregate fission spectra containing approximately 30% forbidden transitions introduces a large uncertainty (about 4%) in the predicted shape of the antineutrino flux emitted from reactors. We have examined the uncertainties in four different ways. If all forbidden transitions are treated as allowed GT transitions, the antineutrino spectra are systematically increased above 2 MeV, as was the conclusion in the earlier papers on the anomaly. However, when the forbidden transitions are treated in various approximations, the shape and magnitude of the spectra are changed significantly. Earlier analyses only looked at one prescription for these transitions. The uncertainty introduced by our lack of knowledge on how to treat these transitions is as large as the size of the anomaly. It should also be noted that there are additional uncertainties due to our oversimplified treatment of the allowed weak magnetism operator (viz., no meson currents and a truncated orbital current), as well as a simplified finite-size treatment for forbidden transitions. These considerations also apply to fission antineutrino fluxes in medium and long baseline reactor experiments, implying a 4% uncertainty on the fission antineutrino flux in those experiments. Reducing the uncertainty within a purely theoretical framework would be difficult. An improvement will require either direct measurements of the antineutrino flux or a substantial improvement in our knowledge of the dominant forbidden beta transitions. ## References * (1) G. Mention, et al., Phys. Rev. D 83 073006 (2011). * (2) Th. A. Mueller, et al., Phys. Rev. C 83, 054615 (2011). * (3) K. Schreckenbach, H. R. Faust, F. von Feilitzsch, A. A. Hahn, K. Hawerkamp, and J. L. Vuilleumier, Phys. Lett. 99B, 251 (1981); F. von Feilitzsch, A. A. Hahn, and K. Schreckenbach, Phys. Lett. 118B, 162 (1982). * (4) K. Schreckenbach, G. Colvin, W. Gelletly, and F. von Feilitzsch, Phys. Lett. 160B 325 (1985). The data from Table 1 and the antineutrino predictions of this reference will be denoted above simply as “Schreckenbach.” * (5) A. A. Hahn, K. Schreckenbach, W. Gelletly, F. von Feilitzsch, G. Colvin, and B. Krusche, Phys. Lett. B218, 365 (1989). * (6) M. B. Chadwick, Nucl. Data Sheets 112, 2887 (2011). * (7) P. Huber, Phys. Rev. C 84, 024617 (2011). * (8) H. F. Schopper, Weak Interactions and Nuclear Beta Decay, North-Holland, Amsterdam, 1966. * (9) A. Sirlin, Phys. Rev. 164, 1767 (1967); A. Sirlin, Phys. Rev. D 84, 014021 (2011). * (10) M. Fallot, S. Cormon, M. Estienne, A. Algora, V. M. Bui, et al., Phys. Rev. Lett. 109, 202504 (2012). * (11) P. Vogel, Phys. Rev. D 29, 1918 (1984) and private communication. * (12) C. Zemach, Phys. Rev. 104, 1771 (1956). * (13) J. L. Friar, Ann. Phys. (N.Y.) 122, 151 (1979). * (14) B. R. Holstein, Phys. Rev. C 9, 1742 (1974); J. L. Friar and I. Sick, Phys. Lett. B579, 285 (2004). Equation (2) of the latter allows conversion between the two results. * (15) C. Lawson and R. Hanson, _Solving Least Squares Problems_ , Prentice-Hall, Englewood Cliffs, New Jersey, 1974. * (16) D. J. Millener, D. E. Alburger, E. K. Warburton, and D. H. Wilkinson, Phys. Rev C 26, 1167 (1982). * (17) Nuclear operators $\vec{J}_{V}$ and $\rho_{A}$ are proportional to a nucleon velocity ($p/M_{N}$), have negative parity, and are therefore first-forbidden. Replacing them by the analogous operators proportional to $E_{0}\,r$ requires only CVC for $\vec{J}_{V}$, but is an approximation for $\rho_{A}$ made exact by introducing an unknown dimensionless nuclear-structure parameter $\lambda$ that differs from case to case. * (18) T. R. England and B. F. Rider: “Evaluation and Compilation of Fission Product Yields,” LA-UR-94-3106 (1994). * (19) Off-equilibrium corrections are needed for fission fragments that decay with half-lives long compared to the timescales involved in the experiments in schreck ; schreck2 ; schreck3 . * (20) A. A. Borovoi, V. I. Kopeikin, L. A. Mikaelyan, and S. V. Tolokonnikov, Sov. Jour. Nucl. Phys. 36, 232 (1982). * (21) V. I. Kopeikin, Phys. Atomic Nuclei 75 143 2012. The mid-cycle combination of burning actinides is 56% 235U, 31% 239Pu, 7% 238U and 6% 214Pu. * (22) P. Vogel, G. K. Schenter, F. M. Mann, and R. E. Schenter, Phys. Rev. C 24, 1543 (1981). * (23) “Schreckenbach-Vogel” means that we used the spectra for 235U, 239Pu, and 241Pu from schreck2 and 238U from vogel-1 .	arxiv-papers	2013-09-17T01:07:29	2024-09-04T02:49:50.982733	{ "license": "Public Domain", "authors": "A.C. Hayes, J.L. Friar, G.T. Garvey, G. Jungman, Guy Jonkmans", "submitter": "Jim Friar", "url": "https://arxiv.org/abs/1309.4146" }
1309.4234	Three-Dimensional Smoothed Particle Hydrodynamics Simulation for Liquid Metal Solidification Process Raden Ahnaf Faqih S.a,b, Christian Fredy Naab aGraduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa 920-1192 Japan, bFaculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132 Indonesia, E-mail: [email protected], [email protected] Abstract.The solidification behavior of liquid metal in a container under rapid cooling process is one of the major concerns to be analyzed. In order to analyze its fundamental behavior, a three-dimensional (3D) fluid dynamics simulation was developed using a particle-based method, known as the smoothed particle hydrodynamics (SPH) method. Governing equations that determine the fluid motion and heat transfer involving phase change process are solved by discretizing their gradient and Laplacian term with the moving particles and calculating the interaction with its neighboring particles. The results demonstrate that the SPH mehod can successfully reproduce the behavior and defect prediction of liquid metal solidification process. Keywords: liquid metal, particle-based method, rapid cooling, solidification, SPH ## 1 Introduction Understanding the solidification behavior of liquid metal is one of the important things for industrial manufacturing field. Sometimes several defects occur if the metal processing is based on the solidification process such as disperse shrinkage, shrinkage cavity, blind cavity, linear cracks, gas porosity, and so on. Various studies are performed for understanding these behaviors. Nowadays, many researchers try to understand it by using numerical study. They make a mathematical model for the multiphase flow, heat transfer, phase transition, and then demonstrate it through a numerical simulation. A review of mathematical modelling of solidification process is discussed in [1]. As to the numerical simulation, it can be divided into two methods, the mesh-based method and particle-based method. Conventional mesh-based methods, in general, encounter difficulties in representing multiphase flow. To overcome this problem, the particle-based method which does not need to generate computational mesh can be used. Several particle-based methods have been developed in recent years. One of the particle-based methods which can be applied to simulate incompressible multiphase flows is smoothed particle hydrodynamics. SPH was originally introduced by Lucy [2] and Gingold & Monaghan [3] to simulate three-dimensional astrophysics problems. However, this method was eventually developed for simulating fluid dynamics and heat transfer problems. SPH method applied to heat transfer for solidification can be seen in [4]. They performed the solidification process for water. In other case, Cleary et al [5] have simulated liquid metal solidification under high pressure die casting. However, they did not explain in detail about the interaction between liquid and solidified metal. The objectives of this paper are to understand and analyze the application of SPH method for liquid metal solidification. We also want to analyze the behavior and defect prediction of liquid metal especially for the case of rapid solidification process. We first apply the SPH formulation for the fluid flow by attaching a temperature dependence of viscosity. We then apply the SPH method for heat transfer which is based on the enthalpy formulation. To verify that the heat transfer works well, we test the heat transfer model by using 1D static particle. For the phase transition model, we consider the nonisothermal phase change problem. We also consider the interaction between different phases and examine it for 3D simulation case. ## 2 Numerical model ### 2.1 The SPH method In SPH, the fluid is represented by a set of particles that can move freely. Each particle carries some fundamental physical properties, such as mass, position, velocity, density, and any other related properties. The value of any function $f$ at a particle can be approximated by summing over the properties of its neighboring particles. The SPH interpolation of the function $f$ of particle $i$ at position $\textbf{r}_{i}$ is approximated by: $f(\textbf{r}_{i})\approx\sum_{j}\frac{m_{j}}{\rho_{j}}f_{j}W(\textbf{r}_{i}-\textbf{r}_{j},h),$ (1) where index $j$ corresponds to any neighboring particle of particle $i$, $m_{j}$ and $\rho_{j}$ are the mass and the density for particle $j$, $f_{j}$ is the value of $f$ for particle $j$, the function $W$ is an interpolation smoothing kernel, and $h$ is a smoothing length that defines the radius of influence around the particle $i$. There are many kinds of smoothing kernel functions in the literature and here we choose cubic spline kernel function as below [6]: $W_{ij}=W(\textbf{r}_{ij},h)=C\begin{cases}\frac{2}{3}-\left(\frac{\|\textbf{r}_{ij}\|}{h}\right)^{2}+\frac{1}{2}\left(\frac{\|\textbf{r}_{ij}\|}{h}\right)^{3}&\text{, }0\leq\frac{\|\textbf{r}_{ij}\|}{h}<1\\\ \frac{1}{6}\left(2-\left(\frac{\|\textbf{r}_{ij}\|}{h}\right)\right)^{3}&\text{, }1\leq\frac{\|\textbf{r}_{ij}\|}{h}<2\\\ 0&\text{, }2\geq\frac{\|\textbf{r}_{ij}\|}{h}\end{cases}$ (2) where $\textbf{r}_{ij}=\textbf{r}_{i}-\textbf{r}_{j}$ and $C=3/(2\pi h^{2})$ for 3D simulation. The radius of influence of this function is $2h$. It means that the particle $i$ will not be affected by any neighboring particles farther than $2h$. However, the closer distance of the neighboring particles to the particle $i$, the greater will be the influence accepted by the particle $i$. The derivative of the function $f$ is obtained by differentiating the interpolation Eq.(1) which is given by: $\nabla f(\textbf{r}_{i})\approx\sum_{j}\frac{m_{j}}{\rho_{j}}f_{j}\nabla_{i}W_{ij}$ (3) where $\nabla_{i}W_{ij}=\nabla W(\textbf{r}_{ij},h)=\frac{1}{h}\frac{\textbf{r}_{ij}}{\|\textbf{r}_{ij}\|}F(\|\textbf{r}_{ij}\|,h)$ and $F$ is the derivative of the kernel function Eq.(2). ### 2.2 Fluid model The governing equations of fluids in SPH method are based on the Navier-Stokes equations in the Lagrangian form. The basic equations are given by [7]: $\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$ $\displaystyle-\rho\nabla\cdot\textbf{v}$ (4) $\displaystyle\rho\frac{d\textbf{v}}{dt}$ $\displaystyle=$ $\displaystyle-\nabla p+\mu\nabla^{2}\textbf{v}+\textbf{F}$ (5) where $t$ is time, v is velocity, $p$ is pressure, $\mu$ is viscosity, and F is external force. Eq.(4) is known as the continuity equation which describes the evolution of the fluid density over time. Eq.(5) is known as the momentum equation which describes the acceleration of the fluid. The first term on the right hand side involving the pressure gradient corresponds to the pressure force and the second term involving viscosities corresponds to the viscous force. The external forces used here are gravitational force and repulsive force from the boundaries. By employing the SPH interpolation of Eq.(3) to the Eq.(4) and Eq.(5) then the SPH representation of the continuity and the momentum equation can be written as [6]: $\displaystyle\frac{d\rho_{i}}{dt}$ $\displaystyle=$ $\displaystyle\sum_{j}m_{j}(\textbf{v}_{i}-\textbf{v}_{j})\cdot\nabla W_{ij},$ (6) $\displaystyle\frac{d\textbf{v}_{i}}{dt}$ $\displaystyle=$ $\displaystyle-\sum_{j}m_{j}\left[\left(\frac{P_{i}}{\rho_{i}^{2}}+\frac{P_{j}}{\rho_{j}^{2}}\right)-\frac{\xi}{\rho_{i}\rho_{j}}\frac{4\mu_{i}\mu_{j}}{\left(\mu_{i}+\mu_{j}\right)}\frac{\textbf{v}_{ij}\textbf{r}_{ij}}{\textbf{r}_{ij}^{2}+\eta^{2}}\right]\nabla_{i}W_{ij}+\textbf{F}.$ (7) The viscous force used here is the laminar viscosity term which was introduced by Morris and Monaghan [8] with $\mu_{i}$ and $\mu_{j}$ the dynamic viscosity of the fluid corresponding to particles $i$ and $j$. Further, $\xi=4.963$ is a viscous scaling factor [9] and $\eta$ is a small parameter to prevent singularity when $\textbf{r}_{ij}$ goes to zero (usually $\eta\sim 0.1h$). An equation of state is required to calculate the pressure in Eq.(7). The equation of state used here is quasi-compressible form which is calculated by using the density information from Eq.(6). The equation of state is given by [9]: $P=\beta\left[\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}-1\right]$ (8) where $\beta=c^{2}\rho_{0}/\gamma$ is the magnitude of the pressure, $\rho_{0}$ is the reference density, $\gamma=7$ for liquid metal, and $c$ is the speed of sound. The value of speed of sound should be large enough to ensure that the density fluctuation is less than 1% or close to the incompressible flow. ### 2.3 Heat transfer and phase transition model The heat transfer process occurs by cooling down the system through the boundaries. Here, the boundaries are modeled by a set of SPH particles which have a certain temperature and thermal conductivity. The temperature and thermal conductivity of the boundary particles are assumed to be constant during the solidification process. We also assume the boundary as adiabatic boundary where there is no flux to the ambient environment. The heat exchange between the particles (liquid, solid, and boundary) occurs by considering the heat conduction. The model of heat conduction is based on the enthalpy method which is given by: $\frac{dH}{dt}=\frac{1}{\rho}\nabla(k\nabla T)$ where $H$ is enthalpy, $k$ is thermal conductivity, and $T$ is temperature. The SPH formulation of this equation is approximated by using the modified SPH approximation for second derivative which is developed by Cleary and Monaghan [10]: $\frac{dH_{i}}{dt}=\sum_{j}\frac{m_{j}}{\rho_{i}\rho_{j}}\frac{4k_{i}k_{j}}{(k_{i}+k_{j})}(T_{i}-T_{j})\frac{\textbf{r}_{ij}\cdot\nabla_{i}W_{ij}}{\textbf{r}_{ij}^{2}+\eta^{2}}.$ (9) This equation ensures that the heat flux is automatically continuous across the different material interfaces, such as between the mold and liquid metal. This also allows multiple phases with different conductivities to be accurately simulated [10]. The temperature in Eq.(9) is calculated by considering the relationship between enthalpy and temperature. In the liquid metal solidification case, the phase change occurs over a range of temperature. The phase transition goes through an intermediate phase which is known as mushy phase. So, the nonisothermal phase change is involved to correlate the enthalpy and temperature. The illustration of the nonisothermal phase change is shown in Figure 1. This relation ensures that the temperature is continuous across the different phases. Figure 1: Relationship between enthalpy and temperature for nonisothermal phase change The relationship between enthalpy and temperature can be written as [11]: $H=\begin{cases}\int_{T_{ref}}^{T}c_{s}(T)dT&\text{, }T\leq T_{s}\text{ $~{}~{}~{}~{}~{}~{}~{}$ (Solid)},\\\ \int_{T_{ref}}^{T_{s}}c_{s}(T)dT+\int_{T_{s}}^{T}\frac{\partial L}{\partial T}dT&\text{, }T_{s}<T\leq T_{l}\text{$~{}~{}$(Mushy)},\\\ \int_{T_{ref}}^{T_{s}}c_{s}(T)dT+L+\int_{T_{l}}^{T}c_{l}(T)dT&\text{, }T>T_{l}\text{$~{}~{}~{}~{}~{}~{}~{}~{}$ (Liquid)},\end{cases}$ (10) where $T_{ref}$ is the reference temperature, $T_{l}$ and $T_{s}$ are respectively the solidus and liquidus temperatures, and $c_{l}$ and $c_{s}$ are the specific heats in the solid and liquid phases, respectively. If we consider the specific heats to be constant then Eq.(10) becomes: $H=\begin{cases}c_{s}T&\text{, }T\leq T_{s},\\\ c_{s}T_{s}+c_{sl}(T-T_{s})&\text{, }T_{s}<T\leq T_{l},\\\ c_{s}T_{s}+c_{sl}(T_{l}-T_{s})+c_{l}(T-T_{l})&\text{, }T>T_{l},\end{cases}$ The temperature of each particle is then calculated by using this relation in the temperature term: $T_{i}=\begin{cases}\frac{H_{i}}{c_{s}}&\text{, }H_{i}\leq H_{s}=c_{s}T_{s}\text{ $~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (Solid)},\\\ T_{s}+\frac{H_{i}-H_{s}}{c_{sl}}&\text{, }H_{s}<H_{i}\leq H_{l}=H_{s}+c_{sl}(T_{l}-T_{s})\text{$~{}$(Mushy)},\\\ T_{s}+\frac{H_{i}-H_{l}}{c_{l}}&\text{, }H_{i}>H_{l}\text{$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (Liquid)},\end{cases}$ where $c_{sl}=L/(T_{l}-T_{s})$. The latent heat $L$ describes the energy released by a particle to change the phase from liquid to solid. ### 2.4 Liquid-solid interaction model The phase transition of a particle from liquid to solid occurs when the particle temperature is below the solidus temperature. Here, we modelled the solid particles as a viscous pseudo-fluid. It means that the behavior of solid particles is like a fluid (liquid or mushy) particles but they move under high viscosity. This approach is used to keep the forces acting on the solid particles and fluid particles maintained. If the solid particle has more than two of its neighbors being solid phase then they act and move together as a solid group (rigid body). The illustration how to create the solid group is shown in Figure 2. First, each solid particle is assigned an initial index, as shown in Figure 2. After that, each particle index is updated by the maximum index of its neighbor particle (Figure 2). This process is repeated until each solid particle index converges to the maximum index in each group (Figure 2). After some steps, we will only have the maximum index on each group and it means the number of the solid group is determined by how many remaining indices we have. For example in the Figure 2, we have two separate solid groups because finally we have two indices that are 13 and 20. Figure 2: Illustration of calculating the number of solid groups Here, we consider the solid group as an undeformable object. In order to prevent the deformation, the solid particle’s movement should be converted to the translational and rotational motion related to the center of mass of its solid group. The center of mass is calculated by using the following equation: $\textbf{r}_{c}=\frac{1}{N}\sum_{j}\textbf{r}_{j}$ where $N$ is the number of solid particle in the solid group. The translational and rotational motion is considered by calculating the moment of inertia and angular velocity of the solid group. The moment of inertia, $I$, and angular velocity, $\omega$, is calculated by the following equations: $I=\sum_{i}\|\textbf{q}_{i}\|^{2}\;\text{and }\omega=\frac{1}{I}\sum_{i}\textbf{q}_{i}\times\textbf{v}_{i}.$ Here, $\textbf{q}_{i}=\textbf{r}_{i}-\textbf{r}_{c}$ is the relative position of particle $i$ to the center of mass. During the simulation, the center of mass and the rotation matrix are updated by using the following equations: $\textbf{r}_{c}(t+\Delta t)=\textbf{r}_{c}(t)+\Delta t\textbf{v}_{c}\left(t+0.5\Delta t\right)\;\text{and }\textbf{R}(t+\Delta t)=\textbf{R}(t)+\Delta t\textbf{Z}\left(t+0.5\Delta t\right)$ where $\textbf{v}_{p}$ is the center of mass velocity and the angular velocity tensor $\textbf{Z}(t)$ in is given by: $\textbf{Z}(t)=\left[\begin{array}[]{ccc}0&-\omega_{z}&\omega_{y}\\\ \omega_{z}&0&-\omega_{x}\\\ -\omega_{y}&\omega_{x}&0\\\ \end{array}\right]\textbf{R}(t).$ The rotation matrix in 3D coordinates system is given by: $\textbf{R}_{x}(\theta)=\left[\begin{array}[]{ccc}1&0&0\\\ 0&\cos\theta&\sin\theta\\\ 0&-\sin\theta&\cos\theta\\\ \end{array}\right],\;\textbf{R}_{y}(\theta)=\left[\begin{array}[]{ccc}\cos\theta&0&-\sin\theta\\\ 0&1&0\\\ \sin\theta&\omega_{x}&\cos\theta\\\ \end{array}\right],\;\textbf{R}_{z}(\theta)=\left[\begin{array}[]{ccc}\cos\theta&\sin\theta&0\\\ -\sin\theta&\cos\theta&0\\\ 0&0&1\\\ \end{array}\right].$ After we get the angular velocity and the center of mass, the relative velocity and position of each solid particle to the center of mass can be written as below: $\displaystyle\textbf{v}_{i}(t+\Delta t)$ $\displaystyle=$ $\displaystyle\textbf{v}_{c}(t)+\omega\times\textbf{q}_{i}(t),$ $\displaystyle\textbf{r}_{i}(t+\Delta t)$ $\displaystyle=$ $\displaystyle\textbf{R}(t+\Delta t)\left[\textbf{r}_{i}(t)-\textbf{r}_{c}(t)\right]+\textbf{r}_{c}(t+\Delta t).$ ## 3 Numerical simulations ### 3.1 One-dimensional sinusoidal temperature We simulate the cooling down process in 1D case to verify that the SPH method for heat conduction works well. We use 1m block with the initial temperature distribution is $T(x,0)=\sin\pi x$. The temperature on the boundaries are constant. We set the boundary temperature to 0oC. The exact solution of this problem is $T(x,t)=e^{\pi^{2}\alpha t}\sin\pi x$ where $\alpha=k/\rho c_{p}=0.1$. Figure 3: Heat conduction for 1D sinusoidal temperature Figure 3 shows the SPH solution for heat conduction fitted with the exact solution for times $t=0,0.1,0.2,...,1$. The result shows that the constant temperature on the boundaries acts as heat sink of the system. ### 3.2 Three-dimensional liquid metal solidification #### 3.2.1 Physical set up and material properties The materials used in this simulation are liquid aluminium as the liquid metal and steel as the mold. The geometry of the mold steel is cube-shaped with the dimension 0.3m$\times$0.3m$\times$0.3m. The temperature of the mold steel is 200oC and the thermal conductivity is $30$ W/mK. The mold is fully filled by the liquid aluminium. The liquid aluminium data is taken from [12]. The solidus and liquidus temperatures are 536.1oC and 589.7oC respectively. The viscosity of the liquid aluminium is temperature dependent. The variation of the viscosity with temperature is shown in the Figure 4. The viscosity above the liquidus temperature is 0.0118 Pa s and the viscosity below the solidus temperature is 52.4 Pa s. The density of the liquid aluminium is also temperature dependent. The liquid metal has a density of 2540 kg/m3 and the solid metal has a density of 2702 kg/m3. In the mushy zone, the variation of density with temperature is close to linear. The variation of the density is shown in the Figure 4. The other data for the aluminium properties can be seen in Table LABEL:table:1. Figure 4: Relation of material properties with temperature: (a) viscosity and (b) density Table 1: Thermal physical properties of aluminium Parameter \| Value \| Unit ---\|---\|--- Thermal conductivity of liquid \| 94.14 \| W/mK Thermal conductivity of solid \| 237.65 \| W/mK Specific heat of liquid \| 1.084 \| kJ/kg/K Specific heat of solid \| 0.963 \| kJ/kg/K Initial temperature \| 600 \| oC Latent heat \| 398 \| kJ/kg #### 3.2.2 Results and discussions In this simulation, we use 15625 liquid metal particles with 0.0106154 m of smoothing length. The simulation starts by arranging the liquid metal particles position in the well-ordered position. Then the particles are solidified rapidly through the side walls and the bottom wall. The results for several time steps are shown in the Figure 5. Actually the domain is fully filled with the liquid metal particles. However, we display the results from the half part, so we can observe the inner part. The results show not only the temperature distribution but also the phase transition. The temperature distribution for the liquid phase and mushy phase are represented by the color bar. The red color represents the liquid phase with temperature above 589.7oC and the blue color represents the lowest temperature for the mushy phase that is 536.1oC. The solid phase is represented by silver color where the temperature is below 536.1oC. Figure 5: Solidification process for several time step: (a) t = 0.0045 s, (b) t = 0.5 s, (c) t = 1 s, (d) t = 1.5 s, (e) t = 2 s, and (f) t = 2.7272 s The solidification process seems very fast. All of the liquid particles become solid after 2.7272 s. It is because we set the initial temperature of a single particle 600oC while the initial temperature of the boundary is 200oC. This value is very close to the liquidus temperature that is 589.1oC. Another reason is that the thermal conductivities of mushy and solid phase are very high. So, if some particles are transformed to the solid phase, the heat transfer will be faster. As an example, the temperature histories of a single particle can be seen in Figure 6. It shows the different temperature drops at different phases. After the particle reaches the mushy phase or solid phase, the heat transfer works faster and leads the temperature to decrease rapidly. Figure 6: Temperature history for a single particle. It reaches mushy phase at $t=1.35455$ s and solid phase at $t=1.99545$ s From the results in Figure 5, it can be seen that the volume of the liquid metal is decreasing every time step and it leads to the shrinkage cavity formation at the final solidification stage. It happened due to the difference in density where the solid phase has a higher density than the liquid phase. This affects the density of the mushy particles to continuously grow during the cooling down process. Hence the equation of states in Eq.(8) gives positive pressure; moreover the gravitational force also contributes to push the particles to the bottom. This simulation seems to approximate nicely the realistic phenomena. However, in the rapid cooling case, other defects such as cracks or voids often appear. Presence of cracks means there are some empty parts inside if we check the inner part of the particles distribution. The particles distribution for different layers can be seen in Figure 7. We got different patterns on different layers. There are some lower densities and higher densities according to the number of particles. In the lower density part some cracks should appear. However, in our cases the cracks do not appear. It means that the model for liquid-solid interaction in 3D simulation should be improved. Our model which assumes the solid group as fluid with high viscosity and acts as undeformable object seems correct. It is because we can produce the shrinkage cavity formation and the simulation is stable until all the particle become solid. However, it is still not enough and needs to be improved such as by applying some particular forces between mushy and solid phases to produce other defects. Figure 7: Density difference according to the number of particles at different layers: (a) 3 $\leq$ z $\leq$ 6 cm, (b) 6 $\leq$ z $\leq$ 9 cm, and (c) 9 $\leq$ z $\leq$ 12 cm ## 4 Summary In this paper, the behavior of liquid metal solidification has been studied and simulated using smoothed particle hydrodynamics. The solidification process was done by installing the enthalpy method in SPH formulation to solve the heat transfer problem. The model of heat transfer process has been tested to static particle grid in one dimension and it gives a relatively good accuracy. The phase transition from liquid to solid was treated by considering the effect of latent heat and nonisothermal phase change to the model of heat transfer. The liquid-solid interaction model for 3D simulation also has been performed. The results showed the appearance of a defect when liquid metal was solidified. We got shrinkage cavity formation in the final solidification stage. However, in the rapid cooling process, other defects often appear, to obtain these defects, the model of liquid-solid interaction should be improved such as by adding a particular force between mushy and solid phases. The improvement of liquid-solid interaction model is left for the further study. ## References * [1] H. Hu and S. A. Argyropoulos (1996). Mathematical modelling of solidification and melting: a review, Modelling Simul. Mater. Sci. Eng., 4, 4, pp. 371-396. * [2] L. B. Lucy (1977), A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 1013 –- 1024. * [3] R. A. Gingold and J. J. Monaghan (1977), Smoothed particle hydrodynamics: theory and application to non–spherical stars, Mon. Not. R. Astr. Soc., 181, 375 – 389. * [4] J. J. Monaghan, H. E. Huppert, and M. G. Worster (2005), Solidification using smoothed particle hydrodynamics, J. Comput. Phys., 206, 684 – 705. * [5] P.W. Cleary, J. Ha, M. Prakash, and T. Nguyen (2010). Short shots and industrial case studies: Understanding fluid flow and solidification in high pressure die casting. Applied Mathematical Modelling. 34, 2018 -– 2033. * [6] M. Liu and G. Liu (2010). Smoothed Particle Hydrodynamics (SPH): an overview and recent developments, Archives of Computational Methods in Engineering, 17, 1, pp. 25 – 76. * [7] J. F. Wendt (2009), Computational fluid dynamics: An introduction. Springer, Berlin. * [8] J.P. Morris, J.J. Monaghan (1997). A switch to reduce SPH viscosity, J. Compt. Phys., 136, 41 –- 50. * [9] P.W. Cleary (1998). Modelling confined multi-material heat and mass flows using SPH. Applied Mathematical Modelling. 22, 981 -– 993. * [10] P.W. Cleary and J.J. Monaghan (1999). Conduction modelling using smoothed particle hydrodynamics, J. Comput. Phys., 148, 1, pp. 227 – 264. * [11] B. Nedjar (2001) An enthalpy-based finite element method for nonlinear heat problems involving phase change, Computers and structures, 80, 9 – 21. * [12] P.W Cleary (2010). Extension of SPH to predict feeding, freezing and defect creation in low pressure die casting. Applied Mathematical Modelling. 34, 3189 -– 3201.	arxiv-papers	2013-09-17T09:07:50	2024-09-04T02:49:50.993943	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Raden Ahnaf Faqih S, Christian Fredy Naa", "submitter": "Christian Naa", "url": "https://arxiv.org/abs/1309.4234" }
1309.4251	# Optimal Distributed Controller Design with Communication Delays: Application to Vehicle Formations Hamid Reza Feyzmahdavian, Assad Alam, and Ather Gattami H. R. Feyzmahdavian is with Electrical Engineering, ACCESS Linnaeus Centre, Royal Institute of Technology, 100 44 Stockholm, Sweden. [email protected]. Alam is with Research and Development, Scania CV AB, 151 87 Södertälje, Sweden. [email protected]. Gattami is with Electrical Engineering, ACCESS Linnaeus Centre, Royal Institute of Technology, 100 44 Stockholm, Sweden. [email protected] work was supported by Scania CV AB, VINNOVA - FFI, and the Swedish Research Council. ###### Abstract This paper develops a controller synthesis algorithm for distributed LQG control problems under output feedback. We consider a system consisting of three interconnected linear subsystems with a delayed information sharing structure. While the state-feedback case of this problem has previously been solved, the extension to output-feedback is nontrivial, as the classical separation principle fails. To find the optimal solution, the controller is decomposed into two independent components. One is delayed centralized LQR, and the other is the sum of correction terms based on additional local information. Explicit discrete-time equations are derived whose solutions are the gains of the optimal controller.111A preliminary version of this work was presented in [1]. ## I INTRODUCTION The systems to be controlled are in many application domains getting larger and more complex. When there is interconnection between different dynamical systems, conventional optimal control algorithms provide a solution where centralized state information is required. However, it is often preferable and sometimes necessary to have a distributed control structure, since in many practical problems, the physical or communication constraints often impose a specific interconnection structure. Hence, it is interesting to design distributed feedback controls for systems of a certain structure and examine their overall performance. The control problem and methodology in this paper is motivated by systems involving a chain of closely spaced heavy duty vehicles (HDVs), generally referred to as vehicle platooning. The objective is to maintain a predefined headway to the vehicle ahead, while maintaining safety and minimizing the fuel consumption. Information technology is paving its path into the transport industry, enabling the possibility of automated control strategies. Governing vehicle platoons by an automated control strategy, the overall traffic flow is expected to improve [2] and the road capacity will increase significantly [3], without endangering safety [4]. By traveling at a close intermediate spacing the air drag is reduced for each vehicle in the platoon. Thereby, the control effort and inherently the fuel consumption can be reduced significantly [5]. This creates a coupling of the dynamics between neighboring vehicles throughout the platoon. However, as the intermediate spacing is reduced the control becomes tighter due to safety aspects; mandating an increase in control action and inherently the fuel consumption through additional acceleration and braking. The fuel consumption constitutes approximately 30 % of the overall cost for a fleet owner [6]. Hence, it is of vast interest for the industry to find a fuel optimal control. Considering the physical constraints in radio, it cannot be assumed that state information is available at every instance in time. Thus, a distributed control strategy is crucial for practical implementation. In recent work [7, 8, 9], distributed control has been studied under the assumption of spatial invariance. Control for chain structures in the context of platoons has been studied through various perspectives, e.g., [10, 11, 12, 13, 14]. It has been shown that control strategies may vary depending on the available information within the platoon. Maintaining a suitable relative distance, stability and robustness of the platoon have been identified to be amongst the main criteria to be considered. However, communication constraints have not in general been considered in control design for platooning applications and the controllers have mainly been ad hoc by tuning the control parameters. In [15, 16], linear quadratic Gaussian (LQG) control under appropriate assumptions on communication delays between the controllers was considered. While a computationally efficient solution was presented for a sequence of vehicles moving in formation, the controller structure is not provided by the corresponding semi-definite programming. A structured sequential design was introduced in [17], where the preceding vehicle’s dynamics along with its states were conveyed through wireless communication. It resulted in a suboptimal control strategy, where physical coupling to a follower vehicle and communication delays were not considered. Mounted radar sensors allows each vehicle to measure the relative distance and velocity of the preceding vehicle. Additional information, providing local information, has lately been introduced through wireless information. However, wireless systems introduce information delays to the system in certain cases due to limitations in radio. Furthermore, varying external environment factors impose process disturbances on the system. In this work, we are primarily concerned with forming a distributed control, that accounts for the interconnection between neighboring vehicles, correlated process disturbances, as well as communication delays. The control is solely based on local model knowledge, over the class of LQG control for chain structured interconnection graphs. The received information is assumed to be common after two time step delays. The main contribution of this paper is to derive an LQG controller, which is easy to implement and optimal under a delayed information sharing pattern for chain structures. In addition to communication delays, the distributed optimal control is based upon systems with interconnected dynamics to both neighboring vehicles and local state information. Derived from the characteristics of actual Scania HDV’s, we present a discrete system model that includes physical coupling with both neighboring vehicles. We also investigate the performance of the proposed controllers, under normal operating conditions for an HDV platoon, with respect to physical constraints that are imposed in a practical set-up. The outline of the paper is as follows. The general system and problem description is given in Sec. II, which in turn determines the structure of the optimal controller. The theoretical premise for the optimal controller is presented in Sec. IV, where it is shown that the problem can be decomposed into two separate optimization problems. Finally, we evaluate the performance of the derived controller through numerical results in Sec. V and give concluding remarks in Sec. VI. ### Notation Throughout the paper, we use the following notation: matrices are written in uppercase letters and vectors in lowercase letters. The $i^{th}$ component of a vector $x$ is denoted by $x_{i}$. Let $[x]_{S}$ be the sub-vector of $x$ containing only those components with indices in set $S$. For instance, if $S=\\{1,3\\}$, then $[x]_{S}$ is given by $[x]_{S}=\begin{bmatrix}x_{1}&x_{3}\end{bmatrix}^{T}$. The sequence $x(0)$, $x(1)$, $\ldots$ , $x(k)$ is denoted by $x(0:k)$. diag$(x)$ denotes a diagonal matrix whose diagonal elements are given by those of the enclosed vector $x$. Let $X$ be a matrix partitioned into blocks. We use $[X]_{ij}$ and $[X]_{i}$ to represent the block in block position $ij$ and $i$th block row, respectively. $[X]_{S_{1}S_{2}}$ denotes the sub-matrix of $X$ containing exactly those rows and columns corresponding to the sets $S1$ and $S2$, respectively. For instance $[X]_{\\{1\\}\\{2,3\\}}=\begin{bmatrix}X_{12}&X_{13}\end{bmatrix}$. The trace of square matrix is denoted by $\textbf{Tr}\\{X\\}$. We use $X^{+}$ and $X^{-}$ to represent $X(k+1)$ and $X(k-1)$ respectively, when appropriate. Given $A\in\mathbb{R}^{m\times n}$, we can write $A$ in terms of its columns as $A=\begin{bmatrix}a_{1}&\cdots&a_{n}\end{bmatrix}$. The operation vec$(A)$ results in a $mn\times 1$ column vector $\textrm{vec}(A)=\begin{bmatrix}a_{1}^{T}&\cdots&a_{n}^{T}\end{bmatrix}^{T}$. We denote by $\textrm{vec}^{\star}(A)$, the sub-vector of $\textrm{vec}(A)$ containing only nonzero elements. Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{r\times s}$, then the operation $A\otimes B\in\mathbb{R}^{mr\times ns}$ denotes the Kronecker product of $A$ and $B$. We denote the expectation of a random variable $x$ by $\textbf{E}\\{x\\}$. The conditional expectation of $x$ given $y$ is denoted by $\textbf{E}\\{x\|y\\}$. ## II System Model and Problem Description In this section we present the physical properties of the system that we are considering. We state the nonlinear dynamics of a single vehicle and the model for the aerodynamics, which induces the physical coupling. Then we present the linear discrete system model for a heterogeneous HDV platoon and its associated cost function. The communication constraints and physical coupling is then used to motivate the structure of the controller. Finally, the problem formulation is given. ### II-A System Model We consider an HDV platoon as depicted in Fig. 1. The state equation of a single HDV is modeled as, $\displaystyle\dot{s}$ $\displaystyle=v,$ (1) $\displaystyle m_{t}\dot{v}$ $\displaystyle=F_{engine}-F_{brake}-F_{airdrag}(v)$ $\displaystyle\hskip 44.3863pt-F_{roll}(\alpha)-F_{gravity}(\alpha),$ $\displaystyle=k_{u}u-k_{b}F_{brake}-k_{d}v^{2}$ $\displaystyle\hskip 33.0052pt-k_{fr}\cos\alpha-k_{g}\sin\alpha,$ where $v$ is the vehicle velocity, $m_{t}$ denotes the accelerated mass and $u\in\mathbb{R}$ denotes the net engine torque. $k_{u},k_{b},k_{d},k_{fr}$, and $k_{g}$ denote the characteristic vehicle and environment coefficients for the engine, brake, air drag, road friction, and gravitation respectively. Figure 1: The figure shows a platoon of $M$ heavy duty vehicles, where each vehicle is able to communicate with its neighbors. The variation in aerodynamics between the vehicles is essential in the analysis of fuel reduction potential for HDVs. For a single HDV it can amount up to 50 % of the total resistive forces at full speed. It is significantly reduced when operating in a platoon formation and a coupling between the vehicles is induced. To account for the aerodynamics, the air drag characteristic coefficient in (1) can be modeled as [18] $\displaystyle\begin{split}\tilde{k}_{d}&=k_{d}(1-\frac{\Phi(d)}{100}-\frac{\phi(d)}{100}),\\\ \Phi(d)&=\alpha_{1}d+\alpha_{2},0\leq d\leq 60\\\ \phi(d)&=\beta_{1}d+\beta_{2},0\leq d\leq 15\end{split}$ where $d$ is the longitudinal relative distance between two vehicles, $\Phi(d)$ and $\phi(d)$ are linear piecewise affine functions of the change in air drag due to a preceding and a follower vehicle respectively, and $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}$ are positive constants. The relative distance reference could be constant or, as in this case, time varying. It is determined by setting a desired time gap $\tau$ s, which in turn determines the spacing policy as $d_{ref}(k)={\tau}v(k)$. Thereby, the vehicles will maintain a larger intermediate spacing at higher velocities. When studying the behavior of an HDV platoon, the velocity does not deviate significantly from the lead vehicle’s velocity. Ideally, all vehicles should maintain a constant speed and intermediate distance. Thus, a linearized model should give a sufficient description of the system behavior under these conditions. By linearizing and applying a one step forward discretization to (1), the discrete model for an HDV platoon with respect to a set reference velocity, an engine torque which maintains the velocity, a fixed spacing between the vehicles, and a constant slope is hence given by $x(k+1)=Ax(k)+Bu(k)+w(k),$ where $\displaystyle A$ $\displaystyle=\begin{bmatrix}{\Theta_{1}}&\gamma_{2}&0&0&0&\cdots&0&0&0\\\ 1&1&-1&0&0&\cdots&0&0&0\\\ 0&\delta_{2}&\Theta_{2}&\gamma_{3}&0&\cdots&0&0&0\\\ 0&0&1&1&-1&\cdots&0&0&0\\\ 0&0&0&\delta_{3}&\Theta_{3}&\cdots&\gamma_{4}&0&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\ 0&0&0&0&0&\cdots&\Theta_{M-1}&\gamma_{M-1}&0\\\ 0&0&0&0&0&\cdots&1&1&-1\\\ 0&0&0&0&0&\cdots&0&\delta_{M}&\Theta_{M}\\\ \end{bmatrix},$ $\displaystyle\begin{split}B&=\begin{bmatrix}k_{u_{1}}&0&0&\cdots&0\\\ 0&0&0&\cdots&0\\\ 0&k_{u_{2}}&0&\cdots&0\\\ 0&0&0&\cdots&0\\\ 0&0&k_{u_{3}}&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&0\\\ 0&0&0&\cdots&k_{u_{M}}\end{bmatrix},\quad x=\begin{bmatrix}{v_{1}}\\\ {d_{12}}\\\ {v_{2}}\\\ {d_{23}}\\\ {v_{3}}\\\ \vdots\\\ {v_{M-1}}\\\ {d_{(M-1)M}}\\\ {v_{M}}\end{bmatrix},\\\ u&=\begin{bmatrix}{u_{1}}\\\ {u_{2}}\\\ {u_{3}}\\\ \vdots\\\ {u_{M}}\end{bmatrix},\begin{array}[]{rl}\Theta_{1}&=1-T_{s}2k_{d}(d_{0})v_{0}/m_{t},\\\ \Theta_{i}&=1-T_{s}2k_{d}\Phi{(d_{0})}v_{0}/m_{t},\quad i=2,\dots,M,\\\ \delta_{i}&=-T_{s}\alpha_{1}k_{d}v_{0}^{2}/m_{t},\\\ \gamma_{i}&=-T_{s}\beta_{1}k_{d}v_{0}^{2}/m_{t},\end{array}\end{split}$ (2) and $T_{s}$ is the sampling time. Thus, the system has a block diagonal structure and can be grouped into subsystems as indicated in (2). The general representation of the derived system can be stated as $\displaystyle\begin{bmatrix}x_{1}(k+1)\\\ x_{2}(k+1)\\\ x_{3}(k+1)\\\ \vdots\\\ x_{M}(k+1)\end{bmatrix}=$ $\displaystyle\begin{bmatrix}A_{11}&A_{12}&0&\cdots&0\\\ A_{21}&A_{22}&A_{23}&\cdots&0\\\ 0&A_{32}&A_{33}&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&A_{MM}\end{bmatrix}\begin{bmatrix}x_{1}(k)\\\ x_{2}(k)\\\ x_{3}(k)\\\ \vdots\\\ x_{M}(k)\end{bmatrix}$ $\displaystyle+$ $\displaystyle\begin{bmatrix}B_{1}&0&0&\cdots&0\\\ 0&B_{2}&0&\cdots&0\\\ 0&0&B_{3}&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&B_{M}\\\ \end{bmatrix}\begin{bmatrix}u_{1}(k)\\\ u_{2}(k)\\\ u_{3}(k)\\\ \vdots\\\ u_{M}(k)\end{bmatrix}+w(k)$ (3) where the corresponding vehicle states for each subsystem are $x_{1}(k)=v_{1}(k),\quad x_{i}(k)=\begin{bmatrix}d_{i-1,i}\\\ v_{i}\end{bmatrix},\quad i=2,\dots,M.$ In practice, many random disturbances are imposed upon a vehicle in motion. The varying road topology has a strong impact due the extensive mass of the HDVs. Weather conditions might vary and traffic conditions might change. Furthermore, variation in wind affects all the vehicles in the platoon and therefore the process noise is considered to be correlated. Hence, the disturbance, $w(k)$ in (3), is assumed to be a Gaussian white noise with a full positive definite covariance matrix $W$. We also assume that the initial state $x(0)$ is uncorrelated with $w(k)$ for all $k$, with zero mean and covariance matrix $P_{0}$. While a general problem was defined, for simplicity, consider an $M=3$ HDV platoon. In this case, the dynamics of the system given in (4) is $\displaystyle x_{1}(k+1)$ $\displaystyle=A_{11}x_{1}(k)+A_{12}x_{2}(k)+B_{1}u_{1}(k)$ $\displaystyle x_{2}(k+1)$ $\displaystyle=A_{21}x_{1}(k)+A_{22}x_{2}(k)+A_{23}x_{3}(k)+B_{2}u_{2}(k)$ $\displaystyle x_{3}(k+1)$ $\displaystyle=A_{32}x_{2}(k)+A_{33}x_{3}(k)+B_{3}u_{3}(k)$ (4) It can be seen in (4) that the state of vehicle $1$ is affected by the states of vehicle $2$ in the next time step. Whereas, the state of vehicle $1$ affects the states of $3$ after two time steps, through vehicle $2$. Vehicle $2$ on the other hand is affected by both vehicle $1$ and $3$ in the next time step. The local models can be conveyed at a single point in time between each subsystem, through wireless communication. However, the system is time critical due to safety aspects and communication should be kept at minimum so the channel is not congested and latency is introduced. Assume that passing information from one vehicle to another vehicle takes one time step, so the available information set of each vehicle at time $k$ can be described as $\displaystyle\mathcal{I}_{1}(k)$ $\displaystyle=\\{x_{1}(k),x_{1}(k-1),x_{2}(k-1),x(0:k-2)\\}$ $\displaystyle\mathcal{I}_{2}(k)$ $\displaystyle=\\{x_{2}(k),x(k-1),x(0:k-2)\\}$ $\displaystyle\mathcal{I}_{3}(k)$ $\displaystyle=\\{x_{3}(k),x_{2}(k-1),x_{3}(k-1),x(0:k-2)\\}$ (5) The three vehicles share all past information with two-step communication delay, as described in (5). The assumptions about the information structure and the sparsity of dynamics guarantee that information propagates at least as fast as the dynamics. This information pattern is a simple case of partially nested information structure. It is shown in [19] that if the information structure is partially nested, then the optimal controller exists, it is unique, and linear. Therefore, the optimal controller for three vehicles under the given information set has the form $\displaystyle u_{1}(k)$ $\displaystyle=$ $\displaystyle f_{11}\bigl{(}x_{1}(k)\bigr{)}+f_{12}\bigl{(}x_{1}(k-1),x_{2}(k-1)\bigr{)}$ $\displaystyle\hskip 49.22314pt+f_{13}\bigl{(}x(0:k-2)\bigr{)}$ $\displaystyle u_{2}(k)$ $\displaystyle=$ $\displaystyle f_{21}\bigl{(}x_{2}(k)\bigr{)}+f_{22}\bigl{(}x(k-1)\bigr{)}$ $\displaystyle\hskip 49.22314pt+f_{23}\bigl{(}x(0:k-2)\bigr{)}$ $\displaystyle u_{3}(k)$ $\displaystyle=$ $\displaystyle f_{31}\bigl{(}x_{3}(k)\bigr{)}+f_{32}\bigl{(}x_{2}(k-1),x_{3}(k-1)\bigr{)}$ (6) $\displaystyle\hskip 49.22314pt+f_{33}\bigl{(}x(0:k-2)\bigr{)}$ where $f_{ij}$ denotes a linear function in all its variables. Consequently, the optimal control $u(k)$ can be expressed as $\displaystyle u(k)=F(k)x(k)+G(k){x(k-1)}+f\bigl{(}{x}(0:k-2)\bigr{)}$ (7) where $f={\begin{bmatrix}f_{13}^{T}&f_{23}^{T}&f_{33}^{T}\\\ \end{bmatrix}}^{T}$ and $\displaystyle F(k)$ $\displaystyle=\begin{bmatrix}F_{11}&0&0\\\ 0&F_{22}&0\\\ 0&0&F_{33}\\\ \end{bmatrix},\;G(k)$ $\displaystyle=\begin{bmatrix}G_{11}&G_{12}&0\\\ G_{21}&G_{22}&G_{23}\\\ 0&G_{32}&G_{33}\\\ \end{bmatrix}.$ ### II-B Cost Function The objective of the lead vehicle is to minimize the fuel consumption and control input, while maintaining a set reference velocity. The objective of the follower vehicles in addition is to follow the preceding vehicles velocity, while maintaining a set intermediate spacing. Hence, similar to what we presented for the continuous LQR in [17], the weights for a $M$ HDV platoon can be set up based upon the performance objectives as $\displaystyle J(u^{})=$ $\displaystyle\min_{u}~{}\sum_{k=0}^{N-1}\Big{(}\sum_{i=2}^{M}w_{i}^{\tau}(d_{(i-1)i}(k)-\tau v_{i}(k))^{2}$ $\displaystyle\hskip 38.41121pt+w_{i}^{\Delta{v}}(v_{i-1}(k)-v_{i}(k))^{2}$ $\displaystyle\hskip 38.41121pt+w_{i}^{d}d_{(i-1)i}^{2}(k)+\sum_{i=1}^{M}w_{i}^{v}v_{i}^{2}(k)+w_{i}^{u_{i}}u_{i}^{2}(k)\Big{)}$ $\displaystyle=$ $\displaystyle\min_{u}~{}\sum_{k=0}^{N-1}\sum_{i=2}^{M}\begin{bmatrix}v_{i-1}(k)\\\ d_{(i-1)i}(k)\\\ v_{i}(k)\end{bmatrix}^{T}Q_{i}\begin{bmatrix}v_{i-1}(k)\\\ d_{(i-1)i}(k)\\\ v_{i}(k)\end{bmatrix}+R_{i}u_{i}^{2}(k)$ $\displaystyle\hskip 38.41121pt+w^{v_{1}}v_{1}^{2}(k)+w^{u_{1}}u_{1}^{2}(k)$ (8) where $\displaystyle\begin{split}&Q_{i}=\begin{bmatrix}w_{i}^{\Delta{v}}&0&-w_{i}^{\Delta{v}}\\\ 0&w_{i}^{d}+w_{i}^{\tau}&-\tau{w}_{i}^{\tau}\\\ -w_{i}^{\Delta{v}}&-\tau{w}_{i}^{\tau}&\tau^{2}w_{i}^{\tau}+w_{i}^{\Delta{v}}+w_{i}^{v}\end{bmatrix},\\\ &Q_{1}=\begin{bmatrix}w^{v_{1}}&0\\\ 0&w^{u_{1}}\end{bmatrix},R_{i}=w_{i}^{u_{i}}.\end{split}$ The weights in (8) give a direct interpretation of how to enforce the objectives for a vehicle traveling in a platoon. The value of $w_{i}^{\tau}$ determines the importance of not deviating from the desired time gap. Hence, a large $w_{i}^{\tau}$ puts emphasis on safety. $w_{i}^{\Delta{v}}$ creates a cost for deviating from the velocity of the preceding vehicle, and $w_{i}^{u_{i}}$ punishes the control effort which is proportional to the fuel consumption. The following terms, $w_{i}^{d},w_{i}^{v}$, put a cost on the deviation from the linearized states. Note that the main objective is to maintain a set intermediate distance, while maintaining a fuel efficient behavior. Therefore, $w_{i}^{\tau},w_{i}^{\Delta{v}}$ and $w_{i}^{u_{i}}$ must be set larger than the remaining weights. ### II-C Problem Formulation We consider a HDV platooning scenario where each vehicle only receives information regarding the relative position and velocity of the immediate neighboring vehicles. The objective is to design a controller that can handle a two time step delay. The aim is to utilize the given structure of the considered system, where we want to minimize the cost function $\displaystyle\begin{split}J=&\textbf{E}\\{x(N)^{T}Q_{0}x(N)\\}\\\ &+\sum_{k=0}^{N-1}\textbf{E}\\{x^{T}(k)Qx(k)+u^{T}(k)Ru(k)\\},\end{split}$ (9) subject to the sparse system dynamics in (3) and the performance objectives in (8). The primary difficulty arises from the imposed information constraints given in (5). Thus, the problem that we solve in this paper is finding an analytical expression for an optimal control input $u_{i}(k)$, which must be a function of the admissible information set $\mathcal{I}_{i}(k)$, where each subsystem control input is unique and a linear function denoted as $\displaystyle u_{i}(k)=\mu_{i}\bigl{(}\mathcal{I}_{i}(k)\bigr{)},\;i=1,\dots,M.$ (10) ###### Assumption 1 The matrices $Q_{0}$ and $Q$ in (9) are positive semi-definite, and $R$ is positive definite. ## III Main Result In this section we present the optimal controller for three-vehicle problem. The proof for this result is presented in the remaining sections. ###### Theorem 1 _Suppose that $W$ is positive definite and that Assumption $1$ holds. Define the matrix $D\triangleq\begin{bmatrix}F&M\end{bmatrix}$ where $M$ has the same sparsity structure as $G$. Let $S$ be the index set of non-zero elements of_ $\text{vec}(D)$, $S\triangleq\left\\{i:\;\textrm{vec}_{i}(D)\neq 0\right\\}.$ _Suppose there exists a stabilizing solution $X$ to the algebraic Riccati equation_ $\displaystyle X=A^{T}XA+Q+A^{T}XB(B^{T}XB+R)^{-1}B^{T}XA$ _We then define_ $\displaystyle H$ $\displaystyle=B^{T}XB+R\vspace{0.1cm}$ $\displaystyle L$ $\displaystyle=(B^{T}XB+R)^{-1}B^{T}XA\vspace{0.1cm}$ _and let_ $\displaystyle Y$ $\displaystyle=\begin{bmatrix}W\otimes(H+B^{T}L^{T}HLB)&-W\otimes B^{T}L^{T}H\\\ -W\otimes HLB&W\otimes H\end{bmatrix}$ $\displaystyle b$ $\displaystyle=\begin{bmatrix}W\otimes H\\\ 0\end{bmatrix}\mbox{vec}(L)+\begin{bmatrix}-W\otimes B^{T}L^{T}H\\\ W\otimes H\end{bmatrix}\textrm{vec}(LA)$ _Then, the optimal controller gains are given by:_ $\displaystyle\textrm{vec}^{\star}(F)$ $\displaystyle=\begin{bmatrix}I&0\end{bmatrix}[Y]_{SS}^{-1}[b]_{S}\vspace{1mm}$ $\displaystyle\textrm{vec}^{\star}(M)$ $\displaystyle=\begin{bmatrix}0&I\end{bmatrix}[Y]_{SS}^{-1}[b]_{S}$ _and the optimal controller has the realization_ $\displaystyle\zeta(k+1)=$ $\displaystyle Ax(k)+Bu(k)$ $\displaystyle\xi(k+1)=$ $\displaystyle A\zeta(k)+BM(x(k-1)-\zeta(k-1))+BL\xi(k)$ $\displaystyle u(k)=$ $\displaystyle F(x(k)-\zeta(k))$ $\displaystyle+M(x(k-1)-\zeta(k-1))+L\xi(k)$ Note that blocks of matrices $F$ and $M$ can be computed from the $\textrm{vec}^{\star}(F)$ and $\textrm{vec}^{\star}(G)$, respectively. For example, $\textrm{vec}^{\star}(F)=\textrm{vec}\left(\begin{bmatrix}F_{11}&F_{22}&F_{33}\end{bmatrix}\right)$. It will be shown that $\xi(k)$ is the minimum-mean square estimate of $x(k)$ given the common information $x(0:k-2)$; that is, $\xi(k)=\textbf{E}\\{x(k)\|x(0:k-2)\\}$. Thus, the optimal controller of three- vehicle problem is the centralized $LQR$ controller under the classical information structure with two-step delay plus correction terms based on the local information at time $k$. ## IV Optimal Controller Derivation In this section, we present the preliminary lemmas that are used to prove the results in Theorem $1$. Before proceeding further, we need to state the following proposition which later permits us to decompose $J$ into two separate parts. ###### Proposition 1 ([20]) _Define the matrices_ $\displaystyle X(k)=$ $\displaystyle A^{T}X^{+}A+Q$ (11) $\displaystyle-(A^{T}X^{+}B)(B^{T}X^{+}B+R)^{-1}(B^{T}X^{+}A)$ $\displaystyle H(k)=$ $\displaystyle B^{T}X^{+}B+R\vspace{0.1cm}$ $\displaystyle L(k)=$ $\displaystyle(B^{T}X^{+}B+R)^{-1}B^{T}X^{+}A\vspace{0.1cm}$ _for $k=0,\cdots,N-1$ and where $X(N)=Q_{0}$. Then the cost function (9) can be written as _ $\displaystyle J=$ $\displaystyle\underbrace{\sum_{k=0}^{N-1}\textbf{E}\left\\{\bigl{(}u(k)-L(k)x(k)\bigr{)}^{T}H(k)\bigl{(}u(k)-L(k)x(k)\bigr{)}\right\\}}_{J_{u}}$ $\displaystyle+\underbrace{x^{T}(0)X(0)x(0)+\sum_{k=0}^{N-1}\textbf{Tr}\\{X(k+1)W\\}}_{J_{w}}$ _where both the zero-mean property of $w(k)$ and independence of $w(k)$ and $(x(k),u(k))$ are exploited. Moreover, $J_{w}$ is independent of $u$._ From Proposition 1, it can be seen that minimizing $J$ is equivalent to minimizing $J_{u}$. Note that, under the Assumption 1, $H(k)$ is positive definite. ### IV-A State Decomposition The first step towards finding the optimal controller is decomposing the state vector into independent terms. ###### Lemma 1 _The state vector can be decomposed as_ $\displaystyle x(k)=$ $\displaystyle\underbrace{w(k-1)+\bigl{(}A+BF(k-1)\bigr{)}w(k-2)}_{x^{1}(k)}$ $\displaystyle+\underbrace{\textbf{E}\left\\{x(k)\|x(0:k-2)\right\\}}_{x^{2}(k)}$ _where $x^{1}(k)$ and $x^{2}(k)$ are independent random variables._ ###### Proof: The term $x^{2}(k)$ is the conditional estimate of state $x(k)$ given the piece of information shared between all vehicles, and $x^{1}(k)$ is the estimation error. The independence between $x(k)-x^{2}(k)$ and $x^{2}(k)$ can be established by Proposition $4$b in the appendix. To calculate $x^{1}(k)$, we proceed in three steps. First consider $x(k-1)=Ax(k-2)+Bu(k-2)+w(k-2)$ Since $x(k-2)$ belongs to the sequence $x(0:k-2)$ and $u(k-2)$ is a deterministic function of $x(0:k-2)$, we have $x(k-1)-\textbf{E}\\{x(k-1)\|x(0:k-2)\\}=w(k-2)$ (12) where we used the zero-mean and independence of $w(k-2)$ and $x(0:k-2)$. The structure of controller is given by equation (7), so $u(k-1)$ can be written as $u(k-1)=F(k-1)x(k-1)+G(k-1){x(k-2)}+f\bigl{(}{x}(0:k-3)\bigr{)}$ Since $G(k-1){x(k-2)}+f\bigl{(}{x}(0:k-3)$ is a deterministic function of $x(0:k-2)$, we have $\displaystyle u(k-1)-\textbf{E}\\{u(k-1)\|x(0:k-2)\\}$ (13) $\displaystyle=$ $\displaystyle F(k-1)\bigl{(}x(k-1)-\textbf{E}\\{x(k-1)\|x(0:k-2)\\}\bigr{)}$ $\displaystyle=$ $\displaystyle F(k-1)w(k-2)$ where we substituted (12) into the second line. Finally, note that $w(k-1)$ and $x(0:k-2)$ are independent. Therefore, $\displaystyle x(k)$ $\displaystyle-\textbf{E}\\{x(k)\|x(0:k-2)\\}$ $\displaystyle=$ $\displaystyle w(k-1)+A\left(x(k-1)-\textbf{E}\\{{x}(k-1)\|x(0:k-2)\\}\right)$ $\displaystyle\hskip 38.41139pt+B\left(u(k-1)-\textbf{E}\\{{u}(k-1)\|x(0:k-2)\\}\right)$ $\displaystyle=$ $\displaystyle w(k-1)+\bigl{(}A+BF(k-1)\bigr{)}w(k-2)$ (14) where we substituted (12) into the second line and (13) into the third line. Thus the result follows. ∎ ### IV-B Controller Decomposition Now that the state has been decomposed into two independent terms, the control input $u(k)$ can be decomposed in a similar fashion. ###### Lemma 2 _The control input $u(k)$ can be decomposed as_ $\displaystyle u(k)$ $\displaystyle=\underbrace{F(k)w(k-1)+M(k)w(k-2)}_{u^{1}(k)}+u^{2}(k)$ _where $u^{1}(k)$ and $u^{2}(k)$ are independent, $u^{2}(k)$ is a linear function of $x(0:k-2)$, and_ $M(k)=F(k)\left(A+BF(k-1)\right)+G(k).$ ###### Proof: Let $u^{2}(k)=\textbf{E}\\{u(k)\|x(0:k-2)\\}$, then $u^{2}(k)$ is a linear function of $x(0:k-2)$ and independent of $u(k)-u^{2}(k)$. Note that $f(x(0:k-2))$ is a linear function of $x(0:k-2)$, so $u^{1}(k)$ is computed as $\displaystyle u^{1}(k)=$ $\displaystyle u(k)-\textbf{E}\\{u(k)\|x(0:k-2)\\}$ $\displaystyle=$ $\displaystyle F(k)\bigl{(}x(k)-\textbf{E}\\{x(k)\|x(0:k-2)\\}\bigr{)}$ $\displaystyle+G(k)\bigl{(}x(k-1)-\textbf{E}\\{x(k-1)\|x(0:k-2)\\}\bigr{)}$ $\displaystyle=$ $\displaystyle F(k)(w(k-1)+\left(A+BF(k-1)\right)w(k-2))$ $\displaystyle+G(k)w(k-2)$ where we used equation (7) in the first line, (14) in the second line and (12) in the third line. The proof is completed by defining $M(k)=F(k)\left(A+BF(k-1)\right)+G(k)$. ∎ ###### Remark 1 Since $B$ and $F$ are diagonal matrices, $G(k)$ and $F(k)A$ have the same sparsity structures. Therefore, sparsity structure of $M(k)$ and $G(k)$ are also the same. From Lemmas 1 and 2, $x^{2}(k)$ and $u^{2}(k)$ are linear functions of $x(0:k-2)$ which is independent of $x^{1}(k)$ and $u^{1}(k)$. As a result the cost function $J_{u}$ can be decomposed as: $\displaystyle J_{u}$ $\displaystyle=\underbrace{\sum_{k=0}^{N-1}\textbf{E}\left\\{\bigl{(}u^{1}(k)-L(k)x^{1}(k)\bigr{)}^{T}H(k)\bigl{(}u^{1}(k)-L(k)x^{1}(k)\bigr{)}\right\\}}_{J^{1}_{u}}$ $\displaystyle+\underbrace{\sum_{k=0}^{N-1}\textbf{E}\left\\{\bigl{(}u^{2}(k)-L(k)x^{2}(k)\bigr{)}^{T}H(k)\bigl{(}u^{2}(k)-L(k)x^{2}(k)\bigr{)}\right\\}}_{J^{2}_{u}}$ The advantage of this decomposition of $J_{u}$ is that we now have two subproblems on the form $\displaystyle\begin{split}\min_{u^{1}}~{}&J^{1}_{u}(x^{1},u^{1})\\\ \mbox{subject to}~{}&u^{1}(k)=F(k)w(k-1)+M(k)w(k-2)\end{split}$ (15) $\displaystyle\min_{u^{2}}~{}J^{2}_{u}(x^{2},u^{2})$ $\displaystyle\mbox{subject to}~{}u^{2}(k)=f\bigl{(}x(0:k-2)\bigr{)}$ (16) ### IV-C Finite Horizon Controller Derivation First consider minimization problem (16). Before proceeding, let us state the following proposition which allows us to find the optimal control $u^{2}(k)$. ###### Proposition 2 ([20]) _Consider the discrete time linear system_ $x(k+1)=Ax(k)+Bu(k)+w(k)$ _where $w(k)$ is a zero mean Gaussian white noise. Assume that $u(k)=\mu\bigl{(}x(0:k)\bigr{)}$. Then the optimal control which minimizes the cost function $J_{u}$, is given by_ $u(k)=L(k)x(k)$ The mapping from $x^{2}(k)$ to $u^{2}(k)$ is given in the following lemma. ###### Lemma 3 _The dynamics of $x^{2}$ can be written as_ $x^{2}(k+1)=Ax^{2}(k)+Bu^{2}(k)+T(k)w(k-2)$ (17) _where $T(k)=A(A+BF(k-1))+BM(k)$._ ###### Proof: See appendix. ∎ The following theorem shows that $u^{2}(k)$ is exactly the optimal controller for centralized information structure with two step delay, where the information set of each vehicle is $\mathcal{I}_{i}(k)=\left\\{x(0:k-2)\right\\}$. ###### Theorem 2 _Given that Assumption $1$ holds, an optimal solution to (16) is given by_ $\displaystyle u^{2}(k)=L(k)x^{2}(k)$ (18) ###### Proof: Consider the system (17) together with the cost function $J^{2}_{u}$. Both $x(0:k-2)$ and $u^{2}(k)$ are linear functions of $x(0:k-2)$ which is independent of $w(k-2)$. Hence, finding the optimal control $u^{2}(k)$ is now a centralized LQR problem. Applying proposition 2, we obtain (18). ∎ We now turn to the optimization problem (15). Recalling the expansions of $x^{1}(k)$ and $u^{1}(k)$ in terms of $w(k-1)$ and $w(k-2)$, the expected value of the $k^{th}$ term of $J^{1}_{u}$ can be expanded as follows: $\displaystyle\textbf{E}\\{$ $\displaystyle\bigl{(}u^{1}(k)-L(k)x^{1}(k)\bigr{)}^{T}H(k)\bigl{(}u^{1}(k)-L(k)x^{1}(k)\bigr{)}\\}$ $\displaystyle=\textbf{Tr}\\{H(k)(F(k)-L(k))W(F(k)-L(k))^{T}\\}$ $\displaystyle\hskip 8.53581pt+\textbf{Tr}\\{H(k)\bigl{(}M(k)-L(k)(A+BF(k-1))\bigr{)}W$ $\displaystyle\hskip 68.28644pt\times\bigl{(}M(k)-L(k)(A+BF(k-1))\bigr{)}^{T}\\}$ where we used Proposition $4$a in the appendix and the fact that $w(k-1)$ and $w(k-2)$ are independent. To minimize $J^{1}_{u}$ with respect to $F(0),\ldots,F(k)$ and $M(1),\ldots,M(k)$, the difficulty is that $F$ and $M$ must satisfy specified sparsity constraints. We use vectorization of matrices to simplify our optimization problem. Let us define the matrix $D(k)$ as follows $\displaystyle D(k)\triangleq\begin{bmatrix}F(k-1)&M(k)\end{bmatrix}\;\;\in\mathbb{R}^{m\times 2p},\;k=1,\ldots,N-1$ and $D(N)\triangleq F(N-1)$. Then $\textrm{vec}\bigl{(}D(k)\bigr{)}$ is given by $\begin{bmatrix}\textrm{vec}\bigl{(}F(k-1)\bigr{)}\\\ \textrm{vec}\bigl{(}M(k)\bigr{)}\end{bmatrix}\;\;\;\in\mathbb{R}^{2mp},\;k=1,\ldots,N-1$ and $\textrm{vec}(D(N)\bigr{)}=\textrm{vec}\bigl{(}F(N-1)\bigr{)}$. Because of the specified sparsity of $F$ and $M$, some components of $\textrm{vec}\bigl{(}D(k)\bigr{)}$ must be zero. Let $S$ be the index set of non-zero elements of $\textrm{vec}\bigl{(}D(k)\bigr{)}$, i.e. $S\triangleq\left\\{i:\;\textrm{vec}_{i}\bigl{(}D(k)\bigr{)}\neq 0\right\\}$ Note that $\textrm{vec}\bigl{(}D(k)\bigr{)}$ and $\textrm{vec}^{\star}\bigl{(}D(k)\bigr{)}$ are related by nonsquare matrix. We define this matrix to be $E$, where dimensions implied by the context, so that $\textrm{vec}\bigl{(}D(k)\bigr{)}=E\textrm{vec}^{\star}\bigl{(}D(k)\bigr{)}$. The columns of $E$ are ${e_{j}}$ for ${j\in S}$ where $e_{j}$ denotes a column vector having all zeros except a $1$ at the $j^{th}$ position. Since exactly one entry in each column of $E$ is equal to $1$, $E^{T}XE$ is a sub-matrix of $X$ containing exactly those rows and columns corresponding to the set $S$. We illustrate the above definition via an example. Let $D=\text{diag}(d_{11},d_{22},d_{33})\in\mathbb{R}^{3\times 3}$. For this matrix, $S=\\{1,5,9\\}$, $E=\begin{bmatrix}e_{1}&e_{5}&e_{9}\end{bmatrix}$, and $\textrm{vec}^{\star}(D)=[d_{11},d_{22},d_{33}]^{T}$. In the following lemma, we show that a vectorization of matrices $F$ and $M$ makes the cost function $J^{1}_{u}$ a sum of quadratic functions without constraints. ###### Lemma 4 _Define_ $\displaystyle Y_{11}(k)$ $\displaystyle=W\otimes(H(k-1)+B^{T}L^{T}(k)H(k)L(k)B)$ $\displaystyle Y_{12}(k)$ $\displaystyle=-W\otimes B^{T}L^{T}(k)H(k)$ $\displaystyle Y_{22}(k)$ $\displaystyle=W\otimes H(k)$ _and let_ $\displaystyle Y_{k}$ $\displaystyle=\begin{bmatrix}Y_{11}(k)&Y_{12}(k)\\\ Y_{12}^{T}(k)&Y_{22}(k)\end{bmatrix}$ $\displaystyle b_{k}$ $\displaystyle=\begin{bmatrix}Y_{22}(k-1)\\\ 0\end{bmatrix}\textrm{vec}\left(L(k-1)\right)+\begin{bmatrix}Y_{12}(k)\\\ Y_{22}(k)\end{bmatrix}\textrm{vec}\left(L(k)A\right)$ _for $k=1,\ldots,N-1$, and_ $\displaystyle Y_{N}$ $\displaystyle=W\otimes H(N-1)$ $\displaystyle b_{N}$ $\displaystyle=\bigl{(}W\otimes H(N-1)\bigr{)}\textrm{vec}\left(L(N-1)\right)$ _Then optimization problem ( 15) is equivalent to_ $\displaystyle\min_{\textrm{vec}^{\star}(D(k))}\sum_{k=1}^{N}$ $\displaystyle\frac{1}{2}\textrm{vec}^{\star}(D(k))^{T}[Y_{k}]_{SS}\textrm{vec}^{\star}(D(k))$ $\displaystyle-\textrm{vec}^{\star}(D(k))^{T}[b_{k}]_{S}$ _Moreover, $Y_{k}$ is positive definite._ ###### Proof: See appendix. ∎ The advantage of this equivalent reformulation of the problem is that we have $N$ quadratic functions without constraints and thus the optimal controller gains can be computed by simply minimizing these functions separately. ###### Theorem 3 _Suppose $W$ is positive definite and Assumption $1$ holds. Then the optimal gains of controllers are given by:_ $\displaystyle\textrm{vec}^{\star}(F(k-1))$ $\displaystyle=\begin{bmatrix}I&0\end{bmatrix}\textrm{vec}^{\star}(D(k))$ $\displaystyle\textrm{vec}^{\star}(M(k))$ $\displaystyle=\begin{bmatrix}0&I\end{bmatrix}\textrm{vec}^{\star}(D(k))$ _for $k=1,\ldots,N-1$ and _$\textrm{vec}^{\star}(F(N-1))=\textrm{vec}^{\star}(D(N))$, _where_ $\textrm{vec}^{\star}(D(k))=[Y_{k}]_{SS}^{-1}[b_{k}]_{S}$. ### IV-D Steady State Controller Derivation Assume that the solution to algebraic Riccati equation (11), $X(k)$, converges to the stabilizing solution as $k$ approaches $\infty$: $\displaystyle X=A^{T}XA+Q+A^{T}XB(B^{T}XB+R)^{-1}B^{T}XA$ Since $H(k)$ and $L(k)$ are specified by $X(k)$, they respectively converge to matrices $H$ and $L$ as follows: $\displaystyle H=B^{T}XB+R,~{}L=(B^{T}XB+R)^{-1}B^{T}XA$ Then $Y_{k}$ and $b_{k}$ will approach the values of $Y$ and $b$ given by $\displaystyle Y$ $\displaystyle=\begin{bmatrix}W\otimes(H+B^{T}L^{T}HLB)&-W\otimes B^{T}L^{T}H\\\ -W\otimes HLB&W\otimes H\end{bmatrix}$ $\displaystyle b$ $\displaystyle=\begin{bmatrix}W\otimes H\\\ 0\end{bmatrix}\textrm{vec}(L)+\begin{bmatrix}-W\otimes B^{T}L^{T}H\\\ W\otimes H\end{bmatrix}\textrm{vec}(LA)$ Thus, the optimal gains are calculated to be $\displaystyle\textrm{vec}^{\star}(F)=\begin{bmatrix}I&0\end{bmatrix}[Y]_{SS}^{-1}[b]_{S}$ $\displaystyle\textrm{vec}^{\star}(M)=\begin{bmatrix}0&I\end{bmatrix}[Y]_{SS}^{-1}[b]_{S}$ ### IV-E Estimation Structure Having determined the optimal controller, we turn now to analyze this result. Define $\zeta(k)=x(k)-w(k-1)$. Hence, we obtain the following state-space system $\zeta(k+1)=Ax(k)+Bu(k)$ with initial condition $\zeta(0)=0$. Note that the assumptions about the information structure and sparsity structure of $A$ and $B$ guarantee that each vehicle can update $\zeta(k)$ at time $k$. For example, consider Vehicle $1$. Since Vehicle $1$ has access to $x_{2}(k-1)$ at time $k$, It can construct $\zeta_{1}(k)=A_{11}x_{1}(k-1)+A_{12}x_{2}(k-1)+B_{1}u_{1}(k-1)$. Letting $\xi(k)=\textbf{E}\\{x(k)\|x(0:k-2)\\}$ the optimal control policy can be written as $\displaystyle u(k)=F(x(k)-\zeta(k))+M(x(k-1)-\zeta(k-1))+L\xi(k)$ In order to fully specify $u(k)$, the conditional estimates $\xi(k)$, as well as the matrices $L$, $F$ and $G$ must be computed. We have $\displaystyle\xi(k+1)$ $\displaystyle=\textbf{E}\\{x(k+1)\mid x(0:k-1)\\}$ $\displaystyle=A\textbf{E}\\{x(k)\mid x(0:k-1)\\}+B\textbf{E}\\{u(k)\mid x(0:k-1)\\}$ $\displaystyle=A\zeta(k)+BM(x(k-1)-\zeta(k-1))+BL\xi(k)$ ## V Numerical Results We evaluate the performance of the system with the controller by giving an example of a realistic scenario that HDV platoons often face on the road. In practice, varying traffic conditions often mandate a deviation in the lead vehicle’s velocity. Therefore, integral action for the lead vehicle is added as a state to the system presented in (2), to model such disturbances. Figure 2: Three HDV platoon, where a disturbance in velocity of the lead vehicle is imposed. The top plot shows the velocity trajectories for the $M=3$ HDV platoon, the bottom plot shows the intermediate spacings, and the bottom plot shows the control inputs. The trajectories are obtained through the optimal distributed control and subindexed $i$, where $i=1,2,3$ denote the platoon position index. We consider a heterogeneous platoon, where the masses are set to $[m_{1},m_{2},m_{3}]=[30000,40000,30000]$ kg. All the vehicles are assumed to be traveling in the steady state velocity $v_{0}=19.44$ m/s ($70$ km/h) at time gap $\tau=0.25$ s, which gives an intermediate distance of $d_{0}=4.86$. The maximum engine and braking torque for a commercial HDV varies based upon vehicle configuration but can be approximated to be 2500 Nm and 60000 Nm/Axle respectively. State disturbances as well as several lead vehicle deviation disturbances are imposed on the system (Fig. 2). The lead vehicle deviation disturbances can be explained by the following scenario. The platoon travels along a road where the road speed is 70 km/h. Suddenly a slower vehicle enters the lane through a shoulder path at a lower speed. The lead vehicle must therefore reduce its speed to 60 km/h, in turn forcing the follower vehicles to reduce their speed and adapt their relative distance accordingly. After a while, the slower vehicle increases its speed to the road speed of 70 km/h and no longer inhibits the platoon. Hence, the lead vehicle again resumes the road speed and the follower vehicles adapt the speed and distance automatically as well. Finally, the platoon arrives at a point where the road speed is changed to 80 km/h. Fig. 2 shows the velocity trajectories of three HDV platoon in the top plot, the corresponding intermediate spacings in the middle plot, and the required control input to handle the disturbances in the bottom plot. The trajectories nearly lie on top of each other, showing that the proposed controller performs a tight control and the disturbances are handled well. There is no overshoot in the velocity or intermediate spacing tracking. Furthermore, the control input is well within the feasible physical range. The weight normalized control input energy required to handle the imposed disturbances is reduced by 15 % for the second vehicle and 14 % for the third vehicle, with respect to the first vehicle. Hence, the controller displays a fuel efficient behavior, since the input energy is directly proportional to the fuel consumption. The theoretical value, in this case, for the cost function with the proposed optimal distributed control is only 0.01 % higher than a fully centralized control with full state information at all times. On the other hand, the proposed controller produces a 67 % lower theoretical cost compared to a centralized control with two step time delays. ## VI SUMMARY AND CONCLUSIONS We have presented an analytical controller, which is optimal under a delayed information sharing pattern for chain structures. 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Cambridge University Press, 1996. ### -A Preliminaries ###### Proposition 3 ([21]) _If $A$, $B$, $C$, $D$ and $X$ are suitably dimensioned matrices, then_ 1. a) $\textrm{vec}(AXB)=(B^{T}\otimes A)\textrm{vec}(X)$, 2. b) $(A\otimes B)(C\otimes D)=(AC)\otimes(BD)$, 3. c) _If $A$ and $B$ are positive definite, then so is $A\otimes B$,_ 4. d) $\textbf{Tr}\\{AXBX^{T}\\}=\textrm{vec}^{T}(X)(B^{T}\otimes A)\textrm{vec}(X)$, 5. e) $(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}$._( $A$ and $B$ are nonsingular)_ ###### Proposition 4 ([20]) _Let $x$ and $y$ be zero-mean random vectors with a jointly Gaussian distribution. Assume $S$ be a symmetric matrix. Then the following facts hold:_ 1. a) $\textbf{E}\\{x^{T}Sx\\}=\textbf{Tr}\bigl{\\{}S\textbf{{E}}\\{xx^{T}\\}\bigr{\\}}$. 2. b) $\textbf{E}\\{x\|y\\}$ and $x-\textbf{E}\\{x\|y\\}$ are independent. ###### Proposition 5 ([21]) _Suppose that a symmetric matrix is partitioned as $\begin{bmatrix}A&B\\\ B^{T}&C\end{bmatrix}$, where $A$ and $C$ are square. This matrix is positive definite if and only if $C$ and $\bigtriangleup=A-BC^{-1}B^{T}$ are positive definite._ ### -B Proof Lemma 3 First note that $x^{2}(k)=x(k)-x^{1}(k)$. Thus, $\displaystyle x^{2}(k+1)=$ $\displaystyle Ax(k)+Bu(k)-\bigl{(}A+BF(k)\bigr{)}w(k-1)$ $\displaystyle=$ $\displaystyle Ax^{1}(k)+Ax^{2}(k)+Bu^{1}(k)+Bu^{2}(k)$ $\displaystyle-\bigl{(}A+BF(k)\bigr{)}w(k-1)$ The proof is completed by substituting $x^{1}(k)=w(k-1)+\bigl{(}A+BF(k-1)\bigr{)}w(k-2)$ and $u^{1}(k)=F(k)w(k-1)+M(k)w(k-2)$ into the second line. ### -C Proof Lemma 4 The equivalence of optimization problems (15) and (4) follows simply by vectorization of matrices. First note that $\textrm{vec}\bigl{(}F(k-1)\bigr{)}=\begin{bmatrix}I&0\end{bmatrix}\textrm{vec}\bigl{(}D(k)\bigr{)}$. Thus $\displaystyle\textrm{vec}(F^{-}-L^{-})$ $\displaystyle=[I\;0]\textrm{vec}\bigl{(}D(k)\bigr{)}-\textrm{vec}(L^{-})$ From Propositions $3$b and $3$d, we have $\displaystyle\textbf{Tr}\left\\{H^{-}(F^{-}-L^{-})W(F^{-}-L^{-})^{T}\right\\}$ $\displaystyle=$ $\displaystyle\textrm{vec}^{T}\bigl{(}D(k)\bigr{)}\begin{bmatrix}W\otimes H^{-}&0\\\ 0&0\end{bmatrix}\textrm{vec}\bigl{(}D(k)\bigr{)}$ $\displaystyle-2\textrm{vec}^{T}(L^{-})\begin{bmatrix}W\otimes H^{-}&0\end{bmatrix}\textrm{vec}\bigl{(}D(k)\bigr{)}$ $\displaystyle+\textrm{vec}^{T}(L^{-})(W\otimes H^{-})\textrm{vec}(L^{-})$ Likewise, $\textrm{vec}\bigl{(}M(k)\bigr{)}=\begin{bmatrix}0&I\end{bmatrix}\textrm{vec}\bigl{(}D(k)\bigr{)}$. Then $\displaystyle\textrm{vec}\bigl{(}M(k)-L(A+BF^{-})\bigr{)}=$ $\displaystyle\hskip 56.9055pt\bigl{(}[0\;\;I]-[LB\;\;0]\bigr{)}\textrm{vec}\bigl{(}D(k))\bigr{)}-\textrm{vec}(LA)$ Therefore, $\displaystyle\textbf{Tr}\bigl{\\{}H\bigl{(}M-L(A+BF^{-})\bigr{)}W\bigl{(}M-L(A+BF^{-})\bigr{)}^{T}\bigr{\\}}$ $\displaystyle=$ $\displaystyle\textrm{vec}^{T}(D)\begin{bmatrix}W\otimes B^{T}L^{T}HLB&-W\otimes B^{T}L^{T}H\\\ -W\otimes HLB&W\otimes H\end{bmatrix}\textrm{vec}(D)$ $\displaystyle-2\textrm{vec}^{T}(LA)\begin{bmatrix}-W\otimes HLB&W\otimes H\end{bmatrix}\textrm{vec}\bigl{(}D(k)\bigr{)}$ $\displaystyle+\textrm{vec}^{T}(LA)(W\otimes H)\textrm{vec}(LA)$ After Substituting these values back into $J_{u}^{1}$, using $\textrm{vec}(D)=E\textrm{vec}^{\star}(D)$, and eliminating constant terms, we arrive at (4). The only part that remains to be proved is that $Y_{k}$ is positive definite. Since $W$ and $H(k)$ are positive definite, $Y_{22}(k)$ is positive definite according to Proposition $3$c. Proposition $3$e then implies that $Y_{22}^{-1}(k)=W^{-1}\otimes H^{-1}(k)$. From Proposition $3$b, we have $\displaystyle Y_{12}(k)Y_{22}^{-1}(k)Y^{T}_{12}(k)=W\otimes B^{T}L^{T}(k)H(k)L(k)B$ Consequently, $\displaystyle\bigtriangleup(k)$ $\displaystyle=Y_{11}(k)-Y_{12}(k)Y_{22}^{-1}(k)Y^{T}_{12}(k)$ $\displaystyle=W\otimes H(k-1)$ Since $\bigtriangleup(k)$ and $Y_{22}(k)$ are positive definite, from Proposition $5$, $Y_{k}$ is positive definite. Finally note that $E^{T}$ has full row rank, so $[Y]_{SS}=E^{T}Y_{k}E$ is positive definite.	arxiv-papers	2013-09-17T10:36:45	2024-09-04T02:49:51.001673	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hamid Reza Feyzmahdavian, Assad Alam, Ather Gattami", "submitter": "Hamid Reza Feyzmahdavian", "url": "https://arxiv.org/abs/1309.4251" }
1309.4290	Simple free-surface detection in two and three-dimensional SPH solver Agra Barecasco1,2, Hanifa Terissa 1,2, Christian Fredy Naa2 1Graduate School of Natural Science and Technology, Kanazawa University Kakuma, Kanazawa 920-1192 Japan 2Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung Jl. Ganesha 10, Bandung 40132 Indonesia E-mail: [email protected], [email protected], [email protected] Abstract. A simple free-surface particle detection method for two and three- dimensional SPH simulation has been implemented. The method uses sphere representation for the SPH particle. The fluid domain is covered by overlapping spheres. A sphere whose surface is not fully covered considered as boundary. To test particle’s boundary status, we used a sum of normalized relative position vectors from neighbouring particles to the test particle. By checking the existence of uncovered sphere surface by this vector sum, boundary status of the test particle can be determined. This boundary detection method can be easily embedded in the SPH solver algorithm. Keywords: Smoothed-particle Hydrodynamics, boundary particle detection ## 1 Introduction In recent years there has been much development of smoothed-particle hydrodynamics (SPH)[1]. Implementation of boundary conditions is not so clear with SPH method compared with mesh based methods. In SPH, the problem that we must first locate the points that are regarded as boundary. For boundary particle detection, Randles and Libersky[1] previously have suggested using the sums of the gradients of the SPH kernels. Ideally, these kernel gradients sum to zero for interior particles. Any particle for which the sum of the gradient kernels is not near zero is presumably an exterior particle. This method gives correct results when the particles are uniformly spaced. Aamer and Dilts[2] developed an algorithm using overlapping spheres as fluid particles representation. The method can geometrically detect free- surface particles in a robust way. However, its extension to three dimensional simulations requires a large amount of calculation. Marrone[3] gave the idea of using a scan cone around the expected normal vector of the fluid surface to make a further check if there is any particle covering the test particle. In this paper, we propose a simple boundary detection method using both the idea of Dilts[2] and Marrone[3]. The aim is to detect uncovered sphere segment on the boundary particle sphere. In our method, we develop a very simple algorithm to estimate the existence of this uncovered sphere segment. ## 2 Numerical Schemes: Smoothed Particle Hydrodynamics Smoothed-particle hydrodynamics is a method to simulate fluids[1]. It is a lagrangian meshless method based on convolution of smoothing function (kernel) $W$ over fluid’s field functions to approximate their values at a point. This field function is discretized by a set of point particles. The kernel approximation is given by the form $f(\mathbf{x})=\int\limits_{\Omega}f(\mathbf{x}^{\prime})W(\mathbf{x}-\mathbf{x}^{\prime})d\mathbf{x}^{\prime},$ where $\Omega$ represents the convolution domain. In our simulation we use the cubic spline kernel: $W(\mathbf{x},h)=\beta\begin{cases}\frac{2}{3}-q^{2}+\frac{1}{2}q^{3},&0\leq q<1\\\ \frac{1}{6}\left(2-q^{3}\right),&1\leq q<2\\\ 0,&q\geq 2,\end{cases}$ where $2h$ is the radius of the kernel function’s support and $q=\frac{\|\mathbf{x}\|}{h}$. To simulate an inviscid fluid, the convolution of this kernel is applied to the following governing Euler equations for inviscid fluid motion: $\displaystyle\frac{\mbox{D}\rho}{\mbox{D}t}$ $\displaystyle=-\rho\nabla\cdot\mathbf{v}$ (1) $\displaystyle\frac{\mbox{D}\mathbf{v}}{\mbox{D}t}$ $\displaystyle=-\frac{1}{\rho}\nabla p+\mathbf{g}$ (2) where $\mbox{D}/\mbox{D}t$ represent material derivative following an infinitesemal fluid element. $\rho,\mathbf{v},p$ represent density, velocity and pressure, respectively. The external acceleration $\mathbf{g}$ is given to simulate gravity. For SPH simulation, the fluid domain is discretized by fluid particles. The discretized SPH approximation form of Euler equations above are given by $\displaystyle\frac{\mbox{D}\rho_{i}}{\mbox{D}t}$ $\displaystyle=$ $\displaystyle\sum\limits_{j}m_{j}\left(\mathbf{v}_{i}-\mathbf{v}_{j}\right)\cdot\nabla W(\mathbf{x}_{i}-\mathbf{x}_{j},h)$ (3) $\displaystyle\frac{\mbox{D}\mathbf{v}_{i}}{\mbox{D}t}$ $\displaystyle=$ $\displaystyle\sum\limits_{j}m_{j}\left(\frac{p_{i}+p_{j}}{\rho_{i}\rho_{j}}\right)\nabla W(\mathbf{x}_{i}-\mathbf{x}_{j},h)+\mathbf{g},$ (4) where $i$ and $j$ denote particle indices and $m_{j}$ represent $j$-th particle mass. To determine the pressure, we used the state equation $p_{i}=c^{2}(\rho_{i}-\rho_{0}),$ (5) where $c$ is the speed of sound constant and $\rho_{0}$ is the reference density. Here $\mathbf{x}$ represents particle position and $j$ sums to all particles within the kernel radius. In the implementation, the sums must check for all particles in the fluid domain if they are within the kernel. To make a faster calculation, we used a linked-list grid method[5] to reduce the calculation amount per particle so $j$ only runs through the nearest neighbouring particles. ### SPH solver algorithm Implementation of eq. (3) and eq. (4) involves many steps. Here we give the general steps of SPH solver that are related to our boundary detection method. For our cases, we used Leap-Frog integrator to advance in time. SPH simulation can be run as follows: 1. 1. Initialization. Set the initial condition of attributes of all SPH particles 2. 2. Do the followings for all fluid particles until the run-time limit has been reached: 1. (a) Calculate the current pressure of all particles by using state equation eq. (5) 2. (b) For each particle, insert it into the linked-list grid[5] and register its nearest neighbour particles 3. (c) For each particle, calculate its momentum change rate (eq. (4)) by using the new calculated pressure, then calculate its new velocity and position 4. (d) For each particle, calculate its density change rate (eq. (3)) by using new calculated velocity, then calculate its new density We see that SPH simulation involves the calculation of summation from all neighbouring particles to determine each particle’s density, pressure and momentum. This is an advantage for our detection method which is explained in the next section. ## 3 Boundary Particle Detection Method ### 3.1 Boundary particle definition We are given a set of spheres in $\mathbb{R}^{3}$, $S=\\{s_{1},s_{2},\dots,s_{n}\\}$. Each sphere $s_{i}$ represents a SPH particle centered at the particle’s position $\mathbf{x}_{i}$. Let $r_{i}$ be the sphere’s radius. A point $\mathbf{x}\in\mathbb{R}^{3}$ is said to be _covered_ by $s_{i}$ if $\|\mathbf{x}-\mathbf{x}_{i}\|<r_{i}$. SPH _fluid domain_ $F_{S}\in\mathbb{R}^{3}$ is a domain where every $\mathbf{x}\in F_{S}$ is covered by one or more SPH spheres of $S$. $F_{S}$ is divided into a number of subspaces by the sphere set $S$. The spheres in $F_{S}$ overlap each other representing the compact fluid domain. Consider two overlapping spheres $s_{i}$ and $s_{j}$. A _spherical cap_ $Cap(i,j)$ of $s_{j}$ on $s_{i}$ is a spherical segment of $s_{i}$’s surface where every point on it is covered by $s_{j}$. Every $Cap(i,j)$ has to be bounded by an intersection circle $Circ(i,j)$ on the surface of $s_{i}$ (see figure 1). The $Circ(i,j)$ is said to be _covered_ if all points in it are covered. Figure 1: Illustration of terminologies Now we consider some special cases. The radius of all spheres in $S$ is the same, so the situation that a sphere is located inside a sphere is impossible. To avoid two spheres or more to coincide, we improved the SPH algorithm with artificial viscousity[1] so that there is no interparticle penetration. Definition 3.1. Let sphere $s_{i}$ be overlapped by $n$ spheres. The sphere $s_{i}$ is said to be _covered_ if every point on sphere $s_{i}$ is covered by at least one of the $n$ intersecting spheres. The particle of the corresponding sphere is then said to be an _interior_ SPH particle. Figure 2: Illustration of a boundary particle Definition 3.2. Let sphere $s_{i}$ be overlapped by $n$ spheres. The corresponding SPH particle is said to be a _boundary_ particle if there exists a point on the corresponding sphere which is not covered by any of the overlapping spheres (see figure 2). The smoothness of the fluid surface depends on the value of $r_{i}$. If we use the spheres of smaller size, the boundary particles have greater resolution, which gives a sharper surface profile (see figure 3). However, a small sized sphere representation bring the risk of geometrical cavity to emerge inside the fluid. This could lead to an error of boundary detection. Figure 3: Drawn in 2D, the two figures show how a smaller representative sphere radius give sharper surface profile. Uncovered sphere segments are shown in red. ### 3.2 Detection method Based on the definitions above, now we explain our method to detect boundary particles. Let $s_{i}$ be a SPH particle represented as a sphere with particle’s position $\mathbf{x}_{i}$ as the center. Within its kernel support, $m$ neighbour particles are present to contribute for $s_{i}$’s SPH attributes. Let these neighbouring particle spheres overlap $s_{i}$. The _cover vector_ $\mathbf{b}_{i}$ is defined as (see figure 4): $\mathbf{b}_{i}=\sum\limits_{j=0}^{n}\frac{\mathbf{x}_{i}-\mathbf{x}_{j}}{\|\mathbf{x}_{i}-\mathbf{x}_{j}\|}$ (6) Let $l_{i}$ be a ray that starts from $\mathbf{x}_{i}$ and parallel to $\mathbf{b}_{i}$. If $l_{i}$ does not intersect any spherical cap, then the corresponding sphere is not covered. Boundary particle detection can be done by checking the existence of $l_{i}-Cap$ intersection. Figure 4: Illustration of cover vector and the scan cone, drawn in 2D. The cone checks if there is any other covering particle within kernel radius. Now we consider the case for particle located at or near the boundary. Because of the non-uniform property of the particles (see figure 5), the direction of $\mathbf{b}_{i}$ only serves as a rough estimation of the normal of the surface. We used a _scan cone_ (see figure 4) which checks if there is any covering particle near $l_{i}$. The cone angle $\theta_{i}$ serves as threshold angle to any particle that lies near $l_{i}$, which detects the existence of a spherical cap intersection with $l_{i}$. Figure 5: The cover vector on each particle differs according to the distribution of neighbouring particles. The value of $\theta_{i}$ depends on the radius of representative sphere used and plays important role in checking the existence of $l_{i}-Cap$ intersection. In our case, we directly chose the value $\theta_{i}=\pi/3$. The scan cone calculation check for particle $i$ is given by $\mbox{IF }\arccos\left(\frac{\mathbf{x}_{j}-\mathbf{x}_{i}}{\|\mathbf{x}_{j}-\mathbf{x}_{i}\|}\cdot\frac{\mathbf{b}_{i}}{\|\mathbf{b}_{i}\|}\right)\leq\frac{\theta_{i}}{2}\mbox{ THEN assign boundary status to the particle $i$}$ (7) In case when the fluid particles join to form a thin jet, a plane or become solitary particle cluster the number of neighbouring particles for each becomes very small. We considered them as boundary particles by giving a threshold number $n_{th}=\begin{cases}4&\mbox{ for 2 dimensional case}\\\ 15&\mbox{ for 3 dimensional case}\end{cases}$ (8) to the number of neighbour particles. A particle having the number of neighbour particles greater than $n_{th}$ must be tested for its boundary status, otherwise it is a boundary particle. ### 3.3 Boundary detection algorithm Our algorithm is embedded within the SPH solver algorithm that has been mentioned in section 2 above. The summation to calculate the cover vector can be done simultaneously with any SPH solver step that involves summation through all neighbouring particles. The simple surface detection algorithm written below uses some steps of SPH solver algorithm simultaneously: 1. 1. At _initialization_ , assign all particles in the fluid domain as boundary particles 2. 2. During the step of registering particles into linked-list grid, for each particle $i$ calculate the cover vector (eq. (6)) by summing $\frac{\mathbf{x}_{i}-\mathbf{x}_{j}}{\|\mathbf{x}_{i}-\mathbf{x}_{j}\|}$ for every detected neighbour particle $j$ 3. 3. During the step of calculating momentum change rate for all particles (eq. (4)), for each particle $i$, if the number of neighbours of $i$ is smaller than or equal to the threshold $n_{th}$ (eq. (8)) then consider the particle $i$ as boundary 4. 4. During the step of calculating density change rate for all particles (eq. (3)), check the existence of $l_{i}-Cap$ intersection by applying the scan cone calculation (eq. (7)). If the intersection exists, assign the current $i$-th particle as interior. ## 4 Test Cases We have applied the detection algorithm to several cases. To visualize particles we used spheres of radius $0.3h_{sm}$, where $h_{sm}$ is the kernel support radius. Red and blue color was used to distinguish between boundary and interior particles. Figure 6: Two dimensional dam break ### 4.1 Two-dimensional dam break We implemented the algorithms to two-dimensional case so the result can be seen clearly. The simulation is a dam break in rectangular tank. The simulation consists of 2997 SPH particles with the initial shape as a rectangular block of fluid at the right side of the tank. Figure 6 shows the result. ### 4.2 Two-dimensional standing wave The simulation is a standing wave in a rectangular tank with periodic boundary condition at the sides. The simulation consists of 3720 SPH particles with the initial condition using a sine function. Figure 7 shows the result. Figure 7: Two dimensional standing wave Figure 8: Three dimensional droplet motion. Figure 9: Three-dimensional standing wave. The upper left figure shows the system before boundary detection. ### 4.3 Three dimensional droplet motion This simulation applied surface tension model used by Zhang[4] to the boundary particles. The droplet consists of 3375 fluid particles. The surface tension depends on the curvature of the surface shape that calculated by applying Moving Least Square fit to the boundary particles as interpolation points. Figure 8 shows how surface tension deforms the initial cube shape into a sphere. ### 4.4 Three-dimensional standing wave This case is a 3-dimensional standing wave with periodic boundary condition at the sides of a cube tank. We used 44191 particles. The initialization of the dome-shaped surface is given by a gaussian function. Figure 9 shows the result. ## 5 Correction Method: Scan Circle The implementation of boundary detection method worked well in determining the deep interior particles, but a drawback occured when determining the boundary particles. Because of the unpredictable behavior of the neighbouring particle distribution, the cover vector of a boundary particle sometimes aims too close to another neighbour particle. The scan cone mistakenly regarded this boundary particle as interior (see figure 10). This impairs the accuracy of this method. Figure 10: The scan cone of boundary particle $i$ regards another boundary particle as a covering particle. Particle $i$ is mistakenly considered as interior. Figure 11: Scan circle To treat this problem, a further research for implementing a _scan circle_ is on going. Consider an intersection given by the scan cone and the sphere of a SPH particle $i$. The intersection is a circle whose center located in ray $l_{i}$ (see figure 11). The existence of uncovered sphere segment can be tested by checking whether this circle is covered or not. ## 6 Summary The simple boundary detection in SPH based on the implementation of cover vector and the scan cone has been successfully implemented to several cases. The algorithm can be embedded in the SPH solver algorithm easily. However, a drawback exists in the boundary detection accuracy because of random behavior of particle distribution. A research on correction method using a scan circle coverage is on going. ## References * [1] P.W. Randles, L.D. Libersky (1996), Smoothed particle hydrodynamics: some recent improvements and applications, Comput. Meth. Appl. Mech. Eng. , 139, 375-408 * [2] H. Aamer, G.A. Dilts (2007), Three-dimensional boundary detection for particle methods, J. Comp. Phys. , 226, 1710-1730 * [3] S, Marrone et.al. (2010), Fast free-surface detection and level-set function definition on SPH solvers, J. Comp. Phys. , 229, 3652-3663 * [4] M. Zhang, (2010), Simulation of surface tension in 2D and 3D with smoothed particle hydrodynamics method, J. Comp. Phys. , 229, 7238-7259 * [5] G.R. Liu, M.B. Liu (1997). Smoothed Particle Hydrodynamics: A Meshfree Particle Method, World Scientific Publishing	arxiv-papers	2013-09-17T12:57:23	2024-09-04T02:49:51.011792	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Agra Barecasco, Hanifa Terissa, Christian Fredy Naa", "submitter": "Christian Naa", "url": "https://arxiv.org/abs/1309.4290" }
1309.4294	# Projectile fragment emission in fragmentation of 56Fe on C, Al, and CH2 targets at 471 A MeV ††thanks: Submitted to Chin. Phys. C Yan-Jing Lia, Dong-Hai Zhangb, Shiwei Yana,c , Li-Chun Wanga, Jin-Xia Chenga, Jun-Sheng Lib, S. Kodairad, and N. Yasudad aCollege of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China bInstitute of Modern Physics, Shanxi Normal University, Linfen 041004, China cBeijing Radiation Center, Beijing 100875, China dFundamental Technology Center, National Institute of Radiological Sciences 4-9-1 Anagawa, Inage-ku, Chiba 263-8555, Japan Corresponding author. Tel: +863572051347; fax: +863572051347. E-mail address:[email protected] Abstract The emission angle and the transverse momentum distributions of projectile fragments produced in fragmentation of 56Fe on CH2, C, and Al targets at 471 A MeV are measured. It is found that for the same target the average value and width of angular distribution decrease with increase of the projectile fragment charge, and for the same projectile fragment the average value of the distribution increases and the width of the distribution decreases with increasing the target charge number. The transverse momentum distribution of projectile fragment can be explained by a single Gaussian distribution and the averaged transverse momentum per nucleon decreases with the increase of the charge of projectile fragment. The cumulated squared transverse momentum distribution of projectile fragment can be well explained by a single Rayleigh distribution. The temperature parameter of emission source of projectile fragment, calculated from the cumulated squared transverse momentum distribution, decreases with the increase of the size of projectile fragment. PACS 25.70.-z, 25.70.Mn, 29.40.Wk ## 1 Introduction The knowledge of heavy ion fragmentation at intermediate and high energy is very important in nuclear physics, astrophysics, and medical physics. Considering the biological effects of space radiation, when astronaut have their mission outside the earth magnetic field, they are suffered from Galactic Cosmic Radiation(GCR) and Solar particle events, e.g., showers of energetic charged particles from the surface of the Sun. These energetic charged particles will be the dominant sources of the radiation dose and affect the health of humans on long-duration spaceflight both inside and outside the station. According to the GCR model developed by Badhwar and O’Neill[1], in unshielded free space in the inner heliosphere, iron ions deliver about $8\%$ of the total dose from the GCR and $27\%$ of the dose equivalent at times near solar maximum, even though they contribute less than $1\%$ of the total GCR flux. Because iron ions are the most densely ionizing particles which are presented in significant numbers in the GCR, there has been considerable interest in understanding their transport through matter and their biological effects. The understanding of the radiobiology of heavy charged particles (HZE) is a subject of great interest due to the complicated dependence of their relative biological effectiveness on the type of ion and its energy, and its interaction with various targets. It has become clear that heavy ions have the largest radiological effects. These effects also appear in regions close to the beam entrance, i.e., in the depth-dose plateau region, where normal tissue is usually situated. In addition, due to the longer ranges of the fragments produced by the fragmentation of the incident ions, the tails of the dose distribution beyond the Bragg peak may be too high for minimizing doses to normal tissue beyond the primary ion range. Finally, recent experimental results[2] have revealed that the fragments are emitted at larger angles than the scattering angles of the beam, which further increases the spread of the beam. Exact information about the fragment emission angular distributions will be especially important in radiotherapy. So far only a few measurements have been performed to analyze fragment emission angles from HZE reactions below 500 A MeV[2-4]. Fragmentation is a term commonly used to specify a nuclear disassembly by force. Hot fragmentation is meant to indicate the most violent of these process, following excitation beyond the limits of nuclear binding, but still ending with bound nuclear fragments of different sizes in the final channels[5]. The formation mechanism of these fragments, whether they the remnants of an incomplete destruction or the products of a condensation out of the disordered matter, has continued to be the topic of very active research. In order to describe the physical process of heavy ion transport, several one- dimensional Monte Carlo codes, such as HZETRN[6], HIBRAC[7], FLUKA[8], NUCFRAG2[9], and three dimensional model[10] are appeared. The improved quantum molecular dynamics model (ImQMD) is a dynamical model which is developed to follow the reaction process on a microscopic level[11, 12]. The properties of 56Fe on various targets at various energies have been studied by many groups[13-25], most of the studies are devoted to the total charge-changing cross sections and the partial cross sections of fragment productions, a little attention is paid to the fragment emission angular distribution study. In this paper, we present the results of the emission angular distribution, transverse momentum distribution and the temperature of emission source of fragment produced in fragmentation of 471 A MeV 56Fe on C, Al, and CH2 targets. CH2 target is used to obtain the cross section on a hydrogen target. The fragmentation cross sections is published in our previous paper[26]. The arrangements of this paper are as follows: we introduce our experimental detail in sec. II. In sec. III, experiment result and discussion are given. At last, we give the conclusion in sec. IV. ## 2 Experimental details ### 2.1 Experimental setup Stacks of C, Al and CH2 targets sandwiched with CR-39 detectors were exposed normally to 471 A MeV 56Fe beams at the Heavy Ion Medical Accelerator in Chiba (HIMAC) at the Japanese National Institute of Radiological Sciences (NIRS). Figure 1 shows the configuration of sandwiched target. A CR-39 sheet, about 0.77 mm in thickness, is placed before and after the targets. The thickness of carbon, aluminum and polyethylene targets is 5, 3, and 10 mm, respectively. Figure 1: Sketch of the target-detector configuration. After exposure, the CR-39 detectors are etched in 7N NaOH aqueous solution at temperature of 70∘C for 15 hours. Then, the beam ions and their fragments manifest in the CR-39 as etch-pit cones on both sides of CR-39 sheets. The images of ion tracks are scanned and analyzed automatically by HSP-1000 microscope system and the PitFit track measurement software, then checked manually. The PitFit software allows us to extract some geometric information, such as the position coordinates, major and minor axes and area of etched track spot on CR-39 surfaces. Image data ($45\times 45mm^{2}$) are acquired for both front and back surfaces of each CR-39 detector. About 2$\times$104 Fe ions are traced from the first CR-39 detector surface in the stack. 56Fe trajectories and the ones of secondary fragments are reconstructed in the whole stack. ### 2.2 Experimental method The spots on the front surface (with respect to the beam direction) are directly scanned firstly, then the CR-39 sheet is turned under the middle line of the sheet and the spots on the back surface are scanned. The trajectories of ion tracks through CR-39 sheets are reconstructed in two steps using the track tracing method[27]: (1) the track position in CR-39 surfaces is corrected by parallel and rotational coordinate translation (except for the track position on upper surface of the first CR-39 sheet), and (2) the difference between the track position of corresponding tracks on both side of the CR-39 sheets and on the surfaces neighboring targets is minimized by a track matching routine. The coordinate of track before the target (or front surface of CR-39 sheet) is $(x,y)$ and of matching track after the target (or back surface of CR-39 sheet) is $(x^{\prime},y^{\prime})$. Following the translation relation, the coordinate of matching track can be calculated as: $\displaystyle x^{\prime}_{th}=$ $\displaystyle ax+by+c,$ (1) $\displaystyle y^{\prime}_{th}=$ $\displaystyle a^{\prime}x+b^{\prime}y+c^{\prime},$ (2) parameters $a,b,c,a^{\prime},b^{\prime}$, and $c^{\prime}$ are determined using the least square method. Then, the coordinate $x^{\prime}_{th}$, $y^{\prime}_{th}$ of matching track is calculated. However, because of the Coulomb scatter etc. contributions, $x^{\prime}_{th}$, $y^{\prime}_{th}$ are certainly different from $x^{\prime},y^{\prime}$, the difference $dx=x^{\prime}_{th}-x^{\prime}$, $dy=y^{\prime}_{th}-y^{\prime}$ is calculated which can help us to determine the matching track. Figure 2: The distribution of dx and dy. (a) and (b) the difference between the front and back surface on a CR-39 sheet, (c) and (d) the difference before and after carbon target. Figure 3: The area distribution of etch-pit spots of all of 56Fe ions and ones of secondary fragments on CR-39 surface after CH2 target. Figure 2 (a) and (b) show the difference $dx$ and $dy$ in the front surface and back surface on a CR-39 sheet, (c) and (d) show the difference $dx$ and $dy$ before and after the target. If the difference are calculated for all combinations of positions for extracted tracks, only the matching combination ought to make a peak which appears in the figures, and the difference $dx$ and $dy$ of other combinations should be randomly distributed. The deviations $\sigma(x^{\prime})$ and $\sigma(y^{\prime})$ give the position accuracies of tracks which are estimated to be $2-4\mu$m for the case of (a) and (b), and $8-20\mu$m for case of (c) and (d). The matching iron ion track is searched within $4\times\sigma(x^{\prime})$ and $4\times\sigma(y^{\prime})$ region of $x^{\prime}_{th}$ and $y^{\prime}_{th}$. The matching projectile fragments are searched within the limited fragmentation angle, which is about $10^{\circ}$ in present experiment. The number of projectile fragments leaving the target is determined from the distribution of the etched area. Figure 3 shows the track base area distribution of 56Fe ions and their fragments in CR-39 sheet. Peaks for 56Fe and its fragments with charge down to $Z=6$ appear clearly. Because of the limitation of CR-39 detector, the tracks of fragments with charge $1\leq{Z}\leq 5$ are not fully etched as a measurable spots. The emission angle $\theta$ of each fragment is calculated by taking readings of the coordinates of the beam track and the fragment track. ## 3 Results and discussion Emission angular distribution and transverse momentum distribution of projectile fragments provide information on the nuclear structure and the mechanism of nuclear interaction through which fragments are produced. These distributions are also very important in designing experiments with radioactive nuclear beams. ### 3.1 Angular distribution Emission angle of each fragment and scattering angle of iron ion is calculated from the coordinates of track positions on the surface of CR-39 sheet after the target. The angular uncertainty is determined using the quadruplet fitting method[4] $\displaystyle\sigma(\theta)=\frac{\sqrt{\sigma^{2}_{z}\sin^{2}\theta+2\sigma^{2}_{p}\cos^{2}\theta}}{2h},$ (3) where $\sigma_{p}$ is the positional uncertainty in $x-y$ plane of the stack coordinate system which is about $3\mu$m for C-target, $\sigma_{z}$ is the positional uncertainty in the z-axis which comes from stack composition and detector thickness measurement and is estimated at $\approx 8\mu$m, $\theta$ represents the emission angle of the fitted line. With a detector thickness of $h\approx 780\mu$m we thus obtain angular uncertainty $\sigma(\theta)\approx 0.16^{\circ}$ for value of $\theta$ up to $8^{\circ}$. Figure 4 shows the angular distribution of primary iron ions and their fragments for different targets. The emission angle of primary iron ion mainly comes from the Coulomb scattering, most of which is less than $0.6^{\circ}$. Most of projectile fragments have the emission angle less than $1.5^{\circ}$, but some of fragments have emission angle up to $8^{\circ}$. The position of the maximum of the fragment angular distribution increases slightly with increase of the target mass, which can be explained that with the increase of target mass the interaction between projectile and target is increased and influence of fragment suffered from target is increased. Figure 4: The angular distribution of primary iron ions and their fragments for different targets, for comparison the counts of fragments is enlarged two times. Figure 5: The emission angle distributions of fragments with charge Z=6, 18, and 24, respectively. Figure 6: The emission angle distributions of fragments with charge Z=24 and 25, respectively. Fig. 5 shows the emission angle distributions of projectile fragments with $Z=6,18$, and 24 produced from the fragmentation of 56Fe on C and CH2 targets, and Fig. 6 shows the emission angle distributions of projectile fragments with $Z=24$ and 25 produced from the fragmentation of 56Fe on C, Al and CH2 targets respectively. For fragments from the fragmentation of 56Fe on C and Al targets, each angular distribution is fitted by a single Gaussian distribution, and for fragments from the fragmentation of 56Fe on CH2 target, each angular distribution is fitted by two Gaussian distribution, the fitting parameters including $\chi^{2}$/DOF are presented in Table 1, where DOF means the degree of freedom of simulation. From Figs. 5 and 6 and results in Table 1 it shows that for the same target the average value and the width of the distribution decrease with increasing the charge number of projectile fragment, and for the same projectile fragment the average value of the distribution increases and the width of the distribution decreases with increasing the target charge number. Table 1. Values of fitting parameters of angular distribution using Gaussian distribution. Z Target percentage(1) mean value(1) error(1) percentage(2) mean value(2) error(2) $\chi^{2}$/DOF 25 CH2 $16.38\pm 5.52$ $0.25\pm 0.02$ $0.13\pm 0.03$ $7.33\pm 2.92$ $0.49\pm 0.17$ $0.28\pm 0.08$ 1.12 C $20.89\pm 1.62$ $0.23\pm 0.02$ $0.21\pm 0.02$ 1.81 Al $33.28\pm 13.27$ $0.29\pm 0.02$ $0.10\pm 0.04$ 1.52 24 CH2 $17.73\pm 1.50$ $0.39\pm 0.02$ $0.21\pm 0.02$ $1.44\pm 0.54$ $1.13\pm 0.11$ $0.18\pm 0.16$ 1.15 C $14.66\pm 1.84$ $0.36\pm 0.03$ $0.26\pm 0.03$ 2.15 Al $21.68\pm 5.69$ $0.36\pm 0.05$ $0.22\pm 0.05$ 0.31 18 CH2 $22.90\pm 3.61$ $0.53\pm 0.06$ $0.25\pm 0.04$ $10.93\pm 4.26$ $1.16\pm 0.08$ $0.18\pm 0.08$ 2.17 C $16.00\pm 3.58$ $0.62\pm 0.12$ $0.54\pm 0.14$ 0.40 6 CH2 $13.71\pm 0.95$ $1.86\pm 0.34$ $1.73\pm 0.31$ $0.30\pm 1.06$ $2.97\pm 1.07$ $0.04\pm 1.44$ 0.12 C $10.74\pm 3.66$ $2.04\pm 0.34$ $1.78\pm 0.81$ 1.01 ### 3.2 Transverse momentum distribution The transverse momentum per nucleon ($p_{t}$) of a projectile fragment was calculated on the basis of its emission angle $\theta$, $p_{t}=p\sin\theta,$ (4) where p is the momentum per nucleon of beam which can be calculated from beam energy per nucleon (E), $p=(E^{2}+2m_{0}E)^{1/2}$. $m_{0}$ is the nucleon rest mass and $\theta$ the emission angle of the projectile fragment with respect to the beam direction. Figure 7 shows the transverse momentum distribution of projectile fragment with charge $Z=6,18$, and 24 produced from the fragmentation of 56Fe on C and CH2 targets. The distributions can be well fitted by a single Gaussian distribution, which is the same as the distributions of fragments produced in reactions of light projectiles[28, 29] and heavy projectiles[30, 31, 32, 33, 34] at relativistic energies. These Gaussians are in good agreement with predictions of the statistical model of Goldhaber[35]. This model assumes that the Fermi momenta of the nucleons in a fragment are statistically distributed as those in the original projectile nucleus. The averaged transverse momentum per nucleon for fragment with charge $Z=6,18$, and 24 are $35.89\pm 35.25$, $11.34\pm 8.74$, and $6.91\pm 4.74$ A MeV/c, respectively, for C target, and $34.38\pm 30.79$, $11.98\pm 7.73$, and $7.08\pm 4.25$ A MeV/c, respectively, for CH2 target. For the same target the averaged transverse momentum per nucleon, and the width of the distribution increase with the decrease of the charge of fragment. This tendency is also observed in Ref.[34]. Projectile fragments come from the directly produced fragments (primary fragments) and the sequential decay fragments from excited primary fragment. However, since the primary fragments are excited, they are deexcited by light particle evaporation. This secondary decay decreases the observed masses and increase the observed momentum widths of the primary fragments[36]. The contribution from sequential decay of primary fragments to the heavy projectile fragments are less than that to the light projectile fragments, the widths of the transverse momentum distributions of light projectile fragments are greater than that of the heavy projectile fragments. So the transverse momentum distribution width deceases with the increase of the charge of the projectile fragment, which is shown in Figure 7. Figure 8 shows relation of the averaged transverse momentum per nucleon and the charge of projectile fragment for interactions of 56Fe and C, Al, and CH2 targets. The averaged transverse momentum per nucleon decreases with the increase of fragment size, no obviously target size dependence is observed in present investigation. Figure 9 shows the transverse momentum distribution of all fragments for interactions of 56Fe and C, Al, and CH2 targets, no obviously target size dependence is also observed. The heavy fragment comes from peripheral collisions, that is collisions with larger impact parameter. The light fragment comes from central and semi-central collisions, that is collisions with smaller impact parameter. According to the participant- spectator model[37], with the increase of impact parameter the overlapped region decreases, the communication between participant and spectator decrease. This results in the decrease of the excitation energy of projectile fragments, so the average of transverse momentum of fragment is decreased. Figure 7: The transverse momentum distributions of fragments with charge Z=6, 18, and 24, respectively. Figure 8: Relation of the averaged transverse momentum and the charge of projectile fragment for interactions of 56Fe and C, Al, and CH2 targets. Figure 9: Transverse momentum distribution of all fragments for interactions of 56Fe and C, Al, and CH2 targets. Figure 10: The cumulative $p_{t}^{2}$ distribution of projectile fragments with charge $Z=6$ (a), $Z=18$ (b), $Z=24$ (c), and $Z=25$ (d). Based on the participant-spectator concept and the fireball model[38], the large number of swept out nucleons combined with an anticipated fairly large number of interactions per particle is presumably responsible for the quasi- equilibrated system, i.e. the fireball which can then be described in term of mean value and statistical (Maxwell-Boltzmann) distribution. If we assume that the emission of projectile fragments is Maxwell-Boltmann distribution in the projectile rest frame with a certain temperature T, then the integral frequency distribution of the square of the transverse momentum per nucleon is $lnF(>p_{t}^{2})=-\frac{A}{2M_{p}T}p_{t}^{2}$ (5) where A is the mass number of fragment, $M_{p}$ is the mass of proton. The linearity of such a plot would be strong evidence for a single temperature of emission source. Figure 10 shows the cumulative plots of $F(>p^{2}_{t})$ as a function of $p^{2}_{t}$ for projectile fragments from the fragmentation of 56Fe on C, Al and CH2 targets. All of the plots can be well fitted by a single Rayleigh distribution of the form $F(p_{t}^{2})=Cexp(\frac{-p_{t}^{2}}{2\sigma^{2}}),$ (6) where $\sigma=\sqrt{2/\pi}<p_{t}>$, which is related to the temperature of fragment emission source, $T=\sigma^{2}A/M_{p}$. The fitting parameters including $\chi^{2}$/DOF and the temperature of the emission source are presented in Table 2. Because CR-39 detector can not identify the mass numbers of projectile fragments, we use the mass number of stable nucleus to calculate the temperature of projectile fragment emission source. The influence from isotope is less than $1\%$ when the abundance of isotope is considered. The dependence of the temperature of emission source on the size of fragment for different targets is shown in figure 11. From the results of Table 2 and figure 11 we can conclude that the temperature of projectile fragment emission source decreases with the increase of the charge of fragment for the same targets. The temperature increases with the increase of target size for emission of fragment with change $Z=6$ and 18, but for emission of fragment with change $Z=24$ and 25 this dependence is not obvious. Table 2. Values of fitting parameters of $p_{t}^{2}$ distribution using Rayleigh distribution and the temperature of the emission source. Z Target C $\sigma^{2}$ ((MeV/c)2) T (MeV) $\chi^{2}$/DOF 25 CH2 $96.52\pm 5.35$ $31.93\pm 1.69$ $1.87\pm 0.10$ 0.831 C $100.16\pm 5.71$ $19.47\pm 1.11$ $1.14\pm 0.07$ 0.235 Al $100.56\pm 16.91$ $14.50\pm 2.78$ $0.85\pm 0.16$ 0.033 24 CH2 $96.77\pm 5.95$ $46.51\pm 3.01$ $2.58\pm 0.17$ 1.398 C $101.72\pm 9.24$ $32.52\pm 2.89$ $1.80\pm 0.16$ 0.575 Al $99.56\pm 17.90$ $27.25\pm 5.00$ $1.51\pm 0.28$ 0.159 18 CH2 $99.45\pm 9.32$ $108.00\pm 9.34$ $4.61\pm 0.40$ 0.142 C $95.86\pm 14.60$ $133.30\pm 22.31$ $5.68\pm 0.95$ 0.310 Al $94.08\pm 33.96$ $276.24\pm 127.42$ $11.78\pm 5.43$ 0.108 6 CH2 $100.76\pm 11.64$ $1168.60\pm 156.12$ $14.95\pm 2.00$ 0.119 C $94.57\pm 12.88$ $1541.40\pm 221.88$ $19.72\pm 2.84$ 0.400 Al $102.55\pm 27.41$ $1783.50\pm 523.46$ $22.82\pm 6.70$ 0.065 Figure 11: The temperature derived from the distribution of $p_{T}^{2}$. The temperature of projectile fragment emission source have been investigated by different collaborations[30,31,39-44]. ALADIN Collaboration studied the slope temperature ($T_{slope}$) for spectator decay as a function of the fragment mass ($A_{frag}$) for spectator decays following 197Au on 197Au collisions, they found that there is a rapid increase of $T_{slope}$ with fragment mass which saturates for $A_{frag}\geq 3$ around $T_{slope}\sim 17$ MeV[39, 45]. EOS Collaboration also studied the variation of remnant temperature with the charged particle multiplicity, they found that the remnant temperature increases with increase of the charged particle multiplicity and the maximum is about $15.6\pm 0.47$ MeV[46]. These maximum temperature is the same as our results within experimental errors for emission of fragment with charge $Z=6$. According to the participant-spectator concept, it is assumed that when the interaction of projectile and target nuclei takes place, the projectile and target sweep out cylindrical cuts through each other. During the separation of the spectators from the participants, there is some intercommunication, which results in the excitation of the spectators. This excitation strongly depends on the contacted area of the colliding system. The heavier fragment is corresponding to the large impact parameter and small contacted areas, the lighter fragment is corresponding to the smaller impact parameter and large contacted areas. So the excitation energy of the heavier fragment is less than that of the lighter fragment, which results in the temperature of emission source of heavier fragment is less than that of the lighter fragment. ## 4 Summary The emission angular distribution and the transverse momentum distribution of projectile fragments produced in fragmentation of 56Fe on C, Al, and CH2 targets are studied in present investigation. It is found that for the same target the average value and width of angular distribution decrease with increase of the projectile fragment charge, and for the same projectile fragment the average value of the distribution increases and the width of the distribution decreases with increasing the target charge number. The transverse momentum distribution of projectile fragment can be explained by a single Gaussian distribution and the averaged transverse momentum per nucleon decreases with the increase of the charge of fragment, and no obvious dependence of transverse momentum on target size is observed. The cumulated squared transverse momentum distribution of fragment can be well explained by a single Rayleigh distribution. The temperature parameter of emission source of projectile fragments decreases with the increase of the size of projectile fragments. ## 5 Acknowledgements This work has been supported by the Chinese National Science Foundation under Grant Nos: 11075100 and 10975019 and the Natural Foundation of Shanxi Province under Grant 2011011001-2, the Shanxi Provincial Foundation for Returned Overseas Chinese Scholars, China (Grant No. 2011-058), the Foundation of Ministry of Personnel of China for Returned Scholars Grant MOP2006138, and the Fundamental Research Funds for the Central Universities. We gratefully acknowledge the staffs of the HIMAC for providing the beam to expose the stacks. ## References * [1] G.D. Badhwar, P.M. O’Neill, Nucl. Tracks Radiat. Meas. 20, 403 (1992) * [2] M. Giacomelli, J. Skvarc, R. Ilic, N. Yasuda, L. Sihver, Radiat. Meas. 36, 329 (2003) * [3] S. Momota, M. Notani, S. Ito, et al., Nucl. Phys. A 701, 150c (2002) * [4] M. Giacomelli, L. Sihver, J. Skvarc, N. 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Yan, L.C. Wang, J.X. Cheng, J.S. Li, S.\n Kodaira, N. Yasuda", "submitter": "Dong-Hai Zhang", "url": "https://arxiv.org/abs/1309.4294" }
1309.4307	# Two rings but no fellowship: LoTr 1 and its relation to planetary nebulae possessing barium central stars. A.A. Tyndall1,2, D. Jones2, H.M.J. Boffin2, B. Miszalski3,4, F. Faedi5, M. Lloyd1, J.A. López6, S. Martell7, D. Pollacco5, and M. Santander-García8 1Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, M13 9PL, UK 2European Southern Observatory, Alonso de Córdova 3107, Casilla 19001, Santiago, Chile 3South African Astronomical Observatory, PO Box 9, Observatory 7935, South Africa 4Southern African Large Telescope. PO Box 9, Observatory 7935, South Africa 5Department of Physics, University of Warwick, CV4 7AL, UK 6Instituto de Astronomía, Universidad Nacional Autónoma de México, Ensenada, Baja California, C.P. 22800, Mexico 7Australian Astronomical Observatory, North Ryde, 2109 NSW, Australia 8Observatorio Astronómico National, Madrid, and Centro de Astrobiología, CSIC- INTA, Spain E-mail: [email protected] (Accepted xxxx xxxxxxxx xx. Received xxxx xxxxxxxx xx; in original form xxxx xxxxxxxx xx) ###### Abstract LoTr 1 is a planetary nebula thought to contain an intermediate-period binary central star system ( that is, a system with an orbital period, P, between 100 and, say, 1500 days). The system shows the signature of a K-type, rapidly rotating giant, and most likely constitutes an accretion-induced post-mass transfer system similar to other PNe such as LoTr 5, WeBo 1 and A70. Such systems represent rare opportunities to further the investigation into the formation of barium stars and intermediate period post-AGB systems – a formation process still far from being understood. Here, we present the first detailed analyses of both the central star system and the surrounding nebula of LoTr 1 using a combination of spectra obtained with VLT-FORS2, AAT-UCLES and NTT-EMMI, as well as SuperWASP photometry.We confirm the binary nature of the central star of LoTr 1 that consists of a K1 III giant and a hot white dwarf. The cool giant does not present any sign of s-process enhancement but is shown to have a rotation period of 6.4 days, which is a possible sign of mass accretion. LoTr 1 also presents broad double-peaked H$\alpha$ emission lines, whose origin is still unclear. The nebula of LoTr 1 consists in two slightly elongated shells, with ages of 17 000 and 35 000 years, respectively, and with different orientations. As such, LoTr 1 present a very different nebular morphology than A70 and WeBo 1, which may be an indication of difference in the mass transfer episodes. ###### keywords: planetary nebulae: individual: LoTr 1, WeBo 1, A66 70 – stars: AGB and post- AGB – stars: binaries: general – stars: chemically peculiar ††pagerange: Two rings but no fellowship: LoTr 1 and its relation to planetary nebulae possessing barium central stars.–References††pubyear: 2013 ## 1 Introduction Figure 1: Deep, narrowband images of PN LoTr 1, in (a) [O iii]$\lambda$5007 Å, and (b) H$\alpha$+[N ii]$\lambda$6584 Å. North is to the top of the image, East is left. The central star is visible at $\alpha$ = 05:55:06.6, $\delta$ = $-$22:54:02.4. Overlaid on image (b) are the slit positions of the spatio- kinematic data presented in section 2.1.2. Slit 1 (E-W) was acquired using NTT-EMMI and is centered on the central star. Slits 2–4 (N-S) and 5–8 (E-W, width of slits 6, 7 and 8 exaggerated to clearly show the positioning due to overlapping with slit 1) were acquired using AAT-UCLES. All slits were taken in [O iii]. The interaction between the progenitor star and a binary or planetary companion is believed to shape the resulting planetary nebula (PN), and in some cases even thought to be almost essential for a PN to form (Moe & De Marco, 2006). The shaping influence of a common-envelope evolution has been studied extensively (see e.g. Tyndall et al., 2012, Jones et al., 2010), and understood in terms of either a collimated fast wind (CFW) carving out an axisymmetric nebula (Soker & Rappaport, 2000) or the ejected common-envelope (CE) forming an equatorial density enhancement (Nordhaus & Blackman, 2006) as required by the ‘Generalised Interacting Stellar Winds’ model (Kahn & West, 1985). However, very little is known about intermediate-period (P=100-1500 days, van Winckel et al., 2009) post-Asymptotic Giant Branch (AGB) binaries, including their effect on PNe formation and morphology, due to lack of observations. The intermediate period binaries fall between post-CE systems (Miszalski et al., 2009) and visually resolved systems (e.g. Ciardullo et al., 1999). Soker (1997) claimed that these interacting systems are the most likely CSPNe (central stars of planetary nebulae) to form the classical ‘butterfly’, or bipolar, morphologies, but with few systems known and limited investigations, this has yet to be confirmed. Only by finding and studying CSPNe with intermediate periods can we substantiate this claim and relate the processes at work in formation of PNe by both post-CE and intermediate-period CSPNe. The planetary nebula (PN) LoTr 1 ($\alpha$ = 05:55:06.6, $\delta$ = $-$22:54:02.4, J2000) was first discovered by A.J. Longmore and S.B. Tritton with the UK 1.2-m Schmidt telescope (Longmore & Tritton, 1980). It is generally noted that LoTr 1 belongs to the so-called ‘Abell 35-type’ group (Bond et al., 1993) of PNe showing evidence of a binary central star system consisting of a cool central star (a rapidly rotating subgiant or giant), and an optically faint hot companion (a white dwarf with effective temperature, $t_{{\rm eff}}$ $\sim$ 100 kK) since these giant stars are too cool to ionise the surrounding nebula. Four PNe fell into this category: Abell 35 (hereafter, A35), LoTr 5 (Thevenin & Jasniewicz, 1997), WeBo 1 (Bond et al., 2003) and Abell 70 (Miszalski et al., 2012, hereafter, A70). However, Frew (2008) determined that A35 is most likely not a true PN, but rather a Strömgren zone in the ambient interstellar medium (ISM). This claim is substantiated by Ziegler et al. (2012), who find that the central star may in fact have evolved directly from the Extended Horizontal Branch to the White Dwarf (WD) phase (a so-called AGB-manqué star). As such, we choose not to consider Abell 35 in our comparisons among this group. Another common factor amongst this particular group of PNe is evidence for the existence of ‘Barium (Ba ii) stars’ (Bidelman & Keenan, 1951) – population i G/K-type AGB stars that show an over-abundance of carbon and s-process elements, in particular barium (Thevenin & Jasniewicz, 1997). A now-canonical model for the formation of these Ba ii stars states that they form not through CE evolution111We note, however, the existence of some exceptional systems that also experience similar enrichment in close binaries, i.e. the Necklace nebula (Miszalski et al., 2013), and which are most likely linked to dwarf carbon stars., as is the case for close binaries found within PNe, but rather via a wind-accretion scenario (Boffin & Jorissen, 1988). Here, the future Ba ii star is polluted whilst on the main sequence by the wind of its companion (Luck & Bond 1991; the companion having dredged up these s-process elements during its thermally pulsing AGB phase), but with the system remaining detached. After the envelope is ejected to form the surrounding nebula, the AGB star evolves into a WD, while the contaminated star retains its chemical peculiarities to form the remnant Ba ii star. One important prediction to come out of the wind-accretion model is that the accreting star, i.e. the future Ba ii star, also accretes angular momentum from the companion to become a rapid rotator (Jeffries & Stevens, 1996; Theuns et al., 1996). Indeed, photometric monitoring of LoTr 5 (Thevenin & Jasniewicz, 1997) and Webo 1 (Bond et al., 2003) has revealed that their cool components are in fact rapid rotators with a rotation period of a few days, thus providing further evidence for this formation scenario. This is further evidenced by the fact that Montez et al. (2010) found that the x-ray emission from the binary central stars of LoTr 5 is most likely due to the chromospheric activity from a spun-up companion. Most Ba ii stars are not observed to be within PNe, almost certainly because the lifetime of the PN is very short with respect to the lifetime of the stellar system. However, this does not completely rule out the possibility that some Ba ii stars may be formed without passing through a PN phase. Indeed, there are a few examples of field stars that have been shown to consist of a rapdidly rotating cool star linked to an optically faint hot component in a similar fashion to the Abell-35 group, but without current evidence for a surrounding PN. As mentioned by Bond et al. (2003), HD 128220 (P = 872 days) is one such system, made up of an O subdwarf and a G0 giant companion. O subdwarfs overlap with CSPN in terms of their log g and $t_{{\rm eff}}$, implying that they too are found to be in a post-AGB phase of evolution (Howarth & Heber, 1990). 56 Pegasi is another example, whereby it possess a system consisting of a K0 giant and a hot white dwarf companion with evidence for an overabundance of Ba ii, and has an estimated orbital period of 111 days (Griffin, 2006). It is possible that both of these systems _have_ gone through a PN phase in the past, but it has since dissipated into the surrounding ISM. This fact, as well as the knowledge that in Ba ii central stars the chemical pollution process happened very recently – either during or immediately prior to the formation of the PN – makes it highly important to study such systems as A70 and WeBo 1, as it will allow us to gain greater insight into both the s-process within AGB stars and mass-transfer mechanisms. In this paper, we present photometric and spectroscopic observations of LoTr 1 and its central star along with complementary data of A70 and WeBo 1, in order to try to relate the evolutionary processes of these three systems222PN K 1-6 (Frew et al., 2011) is another potential candidate for this group of objects, as it also shows evidence of possessing both a G- or K-type giant (inferred from imagery, looking at both optical and 2MASS near-IR colours) and a very hot sub dwarf or white dwarf (inferred from GALEX archival images) at its core. However, no stellar spectroscopy is available as yet to look for signs of chemical enrichment (Frew et al., 2011). More recently, Miszalski et al. (in press) presented evidence for a carbon and s-process enriched giant at the centre of the planetary nebula Hen 2-39. Due to it being a newly investigated system, Hen 2-39 is not included in this study either., and see if they belong to a common “fellowship” of s-process enriched cool CSPN inside ring-like nebulae. Figure 2: PV arrays showing reduced, longslit [O iii]$\lambda$5007 Å NTT-EMMI and AAT-UCLES spectra of LoTr 1, both aligned E-W, to show the overall nebular structure. West is to the top of the array. The velocity axis is heliocentric velocity, $V_{{\rm hel}}$. The display scale has been modified to highlight the spatio-kinematic features referred to in the text. Cross-section 0″ defines where the central star is found. The gap between the two CCDs in figure (a) is visible as a black strip across the frame at cross-section 58″ (see text for details). Figure 3: PV arrays showing reduced [O iii]$\lambda$5007 Å AAT-UCLES spectra of LoTr 1. The velocity axis is heliocentric velocity, $V_{{\rm hel}}$. The display scale has been modified to highlight the spatio-kinematic features referred to in the text. Slits 2–4 are N-S (North is to the top of the array), slit 5 is E-W (West is to the top of the array). Cross-section 0″ defines where the central star is found. The continuum of a field star is visible at cross-section $-$8″ in figures (a) and (b), and at cross-section +18″ in figure (d). ## 2 Observations and Analysis ### 2.1 LoTr1 #### 2.1.1 Imaging The deep [O iii]$\lambda$5007 Å and H$\alpha$\+ [N ii]$\lambda$6584 Å images shown in figures 1(a) and (b) were acquired on 2005 March 03 using the red arm of the European Southern Observatory (ESO) Multi-Mode Instrument (EMMI; Dekker et al. 1986), mounted on the 3.6m New Technology Telescope (NTT) of the La Silla Observatory. EMMI was used with the mosaic of two MIT/LL CCDs, 2048 $\times$ 4096 15 $\mu$m pixels. The exposure time, $t_{{\rm exp}}$, in each filter was 1800s and the binning set to $2\times 2$ ($\equiv$ 0.33″ pixel-1). The seeing was $\sim$0.7″. The images show LoTr 1 to have an apparent double-shell structure, with the central shell having a circular profile with an angular diameter of 47″$\pm$4″, and the outer shell a more irregular, but still roughly circular, appearance, with a diameter of 2′26″$\pm$4″. Furthermore, in [O iii]$\lambda$5007 Å the outer shell appears brighter in the northwest and southeast – this, coupled with the slight deviation from circular symmetry, could be considered evidence for an inclined, elongated structure, where the brighter areas result from a projection effect. Interaction with the ISM is also frequently invoked to explain deviation from symmetry and brightening of the nebular shell (e.g., Jones et al. 2010), however this would not produce the apparent axisymmetry in the nebular brightening (as the nebula would only interact strongly with the ISM in the direction of its motion; Wareing et al. 2007). #### 2.1.2 Nebular spectroscopy Longslit echelle spectroscopy was carried out using AAT-UCLES and NTT-EMMI, focusing on [O iii] emission over eight different slit positions (see figure 1 (b)) in order to gain velocity profiles across a good sample of the nebula. It is important to ascertain the true three-dimensional shape of the surrounding nebula in order to fully understand and constrain the shaping process, and subsequently give a clearer insight into mass-transfer mechanisms. Imaging is insufficient on its own due to a degeneracy between PN inclination and morphology – for example, when the symmetry axis of a bipolar nebula is aligned perpendicular to the plane of the sky, it makes the nebula appear spherical (Kwok, 2010). If the nebular morphology is classified incorrectly then the fraction of aspherical PNe possessing a binary core will also end up being inaccurate, and this information is already exceedingly limited for longer period binaries due to selection biases for those stellar systems possessing wider orbits. Longslit spectroscopy can be used to acquire spatially resolved velocity maps of the constituent parts of the nebula in order to recover the ‘missing’ third dimension of the morphology one cannot gain from imagery alone. The resulting spectra are then plotted as position- velocity (PV) arrays. On 2005 March 03, a spectrum was acquired from the nebula using EMMI in its single order echelle mode, employing grating #10 and a narrowband [O iii]$\lambda$5007 Å filter to prevent contamination from overlapping echelle orders. The maximum slit length of 330″ and a slit width of 1″ was used to give a resolution, R $\sim$ 54,000 (5.5 km s-1). A 1800s exposure was taken at a P.A. of 90∘ crossing the central star (slit position 1 in figure 1(b)), and the data were 2x2 binned to give a spatial scale of 0.33″ and a velocity scale of 3.9 km s-1 per pixel. The seeing was $\sim$0.7″. On 2005 January 14, spectra were acquired from the nebula of LoTr 1 using the 79 lines/mm grating on the UCL Coudé Echelle Spectrograph (UCLES) of the Anglo-Australian Telescope (AAT). UCLES was operated in its longslit mode with a maximum slit length of 56″ and a slit width of 1.97″ to give a resolution R $\sim$ 20,000 (15 km s-1). The EEV2 CCD (2048$\times$4096 13.5 $\mu$m pixels) was used with binning of 2$\times$3, resulting in a pixel scale of 3.88 km s-1 pixel-1 in the spectral direction and 0.48″ pixel-1 in the spatial direction. 1800s exposures were taken at five different slit positions (shown in figure 1(b)) using a narrowband filter to isolate the 45th echelle order containing [O iii] emission line profile. Slits 2–4 were taken at a position angle (P.A.) of 0∘, and slit 5 was taken at a P.A. of 90∘. The seeing during the observations was $\sim$2″. A further three slits (slit positions 6–8 in figure 1(b))were acquired using the same instrument and CCD on 2013 January 3 with a seeing of $\sim$1.5″ and a binning of 2$\times$2 ($\equiv$ 0.32″ pixel-1 in the spatial direction). Here, a slit width of 1″ was employed for slits 6 and 8 (R $\sim$ 45,000 $\equiv$ 6.7 km s-1) and 1.5″ for slit 7 (R $\sim$ 30,000 $\equiv$ 10 km s-1)333The slit width for slit 7 was altered to try to match the seeing conditions during the night. However, as all observations were carried out with the same binning in the spectral direction and the same grating (i.e. approximately the same dispersion), they can all be used to qualitatively assess the spatio-kinematic structure of the nebula (varying on scales $\gg$ the slit-widths employed, due to the angular size of the nebula also being $\gg$ the slit-width).. All the spectra were cleaned of cosmic rays and debiased appropriately. The UCLES spectra were wavelength calibrated against a ThAr emission-lamp, rescaled to a linear velocity scale appropriate for the [O iii]$\lambda$5007 Å emission, and corrected to heliocentric velocity, $V_{{\rm hel}}$. Due to the optical set-up of EMMI, it was necessary to perform the wavelength calibrations using a long-exposure (3600s) ThAr emission lamp at the start and end of the night to gain a good number of arc lines, before cross-correlating with shorter-exposure (200s) Ne lamps taken immediately after each observation, to account for any drift due to telescope and instrument flexure (with small shifts accounted for with a linear correction). The reduced nebular spectra of LoTr 1 are presented in figures 2 and 3, as position-velocity (PV) arrays. In each PV array, cross-section 0″ defines where the central star is found. In figure 2 (a), the bright lines located around the central star at cross-sections +3″ and $-$3″ are most likely artefacts due to the comparative difference in brightness between the central star and the nebula (Jones et al., 2010). The closed velocity ellipses shown in the PV arrays presented in figures 2 (a), 3 (c), and the central PV array of 2 (b) (i.e. representing slit positions 1, 4 and 7), have a major axis which has the same length as the diameter of the inner shell (see section 2.1.1). This indicates that the nebular structure is indeed an isolated, closed shell rather than a projection effect related to a bipolar structure being viewed end-on. No significant asymmetries are observed in these velocity ellipses, which are consistent with a spherical shell or an elongated ovoid viewed directly along the symmetry axis. The expansion velocity, $V_{{\rm exp}}$ for the inner shell is measured to be 17$\pm$4 km s-1 at the location of the central star, while the $V_{{\rm exp}}$ for the outer shell is measured to be 25$\pm$4 km s-1 falling within the typical range for a PN (Weinberger 1989). Assuming typical expansion properties for the nebula (i.e. velocity proportional to distance from the central star), the latter velocity is then the maximum expansion velocity (for an elliptical shell viewed pole-on) or the uniform expansion velocity of the shell (in the case of a sphere). The heliocentric systemic velocity, $V_{{\rm sys}}$ of this central shell was determined to be 14$\pm$4 km s-1. In slit 1 (figure 2(a)), emission is clearly detected from outside of the central shell at cross-sections $\sim$40″ and $\sim-$40″, associated with the outer shell (see section 2.1.1). Here, the emission from the eastern side appears blue-shifted with respect to the nebular $V_{{\rm sys}}$, and the west appears red-shifted; this is indicative of an inclined and extended structure, e.g. an elliptical nebula, where the approaching “lobe” is tilted slightly to the east of its receding counterpart. Consideration of the other slits presented in 3 (b), (c) and (d), confirms this asymmetry in the velocity profile across the nebula, but indicates that the symmetry axis may lie closer to the northeast-southwest direction than east-west. However, any deviation from the line of sight must be rather small given the almost circular appearance of the shell in the images (figure 1). Determination of the exact structure and inclination of the nebula would require a more extensive, higher signal-to-noise (given the outer shells faintness) dataset, covering more of the physical extent of nebula, and a detailed spatio-kinematical model such as that presented in Jones et al., 2012. However, it is clear from both the imaging and spectroscopy presented here that LoTr 1 shows a double-shelled structure with evidence for an elliptical and slightly inclined outer shell, and a morphologically similar inner shell but with a different orientation. Figure 4: SuperWASP photometry of LoTr 1. Upper panel: Periodogram showing a clear detection of periodicity at $6.3967\pm 0.0005$ days. Middle panel: The data points folded on the period obtained from the periodogram. Lower panel: The data binned into phase bins, overlaid is a sinusoidal fit with a peak-to- peak amplitude of 0.061 magnitudes. Figure 5: (a) Flux-calibrated FORS2 spectrum of the central star system of LoTr 1. (b) Flux-calibrated IUE spectra of LoTr 1 (solid line), and of NGC 7293 (dashed line) used to determine the parameters of the white dwarf. Note also the presence of the Mg II emission at 2800 Å which is a sign of chromospheric activity from the cool companion (e.g. Jasniewicz et al., 1996; Montez et al., 2010) #### 2.1.3 Stellar photometry The field of LoTr 1 has been observed by WASP-S between 2006 May 4 and 2012 February 17 with a total of 21407 photometric points obtained with the two cameras DAS 226 and 228. WASP-S is a wide field survey camera situated at SAAO, Sutherland, South Africa, and together with its northerly sister, SuperWASP-N, is designed to obtain extremely accurate photometry of bright stars in order to search for transits from exoplanets (see Pollacco et al. 2006, for a more detailed description of the facility, the data reduction and archive). Data mining tools and the public archive are discussed by Butters et al. (2010). After reduction through the instrument pipeline (Pollacco et al., 2006), the time series were examined with a simple Lomb-Scargle filter for periodic signal detection. From the light curves reproduced in figure 4 taken from Jones et al. (2011), a period of 6.4 days was derived with a peak-to-peak amplitude of 0.061 magnitude. This is close to the value of 6.6 days with an amplitude of 0.1 magnitude in V, initially derived by Bond et al. (1989). Given the period, this amplitude is probably too high to be considered due to irradiation effects (De Marco, 2006), so the most likely explanation is that the periodicity is following spots on a cool star – the signature of a rapid rotator, analogous to the systems of LoTr 5 (5.9$\pm$0.3d, Thevenin & Jasniewicz, 1997) and WeBo 1 (4.69$\pm$0.05, Bond et al., 2003). Figure 6: LoTr 1 FORS2 stellar spectrum (solid line) alongside three synthetic spectra with [Ba/Fe] = 0.1, [Ba/Fe] = 0.2, and [Ba/Fe] = 0.5, smoothed to match the FORS2 resolution and plotted over a wavelength range which includes the Ba ii 4554Å line. Figure 7: H$\alpha$ emission seen in LoTr1. #### 2.1.4 Stellar spectroscopy Figure 8: H$\alpha$+[N ii]$\lambda$6584 Å image of PN A70 showing all slit positions. Slits 1–4 were acquired in [O iii]$\lambda$5007 Å using VLT-UVES, and slit 5 was acquired in H$\alpha$ and [O iii]$\lambda$5007 Å using SPM-MES. The central star is visible at $\alpha$ = 20:31:32.2, $\delta=-$07:05:17.0. On 2012 February 10, a low-resolution spectrum was acquired of the central star of LoTr 1 using the FOcal Reducer and low dispersion Spectrograph (FORS2; Appenzeller et al. 1998) with grism 1200g, on the Antu unit (UT1) of the Very Large Telescope (VLT) based at ESO-Paranal. FORS2 was operated with a maximum slit length of 6′ and a 0.5″ slit width (R $\sim$ 3000), and at a position angle of 45°. The data were 2$\times$2 binned to give a spatial scale of 0.25″ pixel-1, with a dispersion on the spectral axis of 0.3Å pixel-1. A single 600s exposure was taken under seeing of $\sim$ 0.7″. The stellar data were extracted to 1-D and flux-calibrated against standard star Hz 4, and are presented in figure 5 (a). All reductions were carried out using standard starlink routines. The central star of LoTr 1 was classified as a K1 III-type giant by comparing the flux-calibrated FORS2 spectrum with UVES POP standard stars (Bagnulo et al., 2003) that had been rebinned and smoothed to match the resolution of the FORS2 data. To refine the analysis and derive the possible s-process overabundance, we used the stellar spectral synthesis code of R. Gray, spectrum version 2.76444See http://www1.appstate.edu/dept/physics/spectrum/, with models from Castelli & Kurucz (2003). The best fit parameters were determined to be $t_{{\rm eff}}$ $=4750\pm$ 150 K and log $g=2.0\pm 0.5$, which is in agreement with the K1 III spectral type. Using the average value of the absolute magnitude, Mv = +0.7 for a giant of this spectral type as given in Allen’s ‘Astrophysical Quantities’ (2000), the flux for the spectrum shown in figure 5 (a) that allows the derivation of an apparent magnitude, mv = +12.6 (accounting for 20% slit-losses), and extinction $A_{V}$=0.1285 (Schlegel et al., 1998), one can determine a rough distance to the star of 2.6 kpc and a not-unreasonable radius of the giant of 11.5 R⊙. Alternatively, one can use the average value of $V$ derived from SuperWASP (see figure 4), i.e. $V=12.44$, to derive a distance of 2.1 kpc. Note that this observed SuperWASP magnitude is likely to be contaminated by line emission and close field stars because of the broad observing band and large (14″) pixels. Therefore, we think it is preferable to use the distance derived from the flux calibrated spectra. To check that the giant and the planetary nebula are indeed linked and not merely a chance superposition, we computed the radial velocity difference between the nebular lines in the FORS2 spectrum and the giant star’s absorption lines, cross-correlating our spectrum (shifted so that the nebular lines are at zero velocity) with our synthetic K1 III spectrum. The resulting stellar velocity of 4$\pm$2 km s-1 with respect to the nebula implies that the cool central star and nebula are physically related. Figure 9: PV arrays showing reduced VLT-UVES spectra in [O iii]$\lambda$5007 Å from A70. Figures (a) and (c) show emission from the major axis, figures (b) and (d) are from the minor axis. Positive spatial offsets are to the northern (1 and 2) or eastern (3 and 4) ends of the slits.The velocity axis is heliocentric velocity, $V_{{\rm hel}}$. The display scale has been modified to highlight the spatio-kinematic features referred to in the text. Cross-section 0″ defines where the central star is found. The FORS2 spectrum was used to check for signs of Ba ii pollution at 4554Å. Figure 6 shows the FORS2 spectrum of LoTr 1 plotted alongside three synthetic spectra with various barium enhancements, from [Ba/Fe] = 0.1 to [Ba/Fe] = 0.5. An over-abundance of 0.5 or greater classifies the system as a definite Ba ii star, with a value of 0.2–0.5 being possibly a ‘mild’ Ba ii star (Pilachowski, 1977). Due to the relatively low S/N and resolution of the spectrum, we are unable to determine a definitive value for the barium abundance in LoTr 1, but we can clearly state that it is much less than 0.5, and therefore does not show any measurable barium enhancement. This is contrary to both A70 and WeBo 1 (see Miszalski et al. 2012 and Bond et al. 2002, respectively), which both possess definite Ba ii stars. On 2013 January 3, 3$\times$30 minute spectra were acquired from the CSPN of LoTr 1 using AAT-UCLES with the 79 lines/mm grating, operated in full echelle mode with a slit width of 1″ to give a resolution R $\sim$ 45,000. All spectra were reduced using standard starlink routines, corrected to heliocentric velocity and then summed. Unfortunately, because of the large interorders in the spectra555UCLES operated in this mode gave non-continuous wavelength coverage from roughly 5200Å to 8900Å across 19 orders with interorder spacing of 100–200Å., it is not possible to assess the barium abundance, as none of the available orders contain the Baii lines at 6141.7Å and 6496.9Å. We have verified, however, that the iron lines are well fitted with a solar abundance and our preferred model, in agreement with what we derived from the FORS2 spectrum. Looking at the La ii 6390Å and Yii 6222Å lines, it is also clear that these elements are not overabundant, confirming the lack of s-process enhancement in LoTr 1. The AAT spectra allowed us to find that the stellar component of H$\alpha$ is in emission (see figure 7). The line is very broad and is clearly double- peaked, with an equivalent width of about 6Å and the velocity spread is 572 km/s. Using the method of Hodgkin et al. (1995), we derive an H$\alpha$ luminosity $L_{\rm H\alpha}$ = 0.044 L⊙. With the above estimated total luminosity of the star ($L_{}\sim$ 60 $L_{\odot}$), this gives a value $\log L_{\rm H\alpha}/L_{}=-3.12$. Such H$\alpha$ double-peaked emission lines have been found in the other stars we are concerned with here, LoTr 5 (Jasniewicz et al., 1994; Strassmeier et al., 1997) and A35 (Acker & Jasniewicz, 1990; Jasniewicz et al., 1992), but the origin is still unknown. Rapidly rotating giants, such as RS CVn or FK Com stars, are known to have high chromospheric activity which is often revealed by emission cores in some lines. And indeed, Acker & Jasniewicz (1990) find a modulation of the H$\alpha$ emission line with the rotation phase, while Jasniewicz et al. (1992) postulate that the variable double-peaked emission line is the result of an overlap between an absorption and an emission line at H$\alpha$, with the possibility for the absorption component to be formed in the photosphere or through a self-absorption process as in Be stars. However, the FWHM velocity we measure seems too high to be caused by mass motions inside the chromosphere, while the luminosity is too small to be due to accretion. Such double-peaked emission are also sometimes found in symbiotic stars – detached systems which interact via wind accretion. A few of them, out of an outburst event, produce bipolar nebulae very similar to planetary nebulae. It turns out that many symbiotic stars show double H$\alpha$ profiles (e.g. Schild et al., 1996; Burmeister & Leedjärv, 2009), which might be caused by a narrow absorption component from the giant overlaid with very broad H$alpha$ emission from high-velocity jets at the core of the system, or by disc-like structures. It would thus be of interest to further study the possible link between LoTr 1 and symbiotic stars. A spectrum of the central star system of LoTr 1 was acquired using the International Ultraviolet Explorer (IUE) satellite by Bond et al. (1989). Presented here in figure 5 (b), it indicates a strong UV continuum of $t_{{\rm eff}}$ $\geq$ 100 kK. Comparing this IUE spectrum to a known WD within PN NGC 7293 (d = 219${}^{+27}_{-21}$ pc, mv = +13.5; Harris et al., 2007) gave us a reasonable fit: $t_{{\rm eff}}$ $\geq$ 123 kK, and R = 0.017${\rm\ R}_{\odot}$. Based on values of D = 2′22″$\pm$4″, and $V_{{\rm exp}}$ = 25$\pm$4 km s-1 at a distance of 2.1–2.6 kpc, we derived a kinematical age of 33,000$\pm$9,000 years for the outer nebular shell. For the inner shell, given values of D = 47$\pm$2″ and $V_{{\rm exp}}$ = 17$\pm$4 km s-1, the age is derived as 16,000$\pm$6,500 years. Using the white dwarf evolutionary curves from Bloecker (1995) and assuming an average remnant mass of 0.6${\rm\ M}_{\odot}$, we derived a stellar temperature at the age of the PN of approximately 120,000 K. The derived radius and age of LoTr 1 are consistent with Bloecker’s evolutionary curves. ### 2.2 A70 #### 2.2.1 Imaging The H$\alpha$+[N ii]$\lambda$6584 Å image of A70 shown in figure 8 was acquired on 2012 July 08, using FORS2 under program ID 0.89.D-0453(A), with an exposure time of 60s and seeing of 0.8″. At first glance the image shows a general ring-like appearance similar to that of other ring-like PNe (e.g. SuWt 2: Jones et al. 2010); however, just as noted by Miszalski et al. (2012), a closer inspection reveals a “ridged” profile more like that of a bipolar nebula viewed end-on (e.g. Sp 1: Jones et al. 2012). Furthermore, this image shows in detail the low-ionisation knots first identified by Miszalski et al. (2012). Many of these structures seem to be akin to the cometary globules seen in the Helix Nebula (dense condensations of molecular gas embedded in the ionised nebula, see Meaburn et al. 1992), with knotty heads closest to the nebula centre and extended tails reaching out towards the outer rim. Extended material is also visible outside the east-southeasterly edge of the nebular ring (the emission visible to the north of the ring originates from a background field galaxy). Assuming the “ring” is a physical structure rather than a projection effect, one can deduce the inclination of the nebula by deprojection. The angular size of the nebula was determined to be 44″$\times$ 38″$\pm$2″, falling in line with the previously given value of 45.2″x 37.8″ by Tylenda et al. (2003), giving an inclination of $30$°$\pm 10$°. Figure 10: PV arrays showing reduced SPM-MES spectra of A70. Emission is through the horizontal axis (slit 5 in figure 8). The velocity axis is heliocentric velocity, $V_{{\rm hel}}$. The display scale has been modified to highlight the spatio-kinematic features referred to in the text. Cross-section 0″ defines where the central star is found. #### 2.2.2 Nebular Spectroscopy On 2011 June 10-11, high resolution data of the nebula of A70 were acquired in [O iii] using grating #3 on the visual-to-red arm of the Ultraviolet and Visual Echelle Spectrograph (UVES) on Kueyen Unit (UT2) of the VLT (Dekker et al., 2000), under program ID 087.D-0174(A). UVES was operated in its 30″ longslit mode with a 0.6″ slitwidth (R $\sim$ 70,000, 4.3 km s-1 pixel-1) to give a spatial scale of 0.17″ pixel-1. A filter was used to isolate the [O iii] emission lines and prevent contamination from overlapping orders. The seeing was between 0.5″ and 0.7″ for all observations. Four 1200s exposures were taken over four different slit positions. Slits 1 and 2 were taken with VLT-UVES at a P.A. of 160∘ and slits 3 and 4 at a P.A. of 70∘, to line up with the major and minor axis of the nebula, respectively. Slit 5 was acquired in both [O iii] and H$\alpha$ on 2011 May 15 using the Manchester Echelle Spectrograph (MES) mounted on the 2.1-m San Pedro Martir (SPM) telescope based at the Observatorio Astronomico Nacional in Mexico (Meaburn et al., 2003; López et al., 2012). The full slit length of 5′ was used with a slitwidth of 150$\mu$m ($\equiv$ 2″, R $\sim$ 30,000), and taken at a P.A. of 90∘. The data were 2x2 binned to give a spatial scale of 0.75″ pixel-1. The seeing was $\sim$ 3″ The nebular spectra acquired from VLT-UVES shown in figure 9 show two highly filamentary components, one red-shifted and one blue-shifted, joined by bright knots of emission where the slits cross the nebular ring, to form a closed velocity ellipse in both axes. These filamentary and irregular structures are typical of disrupted nebulae, where instabilities have begun to structurally deform the shell (Guerrero & Miranda, 2012), and clearly show that A70 is not simply an inclined ring but instead has “bubbles” extending in the line of sight. It is reasonable to assume that these bubbles form a closed (as the velocity ellipses are closed along both axes) and axisymmetric (the blue- and red-shifted components are roughly symmetrical) structure. The bright emission at the extremes of each slit indicate that the nebula may have a cusped waist, with slits 1 and 2 showing the “crow’s foot”-like structure typical of narrow waisted nebulae viewed along their symmetry axis (Jones et al., 2012). However, there is a clear brightening in these regions and those of slits 3 and 4, consistent with a bright ring. We therefore deduce that A70 comprises of such a bright ring, encircling the waist of a disrupted and faint bipolar shell. Using the same spectra, a polar expansion velocity $V_{{\rm exp}}$ for A70 was calculated to be 39$\pm$10 km s-1. This is in agreement with the value for expansion velocity of $V_{{\rm exp}}$ = 38 km s-1 given by Meatheringham et al. (1988), although no error was quoted. A $V_{{\rm sys}}$ of $-$73$\pm$4 km s-1 was determined for the nebula, which is consistent with the value of $-$72$\pm$3 km s-1 by Miszalski et al. (2012). The kinematical age of A70 was determined to be 2700$\pm$950 years kpc-1. Taking the distance to the nebula to be 5 kpc, as given by Miszalski et al. (2012), this gives an overall kinematical age for A70 of order 13,400$\pm$4,700 years. The limited depth and resolution of the MES-SPM spectra offer little extra to the discussion of the nebular morphology and kinematics, other than to show that there is no major difference between the emission in H$\alpha$ and that in [O iii]$\lambda$5007 Å. They are shown here in figure 10 mainly for comparison with similar data acquired from PN WeBo 1, which will be discussed in section 2.3. ### 2.3 WeBo 1 Figure 11: Images of PN WeBo 1 in (a) H$\alpha$+[N ii]$\lambda$6584 Å, and (b) [O iii]$\lambda$5007 Å, with (b) showing the two slit positions. Both slits were acquired in [N ii] and [O iii] using SPM-MES. The central star is visible at $\alpha$ = 02:40:13.0, $\delta$ = +61:09:17.0. North is to the top of the images, East is left. #### 2.3.1 Imaging The deep H$\alpha$+[N ii]$\lambda$6584 Å image shown in figure 11 (a) is the result of coadding 2 $\times$ 120s exposures, each with seeing better than 1.2″, acquired as part of the IPHAS survey (Drew et al., 2005) using the Wide Field Camera (WFC) on the 2.5m Isaac Newton Telescope based at the Observatorio Roque de los Muchachos, La Palma. The [O iii]$\lambda$5007 Å image shown in figure 11 (b) was acquired using the same instrument on 2010 September 9, with an exposure time of 1200s and under seeing of 1.4″. The images show another ring-like morphology, although structurally different to A70 (see section 2.2), with a pronounced inner edge and fainter, more extended emission around its entire circumference – the ring is particularly diffuse in [O iii]$\lambda$5007 Å, as shown by the lack of a visible inner edge. Similar extended emission is also found in SuWt 2 (Jones et al., 2010) and HaTr 10 (Tajitsu et al., 1999), where the ring is actually the waist of an extended bipolar structure, possibly indicating that WeBo 1 may display the same morphology but with as yet undetected, very faint lobes. In SuWt 2, Jones et al. (2010) attribute this extended material to structural and brightness variations across an irregular toroidal structure; however (particularly in the light of the H$\alpha$+[N ii] spectra acquired - see figure 12), WeBo 1 shows a much more regular and even ring-like shape, indicating that this is more likely an intrinsic structural property (i.e. a tear-drop rather than circular shaped cross-section). Smith et al. (2007) deprojected the ring of WeBo 1, determining that it is seen almost edge-on with an inclination of 75°$\pm$3° with an inner-ring radius of $\sim$ 25″, which is consistent with the dimensions of the ring as cited by Bond et al. 2003 (64″x 22″), and as measured from the images presented here (65″$\times$20″$\pm$4″). #### 2.3.2 Nebular Spectroscopy On 2010 December 10, spectra were acquired of WeBo 1 in both H$\alpha$+[N ii] and [O iii]$\lambda$5007 Å using SPM-MES. The maximum slit length of 5′ was used with a slitwidth of 150$\mu$m ($\equiv$ 2″, R $\sim$ 30000) for each filter. The data were 2x2 binned to give a spatial scale of 0.75″ pixel-1. Slit 1 was taken at a P.A. of 353∘ and slit 2 was taken at a P.A. of 263∘ to cover the major and minor axes of the nebula, respectively. The seeing was $\sim$1.5″. Due to the H$\alpha$ profiles having high galactic background emission, only the background-subtracted [N ii] and [O iii]$\lambda$5007 Å are presented here. The reduced PV arrays are presented in figure 12. Just as in the imagery (see section 2.3.1), the [N ii]$\lambda$6584 Å profiles show well defined emission originating from the ring, while the [O iii]$\lambda$5007 Å is much more diffuse. The [N ii]$\lambda$6584 Å profiles from both slit positions show strong emission at the inner edge of the ring, with slightly fainter material (also with a lower velocity dispersion) reaching out to greater angular extents – consistent with the “tear-drop” cross-section interpretation presented in section 2.3.1. The [N ii]$\lambda$6584 Å profile from slit 1 shows little velocity difference between the emission originating from the two opposing sides of the ring, indicating that the slit is roughly perpendicular to the P.A. of the nebular symmetry axis; therefore, the deprojected velocity difference of the two sides of the ring in slit 2 should offer a good measure of the ring’s expansion velocity. Taking the nebula inclination to be 75°, the deprojected expansion velocity for the ring was calculated to be $V_{{\rm exp}}$ = 22.6$\pm$10 km s-1. A $V_{{\rm sys}}$ of -6$\pm$4 km s-1 was calculated for the nebula. The kinematical age of WeBo 1 was determined to be of order 7300$\pm$3700 years kpc-1. Taking the distance to the nebula to be 1.6 kpc (Bond et al., 2003), this gives an overall age for WeBo 1 of 11,700$\pm$5,900 years.666Bond et al. (2003) assume an expansion velocity of 20 km s-1 to derive a similar age of 12,000$\pm$6,000 yrs. No clear evidence of lobes or extended nebular structure are detected in the spectra, unlike for A70 (see section 2.2.2); however, faint material detected inside the ring on the [N ii]$\lambda$6584 Å PV array of slit 1 could be consistent with such a structure. Deeper, higher resolution spectra are required to confirm the nature of this emission, but it is safe to say that if any lobes are present they are significantly fainter than those of A70, as its lobes were still clearly detected by SPM-MES spectra (see figure 10). Figure 12: PV arrays showing reduced [N ii]$\lambda$6584 Å and [O iii]$\lambda$5007 Å SPM-MES spectra of WeBo 1. Figures (a) and (b) show the [N ii] and [O iii] emission from Slit 1, (c) and (d) from Slit 2 (see figure 11). North is to the top of the array. The velocity axis is heliocentric velocity, $V_{{\rm hel}}$. The display scale has been modified to highlight the spatio-kinematic features referred to in the text. Cross-section 0″ defines where the central star is found. The continuum of a field star is visible at cross-section +15″ in figures (a) and (b). Table 1: Physical parameters of LoTr 1, A 70 and WeBo 1 Nebula \| $v_{exp}$ ( km s-1) \| Kinematical age (yrs) \| Physical size (pc) \| Morphology ---\|---\|---\|---\|--- LoTr 1 (inner) \| 17$\pm$4 \| 17,000$\pm$5,500 \| 0.59$\pm$0.05 \| spherical/elliptical LoTr 1 (outer) \| 25$\pm$4 \| 35,000$\pm$7,000 \| 1.86${}^{+0.05}_{-0.09}$ \| spherical/elliptical A 70 \| 39$\pm$10 \| 13,400$\pm$4,700 \| 1.10${}^{+0.06}_{-0.04}$ \| Ringed-waist with detected lobes WeBo 1 \| 23$\pm$10 \| 11,700$\pm$5,900 \| 0.50$\pm$0.03 \| Ringed-waist, no lobes detected ## 3 Discussion ### 3.1 The Abell-35 group and PN mimics Abell 35 - the archetype of the class of PNe discussed in this paper – has recently been shown to be a PN mimic. It is, therefore, critical to establish the true nature of the objects considered here before beginning a comparison. We restrict our analysis to the three PNe presented in this work, and exclude both Abell 35 (which is no longer considered a true PN) and LoTr 5 (which is a considerably more complex case and discussed in more detail in Frew, 2008; see also Graham et al., 2004). Frew & Parker (2010) present a ‘recipe’ for determining whether we can classify an object as a true PN or not, parts of which we can apply to the nebulae presented here: * • Presence of a hot, blue central star: All three nebulae presented here show evidence of excess UV flux that point towards the existence of a hot companion – see Fig. 5a for LoTr 1, Miszalski et al. 2012 for A70, and Siegel et al. 2012 for WeBo 1. * • Nebular morphology: Each PN possesses what we would classify as a ‘typical’ PN shape, with rings (A70 and WeBo 1) and shells (LoTr 1). A mimic is often more diffuse. * • Systemic velocity: The $V_{{\rm sys}}$ of the central star is consistent with the $V_{{\rm sys}}$ of the nebula for both LoTr 1 and A70 (see section 2.1.4, and Miszalski et al., 2012), and so we can say that the observed emission comes from a true planetary nebula. No $V_{{\rm sys}}$ for the central star of WeBo 1 has been published. * • Nebula expansion: All three nebulae have been shown to have an expansion velocity typical for a PN (see Table 1). * • Nebular diameter: Using the values stated in this paper, the physical diameters of LoTr 1, A70 and WeBo 1 are all of order $\sim$1 pc (see Table 1): a sensible value for a PN. * • Galactic latitude: Two of the three nebulae are found at high galactic latitudes of -22° and -25° for LoTr 1 and A70, respectively (WeBo 1 is at and +1°). PNe are more likely to be found away from the Galactic plane than isolated Strömgren spheres. We can thus be confident that the three objects studied in this paper show the characteristics of bona-fide planetary nebulae, although, as mentioned earlier, their link with the class of symbiotic stars should also be investigated further. ### 3.2 Conclusions From the study conducted in this paper, we have been able to show that LoTr 1 possesses a double-shelled, slightly elliptical morphology of age of 35,000$\pm$7,000 years for the outer shell, and 17,000$\pm$5,500 years for the inner. We have been able to infer the presence of a K1 III-type giant ($t_{{\rm eff}}$ $\sim$ 4500 K) and hot white dwarf ($t_{{\rm eff}}$ $\sim$ 123 kK, R = 0.017${\rm\ R}_{\odot}$) binary system at its core. The cool star has been shown to be kinematically associated with the nebula, and to have a rotation period of 6.4 days. Although it was not possible to accurately determine the [Ba/Fe] value for the central star system, we were able to say with confidence that LoTr 1 does _not_ show any evidence for an over-abundance of Ba ii. LoTr 1 also presents double-peaked emission lines, which have been seen in the other PNe with cool central stars. Unlike LoTr 1, the PNe A70 and WeBo 1 have both been previously confirmed to contain a Ba ii-enriched central star system at their core. The two nebulae are also shown here to display morphologies distinct to that of LoTr 1, with both possessing ring-like waists and possible extended lobes. The similar morphologies and chemical enrichment strongly imply that the two have undergone very similar evolutionary or mass-loss processes. It is possible that the wind-accretion process involved in the formation of Ba ii stars is also responsible for the formation of these ring-like morphologies. Although the CSPN of LoTr 1 does share some common traits with those of A70 and WeBo 1 – namely binarity with a hot- and cool-components, and rapid rotation of the secondary – both the lack of a significant over-abundance of Ba ii and the marked difference in nebular morphology would imply a difference in the evolution of this system. The lack of Ba ii enhancement could be explained by a difference in progenitor mass, metallicity, or simply quantity of mass transferred via the same wind-accretion process (the amount of material accreted is strongly dependent on orbital separation – Boffin & Jorissen, 1988). However, as shown by Boffin & Zacs (1994) only a small amount of matter is needed to be accreted to make a star appear as a barium star and some mass must have been transferred as it is required in order to spin up the secondary to its rapid rotator state. The most obvious explanation, therefore, is that the mass was transferred at an earlier stage in the evolution of the primary, i.e. before the thermally-pulsing AGB phase, when the s-process elements are created and brought to the surface. This would allow us to infer that the AGB evolution of the primary was cut short by this mass-transfer episode, signs of which should be detectable in the properties of the WD. We strongly encourage follow-up observations of the system in order to confirm this hypothesis, and in particular it would be crucial to determine the orbital period of these systems. It is, however, important to note that given the inclination of the LoTr 1 nebula (very close to pole-on), any radial velocity variations of the central star system would be very difficult to detect, particularly for the expected period of $\sim$1–3 years, and this therefore requires very high spectral resolution and stable instruments. Coming back to the nebula, multiple shells are not uncommon in planetary nebulae, with 25-60% found to show outer structures (Chu et al., 1987). However, it is important to distinguish here between ‘halo-like’ shells (Corradi et al., 2003, 2004), which are extended, generally spherical, structures attached to the inner shell, and detached outer shells, with the latter being far less common. This is critical as the haloes are generally understood to be the ionised remnant of mass lost on the AGB which is now being swept up to form the inner shell, while the formation mechanism for multiple, detached shells is still a mystery. Schönberner et al. (1997) show that this may be possible via a combination of photo-ionisation and wind interaction, or, alternatively, a binary evolution might be responsible for rapid changes in mass-loss that could form two distinct shells, such as with Abell 65 (Huckvale et al., 2013). LoTr 1 clearly shows a detached outer shell which may have been produced by rapid changes in mass-loss/transfer in the CSPN. The difference in kinematical ages between the two shells, of roughly 18,000-20,000 years, is much shorter than the single star evolutionary timescales on which these changes might occur. Perhaps the nature of LoTr 5 should also be investigated further, as it too does not have an apparent ring-like morphology despite possessing a rapidly rotating, G5 III-type, Ba ii-rich central star. As mentioned earlier, Montez et al. (2010) have carried out a study into the x-ray emission emanating from this system and concluded that it is most likely chromospheric in origin, implying the presence of a spun-up companion. The results presented here show that we still have some way to go to fully constrain mass-transfer mechanisms in intermediate-period binary and post-AGB systems, and further study of other such similar systems would be highly beneficial, in particular with regards to Ba ii pollution. ## Acknowledgments AAT gratefully acknowledges the support of STFC and ESO through her respective studentships. This work was co-funded under the Marie Curie Actions of the European Commission (FP7-COFUND). Based on observations made with ESO telescopes at the La Silla Paranal Observatory under programme IDs: 0.74.D-0373(A), 087.D-0174(A), 087.D-0205(A), 088.D-0750(A), 0.88D-0573(A), and 0.89D-0453(A). We thank the staff at the ESO La Silla Paranal Observatory and the Anglo-Australian telescope for their support in the acquisition of the observations. 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1309.4501	# A fully automatic problem solver with human-style output M. Ganesalingam and W.T. Gowers ## 1 Introduction and related work ### 1.1 Overview of the paper The main purpose of this paper is to describe a program that solves elementary mathematical problems, mostly but not exclusively in metric space theory, and presents the solutions in a form that is hard to distinguish from solutions that human mathematicians might write. The following two proofs are examples of the program’s output.111The program produces LaTeX output, which we reproduce verbatim here. The first is a proof that if $f:X\to Y$ is a continuous function and $U$ is an open subset of $Y$, then $f^{-1}(U)$ is an open subset of $X$, and the second is a proof that if $f:X\to Y$ is an injection and $A$ and $B$ are subsets of $X$, then $f(A)\cap f(B)\subset f(A\cap B)$. > Let $x$ be an element of $f^{-1}(U)$. Then $f(x)\in U$. Therefore, since $U$ > is open, there exists $\eta>0$ such that $u\in U$ whenever $d(f(x),u)<\eta$. > We would like to find $\delta>0$ s.t. $y\in f^{-1}(U)$ whenever > $d(x,y)<\delta$. But $y\in f^{-1}(U)$ if and only if $f(y)\in U$. We know > that $f(y)\in U$ whenever $d(f(x),f(y))<\eta$. Since $f$ is continuous, > there exists $\theta>0$ such that $d(f(x),f(y))<\eta$ whenever > $d(x,y)<\theta$. Therefore, setting $\delta=\theta$, we are done. > Let $x$ be an element of $f(A)\cap f(B)$. Then $x\in f(A)$ and $x\in f(B)$. > That is, there exists $y\in A$ such that $f(y)=x$ and there exists $z\in B$ > such that $f(z)=x$. Since $f$ is an injection, $f(y)=x$ and $f(z)=x$, we > have that $y=z$. We would like to find $u\in A\cap B$ s.t. $f(u)=x$. But > $u\in A\cap B$ if and only if $u\in A$ and $u\in B$. Therefore, setting > $u=y$, we are done. The ‘human-style’ output demonstrated by these two examples is the most visible novel feature of the program, since the problems it solves are all well within the capacity of existing programs. However, this novelty reflects a less obvious novelty in the way that the program operates. It would not be able to produce human-style output if it did not mirror very closely the way that human mathematicians think. We envisage that the program will be just the first in a sequence of programs, and we hope that as a result of our detailed attention to human thought processes, future programs in the sequence will be able to solve problems that have not previously been solved by fully automatic provers. The structure of this paper is as follows. We begin with a brief discussion of why there has not been more interaction between mathematicians and researchers in automatic theorem proving. Over the following three subsections, we talk about human-oriented theorem proving and explain why we see this tradition as appropriate for what we are trying to do. We then discuss previous work on natural-language output for theorem-proving programs. In the second section of the paper, we explain how our program works. We start by informally presenting an example of the program in action. This is followed by a description of the ‘moves’ that the program makes. We then present a second example, this time discussing more explicitly why the program does what it does. We end the section by explaining how the program writes up its thoughts to produce the kind of output shown above. In the third section, we describe an experiment that we conducted in order to test whether the output of the program was hard to distinguish from genuine human mathematical writing. We conclude with a brief discussion of our future plans. ### 1.2 Mathematicians’ attitudes to automated provers We come to this project as mathematicians who are fascinated by the question of how people manage to think of proofs of difficult theorems. Since this question would seem to be fundamental to the whole enterprise of mathematical research, one might think that large numbers of mathematicians would have studied it. However, although mathematicians certainly have thought about their thought processes – the most famous example being Polya polya1957solve ; polya1954mathematicsi ; polya1954mathematicsii – far less attention is paid to the topic than one might expect, and papers are routinely written in a style that appears to do its best to conceal how the ideas they contain were discovered. The result is that the work of analysing and understanding in detail how mathematicians arrive at proofs has to date been mainly carried out by the automatic theorem proving community. Unfortunately, this work has been largely ignored by mathematicians: regrets about this have been expressed in many places in the literature. In a recent article, Bundy discussed this indifference and suggested several reasons for it bundy2011 . From his reasons one general theme emerges: the mathematical style of current automated provers is very different from the style of human mathematics. This is not just a question of the language such provers use to express problems and their solutions, but also the nature of the solutions themselves, which are often very long and use low-level arguments where a human mathematician would use high-level arguments. Bundy suggests that things are changing (in response to the last point he suggests using ‘hiproofs’, a hierarchical presentation of proofs that allows the reader to look at them at several different levels of granularity) and that it is inevitable that automated provers will in due course become an indispensable tool for mathematicians. We agree with this, and one of our motivations is to do what we can to hasten this development. What could induce mathematicians to be interested in automated provers? In some cases, the answer is easy: there are mathematics problems that require large searches or enormously complicated calculations, and computers are better at these tasks than humans. But most mathematics is not like that, and for the part that is not, there is a difficulty. Current automated provers are still a long way from being able to answer questions (except of the specific kind just mentioned) that might arise in a typical research project, so the task for the moment is to teach computers to solve much easier questions of a kind that mathematicians can easily do by hand. This task, though it has considerable intrinsic interest, has no immediate payoff for mathematical researchers, so if one wants input from mathematicians who are not specialists in automated theorem proving, then the barrier to entry will have to be very low. For this reason it would be highly desirable to have an automated prover with the following properties. 1. 1. (_User-friendly input._) One can input problems without needing to learn a special-purpose formal language. 2. 2. (_User-friendly output._) The program will output solutions expressed in the language that mathematicians customarily use. 3. 3. (_Informative solutions._) The solutions are not just certificates of truth: they are arguments of the kind that a good human mathematician might produce. 4. 4. (_Easy extendability._) One can increase the power of the program by adding data in the form of facts, problem-solving tips, etc., rather than having to rewrite the entire program. In short, it would be wonderful to have an automated theorem prover with which one could interact in much the way one interacts with a human mathematician. This is an ambitious goal that will be difficult to achieve, but if it can be achieved, then it is likely that far more mathematicians will become interested in automatic theorem proving. ### 1.3 Human-oriented theorem proving: some background Since the early days of automated theorem proving, there have been two competing broad approaches. The first, _human-oriented_ , tradition focuses on analysing human methods and replicating them in programs. The second, _machine-oriented_ , tradition instead relies on the brute strength of computers, using extensive search to solve problems in ways that a human would find difficult to replicate. Given our long-term goal of helping to create a program that mathematicians will want to use, it is natural that our own work should belong firmly in the human-oriented tradition. However, this tradition is distinctly out of favour at present; for example, if you look at the range of solutions to the standard TPTP test library automated theorem provers tstp1 ; tstp2 , you could be forgiven for not realizing that the human-oriented tradition had ever existed. Since the human-oriented tradition is less well known these days, we shall briefly describe it here. Our account is not intended to be comprehensive: we focus on the strands that most closely relate to our own work. More detailed accounts of the various periods in theorem proving may be found in bledsoe1988survey , wos1985overview , mackenzie2001 , pp. 464–469 of harrison2009book , the introduction to bundy1983 , bundy1999survey , and cordeschi1996 . The earliest human-oriented prover was Newell and Simon’s 1956 _Logic Theory Machine_ newell1957logic , which proved theorems from Russell and Whitehead’s Principia Mathematica russell1910 , and is often considered to be the first artificial intelligence program. Its methods were directly modelled on those of humans: Newell and Simon studied their own behaviour when proving theorems, and abstracted out _heuristics_ which they then implemented in their program. Their goals were wider than those of contemporary theorem proving: they hoped not just to prove theorems but also to ‘[apply] research on complex information-processing systems … [to] human learning and problem solving’. Indeed, they used the successor to the Logic Theory Machine, their _General Problem Solver_ newell1959gps , as the foundation of a book entitled _Human Problem Solving_ newell1972human . In the subsequent decade, much of the work on automated theorem proving, including the General Problem Solver and Gelernter’s _Geometry Theorem Proving Machine_ , continued in the heuristic, human-oriented tradition, sometimes drawing directly on mathematicians’ analyses of their own methods in works such as polya1957solve ; polya1954mathematicsi ; polya1954mathematicsii . This trend was arrested by Robinson’s highly influential paper _A Machine-Oriented Logic Based on the Resolution Principle_. Robinson’s _resolution_ was a single, simple inference rule which turned out to be dramatically more effective than previously existing methods. Unlike the earlier methods, resolution had no natural human correlate: formulae needed to be converted into a not very readable ‘normal form’ before resolution could be applied, and actual resolution proofs were often difficult or impossible for humans to follow. However, the hugely increased efficiency of resolution meant that these were minor concerns. The next few years saw an explosion of research into resolution. Numerous improvements were rapidly found, and it seemed at the time that a suitably advanced version of resolution might solve the problem of automated theorem proving well enough for all practical purposes. As a result, human-oriented work was more or less abandoned: while a handful of researchers, including Newell and Simon, remained focused on human cognition, the majority were concerned with problem-solving effectiveness and so devoted their attention to resolution. According to mackenzie2001 , the optimism of those times was somewhat dimmed by the discovery of NP-completeness cook1971complexity ; karp1972reducibility , and the associated realization that it was unlikely that there was a complete, polynomial time algorithm for theorem proving. Nevertheless, the majority of researchers continued to focus on machine-oriented methods throughout the 70s. The notable exception to this was a group run by Woody Bledsoe at the University of Texas in Austin. Bledsoe’s first steps in the direction of human-oriented theorem proving may be found in bledsoe1971splitting , which uses top-level heuristic techniques to split a problem into smaller subproblems which are then solved by resolution. This paper was soon followed by a 1972 paper on limit theorems bledsoe1972limit , written by Bledsoe and his students Boyer and Henneman. They took an existing system bledsoe1971splitting and replaced resolution with a new procedure, which he described as ‘[bearing] a closer resemblance to the proof techniques of a mathematician than does resolution.’ Among other results, this procedure was able to prove that differentiability implies continuity and that the limit of a product of functions is the product of the individual limits. In the light of the fourth desirable property mentioned above, one feature of the procedure that is worth highlighting is that it ‘[did] not release its action unless its need is detected’; that is, it did not interfere with those proofs in which was not potentially useful. This is in sharp contrast to resolution, where adding irrelevant axioms can significantly degrade performance. The paper notes that such heuristics ‘should be sought for other areas of mathematics’. Five years later, Bledsoe wrote a seminal paper on mathematical human-oriented proving, _Non-resolution theorem proving_ bledsoe1977nonresolution . In it, he described himself as one of a number of researchers who had ‘made the switch’ after finding that resolution and related techniques had encountered enormous difficulty in solving problems that a human would find easy, and that substituting human techniques resulted in a program which ‘easily succeeded’. Bledsoe expressed the opinion that “purely syntactic” methods such as resolution had reached a plateau, and that further progress would not be made without using more human techniques. He described the key to progress as somehow managing to use ‘the knowledge accumulated by humans over the last few thousand years, to help direct the _search_ for proofs’. The paper went on to outline the main concepts used in human-oriented proving at the time. A number of these are very relevant to the work we describe below, including the use of rewrite rules, forward chaining, typing, a reluctance to expand definitions unless necessary, and the use of ‘natural’ or ‘goal-oriented’ systems. Bledsoe continued to work on human-oriented proving for the remainder of his life; notable achievements include the proof of a number of results using non- standard analysis bledsoe1977automatic , a proof of the intermediate value theorem bledsoe1977setvariables , the use of examples to guide proofs bledsoe1983usingexamples , and analogical proof construction bledsoe1995analogy . Reading through these papers, one has no sense that human-oriented proving had reached a plateau: Bledsoe’s approach to automating the discovery of proofs of mathematical theorems continued to be fertile throughout his life. Unfortunately, the ‘AI winter’ meant that funding for mathematical theorem proving dropped sharply soon after Bledsoe ‘made the switch’. As a result, many of Bledsoe’s students were forced to move to areas with more direct commercial relevance than mathematics, with the result that the Bledsoe tradition in mathematical ATP effectively petered out. Bledsoe’s students did, however, carry his human-oriented tradition into other areas of theorem proving. Of particular note are Robert Boyer and J. Strother Moore, who created a family of provers that are used to this day, primarily for software and hardware verification. Although many features of the Boyer-Moore provers, NQTHM and ACL2, were driven by the needs of software verification as opposed to mathematics, two features of these provers are significant in a mathematical context. First, Boyer and Moore emphasise the importance of creating a single program that could be used to tackle a range of problems, whereas Bledsoe constructed a sequence of loosely related provers, each intended to tackle different problems. Second, based on intensive inspection of their own proof techniques, Boyer and Moore introduced an overall architecture, the ‘waterfall’ architecture, which was much more systematic than the architectures introduced by Bledsoe. The Boyer- Moore waterfall consisted of four heuristics, ordered in terms of their relative attractiveness to humans. A Boyer-Moore prover repeatedly applies the least risky heuristic that can be used, terminating when the result is proved or when no heuristic can be applied. The Boyer-Moore provers partially inspired work in the next chapter of human- oriented theorem proving, the work done by the group run by Alan Bundy at the University of Edinburgh. Bundy had been influenced by real, human mathematics from the beginning of his career bundy1975analysing ; bundy1985discovery , and reports a Bledsoe-like experience of disillusionment with machine oriented methods bundy1983book , but his major achievements in human-oriented proving began with a later attempt to reconstruct the behaviour of the Boyer-Moore prover within a more systematic framework bundy1988proofplans . This led him to formulate the notion of a _proof plan_ , a computational representation of a high-level heuristic used by human mathematicians. Because one application of a proof plan corresponded to many smaller reasoning steps, a theorem prover that used proof plans could operate with a dramatically reduced search space. This often led to a proof discovery process that was very similar to that of a human mathematician. There is an extensive literature on proof plans, which we will not discuss here, since proof plans are not explicitly used in the work described below. A description and evaluation of this work may be found in bundy2000critique . Meanwhile, on the machine side there was a notable triumph: a result with a strong claim to be the first solution by an automated theorem prover of an unsolved problem in mainstream mathematics. The problem was the Robbins conjecture, that an algebra that obeyed a certain axiom must be a Boolean algebra, which had been open for six decades when William McCune proved it in 1996 using an automated theorem prover called EQP (for ‘equational prover’) and eight days of computer time mccune1997solutionof . This might appear to indicate that the two aims of producing more powerful provers and of producing provers that will be easy to use by human mathematicians are in conflict. We do not believe that this is the case, for reasons that we shall now explain. ### 1.4 Doing more with less As we have already said, if one wants to write a program with which mathematicians can interact easily, it is natural to be drawn to the human- oriented tradition. However, as the history of that tradition makes clear, that is by no means the only motivation. There is also a conviction amongst proponents of human-oriented methods that in order to build provers that can solve certain kinds of problems – roughly speaking, the problems that human mathematicians typically deal with, apart from the ones that need a lot of search or heavy calculation – it is _necessary_ to come to grips with human problem-solving techniques. For example, Bundy says the following in his influential _Science of Reasoning_. > Although our science of reasoning might find application in the building of > high performance automatic theorem provers, the two activities are not co- > extensive. They differ both in their motivation and their methodology. > > I take the conventional motivation of automatic theorem proving to be the > building of theorem provers which are empirically successful, without any > necessity to understand why. The methodology is implied by this motivation. > The theorem prover is applied to a random selection of theorems. > Unsuccessful search spaces are studied in a shallow way and crude heuristics > are added which will prune losing branches and prefer winning ones. This > process is repeated until the law of diminishing returns makes further > repetitions not worth pursuing. The result is fast progress in the short > term, but eventual deadlock as different proofs pull the heuristics in > different directions. This description is something of a caricature. No ATP > researchers embody it in its pure form, but aspects of it can be found in > the motivation and methodology of all of us, to a greater or lesser extent. > > Automatic theorem provers based on proof plans make slower initial progress. > Initial proof plans have poor generality, and so few theorems can be proved. > The motivation of understanding proofs militates against crude, general > heuristics with low prescriptiveness and no expectancy. The ‘accidental’ > proof of a theorem is interpreted as a fault caused by low prescriptiveness, > rather then a lucky break. However, there is no eventual deadlock to block > the indefinite improvement of the theorem prover’s performance. If two or > more proof plans fit a theorem then either they represent legitimate > alternatives both of which deserve attempting or they point to a lack of > prescriptiveness in the preconditions which further proof analysis should > correct. > > […] > > Thus, we expect a science of reasoning will help us build better automatic > theorem proving programs in the long term, although probably not in the > short term. Although Bundy’s ‘science of reasoning’ was (at least initially) tightly focused on the notion of a proof plan, we believe that many of his comments apply more generally to the advantages of human oriented proving over machine oriented proving, and especially his conclusion: that in the long term, paying close attention to human methods will pay dividends. There are two main reasons for this. A simple one is that a significant problem with automated theorem provers – some would call it _the_ main problem – is the danger of combinatorial explosion. It is in the nature of solving a complex mathematics problem that one throws up other problems that need solving, which in turn throw up further problems, and so on. If a generous amount of search is permitted, then this recursive nature of problem solving naturally leads to the search being iterated, and thus to a combinatorial explosion. Somehow, in ways that we do not fully understand, humans manage to avoid this difficulty by keeping search strictly under control. Therefore, it seems highly plausible that it will not be possible to design provers that can produce complicated hierarchical proofs without a deep understanding of how humans manage to do so. The second reason is more subtle, and best illustrated by means of an example. Consider the following problem. ###### Problem 1 Let $G$ be a group and let $S$ be a subset of $G$ with the property that if $x\in S$ and $y\in S$, then $xy^{-1}\in S$. Prove that $S$ is closed under taking inverses. This was used as a test problem for a system devised by Reiter in the 1970s reiter1976models . His system was designed to use models to prune search trees: if a statement is false in a simple model, then there is no need to waste time investigating whether it can be proved. It is clear that this reflects an extremely important aspect of human problem solving: before we invest time in proving a statement, we like to feel that that statement has at least some plausibility. Therefore, it is natural to want to incorporate some kind of checking-against-models facility into a theorem proving program. Here, translated for human consumption, is what Reiter’s system does. It begins by taking an element $b\in S$ (if $S$ is empty then the problem is trivial) and sets itself the goal of proving that $b^{-1}\in S$. The one property it has available for use is that $S$ is closed under the operation $(x,y)\mapsto xy^{-1}$, so it uses backwards reasoning to replace the goal by that of finding $x,y\in S$ such that $xy^{-1}=b^{-1}$. At this point, it searches for suitable candidates for $x$ and $y$. It has three statements that these two variables need to satisfy: $x\in S$, $y\in S$ and $xy^{-1}=b^{-1}$. It searches for a solution in a somewhat mechanical way, ordering the conditions that need to be satisfied and then attempting to satisfy them one by one. It begins with the order in which the conditions are presented, so it tries to satisfy the first condition. Since all it knows so far is that $b\in S$, it takes $x=b$. The first statement is now satisfied, so it passes to the second and takes $y=b$ for similar reasons. It is now left needing to prove that $bb^{-1}=b^{-1}$. Instead of wasting time looking for a proof of this false (in general) statement, it checks it against a model such as the Klein four-group, rapidly finding a counterexample. So at this point it backtracks, reorders the statements that need to be satisfied, and ends up attempting to satisfy what used to be the third statement, $xy^{-1}=b^{-1}$, without, for the time being, worrying about whether $x\in S$ and $y\in S$. It finds the simplest solution to this, namely $x=e$ and $y=b$. Now it is left needing to prove that $e\in S$, which is straightforward: it needs $u,v\in S$ such that $uv^{-1}=e$ and the first thing it tries, namely $u=v=b$, works. Thus, Reiter’s system successfully solves the problem, and even does so quite efficiently. However, there are at least two ways in which what it does is strikingly unlike what a human mathematician would do. The first is its mechanical approach to satisfying the conditions on $x$ and $y$. In general, finding an object that satisfies two or more properties is a tricky problem, since it is difficult to focus on more than one condition at a time. Humans will often begin by assessing which is the hardest condition to satisfy, since in practice it often happens that the most natural object that satisfies the hardest condition either already satisfies the easier conditions or can be easily modified to do so. By contrast, Reiter’s program starts by attempting to satisfy the conditions in the order in which they are written down, which leads it to make a wrong guess that no human mathematician would be likely to make. If we are just interested in the performance of one program, this is not much of a problem, since there are only three conditions, so the amount of backtracking it has to do is very slight. However, if we are content with this in the short term, then we miss out on search-reducing techniques that may be useful in the longer term. The second noticeably non-human move made by the computer is to test the statement $bb^{-1}=b^{-1}$ against a model (in Bledsoe’s account of this work, he suggests using the Klein 4-group). To a typical human mathematician, this statement is just _obviously_ false in general. Of course, it is not much help to a theorem proving program to be told that a statement is ‘just obvious’, so let us look more closely at why it is obvious. The answer is that we see very easily that, by the cancellation law, if $bb^{-1}=b^{-1}$ then $b=e$. And even at that point we say to ourselves something like, “Not all groups are trivial,” rather than “The cyclic group of order 2 is a counterexample.” Thus, if we are content with Reiter’s method, then we miss out on two further search-reducing strategies: to simplify statements before deciding whether they are likely to be true, and to make the working assumption that a sufficiently simple general statement that is not obviously true is almost certainly false. That is not all that can be extracted from a close look at how a human would tackle this problem. Reiter’s system always reasons backwards, but a human might well look at the hypothesis that $S$ is closed under the operation $(x,y)\mapsto xy^{-1}$ and reason that since the only element we know to be in $S$ is $b$, there is no hope of solving the problem without making the deduction that $bb^{-1}$ is an element of $S$, and therefore that $S$ contains the identity $e$. And once one has made that deduction, finding $x$ and $y$ in $S$ such that $xy^{-1}\in S$ is significantly easier than it was before. So we have another potentially useful technique: if a statement appears to be essential and can be applied in only one way, then there is no harm in applying it, even if you cannot see what good it will do. These remarks are not intended as a criticism of Reiter’s program, which was successful at what it was trying to do. But they illustrate the general point that the closer the attention one pays to how humans solve problems, the more useful information one can extract. This gives us another reason for trying to make our programs as human as possible. Indeed, we would describe our work as belonging not just to the human-oriented tradition, but as belonging to the extreme human end of the machine-human spectrum. In practice, this means deliberately not allowing ourselves to exploit the speed of computers, for example by letting them carry out large searches or perform very complicated calculations. We hope that by submitting to this restriction, we will force ourselves to develop a number of useful and important techniques while the problems we are tackling are still relatively simple. ### 1.5 Two kinds of proof The arguments of the previous section are intended to show that there are potential advantages to taking a strict human-oriented approach to automated theorem proving. However, there is no denying that there are also advantages to using raw computer power. And the fact that it is the machine-oriented approach that has solved a decades-old unsolved research problem suggests that its advantages outweigh those of the human-oriented approach. Perhaps the situation with theorem proving is like the situation with chess, where the brute force of the computer beats the advanced pattern-recognition skills of the human grandmaster. Before we jump to that conclusion, however, we should consider another possibility: that machine-oriented methods are better for finding certain kinds of proofs, and human-oriented methods are better for finding other kinds. The evidence suggests that this is the case. As Kerber puts it kerber1998 , when talking about machine-oriented programs, “The strength of these systems is truly remarkable and even open mathematical problems, like recently the question whether Robbins algebras are Boolean algebras, can be solved with the assistance of an automated theorem prover. On the other hand, observing the blind search behaviour of such a system as it fails to solve a problem that seems trivial to us as humans can be disappointing.” This phenomenon is not too surprising. Some proofs seem to consist of a succession of somewhat arbitrary and unpredictable steps, while others can be discovered by means of what mathematicians would describe as ‘key ideas’. The proof of the Robbins conjecture comes into the first category: one can think of it as a combinatorial object that belongs to a vast space – the space of sequences of well-formed statements that satisfy certain simple rules – inside which we are looking for a sequence that ends in a certain way. There are no obvious measures of progress that tell us that some initial segments of sequences are ‘obviously right’ or at least ‘getting warmer’ (though it is conceivable that somebody might one day find a more conceptual argument), and in the absence of such clues there is not much for it but to undertake a huge search. It is therefore only to be expected that machine-oriented methods have outperformed human-oriented methods for that problem and for others of a similar kind. However, for the majority of proofs that mathematicians find, there is some kind of ‘story’ to tell of the ideas that give rise to the proof. Typically, such a story will be a high-level overview of the main difficulty and how it is overcome, where ‘overcome’ means that the problem is reduced to one or more problems where that difficulty no longer occurs. Often this reduction is achieved by means of a well-chosen intermediate statement that turns out to follow from the initial assumptions and imply the conclusion. The intermediate statement itself is typically found not by means of a brute-force search but by a process of approximation: one might make a guess, find that it is unhelpful, understand why it is unhelpful, and use that understanding to guide the search for a better intermediate statement. These characteristically human techniques enable mathematicians to penetrate deep into ‘proof space’, but the set of proofs that can be discovered in this way forms a tiny fraction of that space. It seems almost a truism that human methods will be useful for programs that want to find these special proofs that human mathematicians are so mysteriously good at finding. ### 1.6 Systems with natural-language output So far, we have discussed the desirability of a program that will reason in as similar a way as possible to the way that humans reason. Another desirable property that we mentioned earlier is that the input and output of the program should be in the language that mathematicians already use. Several systems have been developed that use natural language to a greater or lesser extent. An early example with some similarity to ours is that of Felty and Miller from 1987 feltymiller1987 . They start with a proof tree and convert it into a more readable form. Their system can also make significant changes to how a proof is presented. The following is an example of output from their system: it is a proof that there are infinitely many primes. The function $f$ mentioned in the proof can be taken to be the function defined by the formula $f(n)=n!+1$: then the beginning of the proof is asserting some properties of this function that are subsequently used in the proof (so any other function with those properties would do just as well). > Assume $\forall x(f(x)>x)\wedge\forall x\forall > y(\mathrm{div}(x,f(y))\supset(x>y))\wedge\forall > x(\neg\mathrm{prime}(x)\supset\exists > y(\mathrm{prime}(y)\wedge\mathrm{div}(y,x))$. We have two cases. Case 1: > Assume $\neg\mathrm{prime}(f(a))$. By modus ponens, we have $\exists > y(\mathrm{prime}(y)\wedge\mathrm{div}(y,f(a)))$. Choose $b$ such that > $\mathrm{prime}(b)\wedge\mathrm{div}(b,f(a))$. By modus ponens, we have > $(b>a)$. Hence, $(b>a)\wedge\mathrm{prime}(b)$. Thus, $\exists > x((x>a)\wedge\mathrm{prime}(x))$. Case 2: Assume $\mathrm{prime}(f(a))$. > Hence, $(f(a)>a)\wedge\mathrm{prime}(f(a))$. Thus, $\exists > x((x>a)\wedge\mathrm{prime}(x))$. Thus, in either case, we have $\exists > x((x>a)\wedge\mathrm{prime}(x))$. Since $a$ was arbitrary, we have $\forall > n(\exists x((x>n)\wedge\mathrm{prime}(x)))$. They describe their mechanism for converting the original tree-structured deductions into readable natural-language text as very simple. It is clear that with some small changes they could have improved the readability. For example, they could have replaced $\mathrm{prime}(x)$ by ‘$x$ is prime’, $\mathrm{div}(x,y)$ by $x\|y$ and the symbols for connectives by English words. However, the result would still have had some slightly odd characteristics – for instance, no human mathematician would bother to write ‘by modus ponens’ – that would have betrayed its mechanical origins. Another program that produced readable text was written by Holland-Minkley, Barzilay and Constable in 1999 hollandminkley1999verbalization . Their aim was to create natural-language output from the Nuprl system. This is an interactive system based on tactics – that is, high-level inference steps – which is designed to mimic human reasoning. The output from the Nuprl system is not at all easy for the untrained mathematician to read. However, they could convert it into language that was considerably closer to what a human mathematician might write, as the following sample demonstrates. > Theorem: For integers $a$ and $b$ and natural number $c$, $(a-- > b)--c=a--(b+c)$. > > Consider that $a$ and $b$ are integers and $c$ is a natural number. Now, the > original expression can be transformed to > $\mathrm{imax}(\mathrm{imax}(a-b;0)-c;0)=\mathrm{imax}(a-(b+c);0)$. From the > add com lemma, we conclude > $\mathrm{imax}(-c+\mathrm{imax}(a+-b;0);0)=\mathrm{imax}(a+-b+-c;0)$. From > the imax assoc lemma, the goal becomes > $\mathrm{imax}(\mathrm{imax}((a+-b)+-c;0+-c);0)=\mathrm{imax}(a+-b+-c;0)$. > There are 2 possible cases. The case $0+-c\leq 0$ is trivial. Consider > $0<0+-c$. Now, the original expression can be transformed to > $\mathrm{imax}((a+-b)+-c;0+-c)=\mathrm{imax}(a+-b+-c;0)$. Equivalently, the > original expression can be rewritten as > $\mathrm{imax}((a+-b)+-c)=\mathrm{imax}(a+-b+-c;0)$. This proves the > theorem. In many ways this looks like the kind of continuous prose that a mathematician would write, though as with Felty and Miller’s system there are a number of telltale signs of the mechanical origins of the text. For instance, the first sentence is not quite grammatical: a human would write, ‘Let $a$ and $b$ be integers and let $c$ be a natural number.’ There is also the trivial point that mathematicians would write ‘max’ rather than ‘imax’ (trivial because it would be very easy to change this). There is also a repetitive quality to the prose that gives it an automatically generated feel: for instance, two sentences open with ‘Now, the original expression can be transformed to’. Another system that is often promoted for the readability of its output is MIZAR. Although this is not an automatic theorem prover, we show an example of its output for the sake of comparison. The following is a formalized version in MIZAR of Euclid’s proof that there are infinitely many primes. [From http://www4.in.tum.de/~wenzelm/papers/romantic.pdf .] reserve n,p for Nat; theorem Euclid: ex p st p is prime & p > n proof set k = n! + 1; n! > 0 by NEWTON:23; then n! >= 0 + 1 by NAT1:38; then k >= 1 + 1 by REAL1:55; then consider p such that A1: p is prime & p divides k by INT2:48; A2: p <> 0 & p > 1 by A1,INT2:def 5; take p; thus p is prime by A1; assume p <= n; then p divides n! by A2,NATLAT:16; then p divides 1 by A1,NAT1:57; hence contradiction by A2,NAT1:54; end; theorem p: p is prime is infinite from Unbounded(Euclid); Although this is quite clearly written in a formal language, it is natural enough that with a bit of effort, one can follow the steps of the argument, at least if one knows the argument already. And if readability were the only concern, then once again there would be simple ways of improving the output. For example, one could remove all the ‘by X’ justifications at the ends of lines. Of course, readability is _not_ the only concern when a proof is written in MIZAR. More generally, there are major differences between our aims and the aims of those who formalize mathematical proofs. Our aim is to write fully automatic programs with output that meets the typical standards of soundness of human mathematicians, whereas the proof-verification community aims to use a great deal of human interaction to produce completely formalized proofs. As we have already mentioned, we would ultimately like to contribute to the creation of a program that mathematicians can use with little or no effort. It therefore makes sense to set very high standards for the readability of our output. Another reason for doing this is that we would like to be confident that our programs really are thinking in a human way. A natural test of this is that it should be able to produce output that is similar to what a human produces. However, passing that test is not by itself sufficient evidence, since in principle one might be able to create a program that discovers human-style proofs with the help of a highly mechanical and non-human discovery process. Such a program could then hide its thought processes and produce nice readable write-ups. We therefore want to add an extra condition: we would like our programs to create human-style write-ups _while being faithful to its thought processes_. By this we mean that the steps that the program takes in order to discover the proofs it discovers should be directly translated into human language and should be presented in the order in which they are made. For example, if a program discovers a short argument after a long search, then the long search should be presented and not just the short argument. There is an obvious objection to this, which is that not even human mathematicians do it. For example, often they will take all sorts of wrong turns when searching for proofs, and these will not be recorded in the write- ups. And often they will present, as if out of nowhere, an object that just happens to make a proof work, hiding from view the process by which they thought of it. This is an important point, but the proofs we will discuss in this paper are not sufficiently difficult to find for the issue to arise in a significant way. For the proofs the program finds, it is possible to produce output that is faithful to the program’s thought processes and also quite close to what a human would write as the final product. When we come to tackle more complex problems, we plan to have two styles of output. One will be a proof as a human might write it, and the other will be more like an account, again as a human might write it, of the proof discovery process. We believe that both styles are potentially useful to mathematicians. It is one thing to claim that a program produces output that is similar to what a human might write; it is another to provide evidence to back up that claim. Later in the paper, we shall describe an experiment that we carried out for this purpose. ### 1.7 Soundness A further motivation for producing human-style write-up is that it gives us a mechanism for checking soundness: if the program produces proofs that a human mathematician finds acceptable, then it is doing what we ask of human mathematicians. This attitude to soundness is in sharp contrast with most research in automated theorem proving, especially of the interactive variety. There what is sought is a _guarantee_ of soundness, which is obtained by carefully building up a corpus of statements from a small set of axioms, using only basic rules of logic and statements that have already been established. There are very good reasons for this: for example, such systems can handle extremely complicated proofs that are very difficult for humans to check by hand, a notable recent example being Gonthier’s machine-checked proof of Thompson’s odd-order theorem gonthier2013machine . They can also check the soundness of computer hardware, where an absolute guarantee is essential. However, for a program that produces proofs of relatively easy theorems, considerations of this kind do not apply. Therefore, we take the attitude to soundness that a typical human mathematician takes: we want it to be clear to a human mathematician that the steps the program takes are sound, and we want the results that the program assumes to be ones that are well established and in some sense ‘prior’ to the results being proved. Although this does not provide a formal guarantee of soundness, there are no examples of well established mathematical results of the level of simplicity we are dealing with that have been found to be fundamentally wrong, so for all practical purposes it is a sufficient assurance of soundness. ### 1.8 The other criteria: user-friendly input and easy extendability The remaining two desirable criteria that a mathematician-friendly program should have were that it should be possible to input problems without having to learn a formal language, and that it should be possible to improve the performance of the program by adding data. It is much harder to write a program that accepts natural-language input than it is to write a program that produces natural-language output, since for any given problem there will be many ways of writing it, and the program has to be able to handle all of them – or at least, enough of them to make the program not too restrictive. For the time being, our input takes a form that is not written in natural language, but is fairly straightforward to write. (For example, to request that the program show that the pre-image of an open set $U$ under a continuous function $f$ is open, we supply hypotheses continuous(f) and open(U) and a target statement open(preimage(f,U)). A more complicated example is forall x epsilon.(in(x,B) => exists y.(in(y,A) & lessthan(d(x,y), epsilon))).) However, Barnet-Lamb and Ganesalingam have recently written a program that is capable of processing many different statements. Roughly speaking, any mathematical statement that is written at the level of formality that is typical of the more formal parts of mathematical textbooks has a good chance of being understood. So it may be that future versions of the program will be able to accept natural-language input as well. As for the possibility of making the program more powerful by adding more data to it, we are optimistic about this, but it would be premature to make any claims in this direction. In general, adding mathematical information to a theorem-proving program can be problematic, because it gives the program more choice. If the program makes significant use of search, then more choice can make the searches bigger, and in practice this has meant that some programs have performed _worse_ when they have more information to go on. The program has not had this problem, but its library is so small that we cannot at this stage claim that it never will. The reason we are optimistic is that our methodology, if we apply it sufficiently rigorously, should ensure that the problem does not arise. If a program threatens to get bogged down searching through all sorts of irrelevant results, then that will be a sign that we have not understood how human mathematicians select relevant results from the large amount that they know. Understanding that may be very challenging, but if we cannot fully understand it, then we will try to reach a partial understanding and make use of that. In other words, we will try to find sufficient conditions for a use of the program’s data to be human-like, and allow uses that satisfy those conditions. If the program only ever uses its data in a human way, then adding data should not degrade its performance, since it does not degrade the performance of human mathematicians. ## 2 The program ### 2.1 Solving routine problems in a fully human way The ‘extreme human’ approach we are trying to adopt can be summarized as follows: _we do not allow our programs to do anything that a good human mathematician wouldn’t do_. A serious difficulty in trying to design a program that satisfies this constraint is that while humans do their very best to avoid search and backtracking, they undoubtedly do at least _some_ search and backtracking when they are trying to solve difficult problems. What is the distinction between the kind of search that humans do and the kind that humans would never do? Indeed, _is_ there a distinction, or is it just a matter of degree? These are difficult questions, and rather than try to answer them immediately, it seems more sensible to try to isolate them by concentrating first on other difficulties and facing up to this one only when we are forced to do so. Accordingly, we define a _routine_ problem to be one that a good human mathematician will typically solve easily without backtracking. If a program is to satisfy the main constraint, then it too will have to solve routine problems without backtracking, so we can simply ban it. Although this is a significant scaling down of ambition, it also makes the project far more realistic in the short term. Furthermore, the class of routine problems is large and diverse enough that the challenges it raises are still very interesting. ### 2.2 Example: closed subsets of complete metric spaces Before we describe our program, let us look at a routine problem and examine how a human mathematician would typically solve it. The problem is to show that a closed subset of a complete metric space is complete. We can state it more formally as follows. ###### Problem 2 Let $X$ be a complete metric space and let $A$ be a closed subset of $X$. Prove that $A$ is complete. The proof discovery process would usually be something like this. 1. 1. _[Clarify what needs to be proved.]_ We must show that every Cauchy sequence in $A$ converges in $A$. 2. 2. _[We must show something about _every_ Cauchy sequence, so pick an arbitrary one.]_ Let $(a_{n})$ be a Cauchy sequence in $A$. 3. 3. _[Clarify what now needs to be proved.]_ We are trying to show that $(a_{n})$ converges in $A$. 4. 4. _[See what we can say about the sequence $(a_{n})$.]_ The sequence $(a_{n})$ is a Cauchy sequence in the space $X$, and $X$ is complete; therefore $(a_{n})$ converges in $X$. 5. 5. _[Give a name to the object that we have just implicitly been presented with.]_ Let $x$ be the limit of the sequence $(a_{n})$. 6. 6. _[See what we can say about $x$.]_ But $A$ is closed under taking limits, so $x\in A$. 7. 7. _[Recognise that the problem is solved.]_ Thus, $(a_{n})$ converges in $A$, as we wanted. Our program is designed to imitate these typical human moves as closely as possible. The result is that what it does can be straightforwardly translated into an account of its thought processes that could pass for an account of human thought processes such as the ones laid out above (without the accompanying commentary). The architecture in which moves are applied is very similar to that of a LCF- style interactive theorem prover gordon1979lcf . In simple cases, the program state consists of a list of statements that can be assumed, which we call _hypotheses_ , and a list of statements to be proved, which we call _targets_. Each statement is basically a formula of the (many-sorted) first-order predicate calculus, but it can also carry annotations that record information that has been accumulated by the program; for example, a statement may be tagged to indicate that it has already being used during the course of the proof. Although information of this kind is logically unnecessary, it is indispensable in human reasoning and therefore plays an important role in our automated theorem prover. The representation diverges from that of a LCF-style prover in more complicated cases, most notably when a conjunctive target needs to be deduced from the hypotheses. In this case a LCF-style prover would use a tactic such as replacing the original goal by two new goals, each of which lists the hypotheses separately. By contrast, our system represents and displays the hypotheses only once. This is not done for reasons of efficiency, but rather to mimic the natural human representation. A human would think in terms of one ambient primary collection of ‘facts that are known’ (the hypotheses) and a number of targets that need to be deduced from these; the program reflects this. In some cases, reasoning will introduce a fact that may be used in the proof of only one of the targets (and the program represents this appropriately), but this is very much the marked (i.e. non-default) case and should not affect the representation of the majority of ‘universally available’ hypotheses. An individual _move_ is an operation that transforms a specific problem state into another state in a sound fashion; thus individual moves correspond to application of tactics to a specific LCF-style prover state. However, we insist that our program’s moves faithfully model cognitive processes of human mathematicians. Thus, many sound transformations of the problem state are not acceptable as moves, because they correspond to operations that humans would never perform. The closest correlate we have to a tactic (as opposed to the application of a tactic to a specific state) is a _generator_ for a _move type_. An example of a _move type_ would be ‘forwards reasoning’. Each move type has an associated _generator_ which, given a state, returns a list of moves from that state. So for example the generator for the ‘forwards reasoning’ move type accepts a state and returns a list of all the ‘forwards reasoning’ moves that can be made from that state. Note that, unlike tactics, generators may return a whole list of moves. Because the prover is fully automated, the list of move types is fixed. Move types are ranked in order of attractiveness or _priority_ , and the basic operation of the program consists of repeatedly choosing the most attractive move type that can be applied, generating the moves of that type, and applying the most attractive one. In practice with the problems we are considering, it is rare for the most attractive move type to generate more than one move, except in cases where the moves generated are related by some simple symmetry. Note that this architecture closely resembles the Boyer-Moore ‘waterfall’ architecture boyermooreacl discussed in §1.3. The waterfall consisted of four heuristics, ranked in order of their relative attractiveness to humans. A Boyer-Moore prover repeatedly applies the most attractive heuristic that can be used, terminating when the result is proved or when no heuristic can be applied. This directly corresponds to the way in which our prover repeatedly chooses the most attractive move type possible, terminating when the result is proved or when no no moves can be made. We shall now describe what our program does when it tackles the example problem given above, and then explain in more detail what the move types are, and what the order of priority is. Like Boyer and Moore, we chose the move types and their priority by examining our own reactions to many different problems. The initial problem state for this problem is as follows. H1. $X$ is a complete space H2. $A$ is closed in $X$ T1. $A$ is complete The first thing the program does is expand the definition of “$A$ is complete”. H1. $X$ is a complete space H2. $A$ is closed in $X$ T2. $\forall(a_{n})$ $(a_{n})$ is Cauchy $\wedge\ (a_{n})$ is a sequence in $A\Rightarrow(a_{n})$ converges in $A$ Next, it does a move that corresponds closely to the human move of picking an arbitrary Cauchy sequence: it gets rid of the universal quantifier and places the premises of the resulting conditional statement above the line, leaving the conclusion below the line. A small technical point is that we insist that it combines these two operations into a single move, because we do not allow “bare” conditionals: that is, statements of the form $P(x)\Rightarrow Q(x)$ that are not universally quantified. (One reason for doing this is that such statements almost always seem very strange and unnatural to humans. However, we have also found situations where it helps the program to avoid doing genuinely bad moves.) H1. $X$ is a complete space H2. $A$ is closed in $X$ H3. $(a_{n})$ is Cauchy H4. $(a_{n})$ is a sequence in $A$ T3. $(a_{n})$ converges in $A$ Next comes an operation that fits under the general heading of _forwards reasoning_ : from the assumptions that $(a_{n})$ is Cauchy and $X$ is complete, it follows that $(a_{n})$ converges. Of course, this deduction can be broken up: one would expand the statement “$X$ is complete” to say that every Cauchy sequence in $X$ converges, then substitute $(a_{n})$ for the universally quantified sequence, and finally use modus ponens to deduce that $(a_{n})$ converges. However, a human mathematician would do all this in one step, and therefore so does our program. Of course, to do it the program must have _access_ to the expansion of “$X$ is complete”, which indeed it does: it has a library of definitions and basic facts that it can make use of whenever it wants. This models the collection of definitions and facts that a human mathematician would store in his or her long-term memory. After the step, the problem state is this (except that there will be tags on the statements H1 and H3 to indicate that they have been used – to keep the problem states easy to read we are not displaying the tags in this discussion). H1. $X$ is a complete space H2. $A$ is closed in $X$ H3. $(a_{n})$ is Cauchy H4. $(a_{n})$ is a sequence in $A$ H5. $(a_{n})$ converges T3. $(a_{n})$ converges in $A$ At this point, it will be clear to a human that the hypothesis H1 has been ‘used up’. That is, it has played its role and will almost certainly not be required again. It is not easy to work out precise necessary and sufficient conditions for when humans judge a statement to be used up in this sense, but we have identified some conditions that appear to be sufficient. The hypothesis H1 satisfies those conditions and is therefore deleted. A human mathematician would also be confident that hypothesis H3 has been used up. Our program does not delete H3 because it does not know that the Cauchy assumption will not be expanded later. This we regard as an imperfection of the program that needs at some point to be corrected. (One possible justification for deleting H3 is that it is implied by H5. Whether this is the ‘right’ reason remains to be worked out.) Deleting statements turns out to have no effect on the performance of this program, so there might seem to be no point in worrying about it. However, we remain of the view that we should try to model _all_ aspects of human problem solving. A potential benefit of thinking about the discarding of hypotheses is that it is a special case of a more general issue that undoubtedly _will_ be important for performance when the problems get harder: dealing with irrelevant statements. Often with complicated problems one is presented with more information than one needs, and it is important to decide which out of all the statements available are likely to be useful. H2. $A$ is closed in $X$ H3. $(a_{n})$ is Cauchy H4. $(a_{n})$ is a sequence in $A$ H5. $(a_{n})$ converges T3. $(a_{n})$ converges in $A$ The next move applied by the program is to expand hypothesis H5. The reason the program likes this move is that the expansion of the hypothesis begins with an existential quantifier, so expanding it gives us a new object to work with. The program automatically removes the existential quantifier, thereby implicitly picking a limit for the sequence $(a_{n})$ – again, this surreptitious removal of quantifiers mirrors the way human mathematicians think and write. After the expansion and quantifier removal, we arrive at the following problem state. H2. $A$ is closed in $X$ H3. $(a_{n})$ is Cauchy H4. $(a_{n})$ is a sequence in $A$ H6. $(a_{n})\to a$ T3. $(a_{n})$ converges in $A$ Now the program does another kind of forwards reasoning, where a result from the library is used. The library result states that a closed set contains its limit points. If we apply this result to the set $A$, the sequence $(a_{n})$ and the limit $a$, then the hypotheses H2, H4 and H5 give us precisely the premises we need, which allows us to conclude that $a\in A$. H2. $A$ is closed in $X$ H3. $(a_{n})$ is Cauchy H4. $(a_{n})$ is a sequence in $A$ H6. $(a_{n})\to a$ H7. $a\in A$ T3. $(a_{n})$ converges in $A$ At this point our deletion rules allow the program to delete H2. We also see why it is important to be cautious about deleting statements: the hypotheses H4 and H6 have been used, but they are going to be used again. Fortunately, our deletion rules do not cause either of them to be deleted. H3. $(a_{n})$ is Cauchy H4. $(a_{n})$ is a sequence in $A$ H6. $(a_{n})\to a$ H7. $a\in A$ T3. $(a_{n})$ converges in $A$ The final step for the program is to look at the expansion of the target, which is $(a_{n})$ is a sequence in $A\ \wedge\ \exists z\ z\in A\ \wedge\ (a_{n})\to z$ and observe that if it sets $z$ equal to $a$, then all three resulting statements occur above the line as hypotheses. Because of this, it declares the problem solved. ### 2.3 Some terminology With the possible exception of our rules for deleting statements, which are not essential to the program, all the move types we use in the program are fairly standard in the field, or are combinations and minor variations of standard move types. Before we say what they are, it will be useful to have some terminology for the different kinds of statements that can occur. We call a statement _atomic_ if it is of the form $P(x_{1},\dots,x_{k})$, where $P$ is a predicate and $x_{1},\dots,x_{k}$ are terms. We do not insist that it is actually written in this form, so for example the statement $d(x,y)<\epsilon$ is an atomic statement, which we could if we wanted to write in the form is_less_than$(d(x,y),\epsilon)$. When a statement is built out of atomic statements using connectives and quantifiers, we classify it according to the operation that appears at the top of its parse tree. For example, the statement $\exists x\ x\in A\ \wedge\ d(x,y)<\epsilon$ is an _existential_ statement, whereas the statement $x\in A\ \wedge\ d(x,y)<\epsilon$ is _conjunctive_. It is often useful to look more than one level down the parse tree. For example, we would call the statement $\forall x\ x\in A\Rightarrow x\in B$ a _universal conditional_ statement. (Recall that we do not allow ‘bare’ conditional statements, so this is a particularly important category.) Similarly, the existential statement above can be further classified as an _existential conjunctive_ statement. Finally, many atomic statements can be expanded into statements that are no longer atomic. For example, the statement $A\subset B$ expands to the universal conditional statement above. It is often useful to know what a statement will become after it is expanded: to specify this we use the prefix ‘pre-’. Thus, the statement $A\subset B$ is pre-universal conditional. We call an expansion _elementary_ if it does not introduce a quantifier. For example, the expansion of $A\subset B$ is not elementary, whereas the expansion of $x\in A\cap B$ as $x\in A\ \wedge\ x\in B$ is elementary. ### 2.4 How the program works The following lines are taken directly from the program’s code: they list, in order of priority, the names of the moves that it can do. In the rest of the section, we shall explain what those moves are. --Deletion deleteDone, deleteDoneDisjunct, deleteDangling, deleteUnmatchable, --Tidying peelAndSplitUniversalConditionalTarget, splitDisjunctiveHypothesis, splitConjunctiveTarget, splitDisjunctiveTarget, peelBareUniversalTarget, removeTarget, collapseSubboxTarget, --Applying forwardsReasoning, forwardsLibraryReasoning, expandPreExistentialHypothesis, elementaryExpansionOfHypothesis, backwardsReasoning, backwardsLibraryReasoning, elementaryExpansionOfTarget, expandPreUniversalTarget, solveBullets, automaticRewrite, --Suspension unlockExistentialUniversalConditionalTarget, unlockExistentialTarget, expandPreExistentialTarget, convertDiamondToBullet, --EqualitySubstitution rewriteVariableVariableEquality, rewriteVariableTermEquality The program repeatedly applies a move of the first type it can from this list. Thus, if a move of type deleteDone can be performed, it performs it. If not, but a move of type deleteDoneDisjunct can be performed, then it performs that. Otherwise, it tries deleteDangling. And so on. #### 2.4.1 Deletion moves The moves can be divided into some broad categories. We begin by discussing moves that delete statements from problem states. Deletion is intended to model the human capacity to recognise that a statement is no longer going to be used and to turn attention away from it. Exactly how mathematicians do this involves several subtleties, which we shall discuss elsewhere. Here we give a brief indication of how the program works. One key point is that we do not normally delete statements unless they have been used as part of a reasoning step. This is because for the problems our program tackles, there is a strong presumption that every statement that appears will at some point be used, so it is risky to delete an unused statement, even if it appears to be unusable. However, once a statement has been used, this risk is greatly diminished, so under suitable conditions we are much happier to delete it. Therefore, the program tags hypotheses as ‘vulnerable’ when they have been used, and its deletion rules are not applied to hypotheses unless they have this tag. #### deleteDone There are several situations where a move results in a target being replaced by the word ‘done’ because it has been proved. Once this has happened, the program immediately deletes it. The aim of the program is to reach a problem state with no targets. #### deleteDoneDisjunct If a target is disjunctive and one of its disjuncts is the word ‘done’, then the entire target is deleted. #### deleteDangling We call a free variable $v$ _dangling_ if it is involved in just one statement. This is a strong sign that the statement cannot be used. It is not a conclusive proof: for example, there might be an existential target that can be solved if we substitute $v$. However, if a statement contains a dangling variable and is vulnerable, then the program deletes it. This is a pragmatic decision on our part rather than one that is fully justified theoretically: there may be problems where deleting statements that have been used and that contain dangling variables is the wrong thing to do, but that situation does not appear to arise for the kinds of problems the program is designed to handle. #### deleteUnmatchable Suppose that we have the statements $x\in A$ and $A\subset B$ as hypotheses. The expansion of $A\subset B$ is $\forall u\ u\in A\implies u\in B$. If we substitute $x$ for $u$, then the premise of this statement becomes $x\in A$, which is identical to the hypothesis. We say that $x\in A$ _matches_ the premise of (the expansion of) $A\subset B$. We call a statement _unmatchable_ if there are no available matches for it. The program is not allowed to substitute the same variable twice into the same hypothesis. (This is partly because no human would ever do so, and partly to avoid getting into a loop.) This can create further circumstances where a hypothesis is unmatchable. For example, suppose we apply forwards reasoning to the statements $x\in A$ and $A\subset B$ to deduce that $x\in B$. Then we can no longer use the match between $x\in A$ and $A\subset B$, so $x\in A$ becomes unmatchable (assuming that there is no other statement that matches it). Since it has been used, it is vulnerable, and will therefore be deleted. If no other statement matches $A\subset B$, then that too will be deleted. #### 2.4.2 Tidying moves Tidying moves are moves that do not substantially change the logic of the problem state, but put it into a more convenient form. #### peelAndSplitUniversalConditionalTarget If the target takes the form $\forall x\ P(x)\Rightarrow Q(x)$, then this move creates a new hypothesis $P(x)$ and replaces the target by $Q(x)$. This corresponds to the human move of saying (or thinking), ‘Let $x$ be such that $P(x)$; we need to show that $Q(x)$.’ We can regard it as a composition of two moves: one to get rid of the quantifier and one to split up the conditional statement. We use the word ‘peel’ to refer to any move that gets rid of a quantifier. If there is more than one target, then this move has to be modified, since we cannot use $P(x)$ to help us prove a different target. In that situation, we create what we call a _box_. A box looks like a problem state within a problem state: it has a line with some statements above it and some statements (or subboxes – box formation can be iterated) below it. If the peel-and-split move creates a box, then that box lives below the main line. That is, if we have a problem state of the form Hypotheses $\forall x\ P(x)\implies Q(x)$ $R$ then after the move we will transform it to Hypotheses $P(x)$ --- $Q(x)$ $R$ The program then knows that it can use $P(x)$ to prove $Q(x)$ but not to prove $R$. The hypotheses above the main line can of course be used to prove both statements. Boxes are regarded as complex targets (in the above case the target corresponding to the box would be to demonstrate that $P(x)$ implies $Q(x)$) and as such can form part of a list of targets. For example, if there are two boxes, one written above the other, that means that the implications within both boxes need to be established. Sometimes one needs to establish just one such implication: in that case one writes the two boxes side by side with a ‘$\vee$’ symbol in between them. In theory, boxes can be nested, though this rarely happens in practice. The rules governing which statements can be used to prove what are as follows. For each statement, there is a minimal box that contains it (counting the entire problem state as a box). Two boxes are _comparable_ if one contains the other in an obvious sense, and two statements are _comparable_ if the minimal boxes containing them are comparable. All deductions that the program makes must involve comparable statements. A quick way to interpret a box is to regard the line in the middle as an implication sign, the lists of statements involved as linked by conjunction, and the box itself as placing brackets round everything. #### splitDisjunctiveHypothesis If there is a hypothesis of the form $P\vee Q$, then the program splits the problem into two, one with $P$ as a hypothesis and one with $Q$ as a hypothesis. Technically, it achieves this by forming two boxes, one displayed above the other to indicate that both must be established. #### splitConjunctiveTarget If there is a target of the form $P\wedge Q$, then it is replaced by two targets $P$ and $Q$. #### splitDisjunctiveTarget If there is a target of the form $P\vee Q$, then it is replaced by two boxes, linked by a $\vee$ symbol. One box has nothing above the line and $P$ below the line, and the other has nothing above the line and $Q$ below the line. This move exists for technical reasons: for example, the program sometimes likes to attach tags to statements, but it has no facility for attaching tags to parts of statements. Therefore, if we want to use a tag to record information about one disjunct of a disjunctive target, we need to ‘split’ the target first. #### peelBareUniversalTarget If the target is of the form $\forall x\ P(x)$ and $P$ is not a conditional statement, then this move replaces the target by $P(x)$. There is a sense in which bare universal targets should never occur, since when we quantify over $x$, we do not quantify over every object in the universe, but rather over some set $X$. So one might argue that the statement $\forall x\ P(x)$ should really be rendered as the universal conditional statement $\forall x\ x\in X\implies P(x)$. What we mean by a ‘bare universal’ statement is one where the ‘premises’ are background statements that we do not want to elevate to ‘substantive’ status. For example, we would regard the expansion of the statement ‘$G$ is Abelian’ as a bare universal, since the only condition needed for two elements to commute is the background information that they are elements of $G$. #### removeTarget This is actually a class of move types, but what they have in common is that under appropriate circumstances they replace a target with the word ‘done’. The most obvious example is when a target equals a hypothesis (and that hypothesis is allowed to be used to prove the target). A more complicated example is when the target is of the form $\exists u\ P(u)\wedge Q(u)$ and there are hypotheses $P(x)$ and $Q(x)$. The other circumstances are similar. #### collapseBoxedTarget If a box B has nothing above its internal line, contains no suspended variables (see below for a definition) and is not joined to another box with the $\vee$ symbol, then the statements below its internal line are listed as targets without the box B. #### 2.4.3 Applying moves An applying move is a move where we apply a hypothesis, result or definition. #### forwardsReasoning This covers a number of closely related steps that the program can take. The most basic example is using hypotheses of the form $P(x)$ and $\forall u\ P(u)\Rightarrow Q(u)$ to obtain the hypothesis $Q(x)$. However, there are several natural variants and generalizations of this. An obvious one is that if there are hypotheses of the form $P_{1}(x),\dots,P_{k}(x)$ and $\forall u\ P_{1}(u)\wedge\dots\wedge P_{k}(u)\implies Q(u)$, then the program generates the hypothesis $Q(x)$. Deductions of this kind are often called _forward chaining_. A variant that is worth highlighting is illustrated by the following simple piece of reasoning: if we know that $x\in A$ and that $A\subset B$ then we can deduce that $x\in B$. Humans will make this deduction in one step rather than first expanding the statement $A\subset B$ as $\forall u\ u\in A\Rightarrow u\in B$. Our program does the same. In general, for each type of reasoning move that involves a universal conditional hypothesis, there is a variant that does essentially the same thing to a pre-universal conditional hypothesis. #### forwardsLibraryReasoning This is reasoning that is ‘mathematical’ rather than ‘purely logical’. We have seen an example of it already: deducing from the statements ‘$(a_{n})$ is a sequence in $A$’, ‘$A$ is closed’ and ‘$a_{n}\to a$’ that $a\in A$. The reason this deduction can be done in one step is that the library contains a general result that says that whenever a sequence in a closed set tends to a limit, then the limit belongs to the closed set as well. Logically speaking, forwards library reasoning is similar to ordinary forwards reasoning, but there are one or two aspects of it that give it a different flavour. The main one is that library results contain no free variables: they are general facts that apply universally. This distinguishes them from hypotheses, which are more contingent. A second difference is that forwards library reasoning is normally used to deduce an atomic hypothesis from other atomic hypotheses. A universal conditional statement is involved, but it is in the library and is not a hypothesis. #### expandPreExistentialHypothesis As its name suggests, this means replacing a pre-existential hypothesis by its expansion. What the name of the move does not reveal is that this expansion is followed immediately by a peeling to get rid of the existential quantifier. So for example the statement ‘$\alpha$ has an inverse’ might be replaced by $\alpha\beta=\beta\alpha=1$. This would be modelling a two-step human thought process. The first step is to note that there exists $\beta$ such that $\alpha\beta=\beta\alpha=1$, and the second step is quietly to forget about the existential quantifier and to refer to $\beta$ as though it has been chosen. Human mathematicians will usually miss out a step that says something like, ‘Let $\beta_{0}$ be such that $\alpha\beta_{0}=\beta_{0}\alpha=1$’, so the program does as well. #### elementaryExpansionOfHypothesis This takes a hypothesis that has an elementary expansion and replaces it by that expansion. This is sometimes combined with some tidying. For example, if the hypothesis in question is $x\in A\cap B$, then the elementary expansion is $x\in A\wedge x\in B$, but this expansion is immediately converted into the two hypotheses $x\in A$ and $x\in B$ and does not itself appear in any problem state. #### backwardsReasoning Given a target $Q(x)$ and a hypothesis $\forall u\ P(u)\Rightarrow Q(u)$, this replaces the target by $P(x)$. More generally, if we have a target $Q(x)$ and a hypothesis $\forall u\ P_{1}(u)\wedge\dots\wedge P_{k}(u)\implies Q(u)$, then it is logically sound to replace the target $Q(x)$ by the $k$ targets $P_{1}(x),\dots,P_{k}(x)$. Deductions of this kind are often called _backward chaining_. The program is allowed to do this more complex backward chaining only under tightly constrained circumstances: it must be that all but one of the statements $P_{1}(x),\dots,P_{k}(x)$ is a hypothesis, so that only one new target is created. This is another pragmatic decision: it is a crude way of deciding whether applying the hypothesis $\forall u\ P_{1}(u)\wedge\dots\wedge P_{k}(u)\implies Q(u)$ is likely to be the right thing to do, and the severity of the test is intended to stop the program making ‘speculative’ deductions that risk leading to combinatorial explosion. As with forwards reasoning, there is a simple variant where the role of the universal conditional hypothesis is played by a pre-universal conditional hypothesis instead. For example, given a target $x\in B$ and a hypothesis $A\subset B$ the program could use this variant to replace the target by $x\in A$. #### backwardsLibraryReasoning This is backwards reasoning that makes use of a general result in the library. However, it is slightly subtler than forwards library reasoning, because it always uses hypotheses as well as a target. The precise rule is that if there are hypotheses $P_{1}(x),\dots,P_{k-1}(x)$, a library result $\forall u\ P_{1}(u)\wedge\dots\wedge P_{k}(u)=Q(u)$ and a target $Q(x)$, then the target can be replaced by $P_{k}(x)$. (The premises of the library result do not have to be stated in the order $P_{1},\dots,P_{k}$.) An example of this kind of reasoning would be to say, “It is sufficient to prove that $B$ is open,” if one wished to prove that $A\cap B$ was open and knew that $A$ was open. This would be making use of the result that an intersection of two open sets is open. #### elementaryExpansionOfTarget This replaces a target by an elementary expansion of that target, if it has one. #### expandPreUniversalTarget This replaces a pre-universal target by its expansion. This move will be followed by one of the tidying moves peelAndSplitUniversalConditionalTarget or peelBareUniversalTarget. It is usually the first move that the program makes when faced with a naturally stated problem. #### solveBullets As we are just about to discuss in more detail, we sometimes mark a variable $w$ with a diamond or a bullet. This indicates that the variable needs at some stage to be chosen in such a way that the problem can be solved. If the variable only ever appears in targets, then one simple way in which this can often be done is to identify another variable $x$ with the property that if we substitute $x$ for $w$, then every target that involves $w$ is equal to a hypothesis. In that situation, all those targets are replaced by ‘done’. This move is what we call ‘solveBullets’. #### automaticRewrite There are a few rewriting rules stored in the library. Two examples are that the statement $x\in f^{-1}(A)$ is rewritten as the statement $f(x)\in A$ and the term $g\circ f(x)$ is rewritten as the term $g(f(x))$. (Of course, these are general rewriting rules and work whatever the variables happen to be called.) The rewriting of statements takes place only when those statements have been isolated as hypotheses or targets, so for example the program would not rewrite the statement $y\in f^{-1}(U)$ when it occurs inside the larger statement $\forall y\ d(x,y)<\delta\Rightarrow y\in f^{-1}(U)$. However, terms can be rewritten as soon as they appear. #### 2.4.4 Suspending moves We now come to the class of moves just alluded to: moves that help us deal with existential targets when it is not immediately clear what to substitute for the existentially quantified variable. A standard technique for this, which is essentially the technique we use, is to form _metavariables_. The rough idea of a metavariable is that one reasons with it as though it had been chosen, deferring the actual choice until later when it becomes clearer what choice will make the argument work. Mathematicians often use this trick: a classic example is the ‘$3\epsilon$-argument’ used to prove that a uniform limit of continuous functions is continuous. When the program ‘pretends that it has chosen’ a variable, it marks that variable with a bullet, and we say that it has been _suspended_. Thus, suspension is the process of converting a variable into a metavariable. However, we found it convenient to introduce two ‘levels of suspension’, to model two styles of reasoning that are logically similar but psychologically quite different. #### unlockExistentialUniversalConditionalTarget To illustrate this, suppose we have a target such as $\exists\delta\ \forall y\ d(x,y)<\delta\Rightarrow f(y)\in B$, and also a hypothesis $\forall u\ u\in A\implies f(u)\in B$. Then it is easy to see that we can reduce the target to $\exists\delta\ \forall y\ d(x,y)<\delta\implies y\in A$. However, this move is not open to the program because it is not allowed to ‘reason inside quantifiers’. This is a matter of convenience: such moves are logically valid, but it is tedious to specify appropriate variants of several of the reasoning moves listed above. Instead, we apply a procedure that we call _unlocking_ , which effectively moves aside the existential quantifier and allows the program to reason as normal with the statements inside it. More precisely, what the program does to ‘unlock’ the statement is create a box. In the example above, it would have no statements above the line, and below the line it would have the statement $\forall y\ d(x^{\blacklozenge},y)<\delta\Rightarrow f(y)\in B$. The diamond on the variable $x$ indicates that $x$ needs to be chosen. It is important for the program not to interchange quantifiers accidentally. For this reason, we tag the box just created with the variable $x^{\blacklozenge}$, to indicate that the existential quantification over $x$ is within that box. After unlocking the statement, the program peels and splits the resulting universal conditional target (so a more accurate name for the move type would be unlockPeelAndSplitExistentialUniversalConditionalTarget). After that, we have a box that looks like this. $d(x^{\blacklozenge},y)<\delta$ --- $f(y)\in B$ Once we have done this, the statement $f(y)\in B$ has become a target and the program is free to apply backwards reasoning to it. #### unlockExistentialTarget This move replaces a target of the form $\exists x\ P(x)$ with a box that has nothing above the line and the statement $P(x^{\blacklozenge})$ below the line. The box is labelled with the variable $x^{\blacklozenge}$. This move will never be applied to an existential universal conditional target, since that will have been dealt with by unlockExistentialUniversalConditionalTarget. The main reason we have two separate moves here is that we prefer to bundle the unlocking together with peeling and splitting when that is possible. To see what unlockExistentialTarget allows the program to do, suppose that we have a target of the form $\exists x\ Q(x)\ \wedge\ R(x)$ and also a hypothesis of the form $\forall u\ P(u)\Rightarrow Q(u)$. In this situation we would like to be able to do backwards reasoning inside the existential quantifier to reduce the target to $\exists x\ P(x)\ \wedge\ R(x)$. However, the program does not have a move for this. Instead, it unlocks the existential target, so that it has a box with the statement $Q(x^{\blacklozenge})\ \wedge\ R(x^{\blacklozenge})$ below the line. The tidying move splitConjunctiveTarget can now turn this new target into two targets, and once it has done that, the applying move backwardsReasoning can be used to replace the target $Q(x^{\blacklozenge})$ by $P(x^{\blacklozenge})$. As another example of the use of unlocking, suppose that we wished to prove that $A\cap B$ is non-empty and had the hypotheses $x\in A$ and $x\in B$. The program cannot see that $x$ is a witness to the non-emptiness of $A\cap B$ without doing some processing. An obvious first step is to expand the target into the statement $\exists u\ u\in A\cap B$. However, the program is not then allowed to do an elementary expansion inside the quantifier. Instead, it unlocks $u$ so that there is a new target $u^{\blacklozenge}\in A\cap B$. This can now be expanded and split into the two targets $u^{\blacklozenge}\in A$ and $u^{\blacklozenge}\in B$, which solveBullets can then match with the hypotheses. This may seem a little circuitous, but it actually models quite closely how humans think. A human might say, ‘I want to show that $A\cap B$ is non-empty, so I need to find some $u$ that belongs to $A\cap B$. In other words, I need $u$ to be in $A$ and in $B$. Aha, I can take $x$.’ The program’s unlocking models the silent disappearance of the existential quantifier before the second sentence of the above. #### expandPreExistentialTarget This does exactly what it says: it replaces a pre-existential target by its expansion. #### convertDiamondToBullet There are certain moves that the program will not do with a ‘diamonded’ variable. In particular, it will not do any reasoning with a hypothesis that involves such a variable: for that it needs a deeper level of suspension, roughly speaking corresponding to the human move of ‘pretending that a variable has been chosen’ and then reasoning with it. Logically this is not an important difference, but it is a useful one for us because it reflects a difference in the way human mathematicians think and write. This helps the program to produce more convincing write-ups. We do not need separate move types for reasoning that involves hypotheses with bulleted variables: we just allow the reasoning moves above to take such variables. An important technicality is that if we postpone the choice of a variable, we must keep track of what other variables it is allowed to depend on. However, what we actually do is note which variables it is _not_ allowed to depend on. This is for two reasons. First, it seems to reflect more accurately how human mathematicians think about such variables, and secondly, it is more economical: there are typically many fewer variables on which a bulleted variable is not allowed to depend than variables on which it is allowed to depend. #### 2.4.5 Equality substitution If we are told that two objects are equal, then we can eliminate all mention of one object in favour of the other. The precise rules governing when and how mathematicians tend to avail themselves of this opportunity are not obvious. The rules below are best regarded as a temporary solution: they do not always result in realistically human choices, and we intend to replace them by more satisfactory rules in the near future. #### rewriteVariableVariableEquality If there is a hypothesis of the form $x=y$, then this move replaces all occurrences of $y$ by $x$ and eliminates the hypothesis. #### rewriteVariableTermEquality If there is a hypothesis of the form $v=t$ or $t=v$, where $v$ is a variable and $t$ is a term, then this move replaces all occurrences of $t$ by $v$. ### 2.5 Justification for the order of priority From examining how humans solve simple problems in the theory of metric spaces, it is not too hard to arrive at the above list of move types. But it is less clear what principles should govern the architecture – that is, the way that the program decides which move type to do in any given situation. One obvious method of choosing an architecture is to work through a large number of problems and try to observe what seems to be the natural approach. After a while, one can make a guess at how the program should work, and if the guess results in strange behaviour for some problems then one can refine it, hoping that the process of refinement will stabilize quickly. An alternative method is to try to devise a theory that explains which move is the best than in each situation: ideally, that will turn out to be the move that humans are naturally drawn to. The second method is harder, but if it works, then the advantage is that the rules are likely to be more robust: without a proper theoretical backing, one cannot be as confident that they will not lead to inappropriate behaviour when the program is presented with an unfamiliar problem. We have used a mixture of the two methods. We have good reasons for some of the choices we have made, but other choices are justified by the fact that they seem to work (in the sense of leading to human-like behaviour). A broad overarching principle that gives a theoretical backing to many of our choices is this: _the program prefers safe moves to dangerous moves_. The picture we have here is one where at any stage there is a choice of moves that can be made, and we have to make an assessment of how likely any given choice is to form part of the argument one is looking for. The greater this likelihood, the safer the move. Because it seems hard to attach probabilities to statements in this way, we have not tried to do so. However, since something like the above picture seems to be what humans do, we bear the picture in mind when planning the program. In particular, if a move is obviously safe, we will assign it a high priority. A good example of a safe move is a tidying move. If, for example, we have a conjunctive hypothesis, then there is nothing to lose by splitting it up into its conjuncts, so that move we do automatically without any hesitation at all. By contrast, expanding a definition is substantially less safe: sometimes it is possible to reason in a high-level way without expanding, and since we do not allow ‘de-expansion’ in this program (and in general allowing it would be highly problematic because of the danger of an infinite loop), expanding a definition is closing off the option of such high-level arguments. As an example, in the problem we discussed earlier, the program does not expand the statements ‘$(a_{n})$ is Cauchy’, ‘$(a_{n})$ is a sequence in $A$’ or ‘$(a_{n})\to a$’. That allows it to do some high-level forwards library reasoning that would no longer be possible if any one of those three statements was expanded. Thus, expansion has a fairly low priority. Having said that, some expansions, such as elementary expansions or expansions of pre-existential hypotheses, are considerably safer, so those ones have higher priority. Somewhere in between are the other reasoning moves. Here it becomes more complicated to apply the general principle (even as an informal guiding principle), because the safety of a move type depends heavily on context. In particular, forwards reasoning is in general fairly unsafe – if you have a lot of information and do not know which statements are relevant, then the probability that any given deduction will form part of the eventual proof may be quite small – but is much safer when it comes to routine problems, which tend not to suffer from the problem of irrelevant information. The psychology literature suggests that _when it is safe_ , humans tend to prefer forwards reasoning to backwards reasoning swelleretal1983 ; owensweller1985 , though this appears to be a question more of style than of problem-solving efficacy: we seem to prefer not to keep track of a moving target if we do not have to. Since forwards reasoning tends to be safe for the highly routine problems our program tackles, we have given all forwards reasoning a higher priority than all backwards reasoning. This also has the beneficial effect of making the program reluctant to switch direction – too much switching from forwards to backwards or vice versa would again be bad mathematical style. This aspect of our program is, however, unstable, in that we know that in order to develop the program we will have to change it. In fact, we even have an example of a rather routine problem where our program performs badly for this very reason. If it is asked to show that the intersection of two subgroups $H$ and $K$ of a group $G$ is itself a subgroup of $G$, then when it is trying to prove that $H\cap K$ is closed under multiplication, it expands, peels and splits the target, obtaining two elements $x$ and $y$ of $H\cap K$ and a target of showing that $xy\in H\cap K$. At that point, there is nothing to stop the program making ‘silly’ deductions such as that $x^{-1}\in H$. There are various easy ways of dealing with this problem and we shall implement these in future versions: for example, we could add restrictions on creating new terms (a human would not think of deducing that $x^{-1}\in H$ because ‘we are not interested in $x^{-1}$’) or we could alter the priorities so that when there are a number of possible forwards moves, so that it is no longer clear that they are all relevant, the program switches to backwards reasoning. One other feature of the ordering of reasoning moves is that we prefer pure reasoning moves to library reasoning moves. That is because in general a hypothesis is more likely to be relevant than a library statement, though if enough of the premises of a library statement are present as hypotheses, that is a fairly strong argument for its relevance. At the bottom of the list of priorities is suspension. That is because humans tend to regard it as a last resort. When mathematicians need to prove statements of the form $\exists x\ P(x)$, then by and large they prefer to simplify the problem until a suitable candidate $x_{0}$ for $x$ becomes obvious and it remains to carry out the relatively easy task of verifying that $P(x_{0})$. Only when this straightforward approach fails do we take the more drastic step of pretending that $x$ has been chosen. We will not say much more here about how we chose the priority order, but we have two brief further points. First, although our reasons are not 100% precise, we found that in practice they were adequate, in the sense that they suggested an order before we started, and we found that we did not have to modify the order when we tried further problems (though, as commented above, there are certain aspects of the architecture that will need to be changed in future versions). Secondly, when it comes to the finer detail of the ordering, there may not be that much to choose between different move types. However, conflicts rarely arise between different move types that are not distinguished by any of the above criteria, so in practice these finer details have little if any effect on what the program actually does. ### 2.6 Example: An intersection of two open sets is open Now that we have discussed how the program works, let us look at another example, which illustrates most of the move types and shows how the order or priority works in practice. The problem to be solved is the following. ###### Problem 3 Let $A$ and $B$ be open subsets of a metric space $X$. Prove that $A\cap B$ is open. The initial problem state is as follows. $A$ is open $B$ is open $A\cap B$ is open No reasoning moves are possible, so we end up having to expand. The highest priority move we can do is expandPreUniversalTarget, which, after the tidying move peelAndSplitUniversalConditionalTarget, has the following effect. $A$ is open $B$ is open $x\in A\cap B$ $\exists\delta\ \forall y\ d(x,y)<\delta\Rightarrow y\in A\cap B$ It may look a little strange that we do not specify that $\delta>0$. The reason for that is that we think of positivity as a ‘background condition’ rather than as a ‘substantive statement’. One reason that matters is connected with safety: if a premise of a universal conditional hypothesis is satisfied by some variable, then that is good evidence that the forwards reasoning one can do as a result is relevant to the problem. Or rather, it is good evidence if the premise is a ‘substantive statement’, but not if it is merely a background condition such as positivity. So the distinction is a useful one. We think of background conditions as similar to, but not the same as, type declarations. The program does, however, know that $\delta$ is a positive real number: that fact is stored as background information when the statement ‘$A\cap B$ is open’ is expanded. At this point, the reluctance of the program to suspend $\delta$ means that it does as much forwards reasoning as it possibly can. It begins with elementaryExpansionOfHypothesis, applied to the third hypothesis. $A$ is open $B$ is open $x\in A$ $x\in B$ $\exists\delta\ \forall y\ d(x,y)<\delta\Rightarrow y\in A\cap B$ This allows it apply two forwardsReasoning twice. After the first application, the problem state is as follows. $A$ is open $B$ is open $x\in A$ $x\in B$ $\forall u\ d(x,u)<\eta[x]\Rightarrow u\in A$ $\exists\delta\ \forall y\ d(x,y)<\delta\Rightarrow y\in A\cap B$ Note that the last hypothesis is in a sense generated by a combination of submoves: the first is forwardsReasoning (using the hypotheses $x\in A$ and ‘$A$ is open’) and the second is a peeling (to get rid of $\exists\eta$ at the beginning of the statement). However, the latter is so automatic that it is not listed as one of our tidying moves: instead, it is considered as part of any other move that potentially generates an existential hypothesis. It is important to keep track of the fact that $\eta$ depends on $x$, which is what is signified by $\eta[x]$. After this, deleteUnmatchable causes the program to delete the statements $x\in A$ and ‘$A$ is open’. This is because both statements have been used, so they are vulnerable, and because it is no longer permissible to substitute $x$ into ‘$A$ is open’. The resulting problem state is as follows. $B$ is open $x\in B$ $\forall u\ d(x,u)<\eta[x]\Rightarrow u\in A$ $\exists\delta\ \forall y\ d(x,y)<\delta\Rightarrow y\in A\cap B$ It then runs through a similar process for $B$ (it does not yet have the capacity to recognise that the problem is symmetric in $A$ and $B$ and say, ‘Similarly …’). After that process, it arrives at the following. $\forall u\ d(x,u)<\eta[x]\Rightarrow u\in A$ $\forall v\ d(x,v)<\theta[x]\Rightarrow v\in B$ $\exists\delta\ \forall y\ d(x,y)<\delta\Rightarrow y\in A\cap B$ It has now reached the point where it must suspend $\delta$. In the first instance, it uses the move unlockExistentialUniversalConditionalTarget, which includes a peeling and splitting. The result is as follows. $\forall u\ d(x,u)<\eta[x]\Rightarrow u\in A$ $\forall v\ d(x,v)<\theta[x]\Rightarrow v\in B$ $d(x,y)<\delta^{\blacklozenge}[\overline{y}]$ --- $y\in A\cap B$ The notation $\delta^{\bullet}[\overline{y}]$ signifies that $\delta$ is not allowed to depend on $y$. The highest priority move the program can do now is elementaryExpansionOfTarget, so it does that, and automatically splits the resulting conjunctive statement (rather than using the move splitConjunctiveTarget). $\forall u\ d(x,u)<\eta[x]\Rightarrow u\in A$ $\forall v\ d(x,v)<\theta[x]\Rightarrow v\in B$ $d(x,y)<\delta^{\blacklozenge}[\overline{y}]$ --- $y\in A$ $y\in B$ This allows it to apply backwardsReasoning twice. After the two deductions it reaches the following state. (It does them separately, so we are jumping a step here.) $\forall u\ d(x,u)<\eta[x]\Rightarrow u\in A$ $\forall v\ d(x,v)<\theta[x]\Rightarrow v\in B$ $d(x,y)<\delta^{\blacklozenge}[\overline{y}]$ --- $d(x,y)<\eta[x]$ $d(x,y)<\theta[x]$ It then uses deleteUnmatchable to delete the two hypotheses it has just used. $d(x,y)<\delta^{\blacklozenge}[\overline{y}]$ --- $d(x,y)<\eta[x]$ $d(x,y)<\theta[x]$ At this point, there is not much that the program can do, because it is not allowed to reason with the diamonded variable $\delta^{\blacklozenge}$. So the highest-priority move it can do is convertDiamondToBullet. Also, since there are no hypotheses above the main line, it may as well remove that line and the box. $d(x,y)<\delta^{\bullet}[\overline{y}]$ $d(x,y)<\eta[x]$ $d(x,y)<\theta[x]$ Now it applies backwardsLibraryReasoning. The result in the library is that if $a<b$ and $b\leq c$, then $a<c$. Applying that with the hypothesis and the first target results in the following problem state. $d(x,y)<\delta^{\bullet}[\overline{y}]$ $\delta^{\bullet}[\overline{y}]\leq\eta[x]$ $d(x,y)<\theta[x]$ The deletion rules do _not_ allow the program to delete the hypothesis we have just used (and this is a good example of a situation where deletion would be a very bad idea). However, it is aware that it cannot use the hypothesis with the new target. (We shall not describe here the precise mechanism by which it gains this awareness.) The program then uses backwardsLibraryReasoning again and this time it does delete the hypothesis, on the grounds that the variable $x$ that appears in the hypothesis is dangling. After that, it has reached the following state. $\delta^{\bullet}[\overline{y}]\leq\eta[x]$ $\delta^{\bullet}[\overline{y}]\leq\theta[x]$ This is a ‘standard’ existence problem, so the solution is in a library of standard solutions and the program declares the problem solved. It is here that the background information that $\delta$, $\eta$ and $\theta$ are positive is used, since the library result is that the minimum of two positive real numbers $a$ and $b$ is a positive real number that is less than or equal to both $a$ and $b$. That concludes the discussion of this example. For more examples of the program’s output, as well as the precise output for this problem (which we have very slightly simplified in one or two places to make the presentation clearer) see http://people.ds.cam.ac.uk/mg262/robotone.pdf . ### 2.7 Writing up In general, natural language generation is a complex process. It involves multiple levels of planning, which draw on both domain knowledge and models of the intended audience, and also a phase when the actual text is generated, which draws on syntactic, morphological and lexical information. An overview of the process may be found in reiter2000building . Because of this complexity, building a fully fledged natural language generation system is a major task. Furthermore, since mathematics contains not just English words but also a large array of distinctive symbols used in distinctive ways, it is not at all straightforward to use off-the-shelf systems. Fortunately, mathematical language has properties that make the task considerably simpler than it is for the English language in general. Foremost among these is the fact that mathematical proofs almost always have a particularly simple rhetorical structure. To some degree this is because the domain of discourse includes only timeless facts, which itself rules out a large proportion of the rhetorical relations found in general text. But the main reason is that there is a strong convention that further constrains the rhetorical structure of proofs. A proof proceeds by the presentation of a sequence of assertions, each of which follows from the premises of the theorem being proved or from previous assertions. This structure is not accidental; it is a direct reflection of the fact that mathematicians process proofs by reading and verifying one sentence at a time, and would not expect the justification of a fact presented in one sentence to be deferred to a later sentence. (We are talking here about proofs of the level of simplicity of the proofs discussed in this paper. For more complicated arguments, facts may sometimes be used before they have been proved, but in good mathematical writing this will be carefully flagged up to make it as easy as possible for the reader to check that the resulting argument is complete and not circular.) This convention gives us an easy way to produce write-ups of our proofs. An obvious strategy is to allow each move to generate some number of sentences (possibly zero), and to concatenate output from the different moves to produce the final text. Note that this strategy is viable only because we are absolutely rigorous about requiring our moves to reflect steps in human reasoning; in effect, the strategy is mimicking a human who is carefully writing down down a proof while coming up with it, which is quite straightforward for an experienced mathematician. (Again, this becomes less true if the proofs are more difficult.) As we shall see below, this simple strategy produces surprisingly good results, though with a weakness that needs to be dealt with by a postprocessing phase, which turns out to be straightforward. Because we have a fixed list of move types, implementing the strategy only requires us to specify which sentences (if any) are produced for the moves of each type. A very simple way to do this is to use _template generation_ : each move type is associated with a _template_ , or ‘piece of text with holes’, and the holes are filled in with concrete information about the facts and objects used in the actual move performed. So, for example, forwards reasoning may be associated with a very simple template ‘since $<\textit{facts}>$, $<\textit{deduced fact}>$’. Instantiating this template would produce text like > Since $A$ is open and $x\in A$, there exists $\eta>0$ such that $u\in A$ > whenever $\textit{d}(x,u)<\eta$. Note that individual facts are expressed in idiomatic ways, rather being displayed in a way that directly reflects the underlying predicate calculus; thus we have ‘$A$ is open’ and ‘$\eta>0$’ rather than ‘$\textit{open}(A)$’ and ‘$\textit{greater\\_than}(\eta,0)$’. The same is true of objects: we display ‘$f\circ g$’ rather than compose(f,g), and so on. Similarly quantification is expressed idiomatically using words like ‘whenever’, where possible, rather than using more stilted phrases like ‘for all’, which would more directly reflect the underlying predicate calculus. An example of the text produced by this method is as follows: > Let $x$ be an element of $A\cap B$. Since $x\in A\cap B$, $x\in A$ and $x\in > B$. Since $A$ is open and $x\in A$, there exists $\eta>0$ such that $u\in A$ > whenever $d(x,u)<\eta$. Since $B$ is open and $x\in B$, there exists > $\theta>0$ such that $v\in B$ whenever $d(x,v)<\theta$. We would like to > find $\delta>0$ s. t. $y\in A\cap B$ whenever $d(x,y)<\delta$. But $y\in > A\cap B$ if and only if $y\in A$ and $y\in B$. We know that $y\in A$ > whenever $d(x,y)<\eta$. We know that $y\in B$ whenever $d(x,y)<\theta$. > Assume now that $d(x,y)<\delta$. Since $d(x,y)<\delta$, $d(x,y)<\eta$ if > $\delta\leq\eta$. Since $d(x,y)<\delta$, $d(x,y)<\theta$ if > $\delta\leq\theta$. We may therefore take $\delta=\min\\{\eta,\theta\\}$. We > are done. The main problem with this text is that it suffers a lack of coherence, in the sense defined in knott1996thesis : the sentences are individually acceptable, but they do not combine to form an idiomatic discourse. The principal reason for this is that the text repeats information unnecessarily. For example, in > Since $x\in A\cap B$, $x\in A$ and $x\in B$. Since $A$ is open and $x\in A$, > there exists $\eta>0$ such that $u\in A$ whenever $\textit{d}(x,u)<\eta$. the repetition of the underlined phrase is awkward. Because it is introduced by the sentence immediately preceding the ‘since’ clause, it is awkward to have it spelt out explicitly within that clause. Similarly, consider: > Since $\textit{d}(x,y)<\delta$, $\textit{d}(x,y)<\eta$ if $\delta\leq\eta$. > Since $\textit{d}(x,y)<\delta$, $\textit{d}(x,y)<\theta$ if > $\delta\leq\theta$. Here, having two identical ‘since’ clauses in consecutive sentences is again awkward: the repetition of material is unwieldy and unidiomatic. We are of the opinion that knott1996thesis correctly diagnoses the underlying problem here. Spelling out rhetorical relations, or aspects of rhetorical relations, that can easily be inferred from the context violates Grice’s maxim of quantity grice1975logic . Often the solution is to substitute an appropriate and less explicit _cue phrase_. For example, ‘since $A$ is open and $x\in A$, …’ is better replaced by ‘therefore, since $A$ is open, …’. The cue phrase ‘therefore’ (which assume the relevant reason has just been given) is less explicit than the cue phrase ‘since’ (which subordinates an explicitly stated reason), so it avoids spelling out information that is clear from the context. In other cases repetition can be avoided by combining sentences; for example, the second example may be changed into > Since $\textit{d}(x,y)<\delta$, $\textit{d}(x,y)<\eta$ if $\delta\leq\eta$ > and $\textit{d}(x,y)<\theta$ if $\delta\leq\theta$. The initial ‘sentence by sentence’ process described above is followed by a series of transformations that manipulate pairs of consecutive sentences in order to resolve the issues just mentioned. (Needless to say, the transformations operate on a structural level rather than on the literal text.) Applying this series of transformations to the example text above yields: > Let $x$ be an element of $A\cap B$. Then $x\in A$ and $x\in B$. Therefore, > since $A$ is open, there exists $\eta>0$ such that $u\in A$ whenever > $d(x,u)<\eta$ and since $B$ is open, there exists $\theta>0$ such that $v\in > B$ whenever $d(x,v)<\theta$. We would like to find $\delta>0$ s. t. $y\in > A\cap B$ whenever $d(x,y)<\delta$. But $y\in A\cap B$ if and only if $y\in > A$ and $y\in B$. We know that $y\in A$ whenever $d(x,y)<\eta$ and that $y\in > B$ whenever $d(x,y)<\theta$. Assume now that $d(x,y)<\delta$. Then > $d(x,y)<\eta$ if $\delta\leq\eta$ and $d(x,y)<\theta$ if $\delta\leq\theta$. > We may therefore take $\delta=\min\\{\eta,\theta\\}$ and we are done. One particular point worth emphasising is that the write-up process is deterministic: it will always produce the same output text for any given proof. This is for two reasons. First, if any non-determinism had been present we would have had to evaluate many outputs for any given proof, which would have made iterative improvement and fine-tuning of the write-ups considerably slower. Secondly, and more importantly, if the process were nondeterministic, our claim that the program produced human-like output would be suspect, in that we would have been able to run the program several times and ‘cherry pick’ output. Unfortunately, this determinism has an undesirable (but fully anticipated) side-effect. When one compare several proofs produced by the program, the write-ups are much more similar than those a human would produce. For example, most proofs produced by the program end with the phrase ‘we are done’. In the long run, we will undoubtedly need to introduce nondeterministic stylistic variation, allowing the program to vary the text generated for a particular step in just the way human would, despite the difficulties that will cause. Finally, it is worth noting that during the evaluation process described in the next section, we collated a wealth of data on how humans write up proofs. We anticipate using this data in combination with carefully chosen natural language processing techniques to create substantially improved versions of the write-up procedure. ## 3 Testing the write-ups Once the program had generated the write-ups for several problems, we wanted to test whether they could pass for write-ups written by a human mathematician. In this section we describe an informal experiment that we carried out for this purpose. We began by asking two mathematicians, one an undergraduate and one a PhD student, to write out proofs for five problems for which our program had generated proof write-ups. We did not tell either of them why we were making this unusual request, and we did not ask them to make their write-ups as good as possible. One of the problems was to show that the inverse image of an open set under a continuous function is open, and one of our volunteers decided to prove the converse, so that he could use the topological definition of continuity to prove another of the assertions – that a composition of continuous functions is continuous. We had to ask him to rewrite the latter and give the epsilon-delta proof, since we wanted the differences between the write-ups to be a matter of style rather than substance. We had another problem of this kind, which was that both our volunteers made frequent use of open balls. For example, their expansion of ‘$A\cap B$ is open’ was ‘for every $x\in A\cap B$ there exists $\delta>0$ such that $B_{\delta}(x)\subset A\cap B$.’ This made some of their arguments neater than the ones produced by our program. We contemplated getting the program to redo the problems using open-balls expansions, but in the end decided that it would be ‘cheating’ to make changes to its output in response to the human write-ups we had solicited, so we left things as they were. The program’s write-ups were not designed to be indistinguishable from human write-ups: we merely wanted them to be acceptable as human write-ups. Therefore, we left in certain features, such as ending every proof with the words, ‘we are done’, that we could with a little trouble have changed. (See the brief discussion of non-determinism at the end of the previous section.) For this reason, we did not want to ask people to guess which write-ups were the by the program. Instead, we presented all fifteen write-ups – two by humans and one by the program for each of the five problems – on the second author’s blog, and asked readers of the blog to comment on them in any way they liked. We also asked them to award points for clarity and style. The orders of the write-ups were chosen randomly and independently. (The precise mechanism was to decide on a one-to-one correspondence between the set $\\{1,2,3,4,5,6\\}$ to the set of permutations of the set $\\{1,2,3\\}$, then to find a website that produced random dice rolls.) So that answers would be as independent as possible, all comments and ratings were sent to the blog’s moderation queue and published only after the experiment was finished and comments on the blog post were closed. The post can be found at http://gowers.wordpress.com/2013/03/25/an-experiment- concerning-mathematical-writing/ , together with all the comments and ratings, but the real point of the experiment was to see whether anybody noticed that not all the write-ups were by humans. Nobody expressed the slightest suspicion of this kind. Having said that, we should also remark that many commenters were highly critical of the program’s output. Three criticisms in particular stand out. First, as we expected, the fact that the program did not use open balls was unpopular: many people commented that this made the write-ups unwieldy. Secondly, several of the human write-ups stated the new target when the initial one had been expanded, peeled and split. Several readers commented that they found this helpful, and criticized our program for not doing it. And thirdly, commenters did not like the way the program spelt out in detail how it thought of the right variable to substitute into existential targets (such as choosing $\min\\{\eta,\theta\\}$ for $\delta$ in the intersection-of-open- sets problem. It would be easy to modify the program so that none of these criticisms apply, so they do not point to fundamentally non-human aspects of how it thinks. To change the first, we would just have to use open-balls expansions of definitions such as ‘$A$ is open’ and ‘$f$ is continuous’. To change the second, we could alter the rule for what the write-up does when we expand, peel and split, so that it states the new target (preceded by a phrase such as ‘We need to show that’). The third criticism would be harder to deal with, but when we switch to having two styles of write up – a ‘proof write-up’ and a ‘proof-discovery account’ – then for the first style we will let the program work out the values of bulleted variables, then simply declare those values when the variable is first mentioned after being suspended. This will correspond to the human practice of writing something like ‘Let $\delta=\min\\{\eta,\theta\\}$’ or ‘Consider the sequence $(b_{n})$ defined by $b_{n}=a_{n}/(1+a_{n})$,’ which ‘magically’ does exactly what it needs to do later in the proof. Although our program’s output came in for quite a bit of criticism, so did the write-ups by the undergraduate and PhD student – it seems that the readers were harsh judges. However, for most of the problems, the human write-ups were found preferable to the program’s. After the success (as we considered it) of this experiment, we dared to try a direct test. We published a new post, this time explaining that one proof was by a program, one by an undergraduate and one by a PhD student, and inviting readers to vote on which one they thought was by the program. For each problem, the write-ups were numbered (a), (b) and (c). There were seven options for the voting: one could choose between (a), (b) and (c), but also choose between ‘The computer-generated output is definitely $$’ and ‘I think the computer-generated output is $$ but am not certain’; the seventh option was ‘I have no idea which write-up was computer generated.’ Again there was the opportunity to comment, for those who wanted to explain the reasons for their choices. We did not reveal the results of the voting so far, or anybody’s comments, until the experiment was ended and the voting was closed. However, there was a different kind of dependence between answers, which was that people had the opportunity to look for clues that two different write-ups were from the same source. Given that we had not tried to remove stylistic ‘tics’ from our program’s write-ups, this put the program at a significant disadvantage. It was clear from the comments that many people had noticed that for each problem exactly one write-up ended with the words ‘we are done’. Despite this, the program did reasonably well at fooling people that it was human. The typical pattern was that roughly half the voters would correctly guess which output was by the program, with slightly under half of that half saying that the output was definitely by the program. The undergraduate would always ‘come second’, and there would always be a fair number of people who said that they had no idea which output was written by the computer. There were surprisingly many votes for ‘The computer-generated output is definitely $$,’ when $$ was the wrong answer. The total number of votes was always at least 300, and for the first problem listed (the intersection of open sets is open) it was over 1000. One slight complication was that after a day or two the post was listed on the front page of Hacker News. The result was that the number of votes doubled in a couple of hours, and it may be that the profile of a typical voter changed. Fortunately, we had noted down the voting numbers just before this happened, so we presented those results as well as the final numbers. In the end, however, the proportions did not change very much. The detailed numbers can be found here: http://gowers.wordpress.com/2013/04/14/answers-results-of-polls-and-a-brief- description-of-the-program/ . One thing this experiment could not tell us, except to a limited extent through the comments, was whether the program was good at fooling _mathematicians_ that it was human. It could be that the more mathematically experienced readers found the program’s output easy to distinguish, while the votes for the human write-ups came from people who were not used to reading mathematical proofs. However, we feel justified in concluding that the program’s output is not _obviously_ written by a computer program, and that was our objective. ## 4 Future work In the long term, we would like to enlarge significantly the set of problems that our program, or some new version of it, is capable of solving. To do this, we will have to enable the program to handle certain kinds of deductions that it currently handles either not at all or only in a rather rudimentary way. In particular, an immediate target is to give the program the means to deal with second-order quantification, which would allow it to solve easy compactness problems, and also problems that require the construction of ‘obvious’ sequences. At a more basic level, the program does not currently solve problems that involve proof by contraposition or contradiction. It is not hard to add moves that allow it to cope with a few problems of this kind, but it is trickier to do so while not letting it apply those moves in inappropriate contexts. More work is needed to understand what triggers the ‘contradiction move’ in human mathematicians, but we expect to be able to add this facility in the near future. The program is also not as good as we would like at handling equality substitutions. 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1309.4602	S. Bhattacharya, P. Chalermsook, K. Mehlhorn, and A. Neumann # New Approximability Results for the Robust $k$-Median Problem Sayan Bhattacharya Max-Planck Institut für Informatik Campus E1 4, 66123 Saarbrücken, Germany {bsayan,parinya,mehlhorn,aneumann}@mpi-inf.mpg.de Parinya Chalermsook Max- Planck Institut für Informatik Campus E1 4, 66123 Saarbrücken, Germany {bsayan,parinya,mehlhorn,aneumann}@mpi-inf.mpg.de Kurt Mehlhorn Max-Planck Institut für Informatik Campus E1 4, 66123 Saarbrücken, Germany {bsayan,parinya,mehlhorn,aneumann}@mpi-inf.mpg.de Adrian Neumann Max-Planck Institut für Informatik Campus E1 4, 66123 Saarbrücken, Germany {bsayan,parinya,mehlhorn,aneumann}@mpi-inf.mpg.de ###### Abstract. We consider a robust variant of the classical $k$-median problem, introduced by Anthony et al. [2]. In the _Robust $k$-Median problem_, we are given an $n$-vertex metric space $(V,d)$ and $m$ client sets $\left\\{S_{i}\subseteq V\right\\}_{i=1}^{m}$. The objective is to open a set $F\subseteq V$ of $k$ facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize $\max_{i}\sum_{v\in S_{i}}d(F,v)$. Anthony et al. showed an $O(\log m)$ approximation algorithm for any metric and APX- hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing $\Omega(\log m/\log\log m)$ approximation hardness, unless ${\sf NP}\subseteq\bigcap_{\delta>0}{\sf DTIME}(2^{n^{\delta}})$. This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances. ###### Key words and phrases: Hardness of Approximation, Heuristics ###### 1991 Mathematics Subject Classification: F.2.2 Nonnumerical Algorithms and Problems ## 1\. Introduction In the classical $k$-median problem, we are given a set of clients located on a metric space with distance function $d:V\times V\rightarrow\mathbb{R}$. The goal is to open a set of facilities $F\subseteq V$, $\|F\|=k$, so as to minimize the sum of the connection costs of the clients in $V$, i.e., their distances from their nearest facilities in $F$. This is a central problem in approximation algorithms, and quite naturally, it has received a large amount of attention in the past two decades [6, 5, 7, 13, 12]. At SODA 2008 Anthony et al. [1, 2] introduced a generalization of the $k$-median problem. In their setting, the set of clients that are to be connected to some facility is not known in advance, and the goal is to perform well in spite of this uncertainty about the future. In particular, they formulated the problem as follows. ###### Definition 1.1 (Robust $k$-Median). An instance of this problem is a triple $(V,{\mathcal{S}},d)$. This defines a set of _locations_ $V$, a collection of $m$ sets of _clients_ ${\mathcal{S}}=\\{S_{1},\ldots,S_{m}\\}$, where $S_{i}\subseteq V$ for all $i\in\\{1,\ldots,m\\}$, and a metric distance function $d:V\times V\rightarrow\mathbb{R}$. We have to open a set of $k$ facilities $F\subseteq V$, $\|F\|=k$, and the goal is to minimize the cost of the most expensive set of clients, i.e. minimize $\max_{i=1}^{m}\sum_{v\in S_{i}}d(v,F)$. Here, $d(v,F)$ denotes the minimum distance of the client $v$ from any location in $F$, i.e. $d(v,F)=\min_{u\in F}d(u,v)$. Note that the Robust $k$-Median problem is a natural generalization of the classical $k$-median problem (where $m=1$). In addition, we can think of this formulation as capturing some notion of fairness. To see this, simply interpret each set $S_{i}$ as a community of clients who would pay $\sum_{v\in S_{i}}d(v,F)$ for getting connected to some facility. Now the objective ensures that no single community pays too much, while minimizing the cost. Anthony et al. [2] gave an $O(\log m)$-approximation algorithm for this problem, and a lower bound of $(2-\epsilon)$ for the best possible approximation ratio by a reduction from Vertex Cover. #### Our Results We give nearly tight hardness of approximation results for the Robust $k$-Median problem. We show that unless ${\sf NP}\subseteq\cap_{\delta>0}{\sf DTIME}(2^{n^{\delta}})$, the problem admits no poly-time $o(\log m/\log\log m)$-approximation, even on uniform and line metrics. Our first hardness result is tight up to a constant factor, as a simple rounding scheme gives a matching upper bound on uniform metrics (see Section 3.1). Our second, and rather surprising, result shows that “Robust $k$-Median” is a rare problem with super-constant hardness of approximation even on line metrics, in sharp contrast to most other geometric optimization problems which admit polynomial time approximation schemes, e.g. [3, 11]. In Section 5 we investigate the performance of some heuristics. Already a very simple greedy strategy provides reasonably good performance on a realistic class of instances. We use an LP relaxation of the problem as a lower bound. #### Our Techniques First, we note that the Robust $k$-Median problem on uniform metrics is equivalent to the following variant of the set cover problem: Given a set $U$ of ground elements, a collection of sets ${\mathcal{X}}=\left\\{X\subseteq U\right\\}$, and an integer $t\leq\|{\mathcal{X}}\|$, our goal is to select $t$ sets from ${\mathcal{X}}$ in order to minimize the number of times an element from $U$ is hit (see Lemma 3.2). We call this problem Minimum Congestion Set Packing (MCSP). This characterization allows us to focus on proving the hardness of MCSP, and to employ the tools developed for the set cover problem. We now revisit the reduction used in proving the hardness of the set cover problem by Feige [8], building on the framework of Lund and Yannakakis [14], and discuss how our approach differs from theirs. Intuitively, they compose the Label Cover instance with a set system that has some desirable properties. Informally speaking, in the Label Cover problem, we are given a graph where each vertex $v$ can be assigned a label from a set $L$, and each edge $e$ is equipped with a constraint $\Pi_{e}\subseteq L\times L$ specifying the accepting pairs of labels for $e$. Our goal is to find a labeling of vertices that maximizes the number of accepting edges. This problem is known to be hard to approximate to within a factor of $2^{\log^{1-\epsilon}\|E\|}$ [4, 15], where $\|E\|$ is the number of edges. Thus, if we manage to reduce Label Cover to MCSP, we would hopefully obtain a large hardness of approximation factor for MCSP as well. From the Label Cover instance, [14] creates an instance of Set Cover by having sets of the form $S(v,\ell)$ for each vertex $v$ and each label $\ell\in L$. Intuitively the set $S(v,\ell)$ means choosing label $\ell$ for vertex $v$ in the label cover instance. Now, if we assume that the solution is well behaved, in the sense that for each vertex $v$, only one set of the form $S(v,\ell)$ is chosen in the solution, we would be immediately done (because each set indeed corresponds to label assignment). However, a solution need not have this form, e.g. choosing sets $S(v,\ell)$ and $S(v,\ell^{\prime})$ would translate to having two labels $\ell,\ell^{\prime}$ for the label cover instance. To prevent an ill-behaved solution, “partition systems” were introduced and used in both [14] and [8]. Feige considers the hypergraph version of Label Cover to obtain a sharper hardness result of $\ln n-O(\ln\ln n)$ instead of $\frac{1}{4}\ln n$ in [14]; here $n$ denotes the size of the universe. Now we highlight how our reduction is different from theirs. The high level idea of our reduction is the same, i.e. we have sets of the form $S(v,\ell)$ that represent assigning label $\ell$ to vertex $v$. However, we need a different partition system and a totally different analysis. Moreover, while a reduction from standard Label Cover gives nearly tight $O(\log n)$ hardness for Set Cover, it can (at best) only give the hardness of $2-\epsilon$ for MCSP. To prove our results, we do need a reduction from the Hypergraph Label Cover problem. This suggests another natural distinction between MCSP and Set Cover. Finally, to obtain the hardness of the Robust $k$-Median problem on the line metric, we embed the instance created from the MCSP reduction onto the line such that the values of optimal solutions are preserved. This way we get the same hardness gap for line metrics. ## 2\. Preliminaries We will show that the Robust $k$-Median problem is $\Omega(\log m/\log\log m)$ hard to approximate, even for the special cases of _uniform metrics_ (see Section 3) and _line metrics_ (see Section 4). Recall that $d$ is a uniform metric iff we have $d(u,v)\in\\{0,1\\}$ for all locations $u,v\in V$. Further, $d$ is a line metric iff the locations in $V$ can be embedded into a line in such a way that $d(u,v)$ equals the Euclidean distance between $u$ and $v$, for all $u,v\in V$. Throughout this paper, we will denote any set of the form $\\{1,2,\ldots,i\\}$ by $[i]$. Our hardness results will rely on a reduction from the _$r$ -Hypergraph Label Cover_ problem, which is defined as follows. ###### Definition 2.1 ($r$-Hypergraph Label Cover). An instance of this problem is a triple $(G,\pi,r)$, where $G=(\mathcal{V},\mathcal{E})$ is a $r$-partite hypergraph with vertex set ${\mathcal{V}}=\bigcup_{j=1}^{r}{\mathcal{V}}_{j}$ and edge set ${\mathcal{E}}$. Each edge $h\in\mathcal{E}$ contains one vertex from each part of ${\mathcal{V}}$, i.e. $\|h\cap{\mathcal{V}}_{j}\|=1$ for all $j\in[r]$. Every set ${\mathcal{V}}_{j}$ has an associated set of _labels_ $L_{j}$. Further, for all $h\in\mathcal{E}$ and $j\in[r]$, there is a mapping $\pi_{h}^{j}:L_{j}\rightarrow C$ that projects the labels from $L_{j}$ to a common set of _colors_ $C$. The problem is to assign to every vertex $v\in{\mathcal{V}}_{j}$ some label $\sigma(v)\in L_{j}$. We say that an edge $h=(v_{1},\ldots,v_{r})$, where $v_{j}\in{\mathcal{V}}_{j}$ for all $j\in[r]$, is _strongly satisfied_ under $\sigma$ iff the labels of all its vertices are mapped to the same element in $C$, i.e. $\pi_{h}^{j}(\sigma(v_{j}))=\pi_{h}^{j^{\prime}}(\sigma(v_{j^{\prime}}))$ for all $j,j^{\prime}\in[r]$. In contrast, we say that the edge is _weakly satisfied_ iff there exists some pair of vertices in $h$ whose labels are mapped to the same element in $C$, i.e. $\pi^{j}_{h}(\sigma(v_{j}))=\pi^{j^{\prime}}_{h}(\sigma(v_{j^{\prime}}))$ for some $j,j^{\prime}\in[r]$, $j\neq j^{\prime}$. For ease of exposition, we will often abuse the notation and denote by $j(v)$ the part of ${\mathcal{V}}$ to which a vertex $v$ belongs, i.e. if $v\in{\mathcal{V}}_{j}$ for some $j\in[r]$, then we set $j(v)\leftarrow j$. The next theorem will be crucial in deriving our hardness result. The proof of this theorem follows from Feige’s $r$-Prover system [8] (see Appendix A). ###### Theorem 2.2. Let $r\in\mathbb{N}$ be a parameter. There is a polynomial time reduction from $n$-variable 3-SAT to $r$-Hypergraph Label Cover with the following properties: * • (Yes-Instance) If the formula is satisfiable, then there is a labeling that strongly satisfies every edge in $G$. * • (No-Instance) If the formula is not satisfiable, then every labeling weakly satisfies at most a $2^{-\gamma r}$ fraction of the edges in $G$, for some universal constant $\gamma$. * • The number of vertices in the graph is $\|{\mathcal{V}}\|=n^{O(r)}$ and the number of edges is $\|\mathcal{E}\|=n^{O(r)}$. The sizes of the label sets are $\left\|L_{j}\right\|=2^{O(r)}$ for all $j\in[r]$, and $\left\|C\right\|=2^{O(r)}$. Further, we have $\|{\mathcal{V}}_{j}\|=\|{\mathcal{V}}_{j^{\prime}}\|$ for all $j,j^{\prime}\in[r]$, and each vertex $v\in{\mathcal{V}}$ has the same degree $r\|\mathcal{E}\|/\|{\mathcal{V}}\|$. We use a _partition system_ that is motivated by the hardness proof of the Set Cover problem [8]. However, we deal with a different problem, and our construction is also different. ###### Definition 2.3 (Partition System). Let $r\in\mathbb{N}$ and let $C$ be any finite set. An $(r,C)$-partition system is a pair $(Z,\left\\{p_{c}\right\\}_{c\in C})$, where $Z$ is an arbitrary (ground) set, and for each $c\in C$, $p_{c}$ is a partition of $Z$ into $r$ subsets, such that the following properties hold. * • (Partition) For all $c\in C$, $p_{c}=\left(A_{c}^{1},\ldots,A_{c}^{r}\right)$ is a partition of $Z$, that is $\bigcup_{j=1}^{r}A_{c}^{j}=Z$, and $A_{c}^{j^{\prime}}\cap A_{c}^{j}=\emptyset$ for all $j,j^{\prime}\in[r],j\neq j^{\prime}$. * • ($r$-intersecting) For any $r$ _distinct_ indices $c_{1},\ldots,c_{r}\in C$ and _not-necessarily distinct_ indices $j_{1},\ldots,j_{r}\in[r]$, we have that $\bigcap_{i=1}^{r}A_{c_{i}}^{j_{i}}\neq\emptyset$. In particular, $A_{c}^{j}\not=\emptyset$ for all $c$ and $j$. In order to achieve a good lower bound on the approximation factor, we need partition systems with _small_ ground sets. The most obvious way to build a partition system is to form an $r$-hypercube: Let $Z=[r]^{\|C\|}$, and for each $c\in C$ and $j\in[r]$, let $A_{c}^{j}$ be the set of all elements in $Z$ whose $c$-th component is $j$. It can easily be verified that this is an $(r,C)$-partition system with $\|Z\|=r^{\left\|C\right\|}$. With this construction, however, we would only get a hardness of $\Omega(\log\log m)$ for our problem. The following lemma shows that it is possible to construct an $(r,C)$-partition system probabilistically with $\|Z\|=r^{O(r)}\log\left\|C\right\|$. ###### Lemma 2.4. There is an $(r,C)$-partition system with $\|Z\|=r^{O(r)}\log\left\|C\right\|$ elements. Further, such a partition system can be constructed efficiently with high probability. ###### Proof 2.5. Let $Z$ be any set of $r^{O(r)}\log\left\|C\right\|$ elements. We build a partition system $(Z,\\{p_{c}\\}_{c\in C})$ as described in Algorithm 1. input : A ground set $Z$, a parameters $r\in\mathbb{N}$, and a set $C$. foreach _$c\in C$_ do /* Construct the partition $p_{c}=(A_{c}^{1},\ldots,A_{c}^{r})$ / Initialize $A_{c}^{j}$ to the empty set for all $j\in[r]$ foreach _ground element $e\in Z$_ do Pick an index $j\in[r]$ independently and uniformly at random and add $e$ to $A_{c}^{j}$ Algorithm 1 A randomized algorithm for constructing an $(r,C)$-partition system. In Algorithm 1, by construction each $p_{c}$ is a partition of $Z$, i.e. the first property stated in Definition 2.3 is satisfied. We bound the probability that the second property is violated. Fix any choice of $r$ _distinct_ indices $c_{1},\ldots,c_{r}\in C$ and _not necessarily distinct_ indices $j_{1},\ldots,j_{r}\in[r]$. We say that a _bad event_ occurs when the intersection of the corresponding sets is empty, i.e. $\bigcap_{i=1}^{r}A_{c_{i}}^{j_{i}}=\emptyset$. To upper bound the probability of a bad event, we focus on events of the form $E_{e,i}$ – this occurs when an element $e\in Z$ is included in a set $A_{c_{i}}^{j_{i}}$. Since the indices $c_{1}\ldots c_{r}$ are distinct, it follows that the events $\\{E_{e,i}\\}$ are mutually independent. Furthermore, note that we have $\Pr[E_{e,i}]=1/r$ for all $e\in Z,i\in[r]$. Hence, the probability that an element $e\in Z$ does not belong to the intersection $\bigcap_{i=1}^{r}A_{c_{i}}^{j_{i}}$ is given by $1-\Pr[\bigcap_{i=1}^{r}E_{e,i}]=1-1/r^{r}$. Accordingly, the probability that no element $e\in Z$ belongs to the intersection, which defines the bad event, is equal to $(1-1/r^{r})^{\|Z\|}$. Now, the total number of choices for $r$ distinct indices $c_{1},\ldots,c_{r}$ and $r$ not-necessarily distinct indices $j_{1},\ldots,j_{r}$ is equal to $\binom{\left\|C\right\|}{r}\cdot r^{r}$. Hence, taking a union-bound over all possible bad events, we see that the second property stated in Definition 2.3 is violated with probability at most $\binom{\left\|C\right\|}{r}\cdot r^{r}\cdot(1-r^{r})^{\|Z\|}\leq(\left\|C\right\|r)^{r}\cdot\exp(-\|Z\|/r^{r})$. If we set $\|Z\|=d\cdot r^{d\cdot r}\log\left\|C\right\|$ with sufficiently large constant $d$, then it is easy to see that the second constraint in Definition 2.3 is satisfied with high probability. ## 3\. Hardness of Robust k-Median on Uniform Metrics First, we define a problem called _Minimum Congestion Set Packing_ (MCSP), and then show a reduction from MCSP to Robust $k$-Median on uniform metrics. In Section 3.2, we will then show that MCSP is hard to approximate by reducing Hypergraph Label Cover to MCSP. ###### Definition 3.1. [Minimum Congestion Set Packing (MCSP)] An instance of this problem is a triple $(U,{\mathcal{X}},t)$, where $U$ is a universe of $m$ elements, i.e. $\|U\|=m$, ${\mathcal{X}}$ is a collection of sets ${\mathcal{X}}=\left\\{X\subseteq U\right\\}$ such that $\bigcup_{X\in{\mathcal{X}}}X=U$, and $t\in\mathbb{N}$ and $t\leq\left\|{\mathcal{X}}\right\|$. The objective is to find a collection ${\mathcal{X}}^{\prime}\subseteq{\mathcal{X}}$ of size $t$ that minimizes $\text{{Cong}}({\mathcal{X}}^{\prime})=\max_{e\in U}\text{{Cong}}(e,{\mathcal{X}}^{\prime})$. Here, $\text{{Cong}}({\mathcal{X}}^{\prime})$ refers to the _congestion_ of the solution ${\mathcal{X}}^{\prime}$, and $\text{{Cong}}(e,{\mathcal{X}}^{\prime})=\|\\{X\in{\mathcal{X}}^{\prime}:e\in X\\}\|$ is the congestion of the element $e\in U$ under the solution ${\mathcal{X}}^{\prime}$. ###### Lemma 3.2. Given any MCSP instance $(U,{\mathcal{X}},t)$, we can construct a Robust $k$-Median instance $(V,{\mathcal{S}},d)$ with the same objective value in $poly(\|U\|,\|{\mathcal{X}}\|)$ time, such that $\|U\|=\|{\mathcal{S}}\|$, $\|{\mathcal{X}}\|=\|V\|$, $d$ is a uniform metric, and $k=\|V\|-t$. ###### Proof 3.3. We construct the Robust $k$-Median instance $(V,{\mathcal{S}},d)$ as follows. For every $e\in U$ we create a set of clients $S(e)$, and for each $X\in{\mathcal{X}}$ we create a location $v(X)$. Thus, we get $V=\\{v(X)\,:\,X\in{\mathcal{X}}\\}$, and ${\mathcal{S}}=\\{S(e)\,:\,e\in U\\}$. We place the clients in $S(e)$ at the locations of the sets that contain $e$, i.e. $S(e)=\\{v(X)\,:\,X\in{\mathcal{X}},e\in X\\}$ for all $e\in U$. The distance is defined as $d(u,v)=1$ for all $u,v\in V,u\neq v$, and $d(v,v)=0$. Finally, we set $k\leftarrow\|V\|-t$. Now, it is easy to verify that the Robust $k$-Median instance $(V,{\mathcal{S}},d)$ has a solution with objective $\rho$ iff the corresponding MCSP instance $(U,{\mathcal{X}},t)$ has a solution with objective $\rho$. The intuition is that a location $v(X)\in V$ is _not_ included in the solution $F$ to the Robust $k$-Median instance iff the corresponding set $X$ is included in the solution ${\mathcal{X}}^{\prime}$ to the MCSP instance. Indeed, let $F$ be any subset of ${\mathcal{X}}$ of size $k$ (= the set of open facilities) and let ${\mathcal{X}}^{\prime}={\mathcal{X}}-F$. Further, let $[X\in{\mathcal{X}}^{\prime}]$ be an indicator variable that is set to $1$ iff $X\in{\mathcal{X}}^{\prime}$. Then $\displaystyle\text{{Cong}}({\mathcal{X}}^{\prime})$ $\displaystyle=\max_{e\in U}\text{{Cong}}(e,{\mathcal{X}}^{\prime})=\max_{e\in U}\sum_{X;e\in X}[X\in{\mathcal{X}}^{\prime}]$ $\displaystyle=\max_{e\in U}\sum_{X;e\in X}\min_{Y\in F}d(X,Y)=\max_{S(e)\in{\mathcal{S}}}\sum_{v(X)\in S(e)}d(v(X),F).$ We devote the rest of Section 3 to the MCSP problem and show that it is $\Omega(\log\|U\|/\log\log\|U\|)$ hard to approximate. This, in turn, will imply a $\Omega(\log\|{\mathcal{S}}\|/\log\log\|{\mathcal{S}}\|)$ hardness of approximation for Robust $k$-Median on uniform metrics. We will prove the hardness result via a reduction from Hypergraph Label Cover. ### 3.1. Integrality Gap Before proceeding to the hardness result, we show that a natural LP relaxation for the MCSP problem [2] has an integrality gap of $\Omega(\log m/\log\log m)$, where $m=\|U\|$ is the size of the universe of elements. In the LP, we have a variable $y(X)$ indicating that the set $X\in{\mathcal{X}}$ is chosen, and a variable $z$ which represents the maximum congestion among the elements. $\displaystyle\min\quad$ $\displaystyle z$ s.t. $\displaystyle\sum_{\mathclap{X\in{\mathcal{X}}:e\in X}}y(X)\leq z\mbox{ for all $e\in U$}$ $\displaystyle\sum_{\mathclap{X\in{\mathcal{X}}}}y(X)=t$ The Instance: Now, we construct a bad integrality gap instance $(U,{\mathcal{X}},t)$. Let $d$ be the intended integrality gap, let $\eta=d^{2}$, and let $U=\left\\{I\,:\,I\subseteq[\eta],\left\|I\right\|=d\right\\}$ be all subsets of $[\eta]$ of size $d$. The collection ${\mathcal{X}}$ consists of $\eta$ sets $X_{1},\ldots,X_{\eta}$, where $X_{i}=\left\\{I\,:\,I\in U\text{ and }i\in I\right\\}$. Note that the universe $U$ consists of $\|U\|=m=\binom{\eta}{d}$ elements, and each element $I$ is contained in exactly $d$ sets, namely $I\in X_{i}$ if and only if $i\in I$. Finally, we set $t\leftarrow\eta/d$. Analysis: The fractional solution simply assigns a value of $1/d$ to each variable $y(X_{i})$; this ensures that the total (fractional) number of sets selected is $\eta/d=t$. Furthermore, each element is contained (fractionally) in exactly one set, so the fractional solution has cost one. Any integral solution must choose $\eta/d=d$ sets, say $X_{i_{1}}\ldots X_{i_{d}}$. Then $I=\left\\{i_{1},\ldots,i_{d}\right\\}\in X_{i_{\lambda}}$ for all $\lambda\in[d]$ and hence the congestion of $I$ is $d$, and this also means that any integral solution has cost at least $d$. Finally, since $\|U\|=m\leq\eta^{d}\leq(d^{2})^{d}$, we have $d=\Omega(\log m/\log\log m)$. Tightness of the result: The bound on the hardness and integrality gap is tight for the uniform metric case, as there is a simple $O(\log m/\log\log m)$-approximation algorithm. Pick each set $X$ with probability equal to $\min(1,2y(X))$. The expected congestion is $2z$ for each element. By Chernoff’s bound [10], an element is covered by no more than $z\cdot O(\log m/\log\log m)$ sets with high probability. A similar algorithm gives the same approximation guarantee for the Robust $k$-Median problem on uniform metrics. ### 3.2. Reduction from r-Hypergraph Label Cover to Minimum Congestion Set Packing The input is an instance $\left(G,\pi,r\right)$ of the $r$-Hypergraph Label Cover problem (see Definition 2.1). From this we construct the following instance $(U,{\mathcal{X}},t)$ of the MCSP problem (see Definition 3.1). • First, we define the universe $U$ as a union of disjoint sets. For each edge $h\in{\mathcal{E}}$ in the hypergraph we have a set $U_{h}$. All these sets have the same size $m^{}$ and are pairwise disjoint, i.e. $U_{h}\cap U_{h^{\prime}}=\emptyset$ for all $h,h^{\prime}\in{\mathcal{E}}$, $h^{\prime}\neq h$. The universe $U$ is then the union of these sets $U=\bigcup_{h\in{\mathcal{E}}}U_{h}$. Since the $U_{h}$ are mutually disjoint, we have $m=\|U\|=\|{\mathcal{E}}\|\cdot m^{}$. Recall that $C$ is the target set of $\pi$. Each set $U_{h}$ is the ground set of an $(r,C)$-partition system (see Definition 2.3) as given by Lemma 2.4. In particular we have $m^{}=r^{O(r)}\log\left\|C\right\|$. We denote the $r$-partitions associated with $U_{h}$ by $\\{p_{c}(h)\\}_{c\in C}$, where $p_{c}(h)=\left(A^{1}_{c}(h),\ldots,A^{r}_{c}(h)\right)$. • Second, we construct the collection of sets ${\mathcal{X}}$ as follows. For each $j\in[r]$, $v\in{\mathcal{V}}_{j}$ and $\ell\in L_{j}$, ${\mathcal{X}}$ contains the set $X(v,\ell)$, where $X(v,\ell)=\bigcup_{h:v\in h}A^{j}_{\pi^{j}_{h}(\ell)}(h)$. In other words, $X(v,\ell)\cap U_{h}$ is empty if $v\not\in h$ and is equal to $A^{j}_{\pi^{j}_{h}(\ell)}(h)$ if $v\in h$. Intuitively, choosing the set $X(v,\ell)$ corresponds to assigning label $\ell$ to the vertex $v$. * • Third, we define $t\leftarrow\|{\mathcal{V}}\|$. Intuitively, this means that each vertex in ${\mathcal{V}}$ gets one label. We assume for the sequel that the $r$-Hypergraph Label Cover instance is chosen according to Theorem 2.2. We assume that the parameter $r$ satisfies $r^{7}2^{-\gamma r}<1$. In the proof of the main theorem, we will fix $r$ to a specific value. ### 3.3. Analysis We show that the reduction from Hypergraph Label Cover to MCSP satisfies two properties. In Lemma 3.4, we show that for a Yes-Instance (see Theorem 2.2), the corresponding MCSP instance admits a solution with congestion one. Second, in case of a No-Instance, we show in Lemma 3.10 that every solution to the corresponding MCSP instance has congestion at least $r$. ###### Lemma 3.4 (Yes-Instance). If the Hypergraph Label Cover instance $(G,\pi,r)$ admits a labeling that strongly satisfies every edge, then the MCSP instance $(U,{\mathcal{X}},t)$ constructed in Section 3.2 admits a solution where the congestion of every element in $U$ is exactly one. ###### Proof 3.5. Suppose that there is a labeling $\sigma$ that strongly satisfies every edge $h\in{\mathcal{E}}$. We will show how to pick $t=\|{\mathcal{V}}\|$ sets from ${\mathcal{X}}$ such that each element in $U$ is contained in exactly one set. This implies that the maximum congestion is one. For each $j\in[r]$ and each vertex $v\in{\mathcal{V}}_{j}$, we choose the set $X(v,\sigma(v))$. Thus, the total number of sets chosen is exactly $\|{\mathcal{V}}\|$. To see that the congestion is indeed one, we concentrate on the elements in $U_{h}$, where $h=(v_{1},\ldots,v_{r})$, $v_{j}\in{\mathcal{V}}_{j}$ for all $j\in[r]$, is one of the edges in ${\mathcal{E}}$. The picked sets that intersect $U_{h}$ are $X(v_{j},\sigma(v_{j}))$, where $j\in[r]$. Since $h$ is strongly satisfied, $\pi_{h}$ maps all labels of the vertices in $h$ to a common $c\in C$, that is $\pi^{j}_{h}(\sigma(v_{j}))=c$ for all $j\in[r]$. Thus $U_{h}\cap X(v_{j},\sigma(v_{j}))=A^{j}_{c}(h)$. By the definition of a partition system (see Definition 2.3), the sets $A^{1}_{c}(h)\ldots A^{r}_{c}(h)$ partition the elements in $U_{h}$. This completes the proof. Now, we turn to the proof of Lemma 3.10. Towards this end, we fix a collection ${\mathcal{X}}^{\prime}\subseteq{\mathcal{X}}$ of size $t$ and show that some element in $U$ has congestion at least $r$ under ${\mathcal{X}}^{\prime}$. The intuition being that many edges in $G=({\mathcal{V}},{\mathcal{E}})$ are not even weakly satisfied, and the elements in $U$ corresponding to those edges incur large congestion. Recall that for a $v\in{\mathcal{V}}$, we define $j(v)\in\mathbb{N}$ to be such that $v\in{\mathcal{V}}_{j(v)}$. ###### Claim 1. For $v\in{\mathcal{V}}$, let ${\mathcal{L}}_{v}=\left\\{\ell\in L_{j(v)}\,:\,X(v,\ell)\in{\mathcal{X}}^{\prime}\right\\}$. For $h\in{\mathcal{E}}$, let $\Lambda_{h}=\left\\{X(v,\ell)\in{\mathcal{X}}^{\prime}\,:\,v\in h\right\\}$ and $\lambda(h)=\left\|\Lambda_{h}\right\|$. If the solution ${\mathcal{X}}^{\prime}$ has congestion less than $r$ then $\left\|{\mathcal{L}}_{v}\right\|<r^{2}$ and $\left\|\Lambda_{h}\right\|<r^{3}$. ###### Proof 3.6. Since $\Lambda_{h}=\bigcup_{v\in h}{\mathcal{L}}_{v}$, it suffices to prove $\left\|{\mathcal{L}}_{v}\right\|<r^{2}$ for all $v$. Assume otherwise, i.e., $\left\|{\mathcal{L}}_{v}\right\|\geq r^{2}$ for some $v\in{\mathcal{V}}_{j}$, $j\in[r]$. Let $h$ be any hyper-edge with $v\in h$. Consider the images of the labels in ${\mathcal{L}}_{v}$ under the projection $\pi^{j}_{h}$. Either we have at least $r$ distinct images or at least $r$ elements in $L_{v}$ are mapped to the same element of $C$. In the former case, we have $r$ pairwise distinct labels $\ell_{1}$ to $\ell_{r}$ in ${\mathcal{L}}_{v}$ and $r$ pairwise distinct labels $c_{1}$ to $c_{r}$ in $C$ such that $\pi^{j}_{h}(\ell_{i})=c_{i}$ for $i\in[r]$. The set $X(v,\ell_{i})$ contains $A^{j}_{c_{i}}(h)$ and $\bigcap_{i\in[r]}A^{j}_{c_{i}}(h)\not=\emptyset$ by property (2) of partition systems (see Definition 2.3). Thus some element has congestion at least $r$. In the latter case, we have $r$ pairwise distinct labels $\ell_{1}$ to $\ell_{r}$ in ${\mathcal{L}}_{v}$ and a label $c$ in $C$ such that $\pi^{j}_{h}(\ell_{i})=c$ for $i\in[r]$. The set $X(v,\ell_{i})$ contains $A^{j}_{c}(h)$ and hence every element in this non-empty set (property (2) of partition systems) has congestion at least $r$. ###### Definition 3.7 (Colliding Edge). We say that an edge $h\in{\mathcal{E}}$ is _colliding_ iff there are sets $X(v,\ell),X(v^{\prime},\ell^{\prime})\in{\mathcal{X}}^{\prime}$ with $v,v^{\prime}\in h$, $v\not=v^{\prime}$, and $\pi^{j(v)}_{h}(\ell)=\pi^{j(v^{\prime})}_{h}(\ell^{\prime})$. ###### Claim 2. Suppose that the solution ${\mathcal{X}}^{\prime}$ has congestion less than $r$, and more than a $r^{4}2^{-\gamma r}$ fraction of the edges in ${\mathcal{E}}$ are colliding. Then there is a labeling $\sigma$ for $G$ that weakly satisfies at least a $2^{-\gamma r}$ fraction of the edges in ${\mathcal{E}}$. ###### Proof 3.8. For each $v\in{\mathcal{V}}$, we define the label set ${\mathcal{L}}_{v}=\\{\ell\in L_{j(v)}\,:\,X(v,\ell)\in{\mathcal{X}}^{\prime}\\}$. Then $\left\|{\mathcal{L}}_{v}\right\|<r^{2}$ by Claim 1. We construct a labeling function $\sigma$ using Algorithm 2. foreach _vertex $v\in{\mathcal{V}}$_ do if _${\mathcal{L}}_{v}\neq\emptyset$_ then Pick a color $\sigma(v)$ uniformly and independently at random from ${\mathcal{L}}_{v}$ else Pick an arbitrary color $\sigma(v)$ from $L_{j(v)}$ Algorithm 2 An algorithm for constructing a labeling function. Now we bound the expected fraction of weakly satisfied edges under $\sigma$ from below. Take any colliding edge $h\in{\mathcal{E}}$. This means that there are vertices $v\in{\mathcal{V}}_{j}$, $v^{\prime}\in{\mathcal{V}}_{j^{\prime}}$ with $j\neq j^{\prime}$, and colors $\ell\in{\mathcal{L}}_{v}$, $\ell^{\prime}\in{\mathcal{L}}_{v^{\prime}}$ such that $v,v^{\prime}\in h$ and $\pi^{j}_{h}(\ell)=\pi^{j^{\prime}}_{h}(\ell^{\prime})$. By Claim 1, $\|{\mathcal{L}}_{v}\|$ and $\|{\mathcal{L}}_{v^{\prime}}\|$ are both at most $r^{2}$. Since the colors $\sigma(v)$ and $\sigma(v^{\prime})$ are chosen uniformly and independently at random from their respective palettes ${\mathcal{L}}_{v}$ and ${\mathcal{L}}_{v^{\prime}}$, we have $\Pr[\sigma(v)=\ell\text{ and }\sigma(v^{\prime})=\ell^{\prime}]\geq 1/r^{4}$. In other words, every colliding edge is weakly satisfied with probability at least $1/r^{4}$. Since more than a $r^{4}2^{-\gamma r}$ fraction of the edges in ${\mathcal{E}}$ are colliding, from linearity of expectation we infer that the expected fraction of edges weakly satisfied by $\sigma$ is at least $2^{-\gamma r}$. ###### Claim 3. Let $\Lambda_{h}=\left\\{X(v,\ell)\in{\mathcal{X}}^{\prime}\,:\,v\in h\right\\}$, and $\lambda(h)=\|\Lambda_{h}\|$. We have $\sum_{h\in{\mathcal{E}}}\lambda(h)=r\|{\mathcal{E}}\|$. ###### Proof 3.9. This is a simple counting argument. Consider a bipartite graph $H$ with vertex set $A\dot{\cup}B$, where each vertex in $A$ represents a set $X(v,\ell)$, and each vertex in $B$ represents an edge $h\in{\mathcal{E}}$. There is an edge between two vertices iff the set $X(v,\ell)$ contains some element in $U_{h}$. The quantity $\sum_{h\in{\mathcal{E}}}\lambda(h)$ counts the number of edges in $H$ where one endpoint is included in the solution ${\mathcal{X}}^{\prime}$. Since ${\mathcal{X}}^{\prime}$ picks $t=\|{\mathcal{V}}\|$ sets and each set has degree $r\|{\mathcal{E}}\|/\|{\mathcal{V}}\|$ in the $H$ (see Theorem 2.2), the total number of edges that are chosen is exactly $\|{\mathcal{V}}\|\times\left(r\|{\mathcal{E}}\|/\|{\mathcal{V}}\|\right)=r\|{\mathcal{E}}\|$. Let ${\mathcal{E}}^{\prime}\subseteq{\mathcal{E}}$ denote the set of colliding edges, and define ${\mathcal{E}}^{\prime\prime}={\mathcal{E}}-{\mathcal{E}}^{\prime}$. Suppose that we are dealing with a No-Instance (see Theorem 2.2), i.e. the solution ${\mathcal{X}}^{\prime}$ has congestion less than $r$ and every labeling weakly satisfies at most a $2^{-\gamma r}$ fraction of the edges in ${\mathcal{E}}$. Then $\lambda(h)\leq r^{3}$ for all $h\in{\mathcal{E}}$ by Claim 1, and no more than $r^{4}2^{-\gamma r}\|{\mathcal{E}}\|$ edges are colliding, i.e. $\left\|{\mathcal{E}}^{\prime}\right\|\leq r^{4}2^{-\gamma r}\left\|{\mathcal{E}}\right\|$, by Claim 2. Using these facts we conclude that $\sum_{h\in{\mathcal{E}}^{\prime}}\lambda(h)\leq r^{7}2^{-\gamma r}\|{\mathcal{E}}\|<\|{\mathcal{E}}\|$, as by assumption $r^{7}2^{-\gamma r}<1$. Now, applying Claim 3, we get $\sum_{h\in{\mathcal{E}}^{\prime\prime}}\lambda(h)=r\|{\mathcal{E}}\|-\sum_{h\in{\mathcal{E}}^{\prime}}\lambda(h)>(r-1)\|{\mathcal{E}}\|$. In particular, there is an edge $h\in{\mathcal{E}}^{\prime\prime}$ with $\lambda(h)\geq r$. Recall that $\Lambda_{h}=\left\\{X(v,\ell)\in{\mathcal{X}}^{\prime}\,:\,v\in h\right\\}$ are the sets in ${\mathcal{X}}^{\prime}$ that intersect $U_{h}$ and note that $\|\Lambda_{h}\|=\lambda(h)\geq r$. Let ${\mathcal{X}}^{}\subseteq\Lambda_{h}$ be a _maximal_ collection of sets with the following property: For every two distinct sets $X(v,\ell),X(v^{\prime},\ell^{\prime})\in{\mathcal{X}}^{}$ we have $\pi^{j(v)}_{h}(\ell)\neq\pi^{j(v^{\prime})}_{h}(\ell^{\prime})$. Hence, from the definition of a partition system (see Definition 2.3), it follows that the intersection of the sets in ${\mathcal{X}}^{}$ and the set $U_{h}$ is nonempty. Now, consider any set $X(v,\ell)\in\Lambda_{h}-{\mathcal{X}}^{}$. Since the collection ${\mathcal{X}}^{}$ is maximal, there must be at least one set $X(v^{\prime},\ell^{\prime})$ in ${\mathcal{X}}^{}$ with $\pi^{j(v)}_{h}(\ell)=\pi^{j(v^{\prime})}_{h}(\ell^{\prime})$. Since $h$ is not colliding, we must have $j(v)=j(v^{\prime})$. Consequently we get $X(v,\ell)\cap U_{h}=X(v^{\prime},\ell^{\prime})\cap U_{h}$. In other words, for every set $X\in\Lambda_{h}-{\mathcal{X}}^{}$, there is some set $X^{\prime}\in{\mathcal{X}}^{}$ where $X\cap U_{h}=X^{\prime}\cap U_{h}$. Thus, $U_{h}\cap(\bigcap_{X\in\Lambda_{h}}X)=U_{h}\cap(\bigcap_{X\in{\mathcal{X}}^{}}X)\neq\emptyset$. Every element in the intersection of the sets in $\Lambda_{h}$ and $U_{h}$ will have congestion $\|\Lambda_{h}\|\geq r$. This leads to the following lemma. ###### Lemma 3.10 (No-Instance). Suppose that every labeling weakly satisfies at most a $2^{-\gamma r}$ fraction of the edges in the hypergragph label cover instance $(G,\pi,r)$, for some universal constant $\gamma$ and that $r^{7}2^{-\gamma r}<1$. Then the congestion incurred by every solution to the MCSP instance $(U,{\mathcal{X}},t)$ constructed in Section 3.2 is at least $r$. We are now ready to prove the main theorem of this section. ###### Theorem 3.11. The Robust $k$-Median problem $(V,{\mathcal{S}},d)$ is $\Omega(\log m/\log\log m)$ hard to approximate on uniform metrics, where $m=\|{\mathcal{S}}\|$, unless $\textsf{NP}\subseteq\bigcap_{\delta>0}\textsf{DTIME}(2^{n^{\delta}})$. ###### Proof 3.12. Assume that there is a polynomial time algorithm for the Robust $k$-Median problem that guarantees an approximation ratio in $o(\log\left\|{\mathcal{S}}\right\|/\log\log\left\|{\mathcal{S}}\right\|)$. Then, by Lemma 3.2, there is an approximation algorithm for the Minimum Congestion Set Packing problem with approximation guarantee $o(\log\left\|U\right\|/\log\log\left\|U\right\|)$. Let $\delta>0$ be arbitrary and set $r=\lfloor n^{\delta}\rfloor$, where $n$ is the number of variables in the $3$-SAT instance (see Theorem 2.2). Then $r^{7}2^{-\gamma r}<1$ for all sufficiently large $n$. We first bound the size of the MCSP instance $(U,{\mathcal{X}},t)$ constructed in Section 3.2. By Lemma 2.4, the size of an $(r,C)$-partition system is $\|Z\|=r^{O(r)}\log\left\|C\right\|$. By Theorem 2.2, we have $\left\|C\right\|=2^{O(r)}$. So each set $U_{h}$ has cardinality at most $r^{O(r)}\cdot r=r^{O(r)}$. Also recall that the number of sets in the MCSP instance is $\|{\mathcal{X}}\|=\sum_{j\in[r]}\|{\mathcal{V}}_{j}\|\cdot\|L_{j}\|=n^{O(r)}$, and that the number of elements is $\|U\|=m=\|{\mathcal{E}}\|\cdot r^{O(r)}\leq(nr)^{O(r)}=n^{O(r)}=n^{O(n^{\delta})}=2^{O(r\log r)}$. Thus $r\geq\Omega(\log m/\log\log m)$. The gap in the optimal congestion between the Yes-Instance and the No-Instance is at least $r$ (see Theorem 2.2 and Lemmas 3.4, 3.10). More precisely, for Yes-instances the congestion is at most one and for No-instances the congestion is at least $r$. Since the approximation ratio of the alleged algorithm is $o(\log m/\log\log m)$, it is better than $r$ for all sufficiently large $n$ and hence the approximation algorithm can be used to decide the satisfiability problem. The running time of the algorithm is polynomial in the size of the MCSP instance, i.e., is $\mathrm{poly}(n^{O(n^{\delta})})=n^{O(n^{\delta})}=2^{O(n^{2\delta})}$. Since $\delta>0$ is arbitrary, the theorem follows. ## 4\. Hardness of Robust k-Median on Line Metrics We will show that the reduction from $r$-Hypergraph Label Cover to Minimum Congestion Set Packing (MCSP) can be modified to give a $\Omega(\log m/\log\log m)$ hardness of approximation for the Robust $k$-Median problem on line metrics as well, where $m=\|{\mathcal{S}}\|$ is the number of client-sets. For this section, it is convenient to assume that the label-sets are the initial segments of the natural numbers, i.e., $L_{j}=\\{1,\ldots,\left\|L_{j}\right\|\\}$ and $C=\\{1,\ldots,\left\|C\right\|\\}$. Given a Hypergraph Label Cover instance $(G,\pi,r)$, we first construct a MCSP instance $(U,{\mathcal{X}},t)$ in accordance with the procedure outlined in Section 3.2. Next, from this MCSP instance, we construct a Robust $k$-Median instance $(V,{\mathcal{S}},d)$ as described below. • We create a location in $V$ for every set $X(v,\ell)\in{\mathcal{X}}$. To simplify the notation, the symbol $X(v,\ell)$ will represent both a set in the instance $(U,{\mathcal{X}},t)$, and a location in the instance $(V,{\mathcal{S}},d)$. Thus, we have $V=\\{X(v,\ell)\in{\mathcal{X}}\\}$. Furthermore, we create a set of clients $S(e)$ for every element $e\in U$, which consists of all the locations whose corresponding sets in the MCSP instance contain the element $e$. Thus, we have ${\mathcal{S}}=\\{S(e)\,:\,e\in U\\}$, where $S(e)=\\{X(v,\ell)\in{\mathcal{X}}\,:\,e\in X(v,\ell)\\}$ for all $e\in U$. This step is same as in Lemma 3.2. * • We now describe how to embed the locations in $V$ on a given line. For every vertex $v\in{\mathcal{V}}_{j},j\in[r]$, the locations $X(v,1),\ldots,X(v,\left\|L_{j}\right\|)$ are placed next to one another in sequence, in such a way that the distance between any two consecutive locations is exactly one. Formally, this gives $d(X(v,\ell),X(v,\ell^{\prime}))=\|\ell^{\prime}-\ell\|$ for all $\ell,\ell^{\prime}\in L_{j}$. Furthermore, we ensure that any two locations corresponding to two different vertices in ${\mathcal{V}}$ are _not close to each other_. To be more specific, we have the following guarantee: $d(X(v,\ell),X(v^{\prime},\ell^{\prime}))\geq 2$ whenever $v\neq v^{\prime}$. It is easy to verify that $d$ is a line metric. * • We define $k\leftarrow\|{\mathcal{X}}\|-t$. Note that as $k=\|{\mathcal{X}}\|-t$, there is a one to one correspondence between the solutions to the MCSP instance and the solutions to the Robust $k$-Median instance. Specifically, a set in ${\mathcal{X}}$ is picked by a solution to the MCSP instance iff the corresponding location is _not_ picked in the Robust $k$-Median instance. ###### Lemma 4.1 (Yes-Instance). Suppose that there is a labeling strategy $\sigma$ that strongly satisfies every edge in the Hypergraph Label Cover instance $(G,\pi,r)$. Then there is a solution to the Robust $k$-Median instance $(V,{\mathcal{S}},d)$ with objective one. ###### Proof 4.2. Recall the proof of Lemma 3.4. We construct a solution ${\mathcal{X}}^{\prime}\subseteq{\mathcal{X}}$, $\|{\mathcal{X}}^{\prime}\|=t$, to the MCSP instance $(U,{\mathcal{X}},t)$ as follows. For every vertex $v\in{\mathcal{V}}_{j},j\in[r]$, the solution ${\mathcal{X}}^{\prime}$ contains the set $X(v,\sigma(v))$. Now, focus on the corresponding solution $F_{{\mathcal{X}}^{\prime}}\subseteq V$ to the Robust $k$-Median instance, which picks a location $X$ iff $X\notin{\mathcal{X}}^{\prime}$. Hence, for every vertex $v\in{\mathcal{V}}_{j}$, $j\in[r]$, all but one of the locations $X(v,1),\ldots,X(v,\left\|L_{j}\right\|)$ are included in $F_{{\mathcal{X}}^{\prime}}$. Since any two consecutive locations in such a sequence are unit distance away from each other, the cost of connecting any location in $V$ to the set $F_{{\mathcal{X}}^{\prime}}$ is either zero or one, i.e., $d(X,F_{{\mathcal{X}}^{\prime}})\in\\{0,1\\}$ for all $X\in V={\mathcal{X}}$. For the rest of the proof, fix any set of clients $S(e)\in{\mathcal{S}}$, $e\in U$. The proof of Lemma 3.4 implies that the element $e$ incurs congestion one under ${\mathcal{X}}^{\prime}$. Hence, the element belongs to exactly one set in ${\mathcal{X}}^{\prime}$, say $X^{}$. Again, comparing the solution ${\mathcal{X}}^{\prime}$ with the corresponding solution $F_{{\mathcal{X}}^{\prime}}$, we infer that $S(e)-F_{{\mathcal{X}}^{\prime}}=\\{X^{}\\}$. In other words, every location in $S(e)$, except $X^{}$, is present in the set $F_{{\mathcal{X}}^{\prime}}$. The clients in such locations require zero cost for getting connected to $F_{{\mathcal{X}}^{\prime}}$. Thus, the total cost of connecting the clients in $S(e)$ to the set $F_{{\mathcal{X}}^{\prime}}$ is at most: $\sum_{X\in S(e)}d(X,F_{{\mathcal{X}}^{\prime}})=d(X^{},F_{{\mathcal{X}}^{\prime}})\leq 1.$ Thus, we see that every set of clients in ${\mathcal{S}}$ requires at most unit cost for getting connected to $F_{{\mathcal{X}}^{\prime}}$. So the solution $F_{{\mathcal{X}}^{\prime}}$ to the Robust $k$-Median instance indeed has objective one. ###### Lemma 4.3 (No-Instance). Suppose that every labeling weakly satisfies at most a $2^{-\gamma r}$ fraction of the edges in the Hypergraph Label Cover instance $(G,\pi,r)$, for some constant $\gamma$. Then every solution to the Robust $k$-Median instance $(V,{\mathcal{S}},d)$ has objective at least $r$. ###### Proof 4.4. Fix any solution $F\subseteq V$ to the Robust $k$-Median instance $(V,{\mathcal{S}},d)$, and let ${\mathcal{X}}^{\prime}_{F}\subseteq{\mathcal{X}}$ denote the corresponding solution to the MCSP instance $(U,{\mathcal{X}},t)$. Lemma 3.10 states that there is some element $e\in U$ with congestion at least $r$ under ${\mathcal{X}}^{\prime}_{F}$. In other words, there are at least $r$ sets $X_{1},\ldots,X_{r}\in{\mathcal{X}}^{\prime}_{F}$ that contain the element $e$. The locations corresponding to these sets are not picked by the solution $F$. Furthermore, the way the locations have been embedded on a line ensures that the distance between any location and its nearest neighbor is at least one. Hence, we have $d(X_{i},F)\geq 1$ for all $i\in[r]$. Summing over these distances, we infer that the total cost of connecting the clients in $S(e)$ to $F$ is at least $\sum_{i\in[r]}d(X_{i},F)\geq r$. Thus, the solution $F$ to the Robust $k$-Median instance has objective at least $r$. Finally, applying Lemmas 4.1, 4.3, and an argument similar to the proof of Theorem 3.11, we get the following result. ###### Theorem 4.5. The Robust $k$-Median problem $(V,{\mathcal{S}},d)$ is $\Omega(\log m/\log\log m)$ hard to approximate even on line metrics, where $m=\|{\mathcal{S}}\|$, unless $\textsf{NP}\subseteq\cap_{\delta>0}\textsf{DTIME}(2^{n^{\delta}})$. ## 5\. Heuristics The Robust $k$-Median problem is a hard to approximate real-world problem and as such heuristic solutions are interesting. In this section, we complement our negative theoretical results with an evaluation of simple heuristics for the Robust $k$-Median problem. In particular we look at two greedy strategies and two variants of a local search approach. We consider a slight generalization of the problem where clients and facilities are at separate locations. This is more realistic and no easier than the original problem, as one can simply place a facility at every client position to solve an instance of the problem as defined in Definition 1.1. Due to space constraints, the full version of this section is deferred to Appendix B. We implemented111Code and data are available at http://resources.mpi- inf.mpg.de/robust-k-median/code-data.7z and compared the following heuristics to the LP relaxation (see B.1). Greedy Upwards. Initialize all facilities as closed. Open the facility that reduces the cost maximally. Repeat until $k$ facilities are open. Greedy Downwards. Initialize all facilities as open. Close the facility that increases the cost minimally. Repeat until $k$ facilities are open. Local Search. Open $k$ random facilities. Compare all solutions that can be obtained from the current solution by closing $\ell$ facilities and opening $\ell$ facilities. Replace the current solution by the best solution found. Repeat until the current solution is a local optimum. In the experiments we use $\ell=2$. Randomized Local Search. Same as Local Search, but instead of considering _all_ solutions in the neighborhood, sample only a random subset. The size of the subset is an additional parameter to the heuristic. In the experiments we use $\ell=3$ and 200 random neighbors. We generate three kinds of 2D-instances. In the first, _uniform_ , the clients are uniformly distributed and all groups have the same size. The other kinds of instances cluster the client groups according to gaussian distributions. The intuition is that in real world instances client groups have something in common, e.g. all come from the same city. They two kinds differ in the number of clients per group. We have _gauss-const_ instances where all groups have the same size and _gauss-exp_ instances where group sizes follow an exponential distribution. Heuristic \| Uniform \| Gauss-Const \| Gauss-Exp ---\|---\|---\|--- Greedy Up \| 1.65 (1.49) \| 5.18 (5.24) \| 6.63 (5.94) Greedy Down \| 1.45 (1.42) \| 2.92 (2.92) \| 2.12 (2.05) Local Search \| 1.13 (1.12) \| 1.63 (1.62) \| 1.41 (1.39) Randomized Local Search \| 1.53 (1.48) \| 2.15 (2.29) \| 2.37 (2.36) Table 1. Mean Performance as a multiple of the LP relaxation value, rounded to three digits. In parentheses we provide the median. 1654 uniform instances, 1009 Gauss-Const instances, and 2029 Gauss-Exp instances of varying sizes were solved. The reported performance is over the instances where the heuristics perform worse that the LP relaxation. Table 1 summarizes the results. The performance differences in Table 2 are statistically significant with a very small two-sided $p$-value, according to a Wilcoxon signed-rank test, except for the difference between Greedy Downward and Randomized Local Search on Uniform and Gauss-Const instances. In these cases the $p$-value is 0.66, respectively 0.08. Since we use an LP relaxation as a comparison point, we do not know whether the instances where the heuristics find a worse solution are actually hard for the heuristics or whether the LP relaxation provides a much too low bound. To investigate this we had a closer look at instances where both Greedy down and Local Search perform badly. For three instances we solved the integer linear program. In these instances at least it was indeed the case that the LP relaxation yielded a bad bound. This suggests that the heuristics work even better than the numbers in Table 2 indicate. As expected instances where the _robust_ nature of the Robust $k$-Median problem are not as important because groups are distributed uniformly are easier than the more realistic instances where groups form clusters. For the two better heuristics, Greedy Downwards and Local Search, also perform better on instances with uneven group sizes. Here too, one can speculate that few groups dominate the problem, and finding a solution that minimizes maximum costs becomes easier. The good performance of these simple heuristics indicate that although the Robust $k$-Median problem is hard to approximate in the worst case, a heuristic treatment can effectively find a very good approximation. ## 6\. Conclusion and Future Work We show a logarithmic lower bound for the Robust $k$-median problem on the uniform and line metrics, implying that there is no good approximation algorithm for the problem. However, the empirical results suggest that real- world instances are much easier, so it is interesting to see whether incorporating real-world assumptions helps reducing the problem’s complexity. For instance, if we assume that the diameter of each set $S_{i}$ is at most an $\epsilon$ fraction of the diameter $\Delta=\max_{u,v}d(u,v)$ of the input instance, can we obtain a constant approximation factor? 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Comput., 27(3):763–803, 1998. ## Appendix A Hypergraph Label Cover An instance of $r$-Hypergraph Label Cover is equivalent to the $r$-Prover system as used by Feige [8] in proving the hardness of approximation for Set Cover. We discuss the equivalence in this section. In the $r$-prover system, there are $r$ provers $P_{1},\ldots,P_{r}$ and a verifier $V$. Each prover is associated with a codeword of length $r$ in such a way that the hamming distance between any pair $P_{i},P_{j}$ is at least ${\sf ham}(P_{i},P_{j})=r/2$; this is possible if $r$ is a power of two because we can use Hadamard code. Given an input 3-SAT formula $\phi$, the verifier selects $r$ clauses uniformly and independently at random. Call these clauses $C_{1},\ldots,C_{r}$. From each such clause, the verifier selects a variable uniformly and independently at random. These variables are called $x_{1},\ldots,x_{r}$. Prover $P_{i}$ receives a clause $C_{j}$ if the $j$th bit of its codeword is $0$; otherwise, it receives variable $x_{j}$. The property of Hadamard code guarantees that each prover would receive $r/2$ clauses and $r/2$ variables. Then each prover $P_{i}$ is expected to give an assignment to all involved variables it receives and sends this assignment to the verifier. The verifier then looks at the answers from $r$ provers and has two types of acceptance predicates. * • (Weak acceptance) At least one pair of answers is consistent. * • (Strong acceptance) All pairs of answers are consistent. Applying parallel repetition theorem [15], Feige argues the following. ###### Theorem A.1. ([8, Lemma 2.3.1]) If $\Phi$ is a satisfiable 3-SAT(5) formula, then there is provers’ strategy that always causes the verifier to accept. Otherwise, the verifier weakly accepts with probability at most $r^{2}2^{-\gamma r}$ for some universal constant $\gamma>0$. Now we show how Theorem 2.2 follows by constructing the instance of Hypergraph Label Cover $(V,E)$ based on the $r$-prover system. For each prover $j$, we create a set $V_{j}$ consisting of vertices $v$ that correspond to possible query sent to prover $j$, so we have $\|V_{j}\|=(5n/3)^{r/2}n^{r/2}$. For each possible random string $x$, we have an edge $h_{x}$ that contains $r$ vertices, corresponding to queries sent to the provers. It can be checked that the total number of possible random strings is $(5n)^{r}$, and the degree of each vertex is $3^{r/2}5^{r/2}=15^{r/2}$; notice that this is equal to $r\|E\|/\|V\|$. A prover strategy corresponds to the label of vertices, and the acceptance probability is exactly the fraction of satisfied edges. Moreover, for each possible query, the number of possible answers is at most $7^{r}$ (for each clause, there are $7$ ways to satisfy it). This implies that $\|L_{j}\|\leq 7^{r}$. ## Appendix B Heuristics The Robust $k$-Median problem is a real-world problem and as such needs to be solved as well as possible despite its hardness of approximation. In this section, we complement our negative theoretical results with an experimental evaluation of different simple heuristics for the Robust $k$-Median problem. In particular we look at two variants of a greedy strategy and two variants of a local search approach. We consider a slight generalization of the problem where clients and facilities are at separate locations. This is more realistic and no easier than the original problem, as one can simply place a facility at every client position to solve an instance of the problem as defined in Definition 1.1. This is by no means an exhaustive exploration of the possible solution space. However, the results we obtain indicate that a heuristic treatment of the Robust $k$-Median problem can yield surprisingly good solutions, even if the heuristics are very naive. For our experiments we consider instances in the plane, as these are closest to the real-world motivation for the problem. We wanted to check how the structure of the instance influences the performance of the heuristics. We suspected that instances where the clients are distributed uniformly are easy, as intuitively a solution that is good for one group of clients is good for all groups. The robust version of the k-median problem is considered because often the exact set of clients is not known before choosing facility locations and one wants to perform well even if the worst set of possible clients turns out to be realized. It is reasonable to assume that every group of clients has something in common, for example that they come from a similar region, like a city. Therefore more realistic instances for the Robust $k$-Median problem have the groups form clusters in space. We also generate such instances for testing our heuristics. ### B.1. Methods Since solving Robust $k$-Median instances to optimality is infeasible for the instances we consider222We attempted solving three instances optimally, see Figure 2, but gave up on the third after nearly half a year of CPU time was consumed., we compare the performance of the various heuristics to the value of a LP-relaxation. We have a variable $x_{j}$ for each possible median location and variables $y_{ij}$ that indicate whether client $i$ is served by facility $j$. The LP is then as follows. $\displaystyle\min\quad$ $\displaystyle T$ $\displaystyle s.t.\quad$ $\displaystyle y_{ij}-x_{j}\leq 0$ $\displaystyle\forall i,j$ $\displaystyle\sum_{j}y_{ij}\geq 1$ $\displaystyle\forall i$ $\displaystyle\sum_{i\in g}d(i,j)\cdot y_{ij}\leq T$ $\displaystyle\forall\text{ groups of clients }g$ $\displaystyle\sum_{j}x_{j}\leq k\quad\text{and}\quad 0\leq x_{j}\leq 1$ $\displaystyle\forall j$ $\displaystyle 0\leq y_{ij}\leq 1$ $\displaystyle\forall i,j$ To solve the LP we use the Gurobi solver [9], version 5.5.0, on a 64-bit Linux system. Note that the assignment of the $y_{ij}$ variables is immediately clear from the assignment of the $x_{j}$. For location $i$, let $j_{1}$, $j_{2}$, …$j_{n}$ be the locations ordered by increasing distance. Then $y_{ij_{\ell}}=\min(x_{j_{\ell}},1-(y_{ij_{1}}+\ldots+y_{ij_{\ell-1}}))$. The constraint $y_{ij_{\ell}}\leq\min(\,,\,)$ is already expressed by the first two constraints. It could however be put into the objective via the big $M$-method. Consider a minimization problem $\min T$ subject to $x=\min(b,c)$. Let $M$ be large integer and consider $\min T+Mt$ subject to $x\leq b$, $x\leq c$, $t\leq b-x$, and $t\leq c-x$. Observe that $t=\min(b,c)-x$ in an optimal solution. One needs to choose $M$ big enough so that $t$ must be zero in an optimal vertex solution. It is however unclear whether this will speed up the solution. We have not tried this method. We implemented and compared the following heuristics: Greedy Upwards. Initialize all facilities as closed. Open the facility that reduces the cost maximally. Repeat until $k$ facilities are open. Greedy Downwards. Initialize all facilities as open. Close the facility that increases the cost minimally. Repeat until $k$ facilities are open. Local Search. Open $k$ random facilities. Compare all solutions that can be obtained from the current solution by closing $\ell$ facilities and opening $\ell$ facilities. Replace the current solution by the best solution found. Repeat until the current solution is a local optimum. In the experiments we use $\ell=2$. Randomized Local Search. Same as Local Search, but instead of considering _all_ solutions in the neighborhood, sample only a random subset. The size of the subset is an additional parameter to the heuristic. In the experiments we use $\ell=3$ and 200 random neighbors. Note that taking the solution of one of the greedy algorithms as starting point for a local search is an obvious improvement, but this would prevent us from comparing the local search algorithm with the greedy heuristic. The local search heuristic is closely related to Lloyd’s algorithm for the k-means problem. In Lloyd’s algorithm, a random set of centers is chosen and iteratively updated by moving the centers to the centroids of the clients that fall in their voronoi cell. This improves the total distance from the centers to all clients in every iteration. In our setting, we want to reduce the cost of the group of clients that currently incurs the maximal cost. This can be done by moving a facility closer to this group of clients, that is, closing one facility and opening another that reduces the objective function. The local search algorithm, by closing and opening more than one facility at a time, does this at least as well. We create instances in the plane and use the euclidean distance. We create two types of instances. In the first type the clients and facilities are uniformly distributed in a $100\times 100$ square. We call these instances the _uniform_ instances. In these instances all groups of clients contain the same number of clients. The $k$ we use for the experiments is 7. The second kind uses random gaussian distributions to sample client positions. To generate the gaussian distributions we sample a matrix $M$ with $v_{1},v_{2}$ on the diagonal, where the two values are chosen uniformly at random from $[0,50]$, the matrix is then rotated by a uniformly random angle. The result is the covariance matrix of the gaussian distribution. The mean is a random point in a $100\times 100$ square. These instances we call _gauss_. We generate two subgroups of instances, in the first subgroup, _gauss-const_ , all groups of clients have the same number of clients, in the second subgroup, _gauss-exp_ , the number of clients in a group is sampled from an exponential distribution. Figure 1 shows examples for the different kind of instances we generate. (a) Uniform (b) Gauss-Const (c) Gauss-Exp Figure 1. Examples for the kind of instances we generate. Circles are clients, squares are facilities, colors indicate group membership. As we didn’t put much effort into optimizing our heuristics for speed (for example we don’t use spatial search structures to find nearest neighbors), we don’t report execution time and focus solely on solution quality. Nevertheless it is clear that the greedy strategies are much simpler to implement and much faster than the local search heuristics. We report average performance on instances where the solution is worse than the LP value, as small, easy instances otherwise skew the results. To conclude relative performance advantages between heuristics we use a Wilcoxon signed- rank test as implemented in SciPy 0.12.0. All computer code we wrote to run the experiments and analyze the results, as well as the instances we solved, is available online at http://resources.mpi- inf.mpg.de/robust-k-median/code-data.7z. ### B.2. Results Heuristic \| Uniform \| Gauss-Const \| Gauss-Exp ---\|---\|---\|--- Greedy Up \| 1.65 (1.49) \| 5.18 (5.24) \| 6.63 (5.94) Greedy Down \| 1.45 (1.42) \| 2.92 (2.92) \| 2.12 (2.05) Local Search \| 1.13 (1.12) \| 1.63 (1.62) \| 1.41 (1.39) Randomized Local Search \| 1.53 (1.48) \| 2.15 (2.29) \| 2.37 (2.36) Table 2. Mean Performance as a multiple of the LP relaxation value, rounded to three digits. In parentheses we provide the median. 1654 uniform instances, 1009 Gauss-Const instances, and 2029 Gauss-Exp instances of varying sizes were solved. The reported performance is over the instances where the heuristics perform worse that the LP relaxation. Clients \| Facilities ---\|--- \| 10 \| 110 \| 210 \| 310 \| 410 \| Greedy \| Search \| GD \| LS \| GD \| LS \| GD \| LS \| GD \| LS 10 \| 1.00 \| 1.00 \| 1.12 \| 1.00 \| 1.31 \| 1.01 \| 1.39 \| 1.02 \| 1.40 \| 1.01 160 \| 1.01 \| 1.01 \| 1.6 \| 1.17 \| 1.63 \| 1.17 \| 1.68 \| 1.15 \| 1.63 \| 1.15 310 \| 1.01 \| 1.01 \| 1.64 \| 1.21 \| 1.69 \| 1.19 \| 1.70 \| 1.19 \| 1.75 \| 1.18 460 \| 1.01 \| 1.01 \| 1.68 \| 1.22 \| 1.73 \| 1.21 \| 1.71 \| 1.21 \| 1.73 \| 1.21 110 \| 1.00 \| 1.00 \| 1.17 \| 1.01 \| 1.22 \| 1.01 \| 1.25 \| 1.01 \| 1.24 \| 1.01 1760 \| 1.0 \| 1.0 \| 1.28 \| 1.06 \| 1.33 \| 1.06 \| 1.34 \| 1.06 \| 1.34 \| 1.06 3410 \| 1.0 \| 1.0 \| 1.3 \| 1.07 \| 1.33 \| 1.07 \| \| \| \| (a) Uniform Clients \| Facilities ---\|--- \| 10 \| 110 \| 210 \| 310 \| 410 \| Greedy \| Search \| GD \| LS \| GD \| LS \| GD \| LS \| GD \| LS 10 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 160 \| 1.0 \| 1.0 \| 2.74 \| 1.64 \| 3.05 \| 1.6 \| 3.33 \| 1.62 \| 3.33 \| 1.57 310 \| 1.0 \| 1.0 \| 2.76 \| 1.70 \| 3.07 \| 1.66 \| 3.32 \| 1.64 \| \| 110 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0∗ \| 1.0∗ \| \| \| \| 3410 \| 1.01 \| 1.0 \| 2.74 \| 1.65 \| 3.02∗ \| 1.63∗ \| \| \| \| (b) Gauss-Const Clients \| Facilities ---\|--- \| 10 \| 110 \| 210 \| 310 \| 410 \| Greedy \| Search \| GD \| LS \| GD \| LS \| GD \| LS \| GD \| LS 10 \| \| \| \| \| 1.0∗ \| 1.0∗ \| 1.0 \| 1.0 \| 1.0 \| 1.0 110 \| 1.0 \| 1.0 \| 1.34 \| 1.16 \| 1.66 \| 1.28 \| 1.65 \| 1.26 \| 1.91 \| 1.34 210 \| 1.0 \| 1.0 \| 1.9 \| 1.41 \| 2.14 \| 1.45 \| 2.31 \| 1.46 \| 2.46 \| 1.49 310 \| 1.0 \| 1.0 \| 2.23 \| 1.48 \| 2.6 \| 1.48 \| 2.69 \| 1.51 \| 2.78 \| 1.50 110 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.0 \| 1.01 \| 1.0 1210 \| 1.0 \| 1.0 \| 1.38 \| 1.21 \| 1.56 \| 1.23 \| 1.73 \| 1.29 \| 1.77 \| 1.29 2310 \| 1.0 \| 1.0 \| 1.94 \| 1.38 \| 2.09 \| 1.44 \| 2.48 \| 1.41 \| 2.29 \| 1.44 3410 \| 1.0 \| 1.0 \| 2.17 \| 1.51 \| 2.48 \| 1.48 \| 2.8 \| 1.55 \| \| (c) Gauss-Exp Table 3. Performance depending on instance size for the Greedy Downwards and Local Search heuristics. All values are averages over 50 instances, except for those marked by ∗. For Gauss-Exp instances the number of clients is the mean of the exponential distribution times the number of groups. Values above the horizontal line come from instances with 10 clients per group, below the line instances have 110 clients per group. Table 2 summarizes the results of the experiments, Table 3 shows the performance for the different instance sizes for the Greedy Upwards and the Local Search heuristic. The performance differences in Table 2 are statistically significant with a very small two-sided $p$-value, except for the difference between Greedy Downward and Randomized Local Search on Uniform and Gauss-Const instances. In these cases the $p$-value is 0.66, respectively 0.08. Since we use an LP relaxation as a comparison point, we do not know whether the instances where the heuristics find a worse solution are actually hard for the heuristics or whether the LP relaxation provides a much too low bound. To investigate this we had a closer look at instances where both Greedy down and Local Search perform badly. For three instances we attempted to solve the integer linear program and succeeded for two of them. In Figure 2 we see different solutions. For these instances at least it was indeed the case that the LP relaxation yielded a bad bound. This suggests that the heuristics work even better than the numbers in Table 2 indicate. (a) Uniform: LP value 2806.4, Greedy value 5982.39, Local Search value 3426.43, OPT 3230.19. (b) Gauss-Const: LP value 1360.26, Greedy value 8307.48, Local Search value 2541.21, OPT 2505.26 (c) Gauss-Exp: LP value 2362.06, Greedy value 10624.4, Local Search value 4354.54, 4192.31 $\leq$ OPT $\leq$ 4354.54 Figure 2. Solutions of the different algorithms on particularly hard instances. From left to right, the LP solution, the Greedy downwards solution, the Local Search solution and the ILP solution. Darkness of facilities indicates "how open" they are in the LP relaxation. In 2c the ILP solver was still running at the time of submission, after having consumed 177 days of CPU time and 46GB of memory. ### B.3. Conclusion Note that all heuristics perform very well on the instances we tried. In accordance with our theoretical results, increasing the number of groups makes the instances harder, more so that increasing the number of facilities or the number of clients. As expected instances where the _robust_ nature of the Robust $k$-Median problem are not as important because groups are distributed uniformly are easier than the more realistic instances where groups form clusters. For the two better heuristics, Greedy Downwards and Local Search, also perform better on instances with uneven group sizes. Here too, one can speculate that few groups dominate the problem, and finding a solution that minimizes maximum costs becomes easier. The good performance of these simple heuristics indicate that although the Robust $k$-Median problem is hard to approximate in the worst case, a heuristic treatment can effectively find a very good approximation.	arxiv-papers	2013-09-18T10:11:31	2024-09-04T02:49:51.078660	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sayan Bhattacharya, Parinya Chalermsook, Kurt Mehlhorn, Adrian Neumann", "submitter": "Adrian Neumann", "url": "https://arxiv.org/abs/1309.4602" }
1309.4612	The Innermost Regions of Relativistic Jets and Their Magnetic Fields 11institutetext: Departamento de Astronomía y Astrofísica, Universidad de Valencia, SPAIN # Numerical study of broadband spectra caused by internal shocks in magnetized relativistic jets of blazars. Jesús M. Rueda-Becerril 11 [email protected] Petar Mimica 11 Miguel A. Aloy 11 Carmen Aloy 11 ###### Abstract The internal-shocks scenario in relativistic jets has been used to explain the variability of blazars’ outflow emission. Recent simulations have shown that the magnetic field alters the dynamics of these shocks producing a whole zoo of spectral energy density patterns. However, the role played by magnetization in such high-energy emission is still not entirely understood. With the aid of _Fermi_ ’s second LAT AGN catalog, a comparison with observations in the $\gamma$-ray band was performed, in order to identify the effects of the magnetic field. ## 1 Introduction Relativistic outflows have been observed extensively in blazars, a class of radio-loud active galactic nuclei (AGNs) whose jets are pointing very close to the line of sight towards the observer Urry:1995aa , and known for showing the most rapid variability of all AGNs. Their remarkable characteristic flares in the X-ray frequency range usually have a duration of the order of one day. Often the internal-shocks (IS) scenario Rees:1994ca is invoked to explain this variability Spada:2001do ; Mimica:2004ay ; Mimica:2005aa . The IS scenario is an idealized model of a variable jet where an intermittently working central engine ejects shells of magnetized plasma which collide due to their velocity differences. As a consequence of the collision internal shocks are formed, particles are accelerated at the shock fronts and the non-thermal, highly variable radiation is produced. Our long-term project is the study of the influence of magnetic fields on the radiation from IS using numerical simulations. In Mimica:2012aa we studied a large number of shell collisions with different magnetization levels. In the present work we focus on a limited number of shell magnetization levels, but vary other parameters such as the jet viewing angle, bulk Lorentz factor of the shells, and their relative Lorentz factor. The data obtained from these simulations is used to categorize the specific effects that variations of each parameter have on average spectra. These synthetic observations are then compared with the second LAT AGN catalog (2LAC) of blazars observed by Fermi Ackermann:2011apj . Our numerical setup is describen in Sec. 2. The results are shown in Sec. 3. Finally we discuss briefly our results in Sec. 4. ## 2 Numerical Setup We use a modified version of the _SPEV_ codeMimica:2009aa ; Mimica:2012aa to compute the non-thermal emission from the IS. We do not consider the full hydrodynamic interaction of colliding magnetized shells (see. e.g. Mimica:2007aa for a detailed study). Instead, we simplify the shell interaction as a one-dimensional Riemann problem and focus our resources on a more detailed treatment of the non-thermal radiation. Our method consists of three phases: 1. 1. _Solution of the Riemann problem._ Making use of an exact RMHD Riemann solver Romero:2005zr we determine the properties of the internal shock waves. We follow the procedure described in Mimica:2010aa to set-up the shells and to extract the information needed for the steps 2 and 3. 2. 2. _Non-thermal particles transport and evolution._ The particles are injected behind the shock fronts following the prescription of Bottcher:2010gn ; Joshi:2011bp ; Mimica:2012aa . We assume that a fraction of the thermal electrons are accelerated to high energies, and that their energy density is a fraction of the internal energy density of the shocked fluid. We assume a cylindrical shell geometry and perform all the calculations in the rest frame of the shocked fluid. In this frame the shocks are propagating away from the initial discontinuity, injecting and leaving non-thermal particles behind. We evolve the energy distribution of non-thermal electrons taking into account synchrotron and inverse-Compton (IC) losses. See Mimica:2012aa for more details. 3. 3. _Radiative transfer._ The total emissivity at each point is assumed to be a combination of the following emission processes: (1) synchrotron radiation, (2) IC upscattering of an external radiation field (EIC) and (3) synchrotron self-Compton (SSC). The details of how they are calculated are given in Mimica:2012aa . The radiative transfer equation is solved taking into account the relativistic effects and time delays. We compute light curves and average spectral energy distribution (SED) for each shell collision. In this work we focus our attention on how parameter variations affect the SEDs. ## 3 Results As mentioned in Sec. 1, the aim of this work is to cover a wider range in the parameter space than was done in Mimica:2012aa . We group our models according to the initial shell magnetization, $\sigma:=B^{2}/4\pi\rho\Gamma^{2}c^{2}$. We denote by letters S, M and W the following families of models: * W: weakly magnetized, $\sigma_{L}=10^{-6},\sigma_{R}=10^{-6}$, * M: moderately magnetized, $\sigma_{L}=10^{-2},\sigma_{R}=10^{-2}$, and * S: strongly magnetized, $\sigma_{L}=1,\sigma_{R}=10^{-1}$. Hereafter the subscripts $L$ and $R$ will denote left (faster) and right (slower) shells, respectively. As parameters to vary we considered both intrinsic and extrinsic ones. Among the intrinsic parameters we choose the Lorentz factor of the slow shell, $\Gamma_{R}$, and the relative Lorentz factor $\Delta g:=\Gamma_{L}/\Gamma_{R}-1$, where $\Gamma_{L}$ is the Lorentz factor of the fast shell. The parameter space covered is shown in Table 1. Parameter \| value ---\|--- $\sigma_{L}$ \| $10^{-6},\ 10^{-2},\ 1$ $\sigma_{R}$ \| $10^{-6},\ 10^{-2},\ 10^{-1}$ $\Gamma_{R}$ \| $10,12,17,20,22,25$ $\Delta g$ \| $0.5,0.7,1.0,1.5,2.0$ $\theta$ \| $5$ Table 1: Parameters of the models. $\Gamma_{R}$ is the Lorentz factor of the slow shell, $\sigma_{L}$ and $\sigma_{R}$ are the fast and slow shell magnetizations and $\theta$ is viewing angle of the observer. For clarity, when we refer to a particular model we label it by appending the values of each of these parameters to the model letter. For instance, S-G10-D1.0-T5 is the strongly magnetized model with $\Gamma_{R}=10$ (G10), $\Delta g=1.0$ (D1.0) and $\theta=5^{\circ}$ (T5). If we refer to a subset of models with one or two parameters fixed we use an abbreviated notation, where we omit the varying parameters from the label. We compute the spectra for a typical source located at $z=0.5$. In the rest of this section we will present some of the final SEDs resulting from our simulations. A larger collection is shown in (Rueda:2013aa, ). The SEDs of each model has been averaged over the time interval $0-10^{6}$ s. ### 3.1 Weakly-magnetized models The SEDs computed for the models W-G10-T5 (varying $\Delta g$) are shown in the left panel of Fig. 1. The spectra show that with increasing $\Delta g$ the IC component also increases, up to three orders of magnitude. In order to see the effects on each emission process, the synchrotron, SSC and EC components for $\Delta g=0.5,2.0$ are shown as dashed, dot-dashed and dot-dot-dashed lines, respectively. As we can see, while the three components of the spectrum (synchrotron, SSC and EC) are around the same order of magnitude for $\Delta g=0.5$, for $\Delta g=2.0$ the SSC is almost two orders of magnitude more luminous than the other two. The inset shows the $\gamma$-ray spectral slope of each model as a function of its $\gamma$-ray flux (see Sec. 3.4). ### 3.2 Moderately-magnetized models The SEDs of the family of models M-D1.0-T5 are presented in the right panel of Fig. 1. Analogous to the left panel of Fig. 1 the synchrotron, SSC and EC components are shown as dashed, dot-dashed and dot-dot-dashed lines, respectively, for $\Gamma_{R}=10,17,25$. The synchrotron component for $\Gamma_{R}=10$ is $\simeq 20$ times brighter than the SSC one, in contrast to the EC which is $100$ times dimmer. For $\Gamma_{R}=25$ the EC is of the same order of magnitude of SSC and synchrotron. The latter two decrease one order of magnitude between M-G10-D1.0-T5 and M-G25-D1.0-T5, while the EC grows by almost one order of magnitude. This is a consequence of the fact that the number of electrons and the comoving magnetic field strength decrease with the increasing $\Gamma_{R}$ Mimica:2012aa , which means that there are less synchrotron photons and less electrons which can scatter them in the SSC process. On the other hand, the radiation field density of the seed photons for the EC is independent of $\Gamma_{R}$, which, in combination with the Doppler boost causes the increase in the EC luminosity. We also see that at $\simeq 10^{23}$ Hz there is a point where all the EC spectra coincide. This is due to the Klein-Nishina cutoff, which we model as a sharp cutoff. The inset shows that the there is no significant change in the flux of $\gamma$-ray photons, although there was for the spectral index, heading towards lower values for increasing $\Gamma_{R}$. Figure 1: Left panel: Averaged spectra resulting from the collision of weakly magnetized shells ($\sigma_{L}=\sigma_{R}=10^{-6}$). For the models $\Delta g=0.5,2.0$, the synchrotron, SSC and EIC contributions (dashed, dot-dashed and dot-dot-dashed lines, respectively) are shown. The inset shows the photon spectral slope $\Gamma_{ph}$ as a function of the photon flux $F_{ph}$ in the $\gamma$-ray band (see Sec. 3.4). Colors of the points correspond to the line colors in the main plot. Right panel: Same as the left panel, but for moderately magnetized shells ($\sigma_{L}=\sigma_{R}=10^{-2}$) and varying $\Gamma_{R}$. For models with $\Gamma_{R}=10,17,25$ we plot the synchrotron, SSC and EIC contributions. ### 3.3 Strongly-magnetized models The third model family consists of the strongly magnetized models where $\sigma_{L}=1$ and $\sigma_{R}=0.1$. The SEDs of the series of models S-D1.0-T5 appear in Fig. 2. As we can see, for $\Gamma_{R}=10$ the synchrotron component is $\simeq 100$ times brighter than the IC. For $\Gamma_{R}=25$ this difference decreases to one order of magnitude. The EIC component rises with rising $\Gamma_{R}$, to the point in which it begins to be comparable to the synchrotron component. These effects are similar to the family M-D1.0-T5, described in Sec. 3.2. The spectra converge due to our treatment of the Klein- Nishina cutoff. In the inset we can see that the flux of $\gamma$-ray photons does not change appreciably in this family of models. Figure 2: Same as right panel in Fig. 1, but for strongly magnetized shells ($\sigma_{L}=1,\sigma_{R}=$ 0.1). ### 3.4 $\gamma$-rays spectral slope A linear mean-squares algorithm is used to deduce the $\gamma$-ray spectral slope $\Gamma_{ph}$. Due to the fact that we are not modeling the Klein- Nishina part of the spectrum, we only performed the calculations of $\Gamma_{ph}$ for those models that do not show a large drop-off in the photon flux. In Fig. 3 we show $\Gamma_{ph}$ as a function of the photon flux for energies $>0.2\,$GeV, where $F_{ph}$ is the photon flux for photon energies $>0.1$ GeV Abdo:2009cb . We compare our results with sources found in 2LAC catalogue Ackermann:2011apj (restricting the comparsion to sources with $0.4\leq z\leq 0.6$). In Fig. 3 we see that weakly and moderately magnetized models overlap with the observations, with more weakly than moderately magnetized models falling within the observed part of the parameter space. Preliminary results of models where the viewing angle, $\theta$, is changed; i. e. SMW-G10-D1.0, appear also in Fig. 3. Figure 3: Spectral slope $\Gamma_{ph}$ as a function of the photon flux for energies $>100$ MeV. $\Gamma_{ph}$ is computed for the photon energies $>200$ MeV Abdo:2009cb . The symbols joined by lines represent our numerical models, while cyan and magenta triangles represent BL Lacs and FSRQs at redshift $z\simeq 0.5$ from 2LAC Ackermann:2011apj . In this figure we also show the preliminary three families of models where we vary the opening angle (filled circles). ## 4 Conclusions In this paper we report on the progress of the study of the influence of the jet magnetization on blazar flares. We vary two parameters of our models: the relative Lorentz factor $\Delta g$ and the bulk Lorentz factor $\Gamma_{R}$. When $\Delta g$ is increased we get a more luminous maximum of the inverse Compton component, which is dominated by the SSC. If $\Gamma_{R}$ is increased we find that the EIC begins to dominate over SSC, as well as becoming comparable to the synchrotron component. In the case of strongly magnetized shells, if $\Gamma_{R}\sim 50$ both synchrotron and EC components are of the same order of magnitude (see Rueda:2013aa ). Among all the models studied here, the weakly magnetized are the ones that best fit _Fermi_ observations Ackermann:2011apj . However, the tendencies of certain models with higher magnetization appear to also be consistent with the observations. ## Acknowledgments JMRB acknowledges the support from the Grisolia fellowship GRISOLIA/2011/041. PM, MAA and CA acknowledge the support from the ERC grant CAMAP-259276 and the grants AYA2010-21097-C03-01 and PROMETEO-2009-103. ## References * (1) C.M. Urry, P. Padovani, PASP 107, 803 (1995) * (2) M.J. Rees, P. Meszaros, ApJL 430, L93 (1994) * (3) M. Spada, G. Ghisellini, D. Lazzati, A. Celotti, MNRAS 325, 1559 (2001) * (4) P. Mimica, M.A. Aloy, E. Müller, W. Brinkmann, A&A 418, 947 (2004) * (5) P. Mimica, M.A. Aloy, E. Müller, W. Brinkmann, A&A 441, 103 (2005) * (6) P. Mimica, M.A. Aloy, MNRAS 421, 2635 (2012) * (7) M. Ackermann et al., ApJ 743, 171 (2011) * (8) P. Mimica, M.A. Aloy, I. Agudo, J.M. Marti, J.L. Gómez, J.A. Miralles, ApJ 696, 1142 (2009) * (9) P. Mimica, M.A. Aloy, E. Müller, A&A 466, 93 (2007) * (10) R. Romero, J. Marti, J.A. Pons, J.M. Ibáñez, J.A. Miralles, JFM 544, 323 (2005) * (11) P. Mimica, M.A. Aloy, MNRAS 401, 525 (2010) * (12) M. Böttcher, C. Dermer, ApJ 711, 445 (2010) * (13) M. Joshi, M. Böttcher, ApJ 727, 21 (2011) * (14) J. Rueda-Becerril, P. Mimica, M.A. Aloy, MNRAS, in preparation (2013) * (15) A.A. Abdo, M. Ackermann, M. Ajello, W.B. Atwood, M. Axelsson, L. Baldini, J. Ballet, B. et al., ApJ 700, 597 (2009), 0902.1559	arxiv-papers	2013-09-18T11:09:03	2024-09-04T02:49:51.090005	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jesus M. Rueda-Becerril, Petar Mimica, Miguel A. Aloy, Carmen Aloy", "submitter": "Jes\\'us Misr\\'ayim Rueda-Becerril", "url": "https://arxiv.org/abs/1309.4612" }
1309.4630	# Radio constraints on Galactic WIMP dark matter Roberto A. Lineros Instituto de Física Corpuscular (CSIC-Universitat de València), Apdo. 22085, E-46071 Valencia, Spain. [email protected] ###### Abstract Synchrotron emission from electron cosmic ray populations can be used to study both cosmic rays physics and WIMP dark matter imprints on radio skymaps. We used available radio data - from MHz to GHz - to analyze the contribution from galactic WIMP annihilations and impose constraints on WIMP observables: annihilation cross section, channel and mass. Depending on the annihilation channel we obtain as competitive bounds as those obtained in FERMI-LAT gamma ray analysis of dwarf satellite galaxies. ###### keywords: dark matter theory , cosmic ray theory , absorption and radiation processes ††journal: NIM A RICAP-2013 ## 1 Introduction During the last decades improvements on cosmological observations have confirmed that around 27% of the matter content of observable Universe is dominated by non-barionic Dark Matter (DM) [1]. The most popular DM candidates are in the form of Weakly Interactive Massive Particles (WIMPs) due to their diverse and complementary ways of detection. Moreover, WIMPs are common to many particle physics models beyond the Standard Model (SM) e.g. Supersymmetry, extra-dimensions, and extended Higgs sector models. All these candidates share similar characteristics: non-colored, electrically neutral and stable on cosmological scales. WIMPs participate actively in the early universe until they freeze out from the thermal bath due to the expansion of the universe. This mechanism provide a natural way to reproduce the observed relic abundance [1], $\Omega_{\rm DM}h^{2}=0.1196\pm 0.0031\,,$ (1) and it requires cross section of the order of electroweak interactions. This cross section is usually referred as the thermal WIMP cross section: $\langle\sigma v\rangle=3\times 10^{-26}\,{\rm cm}^{3}{\rm s}^{-1}.$ One of the expected signatures are the annihilation products from WIMPs around celestial objects like the Milky Way and its dwarf satellite galaxies. The SM particles coming from WIMP annihilations can produce gamma rays, cosmic rays, and neutrinos. Current and projected observatories can explore these new signals thanks to their high sensitivity and resolution. For example, this is the case of the FERMI-LAT gamma ray observatory. On the other hand, radio surveys have been continuously observing and characterizing the galactic activity. One of the most known all–sky survey is the HASLAM map [2] at the frequency of 408 MHz and it is widely used in astrophysics. This work is based on Ref. [3] and describes its main results. ## 2 Constraints from radio data We consider the synchrotron emission at radio frequencies, i.e. from MHz to GHz, from electrons and positrons as result of WIMP annihilations of the DM halo of the Milky Way. In order to cover the whole possibilities, we take into account the uncertainties regarding the production of synchrotron emission: * (i) Galactic magnetic field (GMF) ranging between 1–10 $\mu$G. * (ii) Physically motivated GMF spatial distributions. * (iii) Cosmic ray propagation models. * (iv) DM distributions: isothermal and NFW profile [4]. * (v) DM annihilation into: muons, taus, $W$ bosons and $b$ quarks. * (vi) DM annihilation cross section similar to the thermal value. For most of these scenarios, the radio emission coming from WIMPs is as bright as the available radio surveys. Thus, using the radio survey we set constraints for some of the DM properties e.g. the annihilation cross section. The bounds are then obtained by requiring that the DM signal would not exceed the observed radio emission by 3 standard deviations, i.e. $T_{\rm DM}(\nu)\leq T_{\rm obs}(\nu)+3\,\sigma_{\rm survey}(\nu)\,.$ (2) The radio surveys and the DM emision skymap where previously divided into patches of $\sim 10^{\circ}\times 10^{\circ}$ and then compared. We did not include any astrophysical radio sources meaning that our bounds are conservative. Figure 1: Upper bounds on the annihilation cross section versus DM mass for annihilations into $b$. In dashed line, we show bounds obtained by FERMI-LAT from the analysis on dwarf satellite galaxies [5]. Radio constraints are less competitive with respect to hadronic channels. Further details in Ref. [3]. Figure 2: Upper bounds on the annihilation cross section versus DM mass for annihilations into $W$. Blueish dashed line are the bounds from FERMI-LAT analysis, also we show the bounds from antiproton/proton analysis [6]. Further details in Ref. [3]. Figure 3: Upper bounds on the annihilation cross section versus DM mass for annihilations into muons. Radio constraints extracted from leptonic channel are in better situation than constraints obtained by FERMI- LAT. Further details in Ref. [3]. This procedure mainly constrains the annihilation cross section for a specific annihilation channel. In Figs. 1, 2 and 3, we present some representative results of the analysis carried in Ref. [3]. All of these show upper bounds on the annihilation cross section for different annihilation channels versus the WIMP mass. In Fig. 1, we analyze the case of annihilation into $b$ quarks. We note that our bounds are less restrictive than those obtained in the gamma ray analysis of dwarf satellite galaxies [5]. This is mainly due to annihilations into quarks produce much more gamma rays because of large $\pi^{0}$ production and subsequent decay. Similar situation happens in Fig. 2 when we compare our results with previous works on constraints from antiproton/proton observation [6]. The opposite occurs when we compare the muon channel, Fig. 3. In this case, radio bounds are more competitive than those from [5] because muonic annihilation channel (and in general any leptonic channel) produce larger amount of electrons and positrons. For the $b$ and muon channels, the bounds reach the thermal cross section value at low WIMP masses. This method constrains particle physics models with predicted annihilation channels and cross section higher than our bounds. Our analysis shows the complementarity between radio and gamma rays observations for the study of galactic DM. In principle, a combined analysis in this direction would improve DM indirect searches. ## 3 Conclusions Synchrotron emission from galactic WIMP annihilations presents an interesting and alternative observable to study the WIMP properties. This method is specially efficient for constraining annihilation into leptons. Our bounds are quite competitive with respect to similar analysis and reach the value of WIMP thermal cross section for WIMP masses lower than 15 GeV. ## Acknowledgements We acknowledge N. Fornengo, M. Regis and M. Taoso. This work was supported by the Spanish MINECO under grants FPA2011-22975 and MULTIDARK CSD2009-00064 (Consolider-Ingenio 2010 Programme). Also, it was supported by Prometeo/2009/091 (Generalitat Valenciana), and by the EU ITN UNILHC PITN- GA-2009-237920. ## References * [1] Planck Collaboration, Planck 2013 results. XVI. Cosmological parameters arXiv:1303.5076. * [2] C. G. T. Haslam, C. J. Salter, H. Stoffel, W. E. Wilson, A 408 MHz all-sky continuum survey. II - The atlas of contour maps, A&A Supplement 47 (1982) 1. * [3] N. Fornengo, R. A. Lineros, M. Regis, M. Taoso, Galactic synchrotron emission from WIMPs at radio frequencies, JCAP1 (2012) 5. arXiv:1110.4337, doi:10.1088/1475-7516/2012/01/005. * [4] J. F. Navarro, C. S. Frenk, S. D. M. White, The Structure of Cold Dark Matter Halos, Astrophys. J.462 (1996) 563. arXiv:arXiv:astro-ph/9508025, doi:10.1086/177173. * [5] Fermi-LAT collaboration, Constraining Dark Matter Models from a Combined Analysis of Milky Way Satellites with the Fermi Large Area Telescope, Physical Review Letters 107 (24) (2011) 241302. arXiv:1108.3546, doi:10.1103/PhysRevLett.107.241302. * [6] F. Donato, D. Maurin, P. Brun, T. Delahaye, P. Salati, Constraints on WIMP Dark Matter from the High Energy PAMELA $\overline{p}/p$ Data, Physical Review Letters 102 (7) (2009) 071301. arXiv:0810.5292, doi:10.1103/PhysRevLett.102.071301.	arxiv-papers	2013-09-18T12:50:26	2024-09-04T02:49:51.096290	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Roberto A. Lineros", "submitter": "Roberto Lineros", "url": "https://arxiv.org/abs/1309.4630" }
1309.4665	# Generalizing the self-healing diffusion Monte Carlo approach to finite temperature: a path for the optimization of low-energy many-body bases Fernando A. Reboredo Jeongnim Kim Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA ###### Abstract A statistical method is derived for the calculation of thermodynamic properties of many-body systems at low temperatures. This method is based on the self-healing diffusion Monte Carlo method for complex functions [F. A. Reboredo J. Chem. Phys. 136, 204101 (2012)] and some ideas of the correlation function Monte Carlo approach [D. M. Ceperley and B. Bernu, J. Chem. Phys. 89, 6316 (1988)]. In order to allow the evolution in imaginary time to describe the density matrix, we remove the fixed-node restriction using complex antisymmetric guiding wave functions. In the process we obtain a parallel algorithm that optimizes a small subspace of the many-body Hilbert space to have maximum overlap with the subspace spanned by the lowest-energy eigenstates of a many-body Hamiltonian. We show in a model system that the partition function is progressively maximized within this subspace. We show that the subspace spanned by the small basis systematically converges towards the subspace spanned by the lowest energy eigenstates. Possible applications of this method to calculate the thermodynamic properties of many-body systems near the ground state are discussed. The resulting basis can be also used to accelerate the calculation of the ground or excited states with Quantum Monte Carlo. ###### pacs: 02.70.Ss,02.70.Tt ## I Introduction There is a significant interest in thermodynamical properties observed as $T\rightarrow 0$. Many physical phenomena that cover superconductivity, magnetic and structural transitions, chemical reactions etc. require an adequate treatment of thermal effects. These effects are crucial in systems where there is a large number of low-energy excitations within an energy window $1/\beta=k_{B}T$ above the ground state. Electronic thermal effects are expected to be larger in metals and magnets than in insulators. QTS In metals there is a significant number of excitations with vanishing energy. The magnetic excitations energies frequently go to zero in the long wave limit. A significant fraction of spectroscopic techniques probe the electronic or magnetic excitations near the ground state. The development of first- principles techniques to obtain excitations has historically received a significant theoretical attention. runge84 ; onida02 ; filippi09 Monte Carlo methods used to calculate excitation energiesfilippi09 will be accelerated with basis that retain the physics at the relevant energies. A first-principles finite-temperature description of many-body systems is also relevant to describe chemical reactions.mazzola12 Ionic dynamics are usually calculated within the Born-Oppenheimer approximation. This decouples the wave function of the “quantum” electrons from the wave function of the ions. Within this standard approximation, electrons are at zero temperature while the ions can move with kinetic energies that often exceed the electronic excitations. Even within the Born-Oppenheimer-ground-state approximation, the standard approach based on density functional theory (DFT) shows significant differences with diffusion Monte Carlo (DMC) grossman97 ; saccani13 or Quantum Chemistry benchmarks. At the transition saddle points, when some chemical bonds are broken and new ones are formed, the spacing of the corresponding electronic eigenenergies is minimum, or even zero at the conical intersections. levine07 Electronic thermal effects are seldom included in many-body calculations. morales10 ; mazzola12 In order to routinely include thermal effects, significant improvements in the theory beyond the standard approach are required. Most ab-initio calculations in the literature of condensed matter electronic structure are based in the ground state quantum Monte Carlo calculations of the homogeneous electron gasceperley80 which made possible the first approximations of DFT. hohenberg ; kohn DFT has been extended to finite temperature long ego. mermin65 ; kohn83 ; weinert92 Fermi occupations of Kohn-Sham eigenstates and the addition of an entropy term have been shownweinert92 to provide a variational density functional. However, even nowadays, the zero temperature approximation for the exchange-correlation potential is widely used. This approach has been long known to severely underestimate the critical Curie temperatures of magnetic systems. statunton84 ; gyorffy85 ; staunton85 Including temperature for magnetic systems is possible for cases where the magnetic excitations can be treated classically wang95 ; landau04 and the electrons can be assumed to be in the ground state for constrained configurations of the spins. eisenbach09 But an adequate description of the electronic entropy in the subspace that preserves the spin is still lacking. yin12 Finite temperatures benchmarks of a quality comparable to Ref. ceperley80, are the key ingredients required to parametrize a finite temperature density functional. Without a reliable approximation, most work done under a DFT framework still uses a zero temperature approximation for the exchange correlation functional. Accurate many-body calculations at high temperatures can be performed within the path integral Monte Carlo approach (PIMC). ceperley95 Since the cost of PIMC diverges as $T\rightarrow 0$, it has been mainly used in the hot and dense regime, magro96 ; militzer00 ; driver12 with a temperature comparable to the interaction potential. An alternative approach that could start from the zero temperature limit would be desirable. The most accurate techniques to describe a large number of electrons ($N_{e}>1000$) at zero temperature are based in projection approaches. ceperley80 ; purwanto09 ; booth12 One could potentially extend these methods to finite temperature, limiting the projector $e^{-\beta\mathcal{\hat{H}}}$ to finite $\beta$. Thermodynamical averages can be later obtained from derivatives of the Helmholtz free energy $F(\beta)=-1/\beta\ln[tr(e^{-\beta\mathcal{\hat{H}}})]$, where $\mathcal{\hat{H}}$ is the many-body Hamiltonian operator and $tr(X)$ the trace of $X$ over the complete many-body Hilbert space. The standard diffusion Monte Carlo Method with importance sampling (DMC) ceperley80 ; mfoulkesrmp2001 ; austin12 constrains the sign or the phase of the wave function by imposing the nodes or the phase ortiz93 of a guiding wave function $\Psi_{T}({\bf R})$, where the many-body coordinate ${\bf R}=\\{{\bf r}_{1},{\bf r}_{2},\cdots,{\bf r}_{N_{e}}\\}$ is the set of coordinates of $N_{e}$ electrons. These constraints while enforcing an anti- symmetric fermionic wave function introduce a variational error. The quality of the wave function and its nodes can be improved with several methods within a variational Monte Carlo (VMC) context, mfoulkesrmp2001 ; ortiz95 ; jones97 ; guclu05 ; umrigar07 ; toulouse08 ; petruzielo12 ; rios06 or at the DMC level. ceperley88 ; jones97 ; keystone ; rollingstones ; stonehenge In DMC, the energy of the ground state is exact if the exact nodes or phase are provided. anderson79 ; ortiz93 Improving the nodes is computationally intensive. Avoiding this cost is key for finite temperature calculations. In standard DMC calculations, $\mathcal{\hat{H}}$ is replaced by the fixed- node Hamiltonian $\mathcal{\hat{H}}_{FN}$ or the fixed-phase Hamiltonian $\mathcal{\hat{H}}_{FP}$. The use of the fixed-node or fixed phase approximation can have undesired effects on the calculation of thermal effects. It has been found that many fermionic systems have a ground state with two nodal pockets. bressanini12 That is, if the ground state wave function is real the nodal surface separates the Hilbert space in only two pockets for positive and negative values respectively. It has been conjectured ceperley91 that this is a general property of fermionic ground states. In the fixed-node case, the excitations of $\mathcal{\hat{H}}_{FN}$ are forced to share the nodes of the ground state. To be orthogonal to the fixed-node ground state, the fixed-node excited states have to have at least an additional node. It is known, however, that in many systems there are several fermionic excited states near the ground state with also two nodal pockets.rasch12 Accordingly, $tr(e^{-\beta{\mathcal{\hat{H}}}_{FN}})$ does not describe the low temperature physics. It is easy to see that the same happens in the fixed-phase case. Therefore, if one wishes to use a DMC-like algorithm to obtain thermodynamical properties, one must go beyond the usual fixed-node or fixed-phase approximations. For practical reasons, a parallel approach that can handle a large number of excitations near the ground state would also be beneficial. In this paper, we restart the debate on how to calculate low temperature properties within a many-body ab-initio context taking into account recent theoretical developments. keystone ; rockandroll ; stonehenge A method is derived that introduces temperature within an importance sampling procedure that shares most of the computational tools developed for projection MC approaches. The errors in the evolution operator $e^{-\beta\mathcal{\hat{H}}}$ resulting from the fixed-node restriction are eliminated by using complex linear combinations of eigenstates, which do not have nodes except at the electronic coincidental points (see Fig 1). Instead of optimizing a single many-body wave-function so it better describes the ground state of the system, a basis of several wave functions is optimized to maximize the overlap with the small subspace spanned by the lowest energy eigenstates of the many-body Hamiltonian. We show in a model system that the overlap of the optimized subspace with the lowest energy subspace calculated with a configuration interaction (CI) approach, increases systematically as the iterations increase and that the partition function is maximized. The rest of the paper is organized as follows: In Section II we describe the general formalism; some of the formulae developed in Ref. stonehenge, for complex wave function is repeated here for completeness. Section III outlines the basic algorithm. In Section IV we describe the results for a model calculation; and finally, in Section V we discuss the possible applications and summarize. This paper also has three appendices: A describes how to go beyond the locality and local-time approximations; B describes how to take advantage of the eigenstates when they are complex; Finally, C describes how to work with eigenstate pairs to minimize the variance of the weights of the walkers while keeping the wave function complex. ## II A low-energy expansion of the partition function This section extends the DMC approach ceperley80 for the calculation of the partition function of a many-body system. We first provide background material required to understand the rest of the paper. We generalize the upper bound property of the energy in DMC to an upper bound property of the free energy. We next give the general outline of our approach and describe how to avoid the fixed-node approximation in DMC. Finally, we describe the details: basic formulae and numerical approach. ### II.1 The upper bound property of the truncated Helmholtz free energy Thermal effects can be obtained by calculating all excitations within a thermal energy window above the ground state larger than $\Delta E\sim 1/\beta=k_{B}T$ and then evaluating the density matrix reichl as: $\displaystyle\hat{\rho}(\beta)=$ $\displaystyle e^{-\beta\mathcal{\hat{H}}}$ (1) $\displaystyle=$ $\displaystyle\sum_{n}\left\|\Psi_{n}\right\rangle e^{-\beta E_{n}}\left\langle\Psi_{n}\right\|,$ where$E_{n}<E_{0}+\Delta E$ is the eigenvalue with eigenvector $\|\Psi_{n}\rangle$ of $\mathcal{\hat{H}}$. In general $\mathcal{\hat{H}}$ is given by $\hat{\mathcal{H}}=\sum_{j}^{N_{e}}\frac{(\nabla_{j}+{\bf A}_{j})^{2}}{2}+\hat{V}({\bf R})$ (2) where ${\bf A}_{j}={\bf A}({\bf r}_{j})$ is a vector potential at point ${\bf r}_{j}$ with magnetic field ${\bf B}({\bf r}_{j})=\nabla_{j}\times{\bf A}_{j}$, and $\hat{V}({\bf R})$ includes the electron-electron interaction, the interactions of the electrons spins with the magnetic field and any external potential, local or non-local. In a closed system that can exchange energy with a bath or reservoir (canonical ensemble) all thermodynamical averages can be obtained using the density matrix. The trace of the density matrix $Z(\beta)=tr[\hat{\rho(\beta)}]$ is the partition function, whereas $F(\beta)=-1/\beta\ln[Z(\beta)]$ is the Helmholtz free energy. In general, $\mathcal{\hat{H}}$ has an infinite number of eigenvectors $\|\Psi_{n}\rangle$ that can be ordered with increasing eigenenergy $E_{n}$. If $\beta(E_{n}-E_{0})\gg 1$, the contribution to $Z(\beta)$ of the eigenstate $\|\Psi_{n}\rangle$ becomes negligible. Therefore, a usual approximation is to truncate the trace to a finite matrix with a finite number of eigenstates $M_{S}$. In what follows we defined $tr()$ as the trace of a truncated square matrix with size $M_{S}$. We also relate $Z(\beta)$ and $F(\beta)$ to that truncated trace. Since $e^{-\beta\mathcal{\hat{H}}}$ is positive definite, for a given basis, $Z(\beta)$ increases and $F(\beta)$ decreases as $M_{S}$ increases. The trace of any linear operator is invariant for linear transformations of the form ${\hat{B}}e^{-\beta\mathcal{\hat{H}}}{\hat{B}}^{-1}$ with ${\hat{B}}{\hat{B}}^{-1}=1$. Thus, in principle, one does not need to obtain the eigenstates of $e^{-\beta\mathcal{\hat{H}}}$ or equivalently $\mathcal{\hat{H}}$ to calculate the free energy. Any linearly independent basis that spans the same subspace can be used to obtain $Z(\beta)$. Thermodynamical properties only require us to evaluate $Z(\beta)$ in a linearly independent basis $\\{\|\chi^{S}_{m}\rangle\\}$. However, if statistical methods are used, then each element contributing to the truncated trace also increases the statistical error bar. Therefore, it is computationally more efficient to use the most compact basis, with minimum $M_{S}$, that retains the low-energy properties. Any eigenstate $\|\Psi_{n}\rangle$ can be written in a complete basis ${\|\chi^{S}_{n}}\rangle$ as $\displaystyle\|\Psi_{n}\rangle=e^{-\hat{J}}\left[\sum_{m=0}^{M_{S}-1}\lambda_{n}^{m}\|\chi^{S}_{m}\rangle+\sum_{m=M_{S}}^{N_{B}\rightarrow\infty}\lambda_{n}^{m}\|\chi^{S}_{m}\rangle\right],$ (3) where $\langle{\bf R}\|e^{-\hat{J}}\|{\bf R}\rangle=e^{-J({\bf R})}$ is a Jastrow factor that introduces adequate cusp conditions kato1957 and are $\langle{\bf R}\|\chi^{S}_{m}\rangle=\chi_{m}^{S}({\bf R})$ linear combinations of an infinite orthogonal set $\\{\langle{\bf R}\|n\rangle\\}$ [e.g. Slater determinants, or Pfaffians bajdich08 , or symmetry constrained functions (SCF), etc]. In practice we restrict the Hilbert space to a finite number $N_{B}$. We denote the subspace spanned by $N_{B}\gg M_{S}$ functions $\|m\rangle$ as the big subspace. The big subspace has to be large enough to describe the low temperature physics of the complete Hilbert space, which is in general infinite. We define the small subspace as the subspace spanned by the first $M_{S}$ basis functions $\|\chi^{S}_{n}\rangle$. Within the small basis $\\{\|\chi^{S}_{m}\rangle\\}$, the free energy will be minimum if all $\|\Psi_{n<M_{S}}\rangle$ can be spanned in the small basis, namely, $\lambda_{n\geq M_{S}}^{m}=\lambda_{n}^{m\geq M_{S}}=0$. Errors in the small basis will result in higher values of the free energy. Thus the free energy in the truncated basis is an upper bound to the true Helmholtz free energy. Optimization of the Helmholtz free energy in the small basis is analogous to the variational principle of the ground state. Likewise, the partition function in the truncated basis is a lower bound of the exact partition function. Improved bounds may be obtained with a basis that better describes the lower energy eigenstates. $M_{S}$ has to be large enough to include all the relevant physics for a given temperature. Most of the optimization methods in the QMC literature focus, on optimizing the eigenstates. Several methods have been proposed to obtain low-energy excited states within the linear method Monte Carlofilippi09 ; toulouse12 (LMMC) or diffusion Monte Carlo. ceperley88 ; rockandroll However, since the eigenstates are sometimes difficult calculate, and we only need an average, we argue that one might save computational time by optimizing the many-body basis first as in the correlation function diffusion Monte Carlo (CFDMC) ceperley88 method. Optimizing the basis directly could be more practical than obtaining accurate eigenstates energies, if the number of excitations near the ground state is large (e. g. typically the case in metallic or magnetic systems). ### II.2 Guiding ideas of the finite temperature SHDMC method and definitions Instead of performing the usual projection for infinite imaginary time of a single trial wave-function, we run DMC for multiple guiding wave-functions (forming a linearly independent basis) for finite imaginary time, which is equivalent to finite temperature. Instead of using a single real guiding function with nodes, we use a set of complex antisymmetric guiding functions without nodes. Therefore, the Hamiltonian $\mathcal{\hat{H}}$ is not altered at the nodes as in the standard importance sampling DMC approachceperley80 with the fixed-node approximation.anderson79 As explained in the introduction, extending DMC to finite temperatures requires to go beyond those fixed schemes. Complex-valued antisymmetric wave functions, that do not have nodal pockets, can be constructed as a linear combination of two real wave function with different nodes (see Fig. 1). We go beyond the standard fixed- phase approximation and the local-time approximation. stonehenge As in the SHDMC method for complex wave functions, stonehenge the infamous sign problem is avoided with complex antisymmetric guiding functions. The result is acurate as long as enough statistical information is collected. SHDMCkeystone ; rockandroll ; stonehenge systematically improves a trial wave function by maximizing the overlap with the wave function propagated in imaginary time in DMC. Here instead of maximizing the overlap of a single wave function we will maximize the overlap with the basis. Following the ideas of the SHDMC method, we use a recursive approach. In every iteration $\ell$, importance sampling DMC ceperley80 is performed and statistical data of the evolution in $\beta$ of a set of $M_{S}$ guiding wave functions is projected on the many-body bases $\\{\|\chi^{S,\ell}_{n}\rangle\\}$ and $\\{\|n\rangle\\}$ . The statistical data is used to improve the small basis $\\{\|\chi^{S,\ell+1}_{n}\rangle\\}$ and the guiding functions for the next iteration In what follows, we will omit the iteration index $\ell$ in the notation for clarity, when the basis is not changed. The small basis $\\{\|\chi^{S}_{n}\rangle\\}$ is orthonormal: $\langle\chi^{S}_{n}\|\chi^{S}_{n}\rangle=\delta_{n,m}$. For numerical efficiency, depending on the problem, we choose guiding functions related to the eigenstates of $\mathcal{\hat{U}}=e^{\hat{J}}e^{\mathcal{\hat{H}}}e^{-\hat{J}}$. (i) $\\{\|\chi^{U}_{n}\rangle\\}$ is formed by the Slater expansions of the eigenstates in the small basis. (ii) $\\{\|\chi^{V}_{n}\rangle\\}$ is formed by the Slater expansion of linear combinations of eigenstates pairs. We construct wave functions of the form $\displaystyle\Psi^{T}_{n}({\bf R})=$ $\displaystyle e^{-J({\bf R})}\chi^{X}_{n}({\bf R});$ (4) where the super index $X$ refers to either $S$, $U$, or $V$, depending on the case. To simplify the notation, we omit $X$ in $\Psi^{T}_{n}({\bf R})$. We assume the Jastrow factor operator $e^{-\hat{J}}$ to be diagonal in the many-body configuration space ${\bf R}$, and positive, which implies that it must have an inverse. fn:back_flow The Jastrow factor is fixed in SHDMC but can be optimized variationaly so that the free energy of the system is minimized. In contrast with the CFDMC ceperley88 and the released phase jones97 methods, we use anti-symmetric guiding functions, which are improved recursively with a maximum overlap criterion. Since the exponential growth of the bosonic ground state is prevented by the guiding functions, the free energy obtained is an upper bound. This approach is different to the correlated linear method filippi09 because the wave function is optimized at the DMC level and we use anti-symmetic guiding functions. In variance with the original SHDMC approach for excited states rockandroll multiple wave functions are propagated in parallel. A serial orthogonalization step in the original SHDMC method for excited states rockandroll ; stonehenge is postponed in this new approach until DMC has been run for the all basis functions. fn:complications ### II.3 Working with complex guiding wave functions to avoid the fixed-node approximation Figure 1: (Color online) A schematic representation of the nodes of the real and imaginary parts of a complex antisymmetric function. This type of complex functions must be zero only at the coincidental points. Though real antisymmetric functions must have nodal pockets a complex linear combination must not. Note also that two different antisymmetric real wave functions approaching different eigenstates will have different nodes and produce different fixed-node Hamiltonians. While complex guiding wave functions allow us to avoid the fixed-node approximation, they introduce additional complications stonehenge that are discussed here. Once these complications are dealt with, the sign problem is avoided as in the fixed-node. As in standard SHDMC the result is accurate as long as enough statistics is obtained. Following Refs. ortiz93, and stonehenge, , $\Psi^{T}_{n}({\bf R})=\langle{\bf R}\|\Psi_{n}^{T}\rangle$ can be written as an explicit product of a complex phase and an amplitude $\Psi^{T}_{n}({\bf R})=\Phi_{n}^{T}({\bf R})e^{{\bf i}\phi_{n}({\bf R})}$. The expressions $\displaystyle\Phi_{n}^{T}({\bf R})$ $\displaystyle=\sqrt{\Psi_{n}^{T}({\bf R})\Psi^{T}_{n}({\bf R})},\text{ and }$ (5) $\displaystyle\phi_{n}({\bf R})$ $\displaystyle=\ln[\Psi_{n}^{T}({\bf R})/\Psi_{n}^{T}({\bf R})]/(2i)+\pi m$ (6) allow the computation of all the gradients and Laplacians in terms of those of an arbitrary complex function $\chi_{n}^{X}({\bf R})$ and $e^{-J({\bf R})}$. In Eq. (6) $m$ is an arbitrary integer that changes the Riemann branch of the natural logarithm $\ln$ of a complex number. $m$ only contributes to the gradient or Laplacian at the Reimann cuts. fn:reimann Since the position of the Reimann cuts is an arbitrary mathematical convention, their contribution to gradients and Laplacians is unphysical and ignored. The dependence in $\beta$ of $e^{-\beta\mathcal{\hat{H}}}\Psi_{n}^{T}({\bf R})$ is given by $\displaystyle\Psi_{n}^{T}({\bf R},\beta)=$ $\displaystyle\;e^{-\beta\hat{\mathcal{H}}}\Psi_{n}^{T}({\bf R})$ (7) $\displaystyle=$ $\displaystyle\;e^{-\beta\hat{\mathcal{H}}}\left[\Phi_{n}^{T}({\bf R})e^{{\bf i}\phi_{n}({\bf R})}\right]$ (8) $\displaystyle=$ $\displaystyle\;\Phi_{n}^{T}({\bf R},\beta)e^{{\bf i}\phi_{n}({\bf R})}.$ (9) Equation (9) includes by definition all the temperature dependence in $\Phi_{n}^{T}({\bf R},\beta)~{}=~{}\Psi_{n}^{T}({\bf R},\beta)/e^{{\bf i}\phi_{n}({\bf R})}$, denoted as free-amplitude, since it can be complex, stonehenge with $\Phi_{n}^{T}({\bf R},0)=\Phi_{n}^{T}({\bf R})$. The phase $\phi_{n}({\bf R})$ remains fixed in the interval $[0,\beta)$ as in Ref. ortiz93, Following Ref. stonehenge, we define the quantity $\displaystyle f_{n}({\bf R},\beta)=\Psi_{n}^{T}({\bf R})\Psi^{T}_{n}({\bf R,\beta})e^{\beta E^{X}_{n}},$ (10) where $E^{X}_{n}$ is a reference energy adjusted numerically to satisfy the condition $\langle\chi^{X}_{n}\|e^{\hat{J}}\|\Psi^{T}_{n}({\beta})\rangle~{}=~{}1$. This reference energy is different from the one commonly used to obtain the ground state. In practice, $E^{X}_{n}$ depends on the Slater determinant expansion $\chi^{X}_{n}({\bf R})$ used to construct the guiding wave function and contains the relevant information required to calculate thermodynamical averages. Using Eqs (2) and (7), one can easily obtain $\displaystyle\frac{\partial}{\partial\beta}f_{n}({\bf R},\beta)=$ $\displaystyle\Psi_{n}^{T}({\bf R})\frac{\partial}{\partial\beta}\left[\Psi^{T}_{n}({\bf R,\beta})e^{\beta E^{X}_{n}}\right]$ $\displaystyle=$ $\displaystyle-\Psi_{n}^{T}({\bf R})\left[(\hat{\mathcal{H}}-E^{X}_{n})\Psi^{T}_{n}({\bf R,\beta})e^{\beta E^{X}_{n}}\right]$ $\displaystyle=$ $\displaystyle-\left[E_{n}^{T}({\bf R},\beta)-E^{X}_{n}\right]f_{n}({\bf R},\beta)$ (11) with $\displaystyle E_{n}^{T}({\bf R},\beta)$ $\displaystyle=\frac{\mathcal{\hat{H}}\Psi^{T}_{n}({\bf R},\beta)}{\Psi^{T}_{n}({\bf R},\beta)}$ $\displaystyle=-\frac{1}{2}\\!\\!\sum_{j}^{N_{e}}\frac{\nabla_{j}^{2}\Phi_{n}^{T}({\bf R},\beta)}{\Phi_{n}^{T}({\bf R},\beta)}$ (12) $\displaystyle+\frac{1}{2}\sum_{j}^{N_{e}}\left\|{\bf A}_{j}+{\bf\nabla}_{j}\phi_{n}({\bf R})\right\|^{2}+V({\bf R},\beta)$ $\displaystyle-{\bf i}\sum_{j}^{N_{e}}\left\\{\frac{{\bf\nabla}_{j}\Phi_{n}^{T}({\bf R},\beta)}{\Phi_{n}^{T}({\bf R},\beta)}\cdot\left[{\bf A}_{j}\\!+\\!{\bf\nabla}_{j}\phi_{n}({\bf R})\right]\right.$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+\left.\frac{{\bf\nabla}_{j}\cdot\left[{\bf A}_{j}+{\bf\nabla_{j}}\phi_{n}({\bf R})\right]}{2}\right\\}.$ In order to perform an importance sampling using $\Phi_{n}^{T}({\bf R})$ as a guiding wave function (as in Refs. ceperley80, and ortiz93, ), it is convenient to express the kinetic energy in terms of $\Phi_{n}^{T}({\bf R})$. The term including ${\nabla_{j}^{2}\Phi_{n}^{T}({\bf R},\beta)}/{\Phi_{n}^{T}({\bf R},\beta)}$ can be rewritten using the identityceperley80 ; HLRbook ; mfoulkesrmp2001 $\displaystyle\frac{\nabla^{2}_{j}\Phi^{T}_{n}({\bf R},\beta)}{\Phi_{n}^{T}({\bf R},\beta)}=$ $\displaystyle\frac{\nabla^{2}_{j}f_{n}({\bf R,\beta})}{f_{n}({\bf R,\beta})}+\frac{\nabla^{2}_{j}\Phi^{T}_{n}({\bf R})}{\Phi^{T}_{n}({\bf R})}$ $\displaystyle-\frac{{\bf\nabla}_{j}\cdot\left[f_{n}({\bf R},\beta){\bf F}_{Q}^{j}\right]}{f_{n}({\bf R,\beta})}\;,$ (13) where $\displaystyle{\bf F}_{Q}^{j}={\bf\nabla}_{j}ln\left\|\Phi_{n}^{T}({\bf R})\right\|^{2}\;.$ (14) Note that Eqs. (II.3) and (14) are valid as long as $\Phi^{T}_{n}({\bf R})\neq 0$ and $f_{n}({\bf R},\beta)\neq 0$. In practice, any divergence of ${\bf F}_{Q}^{j}$ at the nodes enforces $f_{n}({{\bf R},\beta})$ to be zero. Figure 1 shows that complex antisymmetric wave functions can be constructed so that they have nodes only at points ${\bf R}$ with ${\bf r}_{i}={\bf r}_{j}$. In this case the nodal error is avoided but errors in the phase introduce a phase shiftortiz93 ; stonehenge and a complex contribution to $E^{L}_{n}({\bf R},\beta)$. However, for complex wave functions without zeros, Eq. (II.3) is always valid, except at the coincidental points (if cusp conditions are not satisfied). To satisfy Eq. (II.3) at the coincidental points, a Jastrow factor is introduced in Eq. (4). While using complex wave functions involves some complications, the advantage is that the evolution in imaginary time $\beta$ describes the thermodynamical properties with $\beta^{-1}=k_{B}T$. However, going beyond the fixed-phase approximation ortiz93 ; meyer13 is required to obtain the thermodynamics. In this work the phase is not “released” in the same sense of Ref. jones97 , it is only free within the small subspace. Replacing Eq. (II.3) into Eq (II.3) and then into Eq. (II.3) one obtains $\displaystyle\frac{\partial f({\bf R},\beta)}{\partial\beta}=$ $\displaystyle\frac{1}{2}\sum_{j}^{N_{e}}\left\\{\nabla_{j}^{2}f_{n}({\bf R},\beta)-{\bf\nabla}_{j}\cdot\left[f_{n}({\bf R},\beta){\bf F}_{Q}^{j}\right]\right\\}$ $\displaystyle-\left[E_{n}^{L}({\bf R},\beta)-E^{X}_{n}\right]f_{n}({\bf R},\beta),$ (15) with $E_{n}^{L}({\bf R},\beta)$ where $\displaystyle E_{n}^{L}({\bf R},\beta)=$ $\displaystyle-\frac{1}{2}\\!\\!\sum_{j}^{N_{e}}\frac{\nabla_{j}^{2}\Phi_{n}^{T}({\bf R})}{\Phi_{n}^{T}({\bf R})}$ (16) $\displaystyle+\frac{1}{2}\sum_{j}^{N_{e}}\left\|{\bf A}_{j}+{\bf\nabla}_{j}\phi_{n}({\bf R})\right\|^{2}+V({\bf R},\beta)$ $\displaystyle-{\bf i}\sum_{j}^{N_{e}}\left\\{\frac{{\bf\nabla}_{j}\Phi_{n}^{T}({\bf R},\beta)}{\Phi_{n}^{T}({\bf R},\beta)}\cdot\left[{\bf A}_{j}\\!+\\!{\bf\nabla}_{j}\phi_{n}({\bf R})\right]\right.$ $\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;+\left.\frac{{\bf\nabla}_{j}\cdot\left[{\bf A}_{j}+{\bf\nabla_{j}}\phi_{n}({\bf R})\right]}{2}\right\\}.$ That is, the local energy now depends on two guiding functions (i) $\Phi_{n}^{T}({\bf R})$, and (ii) $\Phi_{n}^{T}({\bf R},\beta)$ which is an approximation that must be obtained and improved for $\beta\neq 0$. The use of complex valued guiding functions originates the gradients ${\bf\nabla}_{j}\phi_{n}({\bf R})$ that appear in the local energy in Eq. (16). Their contribution prevents the result from reaching the bosonic solution and enforces an upper bound on the fermionic ground state. ortiz93 In addition, the contribution of ${\bf\nabla}_{j}\Phi_{n}^{T}({\bf R},\beta)$ must be taken into account in the presence of a magnetic field (see term between the $\\{\\}$, when ${\bf A}_{j}\neq 0$), even when using a real- bosonic guiding wave function with ${\bf\nabla}_{j}\phi_{n}({\bf R})=0$. A locality approximation locality has been used in the past when a non-local pseudo potential is included in $\mathcal{\hat{H}}$ in the potential term $\hat{V}({\bf R})$. It consists in replacing $V({\bf R},\beta)=\frac{\hat{V}({\bf R})\Psi^{T}_{n}({\bf R},\beta)}{\Psi^{T}_{n}({\bf R},\beta)}$ by $V({\bf R},0)$ in Eq.(II.3). Since $V({\bf R},\beta)\rightarrow V({\bf R},0)$ for $\beta\rightarrow 0$, we will use the locality approximation in the first iteration. However, we will improve it in subsequent iterations (see Appendix A). A local-time approximation, analogous to the locality approximation, locality , was introduced in Ref. stonehenge, to estimate the ratio $\displaystyle\frac{{\bf\nabla}_{j}\Phi_{n}^{T}({\bf R},\beta)}{\Phi_{n}^{T}({\bf R},\beta)}=$ $\displaystyle\frac{{\bf\nabla}_{j}\Phi_{n}^{T}({\bf R})}{\Phi_{n}^{T}({\bf R})}+{\bf\nabla}_{j}\ln\left[\frac{\Phi_{n}^{T}({\bf R},\beta)}{\Phi_{n}^{T}({\bf R})}\right].$ (17) One can neglect the last term in Eq. (17) for $\beta\rightarrow 0$ where ${\Phi_{n}^{T}({\bf R},\beta)}/{\Phi_{n}^{T}({\bf R})}\rightarrow 1$. In the present work we will use the local-time approximation only in the first iteration. For subsequent iterations we improve the evaluation of Eq. (17) by using the sampling of the dependence in $\beta$ of $\Phi_{n}^{T}({\bf R},\beta)$ obtained in the previous iteration. The locality and local-time approximations have little impact in optimization methods that focus on eigenstates because the dependence on $\beta$ of $\Psi^{T}_{n}({\bf R,\beta})$ is minimized when the optimization progresses as $\Psi^{T}_{n}({\bf R})\rightarrow\Psi_{n}({\bf R})$. Going beyond these approximations is required, however, to circumvent the nodes of the eigenstates with complex wave functions. Fortunately, optimization in the small subspace allows an easy sampling of the $\beta$ dependence. The approach is exact if the big basis is large enough and if enough statistical data is collected fn:fixedphase as $\ell\rightarrow\infty$. Circumventing the nodes with complex wave functions is necessary in this case because, in standard DMC calculations using real-valued wave functions with nodes, if any walker crosses a node, it is either killedceperley80 or the move is rejected. umrigar93 This introduces an artificial divergent potential at the nodal surface, which adds a kink at the node (a step for the rejection case). Since there is a one-to-one correspondence between energy of one eigenstate and its nodes, rosetta eigenstates with different energies must have different nodes. As a consequence, two real wave wave functions that approach different eigenstates introduce different nodal potentials. Since the fixed-node Hamiltonian is different for different eigenstates, and affect the dynamics at the node in the evolution in imaginary time, the $\beta$ dependence obtained using the fixed-node approximation will not describe the thermodynamics even if the exact nodes of the ground state are provided. ### II.4 Differences with other DMC-like projection methods The implementation of this method follows essentially the same approach developed for DMC or SHDMC, with some key numerical changes. Equation (15) is identical to Eq. (1) in Ref. ceperley80, except for the local energy, which now has an explicit dependence in $\beta$. Unlike Eq. (13) in Ref. stonehenge, , Eq. (15) is now valid for $\beta\geq 0$. As in Ref ceperley80, , Eq (15) describes the evolution of an ensemble. Each member of the ensemble of configurations ${\bf R}_{i}$ (walker) undergoes (i) a random diffusion and (ii) drifting by the quantum force $\sum_{j}{\bf F}_{Q}^{j}$ (which depends only on $\Phi_{T}({\bf R})$ and not on the phase). Following Ref. stonehenge, , (iii) each walker carries a complex phase factor. In a nonbranching algorithm, the complex weight of the walkers is multiplied by $\exp\\{-\left[E^{L}_{n}({\bf R},\beta)-E^{X}_{n}\right]\delta\beta\\}$ at every diffusion step. In contrast with Ref. jones97, , the phase factor of the walkers starts in 1 and evolves towards the difference between the guiding phase and the phase of $\Psi({\bf R},\beta)$, while in the release phase approach the initial phase of each walker depends on the initial positions of the walkers ${\bf R}_{i}$ but remains constant in $\beta$. If the $\chi_{n}^{X}({\bf R})$ are linear combinations of antisymmetric functions, with arbitrary complex coefficients, the $\Phi_{n}^{T}({\bf R})$ in Eq. (16) do not have nodal surfaces (see Fig. 1). Therefore, ${\bf F}_{q}$ is not divergent, but at the coincidental points. All walkers must add the same inverse temperature $\beta$ after $k$ steps. The standard time-step correction to minimize time step errors at the nodes [Eq. (33) in Ref. umrigar93, ] is not strictly necessary since the divergences in ${\bf F}_{q}$ are removed. If one introduces it, one must readjust $\delta\beta$ during the time evolution so that all the walkers add up to the same $\beta$. For the same reason, the standard accept/reject scheme that enforces detailed balanceHLRbook is modified: other moves are retried after rejection until a move is accepted. ### II.5 Calculation of the partition function This section shows that the partition function can be obtained as $Z(\beta)\simeq\sum_{n=0}^{M_{S}-1}e^{-\beta E^{S}_{n}}$ where the $E^{S}_{n}$ are reference energies instead of eigen energies. Numerically, it is convenient to start the calculation with a distribution of walkers proportional to $f_{n}({\bf R},0)=\|\Phi^{T}_{n}({\bf R})\|^{2}$. In the importance sampling approach ceperley80 setting the second line of Eq. (15) equal to zero, provides an equilibrium distribution proportional to $\|\Phi^{T}_{n}({\bf R})\|^{2}$. As in the DMC and SHDMC methods, the evolution in inverse temperature $\beta$ is discretized into $k$ finite steps $\delta\beta=\beta/k$. Following the SHDMC approachkeystone ; rockandroll ; rollingstones ; stonehenge the weighted distribution of the walkers can be written as $\displaystyle f_{n}({\bf R},k\delta\beta)$ $\displaystyle=\lim_{N_{c}\rightarrow\infty}\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}W_{i}\delta\left({\bf R-R}_{i}\right).$ (18) In Eq. (18), ${\bf R}_{i}$ corresponds to the position of the walker $i$ , and $N_{c}$ is the number of equilibrated configurations. The complex weights $W_{i}$ are given by $\displaystyle W_{i}=e^{-\left[E_{i}^{k}-E^{X}_{n}\right]k\delta\beta}$ (19) with $\displaystyle E_{i}^{k}=\frac{1}{k}\\!\sum_{j=0}^{k-1}E^{L}_{n}({\bf R}_{i}^{-j}),$ (20) Where $k$ is a number of steps and $E_{L}({\bf R}_{i}^{-j})$ is the previous value of the local energy obtained $j$ steps $\delta\beta$ earlier for the walker $i$. The evolution in inverse temperature $\beta$ of the guiding wave function $\Psi_{n}^{T}({\bf R},\beta)$ can be written, without loss of generality, as $\displaystyle\Psi_{n}^{T}({\bf R},\beta)=e^{-\beta E^{X}_{n}}e^{-J({\bf R})}\left[\chi^{X}_{n}({\bf R})+\delta\chi^{X}_{n}({\bf R},\beta)\right].$ (21) That is, the product of an average decay factor $e^{-\beta E^{X}_{n}}$ times the Slater determinant part. The Slater part is given by the the one at $\beta=0$ plus an orthogonal displacement $\delta\chi^{X}_{n}({\bf R},\beta)$. The $X$ in term $\delta\chi^{X}_{n}({\bf R},\beta)$ denotes the explicit dependence on $\chi^{X}_{n}({\bf R})$. At least one overlap $\langle\chi^{X}_{m}\|\delta\chi^{X}_{n},\beta\rangle$ must be different from zero for $n\neq m$ , if $\Psi^{T}_{n}({\bf R})$ is not an eigenstate. Using Eq. (21) we can correct equation (16) beyond the locality and local-time approximations. The displacement $\delta\chi^{X}_{n}({\bf R},\beta)$ can be sampled from the DMC run as follows: From Eqs. (7), (10) and (18), one can obtain $\displaystyle e^{-J({\bf R})}\delta\chi^{X}_{n}({\bf R},\beta)=$ $\displaystyle\langle{\bf R}\|\left[e^{\beta E^{X}_{n}}\|\Psi^{T}_{n}(\beta)\rangle-\|\Psi^{T}_{n}\rangle\right]$ $\displaystyle=$ $\displaystyle\;\langle{\bf R}\|\left[e^{-\beta(\hat{\mathcal{H}}-E^{X}_{n})}-1\right]e^{-\hat{J}}\|\chi^{X}_{n}\rangle\;$ (22) $\displaystyle=$ $\displaystyle\;\frac{\left[f_{n}({\bf R},\beta)-f_{n}({\bf R},0)\right]}{\Psi^{T}_{n}({\bf R})}$ (23) $\displaystyle\simeq$ $\displaystyle\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}e^{J({\bf R})}\frac{[W_{i}-1]}{\chi^{X}_{n}({\bf R})}\delta({\bf R}-{\bf R}_{i}).$ (24) Within the subspace spanned by the basis $\\{\chi^{S}_{n}({\bf R})\\}$, the identity operator $\hat{E}$ is given by $\displaystyle\langle{\bf R}^{\prime}\|\hat{E}\|{\bf R}\rangle$ $\displaystyle=\sum_{m=0}^{M_{S}-1}e^{-J({\bf R^{\prime}})}\chi^{S}_{m}({\bf R^{\prime}})\chi^{S}_{m}({\bf R})e^{J({\bf R})}.$ (25) Applying Eq. (25) to both sides of Eq. (24), and integrating over ${\bf R}$, one can easily obtain an expression of the diffusion displacement within the basis $\\{\|\chi^{S}_{n}\rangle\\}$ as $\displaystyle\delta\chi^{X}_{n}({\bf R},\beta)$ $\displaystyle=\sum_{m=0}^{M_{S}-1}\lambda_{n}^{m}(\beta)\chi^{S}_{m}({\bf R})$ (26) with $\displaystyle\lambda_{n}^{m}(\beta)=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}e^{2J({\bf R}_{i})}\frac{\chi^{S}_{m}({\bf R}_{i})}{\chi^{X}_{n}({\bf R}_{i})}[W_{i}-1]$ where $N_{c}=\sum_{i=1}^{N_{c}}W_{i}$. Using $X=S$ in Eqs. (18) -(21) one can prove that $\displaystyle\langle\chi^{S}_{n}\|\mathcal{\hat{U}}\|\chi^{S}_{m}\rangle=$ $\displaystyle\int{\bf dR}e^{J({\bf R})}\chi^{S}_{n}({\bf R})\Psi_{m}^{T}({\bf R},\beta)$ $\displaystyle=$ $\displaystyle e^{-\beta E^{S}_{n}}\left[\delta_{n,m}+\int{\bf dR}\chi^{S}_{n}({\bf R})\delta\chi^{S}_{m}({\bf R},\beta)\right]$ $\displaystyle=$ $\displaystyle e^{-\beta E^{S}_{n}}(\delta_{n,m}+\lambda_{n}^{m})$ (28) with $\mathcal{\hat{U}}$ having the structure of the transcorrelated method ten-no02 $\displaystyle\mathcal{\hat{U}}=$ $\displaystyle e^{\hat{J}}e^{-\beta\mathcal{\hat{H}}}e^{-\hat{J}}.$ (29) We use condition $\lambda_{n}^{n}=0$ [See Eq (LABEL:eq:deltalambda)] to determine the value of $E^{S}_{n}$. In practice, we adjust the reference energy of the guiding functions every iteration as $e^{-\beta E^{X,\ell+1}_{n}}=e^{-\beta E^{X,\ell}_{n}}(\delta_{n,m}+\lambda_{n}^{m})$. Since $tr(e^{-\beta\mathcal{\hat{H}}})=tr(\mathcal{\hat{U}})$, the contribution to the Helmholtz free energy of the small subspace is given by $\displaystyle F(\beta)=-\frac{1}{\beta}\ln{\left[\sum_{n=0}^{M_{S}-1}e^{-\beta E^{S}_{n}}\right]},$ (30) where the expression inside the brackets is the partition function $Z(\beta)$. In general for an arbitrary guiding function $\Psi_{m}^{T}({\bf R})$, the variance will grow with $M_{S}$. An energy span larger than $\beta^{-1}$ must be retained in the basis to calculate thermodynamical properties. Arbitrary trial wave functions spanned by this space might have significant variance in the walkers weights. To reduce the variance we use guiding functions that are approximately linear combinations of a pair of neighboring eigenstates. When using guiding functions that are different from the small basis functions, the contributions to the trace of the density matrix in the small basis can be obtained with $\displaystyle e^{-\beta E^{S}_{n}}=\sum_{i}\left[\|\langle\chi^{X}_{i}\|\chi^{S}_{n}\rangle\|^{2}+\langle\chi^{X}_{i}\|\chi^{S}_{n}\rangle\langle\chi^{S}_{n}\|\delta\chi^{X}_{i}\rangle\right]e^{-\beta E^{X}_{i}}\;.$ (31) The details of the derivation are in Appendix B. In a recent paper, Mazzola, Zen and Sorellamazzola12 proved that $\displaystyle\langle\Psi^{T}_{n}\|e^{-\beta\hat{\mathcal{H}}}\|\Psi^{T}_{n}\rangle\geq e^{-\beta\langle\Psi^{T}_{n}\|\hat{\mathcal{H}}\|\Psi^{T}_{n},\rangle}.$ (32) Ref. mazzola12, used the righthand side of Eq. (32) to approximate the free energy obtaining a lower bound for $F(\beta)$. Reference mazzola12, can be considered a VMC approach to the evaluation of the free energy. That approximation becomes exact if all the $\|\Psi^{T}_{n}\rangle$ are eigenstates of $\hat{\mathcal{H}}$. However, that method is very poor for an arbitrary random guiding function. In the present approach, we go beyond Ref. mazzola12, by evaluating the lefthand side of Eq. (32) directly using DMC. In many situations, the excitations of a mean field method based on approximations DFT might be good enough to obtain the low energy thermodynamical properties using Eq. (30). If that were the case, at least two DMC runs for each function of the basis are required. One to obtain the $\beta$ dependence and a second to evaluate the reference energies beyond the local-time approximation. However, in the so-called highly correlated materials, usual approximations of DFT fail to describe the low energy physics. In those cases a method that could optimize the basis is more important. That method is described in the following subsections. ### II.6 The first iteration: Construction of the first small basis $\\{\|\chi^{S,1}_{n}\rangle\\}$ While the present approach will optimize the basis from any starting basis set, the calculation will be more efficient starting from a good basis. A procedure to generate a good starting set is described here. The only restriction for the small basis $\\{\|\chi^{S}_{n}\rangle\\}$ is to avoid the nodes associated with real wave functions. In this work we choose the initial basis with a Lanczos-like procedure combined with the SHDMC approach. The big subspace basis set ${\|m\rangle}$ is constructed by symmetry constrained functions (linear combinations of Slater determinants with the same symmetry of the ground state) ordered with increasing mean field energy. We choose the first basis function of the small subspace to be $\displaystyle\|\Psi_{0}^{S}\rangle=\frac{1}{\sqrt{2}}(\|0\rangle+{\bf i}\|1\rangle)$ (33) being $\|0\rangle$ and $\|1\rangle$the ground and first excited states of a non- interacting solution of the system. Using Eq. (33) as guiding function in Eqs. (II.5)-(24) and replacing $\chi^{S}_{m}({\bf R})$ by $\langle{\bf R}\|m\rangle$ in Eq. (25), but not on $\Psi^{T}_{n}({\bf R})$, we obtain an expression similar to Eq. (26) $\displaystyle\delta\tilde{\chi}_{n}^{S}({\bf R})=\langle{\bf R}\|\delta\tilde{\chi}^{S}_{n}\rangle=\tilde{\sum}_{m=0}c_{n}^{m}\langle{\bf R}\|m\rangle.$ (34) The tilde in $\delta\tilde{\chi}_{n}^{S}({\bf R})$ means that the expansion is in the big basis $\\{\|m\rangle\\}$ with $\displaystyle c_{n}^{m}(\beta)=\frac{1}{N_{c}}\sum_{i=1}^{N_{c}}e^{2J({\bf R}_{i})}\frac{\langle m\|{\bf R}_{i}\rangle}{\chi^{S}_{n}({\bf R}_{i})}[W_{i}-1].$ The symbol $\tilde{\sum}$ in Eq. (34) means that the sum is restricted to the coefficients $c_{n}^{m}$ with an error bar smaller than $25\%$ of the absolute value (this is the standard recipe of the SHDMC algorithm keystone ). We define the next basis function $\|\chi^{S}_{n+1}\rangle$ recursively as $\displaystyle\|\chi^{S}_{n+1}\rangle=$ $\displaystyle\frac{1}{\mathcal{N}_{n+1}}\hat{P}_{n+1}\|\delta\tilde{\chi}^{S}_{n}\rangle\text{ with }$ (36) $\displaystyle\hat{P}_{n}=$ $\displaystyle 1-\sum_{m=0}^{n-1}\|\chi^{S}_{m}\rangle\langle\chi^{S}_{m}\|$ (37) where $\mathcal{N}_{n+1}$ is a normalization constant. Equations (37) and (36) mean that $\|\chi^{S}_{n}\rangle$ is the projection of the displacement $\|\delta\tilde{\chi}^{S}_{n+1}\rangle$ orthogonal to the subspace spanned by the $n$ basis functions found previously. One repeats this procedure until a basis of $M_{S}$ functions $\\{\|\chi^{S}_{n}\rangle\\}$ is constructed. This Lanczos-like procedure grants that the initial small subspace basis times the Jastrow factor has a large projection onto the lowest-energy eigenstates of $\mathcal{\hat{H}}$ or the largest eigenstates of $e^{-\beta\mathcal{\hat{H}}}$. Since the evolution in inverse temperature $\beta$ is not known during the initialization step, we use the local-time approximation discussed in the previous section. However, once a basis is generated, we can go beyond the local-time approximation in successive iterations. Note that by construction any $\chi^{S}_{n}({\bf R},\beta)$ can be approximated as a linear combination of the basis function $\chi^{S}_{m}({\bf R})$ with $m<n+1$, since the finite temperature projection of one wave function of the basis into the other was used to construct the small basis. The details on how to approximate the evolution in $\beta$ are in Appendix A. ### II.7 Systematic improvement of the small basis $\\{\|\chi^{S}_{n}\rangle\\}$ One of the goals of this work is to obtain a much smaller basis of $M_{S}$ functions $\\{\|\chi^{S}_{n}\rangle\\}$ than $\\{\|m\rangle\\}$, the big set of $N_{B}$ basis functions. The small basis $\\{\|\chi^{S}_{n}\rangle\\}$ should retain the lowest energy physics of $\mathcal{\hat{H}}$ and $e^{-\beta\mathcal{\hat{H}}}$. While for some purposes (e.g. the ground state calculation), the initial basis set described in the previous section might be enough, in this subsection we describe how to further optimize the small basis so it better describes the low-energy excitations of $\mathcal{\hat{H}}$ and thermodynamical properties. Note that $\chi^{S}_{n}({\bf R},\beta)=e^{\hat{J}}e^{-\beta\mathcal{\hat{H}}}e^{-\hat{J}}\chi^{S}_{n}({\bf R})$ will converge to the antisymmetric part of the ground state wave-function as $\beta\rightarrow\infty$. In order to avoid every state in the basis collapsing to the same function we (i) remove the projection into the other states of the basis, (ii) add the diffusion displacement orthogonal to the small subspace, and (iii) perform a Gram Schmidt orthogonalization as follows: $\displaystyle\|\chi^{S,\ell+1}_{0}\rangle=$ $\displaystyle\|\chi^{S,\ell}_{0}\rangle+\|\delta\tilde{\chi}_{0}^{S}\rangle-\|\delta\chi^{S}_{0}(\beta)\rangle$ (38) $\displaystyle\|\chi^{S,\ell+1}_{n}\rangle=$ $\displaystyle\hat{P}^{\ell+1}_{n}\left[\|\chi^{S,\ell}_{n}\rangle+\|\delta\tilde{\chi}_{n}^{S}\rangle-\|\delta\chi^{S}_{1}(\beta)\rangle\right]\;\;n>0,$ with $\hat{P}^{\ell+1}_{n}$ given by Eq. (37) replacing $\chi^{S}_{m}$ by $\chi^{S,\ell+1}_{m}$. Note in Eq. (38) that $\|\delta\tilde{\chi}_{n}^{S}\rangle$ given by Eq. (34) is a direct projection of the diffusion displacement into the big basis of SCFs $\\{\|m\rangle\\}$, whereas $\|\delta\chi^{S}_{n}(\beta)\rangle$ given by Eq. (26) is an indirect projection (since the $\|\delta\chi^{S}_{n}(\beta)\rangle$ are projected into the small basis $\\{\|\chi^{S}_{n}\rangle\\}$ which in turn are linear expansions of functions that belong to $\\{\|m\rangle\\}$). The difference $\|\Delta\tilde{\chi}^{S}_{n}\rangle=\|\delta\tilde{\chi}^{S}_{n}\rangle-\|\delta\chi^{S}_{n}(\beta)\rangle$ is by construction orthogonal to the small subspace. Accordingly, it describes the decay of the small subspace basis into the eigenstates with the lowests energies. ## III Algorithm The goal of this algorithm is to optimize a minimal basis $\\{\|\chi^{S}_{n}\rangle\\}$ to span the lowest energy excitations of $\mathcal{\hat{H}}$ (equivalently the eigenstates of $e^{-\beta\mathcal{\hat{H}}}$ with the largest eigenvalues). That basis can be used to calculate finite temperature expectation values of thermodynamical properties and accelerate the calculation of the ground and lower excited states. In this section we summarize how the theory described in detail earlier can be implemented. Initialization: the small basis $\\{\|\chi^{S}_{n}\rangle\\}$, a set of orthogonal linear combinations of many-body functions $\|m\rangle$ is constructed using the procedure described in II.6. Once this procedure is concluded, a set of linearly independent guiding functions $e^{-\hat{J}}\|\chi^{V}_{n}\rangle$ is constructed as linear combinations of pairs of approximated eigenstates of $\mathcal{\hat{U}}$. Subsequently, we can use for the evaluation of the local energy $E^{L}_{n}({\bf R},\beta)$ an approximate dependence of the guiding functions in $\beta$ given by Eq. (47). Basis update iteration: each iteration $\ell$ is composed by (1) a parallel diffusion of intermediate functions $\|\chi^{V}_{n}\rangle$. (2) A linear transformation to obtain $\|\delta\tilde{\chi}^{S}_{n}\rangle$ and $\|\delta\chi^{S}_{n}\rangle$. (3) Recalculation of $\mathcal{\hat{U}}$ in the small basis. (4) Update of the small basis for the next iteration $\|\tilde{\chi}^{S,\ell+1}_{n}\rangle$. (5) Update of the intermediate functions $\|\chi^{V,\ell+1}_{n}\rangle$. (6) Finally, the algorithm decides to increase the number of samples in the next iteration or not. These six steps are repeated recursively. The individual steps of the iteration are described below in more detail: (1) Parallel diffusion: Each displacement $\displaystyle\|\delta\chi^{V}_{n}\rangle=e^{\hat{J}}\left[e^{-\beta\mathcal{\hat{H}}}-1\right]e^{-\hat{J}}\|\chi^{V}_{n}\rangle$ (39) is projected into the small basis $\\{\|\chi^{S}_{n}\rangle\\}$ and the big basis $\\{\|m\rangle\\}$ using Eqs. (LABEL:eq:deltalambda) and Eq. (LABEL:eq:cnm) respectively replacing $\chi^{X}_{n}({\bf R})$ by $\chi^{V}_{n}({\bf R})$. Each diffusion contains $M_{b}$ sampling subblocks. For each sampling subblock, uncorrelated walker positions are generated from the previous one with a VMC algorithm. Next, DMC is run for $k$ steps with a shorter time step $\delta\beta$. The coefficients of $\|\delta\tilde{\chi}^{V}_{n}\rangle$ and $\|\delta\chi^{V}_{n}\rangle$ are sampled at the end of each subblock using Eqs. (LABEL:eq:deltalambda) and (LABEL:eq:cnm). Statistical data is collected for $M_{b}$ subblocks for each parallel diffusion before an update of the small basis. (2) Linear transformation: The $\|\delta\tilde{\chi}^{S}_{n}\rangle$ and $\|\delta\chi^{S}_{n}\rangle$ can be constructed in terms of $\|\delta\tilde{\chi}^{V}_{n}\rangle$ and $\|\delta\chi^{V}_{n}\rangle$ using Eq. (B) replacing the super index $X$ by $V$. (3) Calculation of $\mathcal{\hat{U}}$, $\\{\|\Psi^{U,\ell+1}_{n}\rangle\\}$, and $\\{\|\Psi^{V,\ell+1}_{n}\rangle\\}$ : A matrix representation of $\mathcal{\hat{U}}$ is obtained in the small subspace $\\{\|\Psi^{U,\ell}_{n}\rangle\\}$, using Eqs. (LABEL:eq:deltalambda), (II.5) and (B). The left and right eigenvectors are obtained by diagonalizing $\mathcal{\hat{U}}$. (4) Update of the small Basis: Equation (38) is used to improve the small basis. (5) We perform the correspondence $\chi^{S,\ell}_{n}\rightarrow\chi^{S,\ell+1}_{n}$ (see Appendix A ) and construct the basis $\\{\|\chi^{U,\ell+1}_{n}\rangle\\}$, and $\\{\|\chi^{V,\ell+1}_{n}\rangle\\}$ with the coefficients of the eigenvectors of $\mathcal{\hat{U}}$ in the iteration $\ell$. (6) Updating $M_{b}$: At first, the number of sampling subblocks $M_{b}$ is set to a small number (e.g. $M_{b}=3$). When the noise is dominant $\sum_{n}\langle\Delta\tilde{\chi}^{S,\ell+1}_{n}\|\Delta\tilde{\chi}^{S,\ell}_{n}\rangle\leq 0$, we increase $M_{b}$ by a factor larger than 1. As a result, the total number of configurations $N_{c}$ sampled increases as the iteration $\ell$ increases and the statistical error is reduced. Hence, as the statistics are improved the number of basis functions retained in the expansion Eq. (25) increases over time. (7) We use Eq. (30) to calculate thermodynamical properties. ## IV Results in model calculations This section describes the results obtained for a model system with an applied magnetic field. The results are compared with configuration interaction (CI) calculations in the same model used in Refs. rosetta, ; keystone, ; rockandroll, and stonehenge, . The lowest-energy eigenstates were found for two polarized electrons ($J=1$) moving in a two-dimensional square with a side length $1$ and a repulsive interaction potential of the form $\hat{V}({\bf r},{\bf r^{\prime}})=8\pi^{2}\gamma\cos{[\alpha\pi(x-x^{\prime})]}\cos{[\alpha\pi(y-y^{\prime})]}$ with $\alpha=1/\pi$ and $\gamma=4$. The main advantage of the model is that fully converged CI calculations can be performed which are nearly analytical. In order to perform the CI calculations the many-body wave function of the small basis $\\{\chi^{S}_{n}\rangle\\}$ are spanned in a basis of functions $\\{\|m\rangle\\}$ that are eigenstates of the noninteracting system. They are linear combinations of functions of the form $\prod_{\nu}\sin(m_{\nu}\pi x_{\nu})$ with $m_{\nu}\leq 7$. Converged CI calculations were performed to obtain a nearly exact expression of the lowest energy states of the system $\|\Psi_{n}\rangle=\sum_{m}a_{m}^{n}\|m\rangle$. The matrix elements involving the magnetic vector potential ${\bf A}$ (in the symmetric gauge) were calculated analytically. The result of the CI calculations were used to evaluate the partition functions and to quantify the convergence of the basis. The same basis used to construct the CI Hamiltonian is used as the big basis to test our finite temperature version of SHDMC. All the calculations reported are with $J({\bf R})=0$, which increases the statistically noise, makes the test more difficult and facilitates the comparison with the CI results. The results presented here are a proof of principle on the validity of the algorithm, which is necessary before requesting and using the massive amount of computing time required for realistic finite temperature calculations in solids. While clearly a demonstration in a realistic system is required in the future, a comparison with an exact model is the first essential step to validate the scheme. This includes not only the value obtained for the partition function but also a detailed analysis of the convergence of the basis. In the absence of magnetic fields there are two degenerate solutions: one that transforms line $x$, and the other that transform like $y$. This degeneracy is broken with a magnetic field. The eigenstates transform like $x+{\bf i}y$ and $x-{\bf i}y$. Figure 2 shows the evolution of $Z_{+}(\beta)=tr_{+}(\mathcal{\hat{U}})$ of the model system with $M_{S}=20$. The subindex “$+$” in $Z_{+}$ and $tr_{+}$ means that the results of Fig. (2) were obtained considering only the subspace of the Hamiltonian that transforms like $x+{\bf i}y$. The calculation of thermodynamical properties requires, however, the inclusion of all possible symmetries of the wave function, which implies that $Z_{-}$, the trace in a small basis that transforms as $x-{\bf i}y$, should also be added. In order to calculate $Z_{+}(\beta)$ we have defined the zero of energy to be the ground state of the CI. The calculations were run using $\delta\beta=0.00002$ and $\beta=0.004$. We have used a magnetic field of $B=0.6283$. Figure 2 shows the value of $Z_{+}(\beta)$, calculated with different methods, relative to the full CI value obtained with the $M_{S}=20$ lowest eigenvalues. The blue cycles were obtained with SHDMC using Eq. (30). The red rhombi were obtained by evaluating $Z_{+}(\beta)=\sum_{n,m}\|\langle\Psi^{T}_{m}\|\Psi_{n}\rangle\|^{2}e^{-\beta E_{n}}$, being the $\|\Psi_{n}\rangle$ and $E_{n}$ the eigenvectors and eigenvalues of the full CI. The empty squares were obtained as $Z_{+}(\beta)\simeq\sum_{n}e^{-\beta\langle\|\Psi^{T}_{n}\|\hat{\mathcal{H}}\|\Psi^{T}_{n}\rangle}$ being the $\|\Psi^{T}_{n}\rangle$ linear combinations of pairs of approximated eigenstates obtained with SHDMC. Therefore, the squares correspond to the result that one would had obtained using the approximation of Ref. mazzola12, in Eq. (32) for a very good set of functions. Finally, the up triangles mark the result obtained with $Z_{+}(\beta)\simeq\sum_{n}e^{-\beta\langle\|\Psi^{T}_{n}\|\hat{R^{-1}}\hat{\mathcal{H}}\hat{R}\|\Psi^{T}_{n}\rangle}$, being $\hat{R}$ a random rotation defined in the small subspace. Figure 2 shows that as the iteration $\ell$ increases, the $Z_{+}(\beta)$ obtained with all methods increases. Note, that the scale of the $y$ axis does not start from zero. The initialization scheme already produces a basis that retains 90 % of the exact value of the truncated partition function. Similar to previous SHDMC methods, the present generalization optimizes the small basis overlap, not their average energy. The trace of $\mathcal{\hat{U}}$ increases indirectly as the small basis approaches to the subspace of the eigenvectors with lowest energy. Note that the SHDMC results are within $\sim 1.5$% below the values obtained analytically by projection into the CI data (red rhombi). This difference is due to the remaining errors in the evolution phase which neglects the projection orthogonal to the small subspace. The method used in Ref. mazzola12, applied to approximated pairs of eigenstates (empty squares) gives results only slightly below the SHDMC values, because the energy difference between eigenstates is much smaller that $\beta^{-1}$. However, a random rotation of the basis that spans the same subspace (up triangles) would had produced a significantly worse result. This shows that the approach described in Ref. 12 is significantly worse if each element of the basis does not have a large projection on each eigenstate. Note in the inset of Fig. 2 that the number of sampling blocks remains very low ($M_{b}=3$) for the first 25 iterations and starts increasing when the noise becomes dominant around $\ell=32$. If one considers only the first 32 iterations a significant improvement of the partition function is achieved with little computational cost as compared with the calculation of individual eigenstates. Figure 2: (Color online) Evolution of the trace of $\mathcal{\hat{U}}=e^{\hat{J}}e^{-\beta\mathcal{\hat{H}}}e^{-\hat{J}}$ in the subspace defined by the small basis $\\{\|\chi^{S,\ell}_{n}\rangle\\}$ as a function of the iteration index $\ell$ for $M_{S}=20$ relative to the truncated trace obtained with full CI eigenvalues. Note that the scale starts from 86% of the full CI value. The blue circles are results obtained with SHDMC using Eq. (30). The red rhombi denote the exact evaluation of $\mathcal{\hat{U}}=e^{\hat{J}}e^{-\beta\mathcal{\hat{H}}}e^{-\hat{J}}$ in the small basis, using full CI data. The empty squares (up triangles) were obtained by evaluating $\langle\Psi^{T}_{n}\|\hat{\mathcal{H}}\|\Psi^{T}_{n},\rangle$ for linear combination of eigenstate pairs (random rotation within the small subspace) in the full CI basis while $Z(\beta)$ was obtained using an lower bound approximation. mazzola12 The inset shows the computational cost, the number of sampling subblocks $M_{b}$, as a function of the iteration. Figure 3 quantifies the convergence of the small basis $\\{\|\chi^{S,\ell}_{n}\rangle\\}$ towards the basis defined by the eigenstates of the CI $\\{\|\Psi_{n}\rangle\\}$. For that purpose we define the logarithm of the residual subspace projection as $\displaystyle LR_{sp}=\ln\left\\{1-\|Det\left[\langle\Psi_{n}\|\chi^{S,\ell}_{m}\rangle\right]\|^{1/M_{S}}\right\\}.$ (40) In Eq. (40) $Det\left[\langle\Psi_{n}\|\chi^{S,\ell}_{m}\rangle\right]$ is the Determinant of a square matrix of size $M_{S}$ formed by the overlap $\langle\Psi_{n}\|\chi^{S,\ell}_{m}\rangle$. The determinant of the matrix is a complex number of modulus $1$ in the limit when any eigenstate $\|\Psi_{n<M_{S}}\rangle$ can be written as a linear combination of $\|\chi^{S,\ell}_{m}\rangle$. Any error in any member of the small basis $\\{\|\chi^{S,\ell}_{m}\rangle\\}$ reduces the modulus of the determinant by a factor. The exponent $1/M_{S}$ in Eq. (40) is a standard geometric average. A large negative value in Eq. (40) indicates a very good small basis with a determinant that is approaching $1$. Figure 3 shows the evolution of $LR_{sp}$ given by Eq. (40) as a function of the iteration index $\ell$ for the same system described in Fig. 2. Note that $LR_{sp}$ becomes increasingly negative as a function of $\ell$, which implies a global improvement of the basis approaching to the one described by the eigenstates of the full CI. Figure 3: Logarithm of the residual subspace projection $LR_{sp}$ [see Eq. 40] of the small basis $\\{\|\chi^{S,\ell}_{n}\rangle\\}$ into the basis $\\{\|\Psi_{n}\rangle\\}$ defined by the eigenstates of the full CI as a function of the iteration index $\ell$ for the same system described in Fig 2. A larger negative number correspond to in improved small basis. We next need to characterize how well an individual eigenstate can be described by the small basis. To measure this we define the logarithm of the residual projection $LR_{n}$ as $\displaystyle LR_{n}=\ln\left[1-\sum_{m=0}^{M_{S}-1}\|\langle\Psi_{n}\|\chi^{S,\ell}_{m}\rangle\|^{2}\right].$ (41) Note in Eq. (41) that, if the normalized eigenstate can be written as a linear combination of the small basis $\\{\|\chi^{S,\ell}_{m}\rangle\\}$, the expression in the brackets should be zero. A large negative number in $LR_{n}$ implies that the eigenstate $\|\Psi_{n}\rangle$ is very well described in the small basis. Figure 4 describes the evolution of $LR_{n}$ for different eigenstates of the CI as a function of the iteration index $\ell$. The blue (red) contribution to the color decreases (increases) as the index $n$ increases. The continuous line follows the ground state. One can clearly observe that the ground state of the CI is already very well described at the initialization stage within the Lanczos-like setup. As the iteration $\ell$ increases, the small basis describes the lowest-energy excitations better while higher excitations require more iterations. Note that for 25 iterations 15 eigenvectors are very well described within a basis of 20 states. The total cost at this point is 300 000 individual DMC steps. The calculations of 15 eigenstates with the original SHDMC algorithm for excited statesrockandroll would had cost at least twice as much. The current algorithm becomes competitive, in addition, if one considers that it is parallel, which allows to distribute this cost in multiple tasks ($M_{S}$) reducing the time to solution to 2% as compared with the original SHDMC algorithm for excited states. Figure 4: (Color online) Evolution of the logarithm of residual projection $LR_{n}$ as a function of the iteration index $\ell$. Blue (red) color denotes lower (higher) values of $n$. The continuous line follows the ground state eigenstate. Finally, for infinite statistics one could in principle obtain the eigenenergies of $\mathcal{\hat{H}}$ from the eigenvalues of $u_{n}$ as $E_{n}\approx-\ln(u_{n})/\beta$. This procedure is known to be inefficient to obtain the eigenenergies which are better described by sampling $\mathcal{\hat{H}}$ as in the CFDMC approach. The off diagonal noise in the matrix elements of $\mathcal{\hat{U}}$ has a perverse effect on the magnitude of the eigenvalues. Therefore, while this method is an efficient one to optimize the basis, it should be combined with other methods to obtain the eigenvalue spectra. ## V Summary and discussions In this paper we have presented a general framework aimed to calculate thermodynamical properties of many-body system in an importance sampling DMC context. ceperley80 We showed that a many-body basis can be optimized to describe a small subspace maximizing the overlap with the subspace described by the lowest eigenstates of the Hamiltonian. The Helmholtz free energy obtained within this truncated basis is an upper bound of the exact free energy of a system. The corresponding partition function is a lower bound of the exact partition function. This generalization of the SHDMC method for finite temperature takes advantage of complex wave functions that do not have nodal pockets. Accordingly, we avoid the appearance of the artificial potentials in the standard fixed-node approximation when the amplitude of the importance sampling guiding function is zero. The antisymmetric properties of the wave-function are enforced by a complex phase. This introduces a complex contribution to the local energy. The complex local energy is handled in the complex weight of the walkers. Going beyond the local-time approximation used in Ref. stonehenge, , the evolution of the complex phase factor for $\beta>0$ is now approximately taken into account in the evaluation local energy. The evolution in $\beta$ becomes exact as the statistical error is reduced as the sampling increases. In variance CFDMC, ceperley88 the solution remains an upper bound of the fermionic free energy. While the CFDMC approach uses a bosonic-trial wave function without nodes, SHDMC uses a complex linear combination of anti- symmetric functions without nodal pockets. The walker distribution is prevented to fall into the bosonic ground state solution by the phase factor of the guiding function, which remains antisymmetric, and introduces an effective potential in the local energy. ortiz93 ; stonehenge The present approach shares many aspects of the SHDMC method for complex wave functions, but it also incorporates a key advantage of the CFDMC ceperley88 approach: several wave functions are optimized at the same time. In systems where many excitations can be approximated by a single amplitude $\Phi({\bf R})$ and a different phase factor $e^{{\bf i}\phi_{n}({\bf R})}$, a correlated sampling approach that reweighs the walkers in Eq. (18) as $W_{i}\rightarrow W_{i}\Phi^{T}_{n}({\bf R})/\Phi({\bf R})$ and changes the phase contribution to the local energy in Eq. (16), would save significant time. If that approximation were used, this generalization of the SHDMC method would look very similar in spirit with CFDMC, the main difference being the use of a complex guiding functions that prevents the exponential growth of the bosonic ground state. The displacement of each wave function in the small basis during the DMC process is decomposed into a displacement within the subspace already described by the other elements of the small basis plus a contribution orthogonal to the small subspace. The displacement included within the basis is used to improve upon the locality locality and local-timestonehenge approximations. The displacement orthogonal to the small subspace is used to correct the small basis used in the next iteration. The serial orthogonalization step required in the original SHDMC algorithm for excited states rockandroll ; stonehenge is avoided with a method that allows the calculation of multiple wave functions in parallel. In addition, the complications of inequivalent nodal pockets of excited states rockandroll is avoided using complex trial wave functions without nodes. The scaling of the cost of an individual iteration of this method is proportional to $M_{S}\times N_{c}$; $M_{S}$ the size of the small basis and $N_{c}$ the number of configurations of the DMC run. The cost of and individual DMC step is dependent of the number of basis functions $N_{b}$ and electrons. It is well known that as the size of the system increases, the number of basis functions $N_{b}$ required to maintain a fixed error bar for a given eigenstate must increase factorially. But in practice, the error required for evaluation of thermodynamical properties is determined by $\beta^{-1}$: errors must be much smaller than the temperature of the system. Therefore, as the temperature increases and averages of multiple eigenstates are obtained, the detail required by calculations of the ground state energies with chemical accuracy is no longer necessary. The present approach can take advantage of the acceleration of the algorithms used to evaluate large numbers of determinants. nukala09 ; clark11 For a very large $N_{b}$, the cost of these algorithms scales as $N_{e}\times N_{b}$. The total cost is dependent on the physical system and the goal of the calculation. If the goal is to converge the entire basis or to optimize the Free energy, the bottleneck for convergence is the energy gap $E_{M_{S}+1}-E_{M_{S}}$ which determines the convergence of the basis towards the highest eigenstates considered. Accordingly, in this case, the ideal situation for this method would be a system with ($M_{S}$) nearly degenerate eigenstates well-separated from the rest of the spectra within an energy scale of $k_{b}T$. If the goal, instead, is to converge the small basis so as only the lowest $M_{L}$ eigenstates are well described, the convergence of the algorithm is much faster and it is limited by the number of statistical samples and the exponential decay $e^{-\beta(E_{M_{S}+1}-E_{M_{L}})}$. The cost is reduced as compared with the calculation of eigenstates if one accepts an error in the higher excitations. If one wishes to retain the physics of higher eigenstates in the basis, it is computationally more efficient to increase $M_{S}$ (which increases the cost linearly), instead of improving the basis for the higher excitations which increases the cost exponentially. Comparisons of the method with full CI calculations show that SHDMC can be used to optimize many-body basis sets to maximize the overlap with the lowest energy excitations of the Hamiltonian. Each eigenenergy obtained with this method has lower quality than those obtained with alternative approaches such as LMMC or the standard SHDMC for excited states. However, this method could be a useful tool to optimize the basis, minimizing the size of the matrices used in LMMC and thus reducing the effects of numerical noise in LMMC. ###### Acknowledgements. The authors would like to thank J. Krogel and P. R. C. Kent for a critical reading of the manuscript and discussions. This work has been supported by the grant ERKCS92 Materials Science and Engineering division of Basic Energy Sciences, Department of Energy. ## Appendix A Going beyond the locality and local-time approximations Approximate coefficients for the Slater determinant expansion of the eigenstates of $\mathcal{\hat{H}}$ and $e^{-\beta\mathcal{\hat{H}}}$ can be obtained from the eigenstates of $\mathcal{\hat{U}}=e^{\hat{J}}e^{-\beta\mathcal{\hat{H}}}e^{-\hat{J}}$ in the $\\{\chi^{S}_{n}\\}$ basis [see Eq. (II.5)]. But, $\mathcal{\hat{U}}$ is not hermitian, since $\mathcal{\hat{U}}^{\dagger}=e^{2\hat{J}}\mathcal{\hat{U}}e^{-2\hat{J}}$. Nevertheless, as long as the Jastrow factor operator $e^{-\hat{J}}$ has an inverse, $\mathcal{\hat{U}}$ has a set of right eigenvectors $\|\chi^{U}_{i}\rangle=e^{\hat{J}}\|\Psi_{i}\rangle$ and left eigenvectors $\langle\chi^{U}_{i}\|=\langle\Psi_{i}\|e^{-\hat{J}}$. Since $\mathcal{\hat{H}}$ is Hermitian, its eigenstates $\|\Psi_{i}\rangle$ are orthogonal, which implies that $\langle\chi^{U}_{i}\|\chi^{U}_{j}\rangle=\delta_{i,j}$. Within statistical error bars, in the small subspace, the matrix elements of $\mathcal{\hat{U}}$ obtained with the basis $\|\chi^{S,\ell}_{n}\rangle$ of the iteration $\ell$ are given by Eq. (II.5). In the first iteration the matrix elements of $\mathcal{\hat{U}}$ can be obtained directly from the Lanczos-like procedure. Within the small subspace, $\mathcal{\hat{U}}$ can be written as $\displaystyle\mathcal{\hat{U}}=\sum_{i}u_{i}\|\chi^{U}_{i}\rangle\langle\chi^{U}_{i}\|.$ (42) Since the $u_{i}$ are also the eigenvalues of $e^{-\beta\mathcal{\hat{H}}}$ their dependence with $\beta$ is exponential. Thus for an arbitrary $\beta^{\prime}$ the eigenvalue will be $u_{i}^{\beta^{\prime}/\beta}$. Provided that the difference $\|\Delta\tilde{\chi}^{S,\ell+1}_{n}\rangle=\|\chi^{S,\ell+1}_{n}\rangle-\|\chi^{S,\ell}_{n}\rangle$ is small [which is always valid for $\beta\rightarrow 0$ see Eq. (38)], the dependence in $\beta$ of $\|\chi^{S,\ell+1}_{n}(\beta)\rangle$ can be approximated as follows: Let’s first define the operator $\hat{R}=\sum_{n=0}^{M_{S}-1}\|\chi^{S,\ell}_{n}\rangle\langle\chi^{S,\ell+1}_{n}\|$ and its inverse within the small subspace $\hat{R}^{\dagger}=\sum_{n=0}^{M_{S}-1}\|\|\chi^{S,\ell+1}_{n}\rangle\langle\chi^{S,\ell}_{n}\|$. Accordingly, the dependence in $\beta^{\prime}$ of the new basis can be approximated as $\displaystyle\left\|\chi^{S,\ell+1}_{n}\left(\beta^{\prime}\right)\right\rangle=$ $\displaystyle\hat{R}^{\dagger}\mathcal{\hat{U}}^{{\beta^{\prime}}/{\beta}}\hat{R}\left\|\chi^{S,\ell+1}_{n}\right\rangle,$ (43) with $\mathcal{\hat{U}}^{{\beta^{\prime}}/{\beta}}$ given by Eq. (42) replacing $u_{i}$ by $u_{i}^{\beta^{\prime}/\beta}$. ## Appendix B Working with eigenstates of $\mathcal{\hat{U}}$ While in some systems eigenstates of $\mathcal{\hat{H}}$ are always real (e.g. confined systems with time reversal symmetry), in many cases the wave function of the eigenstates is known to be complex, without nodal pockets. In those cases Eq. (II.3) is valid and no walker needs to be killed or rejected because the eigenstate wave function does not have a nodal surface. stonehenge In these cases it might be advantageous to propagate single eigenstates of $\hat{R}^{\dagger}\mathcal{\hat{U}}^{\frac{j}{k}}\hat{R}$, since the variance of the weights is minimized and lower (larger) $T$ ($\beta$) can be reached with less statistical data. An additional advantage of working with functions that are closer to the eigenstates is that the locality and local-time approximations can be used. Since any $\|\chi^{S}_{n}\rangle=\sum_{i=0}^{M_{S}-1}\langle\chi^{X}_{i}\|\chi^{S}_{n}\rangle\|\chi^{X}_{i}\rangle$ can be written as a linear combination of $\|\chi^{X}_{i}\rangle$ and vise versa, when eigenstates are complex one can use as trial wave function $e^{\hat{J}}\|\chi^{U}_{i}\rangle$ in the SHDMC propagation and sampling. Then the propagation of $\|\chi^{S}_{n}\rangle$ can be written as a linear combination of the propagation of the eigenstates of $\mathcal{\hat{U}}$. as: $\displaystyle\|\delta\chi^{S}_{n}\rangle$ $\displaystyle=\sum_{i=0}^{M_{S}}\left\\{\langle\chi^{X}_{i}\|\chi^{S}_{n}\rangle e^{-\beta(E^{X}_{i}-E^{S}_{n})}\left[\|\chi^{X}_{i}\rangle+\|\delta\chi^{X}_{i}\rangle\right]\right\\}-\|\chi^{S}_{n}\rangle\text{ and }$ $\displaystyle\|\delta\tilde{\chi}^{S}_{n}\rangle$ $\displaystyle=\sum_{i=0}^{M_{S}}\left\\{\langle\chi^{X}_{i}\|\chi^{S}_{n}\rangle e^{-\beta(E^{X}_{i}-E^{S}_{n})}\left[\|\chi^{X}_{i}\rangle+\|\delta\tilde{\chi}^{X}_{i}\rangle\right]\right\\}-\|\chi^{S}_{n}\rangle$ (44) $E^{S}_{n}$ results from the condition $\langle\chi^{S}_{n}\|\delta\chi^{S}_{n}\rangle=0$ and it is given by $\displaystyle e^{-\beta E^{S}_{n}}=\sum_{i}\left[\|\langle\chi^{X}_{i}\|\chi^{S}_{n}\rangle\|^{2}+\langle\chi^{X}_{i}\|\chi^{S}_{n}\rangle\langle\chi^{S}_{n}\|\delta\chi^{X}_{i}\rangle\right]e^{-\beta E^{X}_{i}}\;,$ (45) replacing $X$ by $U$ in Eqs. (B) and (45). The brackets $\langle\chi^{U}_{i}\|\chi^{S}_{n}\rangle$ are obtained by diagonalizing $\mathcal{\hat{U}}$: the coefficient $n$ of the left eigenvectors $i$ in the small basis $\\{\langle\chi^{S}_{n}\|\\}$. The disadvantages is that complex eigenstates appear only in certain Hamiltonians or boundary conditions. Albeit without nodes, they might have large variation in probability density, in particular for small magnetic fields or twist boundary conditions close to high symmetry points. Large variations in the probability density hinder correlating sampling. ## Appendix C Working with eigenstates pairs It is well known that in many physical systems the energy spacing between eigenstates decreases as the size of the system increases. It is also known that the variance of the local energy, which is related to the statistical error in the energy, increases as the size of the system increases. nemec10 Accordingly as the size of the system increases, it becomes more difficult to obtain eigenstates. As the size of the system increases the error in the variance introduced by a linear combination of eigenstates in the Slater part of the wave function becomes smaller than the variance introduced by short range correlations. These short range correlations cannot be accounted by the Slater part, even with a very large basis $\\{\|n\rangle\\}$, or with simple Jastrow factors. Accordingly, in large systems little is lost by using a linear combination of eigenstates, since their contribution to the variance is proportional to the energy separation that decreases as the system become larger. In contrast, much is gained avoiding the divergences at the nodes by using complex linear combinations of eigenstates, in particular, to obtain average of thermodynamical properties. However, to propagate for larger $\beta$ with a branching algorithm, it will be necessary to minimize the variance of the local energy. If the eigenstates are real, one must use a linear combination of eigenstates. The minimum variance will be reached by constructing the Slater part of the guiding wave function with linear combinations of eigenstates with consecutive eigenvalues of $u_{i}$. In this work we use $\displaystyle\|\chi^{V}_{2j-1}\rangle$ $\displaystyle=\frac{1}{\sqrt{2}}\left(\|\chi^{U}_{2j-1}\rangle+e^{{\bf i}\phi}\|\chi^{U}_{2j}\rangle\right)$ (46) $\displaystyle\|\chi^{V}_{2j}\rangle$ $\displaystyle=\frac{1}{\sqrt{2}}\left(\|\chi^{U}_{2j-1}\rangle-e^{{\bf i}\phi}\|\chi^{U}_{2j}\rangle\right),$ where $e^{{\bf i}\phi}$ is a complex phase which can be adjusted to minimize the variance of the amplitude of the complex wave function $\Phi^{T}_{n}({\bf R})$. The conjugate vectors $\langle\chi^{V}_{2j-1}\|$ are constructed using complex conjugate coefficients and the left eigenvectors of $\mathcal{\hat{U}}$ in the small basis $\\{\langle\chi^{S}_{n}\|\\}$. 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1309.4741	# Overlap Cycles for Permutations: Necessary and Sufficient Conditions Victoria Horan Air Force Research Laboratory Rome, NY 13441 USA [email protected] ###### Abstract Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1992. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, permutations, partitions, vector spaces, and designs. One generalization of universal cycles, which require almost complete overlap of consecutive words, is $s$-overlap cycles, which relax such a constraint. In this paper we study permutations and some closely related class of strings, namely juggling sequences and functions. We prove the existence of $s$-overlap cycles for these objects, as they do not always lend themselves to the universal cycle structure. ## 1 Introduction ††Approved for public release; distribution unlimited: 88ABW-2013-4003, 9 Sep 13 Listing structures for combinatorial objects are quickly becoming useful in more and more interesting applications. Gray codes, first defined in 1947 by Frank Gray [6], are used in many different surprising places from position encoders [11] to genetic algorithms [4]. More recently universal cycles are being used in areas such as rank modulation for multilevel flash memories [12], however overlap cycles are still being explored and have the potential to be useful in many applications. An $s$-overlap cycle, or $s$-ocycle, is an ordering of a set of objects $\mathcal{C}$, each represented as a string of length $n$. The ordering requires that object $b=b_{0}b_{1}\ldots b_{n-1}$ follow object $a=a_{0}a_{1}\ldots a_{n-1}$ only if $a_{n-s}a_{n-s+1}\ldots a_{n-1}=b_{0}b_{1}\ldots b_{s-1}$. Ocycles were introduced by Godbole, Knisley, and Norwood in 2010 [5]. Universal cycles, or ucycles are $(n-1)$-ocycles and were originally introduced in 1992 by Chung, Diaconis, and Graham [2]. To find $s$-ocycles and ucycles on a set of strings, most proofs employ the same method. For a given string $X=x_{1}x_{2}\ldots x_{n}$, let $X^{s-}=x_{1}x_{2}\ldots x_{s}$ denote the $s$-prefix of $X$ and $X^{s+}=x_{n-s}x_{n-s+1}\ldots x_{n}$ denote the $s$-suffix of $X$. The first step in the proof is to construct the transition digraph for the set of strings as follows. Vertices represent $s$-prefixes and $s$-suffixes of strings (the overlaps), while each edge represents a string, traveling from its $s$-prefix to its $s$-suffix. Note that the transition digraph is a directed multigraph in which an Euler tour (a closed walk in which every edge is traversed exactly once) corresponds bijectively to an $s$-ocycle. To prove the existence of an Euler tour, we use the following well-known theorem. ###### Theorem 1.1. _([13], p. 60)_ A directed graph $G$ is eulerian if and only if it is both balanced and weakly connected. In this paper, we consider using the ocycle listing structure for permutations, as well as functions and juggling sequences. We represent permutations of an $n$-set $\\{0,1,\ldots,n-1\\}$ as a string $\Pi=\pi_{0}\pi_{1}\ldots\pi_{n-1}$, where the functional representation is used, i.e. $\pi(0)=\pi_{0}$. Closely related to permutations, juggling sequences have been an active research area since the 1980’s [1]. These sequences are used to determine what patterns a fixed set of balls can be juggled in, where only one ball may be thrown at a time. ###### Definition 1.2. _[3]_ A juggling sequence is a string $T=t_{0}t_{1}\ldots t_{n-1}$ where each $t_{i}\geq 0$ such that $\left\|\\{i+t_{i}\pmod{n}\mid 0\leq i\leq n-1\\}\right\|=n.$ This sequence illustrates that at time $i$, we should throw a ball high enough that it is in the air for $t_{i}$ beats, or to height $t_{i}$. The number of balls used in a given juggling sequence $T=t_{0}t_{1}\ldots t_{n-1}$ is given by $b=\frac{1}{n}\sum_{i=0}^{n-1}t_{i}.$ The period of a juggling sequence $t_{0}t_{1}\ldots t_{n-1}$ is $n$, the length of the string. An alternative definition considers the corresponding permutation sequence for a juggling sequence. Given a string $T=t_{0}t_{1}\ldots t_{n-1}$, the permutation sequence is the string $\Pi_{T}=\pi_{0}\pi_{1}\ldots\pi_{n-1}$ where $\pi_{i}=t_{i}+i\pmod{n}$. Then $\Pi_{T}$ is a permutation of the $n$-set $\\{0,1,\ldots,n-1\\}$ if and only if $T$ is a valid juggling sequence. From this definition it is clear that there is a very close relationship between juggling sequences and permutations, which is further illustrated by their similar results explored in this paper. The permutation sequence clarifies the point that a juggler cannot catch or throw two objects simultaneously. For example, when $n=3$ and $b=2$ the pattern $015$ is valid. We can see this by drawing the juggling diagram, shown in Figure 1. However, the sequence $105$ is not valid, as shown in Figure 2. In this second example, note that on the second beat we are required two catch two balls simultaneously - an operation that is not allowed. 015015015 Figure 1: Juggling Diagram for Sequence $015$ 105105105 Figure 2: Juggling Diagram for Sequence $105$ In [3], Chung and Graham show that it is not always possible to find a single universal cycle that contains all juggling sequences of period $n$ and at most $b$ balls. However they do prove that we can use several disjoint universal cycles to cover each sequence exactly once. This result is closely related to the long studied problem of universal cycles for permutations. Since ucycles for permutations are not possible using the standard representation, a modification of the representation can often provide an effective solution [9, 10]. An alternative that has only recently been explored is to modify the listing structure rather than the object representation. By utilizing $s$-ocycles rather than ucycles, partial results for permutations have been obtained in [7], however we obtain a more complete solution in this paper. In Section 2 we establish a complete result on the existence of $s$-ocycles for permutations, and in Section 3 we explore similar results for functions and juggling sequences. ## 2 Permutations We begin with a general lemma that will be used for both permutations and juggling sequences. For a string $X=x_{0}x_{1}\ldots x_{n-1}$, define the rotation function as follows. A rotation by $s$ is given by: $\rho^{s}(X)=x_{s}x_{s+1}\ldots x_{n-1}x_{0}x_{1}\ldots x_{s-1}.$ The following lemma shows that for a string of length $n$, rotations by $s$ partition the string into blocks of length gcd$(n,s)$, and we can always perform repeated $s$-rotations to start $X$ with any block desired. ###### Lemma 2.1. Let $n,s\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$ and gcd$(n,s)=d$. Consider a string $X=x_{0}x_{1}\ldots x_{n-1}$, also written as $X=Y_{0}Y_{1}\ldots Y_{m-1}$ where $n=md$ and $Y_{i}=x_{id}x_{id+1}\ldots x_{id+d-1}$. Then for any $i\in\\{0,1,\ldots,m-1\\}$ there is some $j\in\\{0,1,\ldots,m-1\\}$ such that: $Y_{i}Y_{i+1}\ldots Y_{m-1}Y_{0}Y_{1}\ldots Y_{i-1}=\rho^{js}(X).$ ###### Proof. Suppose that $s=kd$. Then each rotation $\rho^{s}(X)$ advances through $k$ blocks of $X$. Thus if we can show that gcd$(m,k)=1$, we are done. However we note that gcd$(n,s)=d$ implies that there are integers $p,q$ such that $pn+qs=d$. This implies the following. $\displaystyle pn+qs$ $\displaystyle=$ $\displaystyle d$ $\displaystyle pmd+qkd$ $\displaystyle=$ $\displaystyle d$ $\displaystyle pm+qk$ $\displaystyle=$ $\displaystyle 1$ Thus the same integers $p,q$ provide confirmation that gcd$(m,k)=1$. ∎ ###### Theorem 2.2. Let $n,s\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$ and let $M$ be a multiset of size $n$. There exists an $s$-ocycle on permutations of $M$ if and only if $n-s>\hbox{gcd}(n,s)$. ###### Proof. Define gcd$(n,s)=d$, and suppose that $n-s=kd>d$. Construct the transition digraph $D$ with vertices representing $s$-permutations of $M$ and edges representing permutations of $M$. We will show that this graph is balanced and connected, and hence eulerian. Recall that an Euler tour in this graph corresponds to an $s$-ocycle for permutations of $M$. Balanced: Let $X^{s-}=x_{0}x_{1}\ldots x_{s-1}$ be an arbitrary vertex in the transition graph. For each possible out-edge corresponding to an $n$-permutation $X=x_{0}x_{1}\ldots x_{n-1}$, we also have the in-edge $X^{\prime}=x_{s}x_{s+1}\ldots x_{n-1}x_{0}x_{1}\ldots x_{s-1}$. Thus there is a one-to-one correspondence between in- and out-edges at each vertex, and so D is balanced. Connected: Consider an arbitrary vertex $X^{s-}=x_{0}x_{1}\ldots x_{s-1}$, which is an $s$-prefix of some permutation $X=x_{0}x_{1}\ldots x_{n-1}$. We will consider a partition of $X$ into blocks of size $d$, i.e. $X=X_{0}X_{1}\ldots X_{m-1}$ where $X_{i}=x_{id}x_{id+1}\ldots x_{(i+1)d-1}$. Our goal is to show that we can permute elements in any $k$ consecutive blocks $X_{i}X_{i+1}\ldots X_{i+k-1}$, which illustrates that any adjacent transpositions are possible, and hence we can reach all permutations. In D, rotations of $X$ correspond to the following cycle, where subscripts are computed modulo $n$. $\displaystyle x_{0}x_{1}\ldots x_{s-1}$ $\displaystyle\rightarrow$ $\displaystyle x_{n-s}x_{n-s+1}\ldots x_{n-1}$ $\displaystyle\rightarrow$ $\displaystyle x_{n-2s}x_{n-2s+1}\ldots x_{n-s-1}$ $\displaystyle\rightarrow$ $\displaystyle x_{n-3s}x_{n-3s+1}\ldots x_{n-2s-1}$ $\displaystyle\vdots$ $\displaystyle\rightarrow$ $\displaystyle x_{n-is}x_{n-is+1}\ldots x_{n-(i-1)s-1}$ $\displaystyle\vdots$ $\displaystyle\rightarrow$ $\displaystyle x_{0}x_{1}\ldots x_{s-1}$ Clearly from the vertex $X^{s-}$ we can permute all elements in $\\{x_{s},x_{s+1},\ldots,x_{n-1}\\}$ to determine an out-edge, and at any point we can permute the $(n-s)$-suffix of a permutation. We would like to show that, through rotations, the $(n-s)$-suffix may contain any $k$ consecutive blocks from $\\{X_{0},X_{1},\ldots,X_{m-1}\\}$. Consider the vertex $x_{n-is}x_{n-is+1}\ldots x_{n-(i-1)s-1}$. How does the starting index $n-is$ correspond to $d$? $\displaystyle n-is$ $\displaystyle\equiv$ $\displaystyle in-is\pmod{n}$ $\displaystyle\equiv$ $\displaystyle i(n-s)\pmod{n}$ $\displaystyle\equiv$ $\displaystyle ikd\pmod{n}$ Thus the vertices in this rotation cycle always begin with a multiple of $kd$. Our final step is to show that any multiple of $d$, say $id$, may be written as a multiple of $kd$ modulo $n$, which is done by Lemma 3.8. Then we can rotate to start with any block desired, which is equivalent to pushing any $k$ consecutive blocks to the $(n-s)$-suffix. To summarize, we can perform adjacent transpositions $x_{i}\leftrightarrow x_{i+1}$ within $X^{s-}$ by rotating $X$ until $x_{i}$ and $x_{i+1}$ fall into the $k$ blocks in the $(n-s)$-suffix and then transposing. Finally, by continuing along through rotations we will arrive at the vertex $x_{0}x_{1}\ldots x_{i-1}x_{i+1}x_{i}x_{i+2}\ldots x_{s-1}$. Thus adjacent transpositions are always connected, hence all permutations can be reached. For the converse, suppose that $n-s=d=\hbox{gcd}(n,s)$. In this case, rotations of the permutation $X=x_{0}x_{1}\ldots x_{n-1}$ provide an $(n-s)$-suffix of length $d$ – just one block from $X=X_{0}X_{1}\ldots X_{m-1}$. Thus we can always permute elements within each block, however the cyclic order of the blocks is fixed and we can perform the swap $x_{i}\leftrightarrow x_{j}$ if and only if $x_{i}$ and $x_{j}$ are in the same block. Thus we are able to permute elements within blocks, but not permute blocks (only rotate the block order). Hence the transition digraph connects only permutations with the same block order (rotations allowed). Permutations with block order that are not simple rotations are not connected, so the graph is disconnected and no Euler tour exists. ∎ Note that Theorem 2.2 agrees with the following well-known fact about universal cycles, or $(n-1)$-ocycles. ###### Corollary 2.3. There is no universal cycle for permutations of $M$. ###### Proof. In this case, we apply Theorem 2.2 with $s=n-1$. Then gcd$(n,s)=1=n-s$, so no $(n-1)$-ocycle exists. ∎ ## 3 Related Objects ### 3.1 Functions Many times injective, surjective, and bijective functions are represented by permutations. We have the following facts about functions and theorems corresponding to their alternate representations. ###### Fact 3.1. An injective function $f:[k]\rightarrow[n]$ may be represented by $x_{1}x_{2}\ldots x_{k}$, the $k$-permutation of $[n]$ defined so that $x_{i}=f(i)$. ###### Theorem 3.2. _[7]_ For all $n,s,k\in\mathbb{Z}^{+}$ with $1\leq s<k<n$, there is an $s$-ocycle for $k$-permutations of $[n]$. ###### Fact 3.3. A surjective function $f:[n]\rightarrow[h]$ may be represented by $x_{1}x_{2}\ldots x_{n}$, the string with ground set $[h]$ defined so that $x_{i}=f(i)$. In [8], it is shown that surjective functions are also represented by weak orders of $[n]$ with height $h-1$. This observation is followed by the following theorem. ###### Theorem 3.4. _[8]_ For all $n,s,h\in\mathbb{Z}^{+}$ with $1\leq s\leq n-2$, gcd$(s,n)=1$, and $0\leq h\leq n-1$, there is an $s$-ocycle for $\mathcal{W}(n,h)$. We are able to improve this theorem as follows. ###### Theorem 3.5. For all $n,s,h\in\mathbb{Z}^{+}$ with $1\leq s\leq n-2$ and $h\leq n-1$ there is an $s$-ocycle for strings with ground set $[h]$. ###### Proof. We will show that the corresponding transition graph is eulerian. Balanced: Consider a vertex $X^{s-}=x_{1}x_{2}\ldots x_{s}$. $X^{s-}$ is an $s$-prefix of the string $X=x_{1}x_{2}\ldots x_{n}$ where $X$ has ground set $[h]$. Since $x_{s+1}x_{s+2}\ldots x_{n}x_{1}x_{2}\ldots x_{s}$ is also a string with ground set $[h]$, there is a bijection between in- and out-edges at $X^{s-}$. Hence the graph is balanced. Connected: Define the minimum vertex $V^{s-}$ to be the $s$-prefix of the permutation $V=12\cdots mm\cdots m$. Let $X=x_{1}x_{2}\ldots x_{n}$ be an arbitrary multiset permutation with ground set $[h]$, and let $X^{s-}$ be the $s$-prefix. We will show a path from $X^{s-}$ to $V^{s-}$ exists in the transition graph. Compare $X^{s-}$ and $V^{s-}$, and let $i$ be the least index such that $X^{s-}(i)\neq V^{s-}(i)$. Note that since $h\leq n-1$, some element from $[h]$ must appear at least twice in $X$. We will refer to any element appearing more than once as a duplicate. We have two cases. 1. 1. If the letter $x_{i}\in[h]$ appears twice in $X$: Let $d=\hbox{gcd}(n,s)$, and rotate $X$ until the $d$-block containing $x_{i}$ is first. If we rotate $X$ again by following an out-edge in the graph, we have arrived at a vertex $A$ representing an $s$-substring of $X$ without $x_{i}$. Since $x_{i}$ also appears elsewhere in $X$, we can follow (backwards) the in-edge that is identical to $A$ except with $x_{i}$ replaced by $v_{i}$. Now we are at an edge corresponding to an $s$-substring of $x_{1}x_{2}\ldots x_{i-1}v_{i}x_{i+1}\ldots x_{n}$, so we are one step closer to the minimum vertex. 2. 2. If the letter $x_{i}\in[h]$ appears exactly once in $X$: Since $x_{i}$ is not a duplicate in $X$, some other letter $x_{j}\in[h]$ is a duplicate. In this case, we proceed as in case 1 to replace $x_{j}$ with the letter $x_{i}$. Then $x_{i}$ is a duplicate so we may follow case 1 again to replace $x_{i}$ with $v_{i}$. At this point we have moved closer to the minimum vertex. Continuing until we have transformed $X^{s-}$ to $V^{s-}$ produces a path from $X^{s-}$ to the minimum vertex. Hence the graph is connected. As the transition graph is balanced and connected, it is eulerian by Theorem 1.1. ∎ ###### Fact 3.6. A bijective function $f:[n]\rightarrow[n]$ may be represented by $x_{1}x_{2}\ldots x_{n}$, the permutation of $[n]$ defined so that $x_{i}=f(i)$. ###### Theorem 3.7. Let $n,s\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$. There exists an $s$-ocycle on permutations of $[n]$ if and only if $n-s>\hbox{gcd}(n,s)$. ###### Proof. Use Theorem 2.2 with $M=[n]$. ∎ ### 3.2 Juggling Sequences We begin with some lemmas that will help us to prove our main result. ###### Lemma 3.8. Let $R=r_{0}r_{1}\ldots r_{n-1}$ be a string, and let $1\leq s\leq n$. Then $R^{\prime}=\rho^{s}(R)=r_{0-s}r_{1-s}\ldots r_{n-1-s}$ (where addition is modulo $n$) is a valid juggling sequence if and only if $R$ is a valid juggling sequence. ###### Proof. We will check the corresponding permutation sequence for $R^{\prime}$ and show that it is valid. Suppose for a contradiction that there are $i,j\in\\{0,1,\ldots,n-1\\}$ with $r^{\prime}_{i}+i\equiv r^{\prime}_{j}+j\pmod{n}.$ Then we have: $\displaystyle r^{\prime}_{i}+i$ $\displaystyle\equiv$ $\displaystyle r^{\prime}_{j}+j\pmod{n}$ $\displaystyle r_{i-s}+i$ $\displaystyle\equiv$ $\displaystyle r_{j-s}+j\pmod{n}$ $\displaystyle r_{i-s}+i-s$ $\displaystyle\equiv$ $\displaystyle r_{j-s}+j-s\pmod{n}$ $\displaystyle r_{k}+k$ $\displaystyle\equiv$ $\displaystyle r_{\ell}+\ell\pmod{n}$ Thus $\Pi_{R^{\prime}}$ is valid if and only if $\Pi_{R}$ is valid. ∎ ###### Lemma 3.9. Let $T=t_{0}t_{1}\ldots t_{n-1}$ be a juggling sequence, and let $0\leq i,s\leq n-1$. Then $\Pi_{\rho^{s}(T)}(i)=\rho^{s}(\Pi_{T})(i)-s\pmod{n}.$ ###### Proof. $\displaystyle\Pi_{\rho^{s}(T)}(i)$ $\displaystyle=$ $\displaystyle\rho^{s}(T)(i)+i\pmod{n}$ $\displaystyle=$ $\displaystyle T(i+s)+i\pmod{n}$ $\displaystyle=$ $\displaystyle T(i+s)+i+s-s\pmod{n}$ $\displaystyle=$ $\displaystyle\Pi_{T}(i+s)-s\pmod{n}$ $\displaystyle=$ $\displaystyle\rho^{s}(\Pi_{T})(i)-s\pmod{n}$ ∎ ###### Lemma 3.10. Fix $n,s,b\in\mathbb{Z}^{+}$ with $1\leq s\leq n-1$. Define $D$ to be the $s$-ocycle transition digraph for juggling sequences of length $n$ using at most $b$ balls. From any vertex $v_{0}v_{1}\ldots v_{s-1}$, there exists a path to any vertex $v_{0}^{\prime}v_{1}^{\prime}\ldots v_{s-1}^{\prime}$ where $v_{i}^{\prime}\equiv v_{i}\pmod{n}$. ###### Proof. Since adding/subtracting $n$ to any digit in a juggling sequence of length $n$ does not invalidate the sequence, if $v_{i}\geq n$ we can simply rotate some juggling sequence $X$ with $s$-prefix $v_{0}v_{1}\ldots v_{s-1}$ until $v_{i}$ is in the $(n-s$)-suffix, replace with $v_{i}$ with $v_{i}-n$, and then rotate back to our original $s$-prefix with $v_{i}$ replaced by $v_{i}-n$. Repeating this eventually will find a path to $v_{1}^{\prime}v_{2}^{\prime}\ldots v_{s}^{\prime}$. ∎ ###### Theorem 3.11. Fix $n,s,b\in\mathbb{Z}^{+}$ such that $1\leq s\leq n-1$. There exists an $s$-ocycle for the set of juggling sequences with period $n$ and at most $b$ balls if and only if $n-s>\hbox{gcd}(n,s)$. ###### Proof. We prove the forward direction by showing that the $s$-ocycle transition digraph $D$ has an Euler tour. By Theorem 1.1 this is done by showing that the graph is balanced and connected. Balanced: Consider a vertex $R=r_{0}r_{1}\ldots r_{s-1}$ in $D$. We will show that any $(n-s)$-string $Q=q_{0}q_{1}\ldots q_{n-s-1}$ that is a valid $s$-suffix for $R$ is also a valid $s$-prefix for $R$. Note that strings $RQ$ and $QR$ are simply rotations of each other, so by Lemma 3.8 either both strings are valid juggling sequences or neither string is. In this manner, there is a bijection between in- and out-edges, hence all vertices are balanced. Connected: Consider an arbitrary vertex $T^{s-}=t_{0}t_{1}\ldots t_{s-1}$ that is an $s$-prefix to some juggling sequence $T=t_{0}t_{1}\ldots t_{n-1}$. First, by Lemma 3.10 we may assume that $t_{i}\in\\{0,1,\ldots,n-1\\}$ for all $i\in\\{0,1,\ldots,n-1\\}$. We will show that this arbitrary vertex is connected to the minimum vertex, which we define to be $V^{s-}=0^{s}$ (a prefix of $V=0^{n}$). In doing so, we will have shown that every vertex is connected to $V^{s-}$ and hence the graph is connected. Compare permutation sequences $\Pi_{T}$ and $\Pi_{V}$ corresponding to juggling sequences $T$ and $V$, respectively. Note that $\Pi_{V}=012\ldots(n-1)$, so suppose that for all $x\in\\{0,1,\ldots,i-1\\}$ we have $\Pi_{T}(x)=\Pi_{V}(x)=x$, but that $\Pi_{T}(i)=j$ for some $j\in\\{i+1,i+2,\ldots,s-1\\}$. We will find a path from the $s$-prefix of $T$ to the $s$-prefix of some juggling sequence $T^{\prime}$ with permutation sequence $\Pi_{T^{\prime}}$ that agrees with $\Pi_{V}$ in the first $i+1$ positions. Repeating this procedure until $i=n$ will construct a path through $D$ from $T^{s-}$ to $V^{s-}$. Assume $n-s=kd$ and $n=md$ for integers $m,k$ where $d=\hbox{gcd}(n,s)$, and let $T=Y_{0}Y_{1}\ldots Y_{m-1}$ be a partition of $T$ into $d$-blocks, i.e. $Y_{a}=t_{ad}t_{ad+1}\ldots t_{ad+d-1}$. Suppose that $t_{i}\in Y_{a}$ and $t_{j}\in Y_{b}$. We have two cases. 1. 1. If $b-a<k$: In this case, from Lemma 3.8 we may perform $s$-rotations on $T$ so that we arrive at a vertex with both $t_{i}$ and $t_{j}$ in the $(n-s)$-suffix of $T$. Suppose that we performed a total rotation of size $R$, i.e. we are now at the vertex that represents the $s$-prefix of $\rho^{R}(T)$. At this point, the values $t_{i}$ and $t_{j}$ are located in positions $i-R$ and $j-R$ of $\rho^{R}(T)$, respectively. By Lemma 3.9 the corresponding permutation sequence entries are: $\displaystyle\Pi_{\rho^{R}(T)}(i-R)$ $\displaystyle=$ $\displaystyle\rho^{R}(\Pi_{T})(i-R)-R\pmod{n}$ $\displaystyle=$ $\displaystyle\Pi_{T}(i)-R\pmod{n}$ and $\displaystyle\Pi_{\rho^{R}(T)}(j-R)$ $\displaystyle=$ $\displaystyle\Pi_{T}(j)-R\pmod{n}.$ We now note that the vertex $(\rho^{R}(T))^{s-}$, defined as the $s$-prefix of $\rho^{R}(T)$, is also the $s$-prefix of the juggling sequence $\rho^{R}(T^{\prime})$ that is obtained from $\rho^{R}(T)$ by performing the swap $\Pi_{\rho^{R}(T)}(i-R)\leftrightarrow\Pi_{\rho^{R}(T)}(j-R)$ and adjusting the values $\rho^{R}(T)(i-R)$ and $\rho^{R}(T)(j-R)$ appropriately. Then rotating backwards by $R$ we reach a juggling sequence $T^{\prime}$ with $\Pi_{T^{\prime}}(x)=x$ for all $x\in\\{0,1,\ldots,i-1,i\\}$. This means that we have found a path from $T^{s-}$ to $T^{\prime s-}$, which is one step closer to the minimum vertex. 2. 2. If $b-a\geq k$: In this case the blocks containing $t_{i}$ and $t_{j}$ are more than $k$ apart, so we cannot rotate to have both $t_{i}$ and $t_{j}$ in the $(n-s)$-suffix of $T$ simultaneously. Instead, we pick a point $t_{z}$ in the block $Y_{c}$ that is $k$ blocks preceding $Y_{b}$ and follow case (1) with $t_{z}$ in place of $t_{i}$. This produces a juggling sequence $T^{\prime}$ with $\Pi_{T^{\prime}}(z)=i$, where $i\leq z<j$. Repeating, we will eventually have moved the permutation sequence value $i$ to a block that is close enough to block $Y_{a}$ to apply Case (1). By repeating the above procedures, we will eventually have transitioned to the vertex corresponding to a juggling sequence with $s$-prefix $01\ldots(s-1)$. At this point we have reached the minimum vertex and we are done. For the reverse direction, suppose that $n-s=\hbox{gcd}(n,s)=d$. Recall that the number of balls $b\in\mathbb{Z}^{+}$ is given by: $b=\frac{1}{n}\sum_{i=0}^{n-1}t_{i}.$ Thus juggling sequences of period $n$ must always satisfy $\sum_{i=0}^{n-1}t_{i}\equiv 0\pmod{n}$. Equivalently, when we partition a juggling sequence $T=t_{0}t_{1}\ldots t_{n-1}$ into blocks of length $d$, i.e. $T=Y_{0}Y_{1}\ldots Y_{m-1}$, where $n=md$, we must have: $\sum_{i=0}^{m-1}w(Y_{i})=\sum_{i=0}^{m-1}\sum_{j=0}^{d}t_{id+j}=\sum_{i=0}^{n-1}t_{i}\equiv 0\pmod{n},$ where, for a given block $Y_{i}$, we call $w(Y_{i})=\sum_{j=0}^{d}t_{id+j}$ the weight of block $Y_{i}$. Now since $n-s=\hbox{gcd}(n,s)$, from vertex $T^{s-}=t_{0}t_{1}\ldots t_{s-1}$ in $D$ we may only move to vertices in which the $(n-s)$-suffix has weight equivalent to $w(Y_{m-1})$ mod $n$. Thus if we can show that, for any $n,s,$ and $b$, there exists a juggling sequence with a block with weight $w\not\equiv 0\pmod{n}$, then we are done. This is witnessed by the juggling sequence $T=\begin{array}[]{ccccccccc}d&0&0&\ldots&0&(n-d)&0&\ldots&0,\end{array}$ with permutation sequence $\Pi_{T}=\begin{array}[]{cccccccccc}d&1&2&\ldots&(d-1)&0&(d+1)&(d+2)&\ldots&(n-1).\end{array}$ This juggling sequence utilizes one ball (recall we required $b\in\mathbb{Z}^{+}$) and the weight of the first block of length $d$ is $d\not\equiv 0\pmod{n}$. Thus this first block must always have weight equal to $d$ modulo $n$ if $n-s=d$, and so the vertex $T^{s-}$ that represents the $s$-prefix of $T$ is not connected to the vertex $0^{s}$. Hence no Euler tour can exist and so no $s$-ocycle exists. ∎ While Theorem 3.11 completes the question of when $s$-ocycles for juggling sequences of period $n$ and at most $b$ balls exist, several variations remain open. For example, one might consider juggling sequences with: * • exactly $b$ balls, * • at least $b$ balls, * • fixed minimum period, or * • period $n$ with no restriction on number of balls. ## References * [1] J. Buhler, D. Eisenbud, R. Graham, and C. Wright, _Juggling Drops and Descents_ , Amer. Math. Monthly 101:6, 507-519. * [2] F. Chung, P. Diaconis, and R. Graham, _Universal Cycles for Combinatorial Structures_ , Discrete Mathematics 110 (1992), 43-59. * [3] F. Chung and R. Graham, _Universal Juggling Cycles_ , Integers: Electronic Journal of Combinatorial Number Theory 7:2 (2007), #A08. * [4] S. Forrest, _Genetic Algorithms: Principles of Natural Selection Applied to Computation_ , Science 261:5123 (1993), 872-878. * [5] A. Godbole, D. Knisley, and R. Norwood, _On $\alpha$-Overlap Graphs_, Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congr. Numer. 204 (2010), 161-171. * [6] F. Gray, _Pulse code communication_ , March 17, 1953 (filed Nov. 1947), U.S. Patent 2,632,058. * [7] V. Horan and G. Hurlbert, _$s$ -Overlap Cycles for Permutations,_ to appear in Bull. Inst. Combin. Appl. * [8] V. Horan and G. Hurlbert, _Universal Cycles for Weak Orders,_ To appear in SIAM Journal of Discrete Math. * [9] G. Hurlbert and G. Isaak, _Equivalence class universal cycles for permutations_ , Discrete Math. 149 (1996), pp. 123–129. * [10] J. R. Johnson, _Universal cycles for permutations_ , Discrete Math. 309 (2009), pp. 5264–5270. * [11] E. A. Ragland, V. Nuys, and H. B. Scultheis, Jr., _Direction-Sensitive Binary Code Position Control System_ , Feb. 11, 1958 (filed Oct. 1953), U.S. Patent 2,823,345. * [12] Z. Wang and J. Bruck, _Partial Rank Modulation for Flash Memories_ , IEEE International Symposium on Information Theory Proceedings (2010), 864-868. * [13] D. B. West, _Introduction to Graph Theory_. Prentice Hall, Second Edition, 2001\.	arxiv-papers	2013-09-18T18:39:30	2024-09-04T02:49:51.130395	{ "license": "Public Domain", "authors": "Victoria Horan", "submitter": "Victoria Horan", "url": "https://arxiv.org/abs/1309.4741" }
1309.4953	# A Near Optimal Approximation Algorithm for Vertex-Cover Problem Deepak [email protected] Department of Computer Science & Engineering National Institute of Technology Silchar Silchar, Assam, India ###### Abstract Recently, there has been increasing interest and progress in improvising the approximation algorithm for well-known NP-Complete problems, particularly the approximation algorithm for the Vertex-Cover problem. Here we have proposed a polynomial time efficient algorithm for vertex-cover problem for more approximate to the optimal solution, which lead to the worst time complexity $\Theta(V^{2})$ and space complexity $\Theta(V+E)$. We show that our proposed method is more approximate with example and theorem proof. Our algorithm also induces improvement on previous algorithms for the independent set problem on graphs of small and high degree. Keywords:Approximation algorithm Vertex-Cover Problem Complexity Adjacency list. ## 1 Introduction A graph G represents as G = (V, E): V is number of vertices in graph and E is number of edges in graph and impliment as an adjacency lists or as an adjacency matrix for both directed and undirected graphs. There are two types of graph i.e. (I) Sparse graphs-those for which $\|E\|$ is much less than $\|V\|^{2}$ $(E<<V^{2})$. (II) Dense graphs-those for which $\|E\|$ is close to $\|V\|^{2}(E\simeq V^{2})$. Here we presented the graph as the adjacency-list for the evaluation of our algorithm. The vertex-cover problem is to find a minimum number of vertex to cover a given undirected graph. We call such a vertex cover an optimal vertex cover. This problem is the optimization version of an NP-complete decision problem. Proposed algorithm is polynomial time algorithm in order to find the set of vertex to cover the graph. Which shows the better performance than the traditional algorithm for vertex cover [1]. ## 2 The vertex-cover problem As it is NP-Hard problem so it is hard to find an optimal solution of a graph G, but not difficult to find a near optimal solution. Our propose method gives very near optimal solution for Vertex-cover problem. The following approximation algorithm takes an undirected graph G as input [1] and returns a set of vertex to cover the graph and whose size is less than the previous method. All graphs mentioned here are simple undirected graph. We follow [2] for definitions. Our proposed method (See algorithm and Fig.2) is on undirected graph. Here we used the adjacency list to represent graph G. We introduce a new field weight in the list to store the degree of each individual vertex. i.e. struct list { char vertex; int weight; struct node next; struct node ref; }; Algorithm 1 Approximate Vertex-Cover Algorithm 0: In the List we introduce another field weight The value of weight is number of node in reference (ref) 1: $C^{+}\leftarrow\emptyset$ 2: L = List 3: L[w] = Reference weight 4: (h, v) = highest weight of the list and respective vertex 5: if $h\neq 0$ then 6: $C^{+}\leftarrow C^{+}\cup{v}$ 7: v[w] $\leftarrow 0$ 8: for all vertex of List L[ref] $\in\\{v\\}$ do 9: L[w] $\leftarrow$ L[w]-1 10: end for 11: go to 4 12: else 13: return $C^{+}$ 14: end if It’s space complexity is $\Theta(V+E)$ [1]. For step 4 search in the list is O(V). In the for loop i.e. step 8 to 10 for each individual vertex need to search its reference vertices. So worst time complexity is $O(V(V-1))$= $O(V^{2})$. So the worst time complexity of the graph is $\Theta(V^{2})$. ###### Theorem 1. _(Thomas H. Cormen et.al.[1])_ APPROX-VERTEX-COVER is a polynomial-time 2-approximation algorithm. ###### Theorem 2. Proposed Approximate Vertex-Cover is a polynomial-time $(2-\varepsilon)$-approximation algorithm. ###### Proof. In Theorem 1 $C$ is the set of vertex for APPROX-VERTEX-COVER and $C^{}$ in the optimal vertex cover $i.e.\|C\|\leq 2\|C^{}\|$. In our approach we pick one vertex and remove the edges connected to that vertex. So most of the times we don’t consider both end point of one edges, which followed in Theorem 1. For our proposed method we consider the resultant set of vertex is $C^{+}$, then $\|C^{+}\|=\|C\|-\varepsilon$. $\Rightarrow\|C^{+}\|=(2-\varepsilon)\|C^{}\|$, $0\leq\varepsilon\leq 1$. ∎∎ In some cases proposed method$(C^{+})$ approaches to optimal solution when $\varepsilon$ value is 1. Our method is shown in Fig. 1, algorithm (Approximate Vertex-Cover), and proved in Theorem 2. The comparison of the optimal vertex-cover, previous vertex-cover and proposed vertex-cover shown in Fig. 2. Figure 1: The operation of Approximate Vertex-Cover. (a) The input graph G, which has 7 vertices and 8 edges. (b) The vertex D has the highest weight, is the first vertex chosen by Approximate Vertex-Cover. The weight of the vertex initiate to zero and the vertices directly connected from vertex D weight value decrees by 1. The vertex D is shaded and the edges is dashed incident from vertex D. The vertex D add to the set $C^{+}$. (c) There are three vertex with highest weight (B, C, E); Arbitrary vertex C is chosen and add to the set $C^{+}$. (d) Vertex A is chosen; added to $C^{+}$. (e) Vertex F is chosen; added to $C^{+}$. (f) The resultant vertex cover with our proposed method. Approximate Vertex-Cover, contains the four vertices a, c, d, f. Figure 2: Comparison of optimal, Traditional and proposed vertex cover result. (a) The inputted original graph G. (b) The optimal result of input graph. (c) The result with traditional method. (d) The proposed algorithm’s result for vertex-cover algorithm. ## 3 Conclusion Here in our proposed technique we produce the set of vertex for vertex-cover problem. Which is more near optimal solution and better than the previous technique. ## References * [1] Thomas H. Cormen et.al. Introduction to Algorithms, Second Edition. The MIT Press, 2001. ISBN 0-262-03293-7 * [2] Liu, Chung Laung. Elements of discrete mathematics. Vol. 101. New York: McGraw-Hill, 1985.	arxiv-papers	2013-09-19T12:27:27	2024-09-04T02:49:51.151682	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Deepak Puthal", "submitter": "Deepak Puthal", "url": "https://arxiv.org/abs/1309.4953" }
1309.4986	# Four generated, squarefree, monomial ideals Adrian Popescu Adrian Popescu, Department of Mathematics, University of Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany [email protected] and Dorin Popescu Dorin Popescu, Simion Stoilow Institute of Mathematics of Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest 014700, Romania [email protected] ###### Abstract. $I\supsetneq J$ be two squarefree monomial ideals of a polynomial algebra over a field generated in degree $\geq d$, resp. $\geq d+1$ . Suppose that $I$ is either generated by three monomials of degrees $d$ and a set of monomials of degrees $\geq d+1$, or by four special monomials of degrees $d$. If the Stanley depth of $I/J$ is $\leq d+1$ then the usual depth of $I/J$ is $\leq d+1$ too. Monomial Ideals, Depth, Stanley depth. 2010 Mathematics Subject Classification: Primary 13C15, Secondary 13F20, 13F55, 13P10. The support of the first author from the Department of Mathematics of the University of Kaiserslautern and the support of the second author from grant PN-II-RU-TE-2012-3-0161 of Romanian Ministry of Education, Research and Innovation are gratefully acknowledged. ## Introduction Let $K$ be a field and $S=K[x_{1},\ldots,x_{n}]$ be the polynomial $K$-algebra in $n$ variables. Let $I\supsetneq J$ be two squarefree monomial ideals of $S$ and suppose that $I$ is generated by squarefree monomials of degrees $\geq d$ for some positive integer $d$. After a multigraded isomorphism we may assume either that $J=0$, or $J$ is generated in degrees $\geq d+1$. By [5, Proposition 3.1] (see [12, Lemma 1.1]) we have $\operatorname{depth}_{S}I/J\geq d$. Depth of $I/J$ is a homological invariant and depends on the characteristic of the field $K$. The purpose of our paper is to study upper bound conditions for $\operatorname{depth}_{S}I/J$. Let $B$ (resp. $C$) be the set of the squarefree monomials of degrees $d+1$ (resp. $d+2$) of $I\setminus J$. Suppose that $I$ is generated by some squarefree monomials $f_{1},\ldots,f_{r}$ of degrees $d$ for some $d\in{\mathbb{N}}$ and a set of squarefree monomials $E$ of degree $\geq d+1$. If $d=1$ and each monomial of $B\setminus E$ is the least common multiple of two $f_{i}$ then it is easy to show that $\operatorname{depth}_{S}I/J=1$ (see Lemma 3). Trying to extend this result for $d>1$ we find an obstruction given by Example 2. Our extension given by Lemma 4 is just a special form, but a natural condition seems to be given in terms of the Stanley depth. More precisely, let $P_{I\setminus J}$ be the poset of all squarefree monomials of $I\setminus J$ with the order given by the divisibility. Let $P$ be a partition of $P_{I\setminus J}$ in intervals $[u,v]=\\{w\in P_{I\setminus J}:u\|w,w\|v\\}$, let us say $P_{I\setminus J}=\cup_{i}[u_{i},v_{i}]$, the union being disjoint. Define $\operatorname{sdepth}P=\operatorname{min}_{i}\operatorname{deg}v_{i}$ and the Stanley depth of $I/J$ given by $\operatorname{sdepth}_{S}I/J=\operatorname{max}_{P}\operatorname{sdepth}P$, where $P$ runs in the set of all partitions of $P_{I\setminus J}$ (see [5], [20]). Stanley’s Conjecture says that $\operatorname{sdepth}_{S}I/J\geq\operatorname{depth}_{S}I/J$. The Stanley depth of $I/J$ is a combinatorial invariant and does not depend on the characteristic of the field $K$. Stanley’s Conjecture holds when $J=0$ and $I$ is an intersection of four monomial prime ideals by [8], [10], or $I$ is such that the sum of every three different of its minimal prime ideals is a constant ideal by [11] (see also [14]), or $I$ is an intersection of three monomial primary ideals by [22], or a monomial almost complete intersection by [4]. ###### Theorem 1. (D. Popescu [12, Theorem 4.3]) If $\operatorname{sdepth}_{S}I/J=d$ then $\operatorname{depth}_{S}I/J=d$, that is Stanley’s Conjecture holds in this case. Next step in the study of Stanley’s Conjecture is to show the following weaker conjecture. ###### Conjecture 1. Suppose that $I\subset S$ is minimally generated by some squarefree monomials $f_{1},\ldots,f_{r}$ of degrees $d$, and a set $E$ of squarefree monomials of degrees $\geq d+1$. If $\operatorname{sdepth}_{S}I/J=d+1$ then $\operatorname{depth}_{S}I/J\leq d+1$. Set $s=\|B\|$, $q=\|C\|$. In the study of the above conjecture very useful seem to be the following two particular results of [13, Theorem 1.3] and [19, Theorem 2.4]. ###### Theorem 2. (D. Popescu) If $s>q+r$ then $\operatorname{depth}_{S}I/J\leq d+1$. ###### Theorem 3. (Y. Shen) If $s<2r$ then $\operatorname{depth}_{S}I/J\leq d+1$. These results were hinted by Stanley’s Conjecture since it is obvious that $s>q+r$, or $s<2r$ imply $\operatorname{sdepth}_{S}I/J\leq d+1$. The proof of Theorem 2 uses Koszul homology (see [1, Section 1.6]). Shen’s proof of the above theorem as well of Theorem 2 is easy and uses the Hilbert depth considered by Bruns-Krattenhaler-Uliczka [2] (see also [21], [6]). An equivalent definition for the Stanley depth is: $\operatorname{sdepth}(M)=\operatorname{max}\\{\operatorname{sdepth}\mathcal{D}\ \|\ \mathcal{D}\textnormal{ is a Stanley decomposition of }M\\},$ where a Stanley decomposition of a $\mathbb{Z}-$graded (resp. $\mathbb{Z}^{n}-$graded) $S-module$ $M$ is $\mathcal{D}=(S_{i},u_{i})_{i\in I}$, where $u_{i}$ are homogenous elements of $M$ and $S_{i}$ are graded (resp. $\mathbb{Z}^{n}-$graded) $K-$algebra retracts of $S$ and $S_{i}\cap\operatorname{Ann}(u_{i})=0$ such that $M=\oplus_{i}S_{i}u_{i}$; and $\operatorname{sdepth}\mathcal{D}$ is the $\operatorname{depth}$ of the $S-$module $\oplus_{i}S_{i}u_{i}$. A more general concept is the one of Hilbert depth of a $\mathbb{Z}-$graded module $M$, denoted by $\operatorname{hdepth}_{1}(M)$. Instead of considering equality, we only assume that $M\cong\oplus S_{i}(-s_{i})$, where $s_{i}\in\mathbb{Z}$. One can also construct $\operatorname{hdepth}_{n}$ analogously if $M$ is a multigraded (that is $\mathbb{Z}^{n})$ module. In [9] is presented (and implemented) an algorithm that computes $\operatorname{hdepth}_{1}(M)$ based on a Theorem of Uliczka [21]; and in [7] was presented an algorithm that computes $\operatorname{hdepth}_{n}(M)$. Meanwhile, another algorithm that computes $\operatorname{hdepth}_{1}$ and more was given in [3]. [9, Proposition 1.9] gives a partial answer to a question of Herzog asking whether $\operatorname{sdepth}m=\operatorname{sdepth}(S\oplus m)$, where $m$ is the graded maximal ideal of $S$. More precisely, for $n\in\\{1,2,3,4,5,7,9,11\\}$ one obtains $\operatorname{hdepth}_{1}m=\operatorname{hdepth}_{1}(S\oplus m)$, which gives $\operatorname{sdepth}m=\operatorname{sdepth}(S\oplus m)$ (again Hilbert depth helps the study of Stanley depth). For $n=6$ we have $\operatorname{hdepth}_{1}m\neq\operatorname{hdepth}_{1}(S\oplus m)$, which means that in general Herzog’s question could have a negative answer. Later Ichim and Zarojanu checked the case $n=6$ and found indeed a counterexample to Herzog’s question, which will be included in the new version of [7]. An important step in proving Conjecture 1 is the following theorem. ###### Theorem 4. (D. Popescu-A. Zarojanu [16], [17, Theorem 1.5]) Conjecture 1 holds in each of the following two cases: 1. (1) $r=1$, 2. (2) $1<r\leq 3$, $E=\emptyset$. Next theorem is the main result of this paper. ###### Theorem 5. Conjecture 1 holds in each of the following two cases: 1. (1) $r\leq 3$, 2. (2) $r=4$, $E=\emptyset$ and there exists $c\in C$ such that $\operatorname{supp}c\not\subset\cup_{i\in[4]}\operatorname{supp}f_{i}$. This follows from our Theorems 6, 8. The proof of 6 extends the proof of [17, Theorem 2.3]. We owe thanks to A. Zarojanu, who noticed some small mistakes in a previous version of this paper and gave us the bad example 5. ## 1\. Depth and Stanley depth Let $I\supsetneq J$ be two squarefree monomial ideals of $S$. We assume that $I$ is generated by squarefree monomials $f_{1},\ldots,f_{r}$ of degrees $d$ for some $d\in{\mathbb{N}}$ and a set of squarefree monomials $E$ of degree $\geq d+1$. We may suppose that either $J=0$, or is generated by some squarefree monomials of degrees $\geq d+1$. $B$ (resp. $C$) denotes the set of the squarefree monomials of degrees $d+1$ (resp. $d+2$) of $I\setminus J$. ###### Lemma 1. Let $J\subset I$ be square free monomial ideals and $j\in[n]$ be such that $(J:x_{j})\not=(I:x_{j})$. Then $\operatorname{depth}_{S}(I:x_{j})/(J:x_{j})\geq\operatorname{depth}_{S}I/J$. ###### Proof. We have $\operatorname{pd}_{S}I/J\geq\operatorname{pd}_{S_{x_{j}}}(I/J)\otimes S_{x_{j}}=\operatorname{pd}_{S_{x_{j}}}((I:x_{j})/(J:x_{j}))\otimes S_{x_{j}}=\operatorname{pd}_{S}((I:x_{j})/(J:x_{j}))$ the last equality holds since $x_{j}$ does not appear among the generators of $(I:x_{j})$ and $(J:x_{j})$. Now it is enough to apply the Auslander-Buchsbaum Theorem. ###### Lemma 2. Let $t\in[n]$. Suppose that $I\not=J+I\cap(x_{t})$ and $\operatorname{depth}_{S}I/(J+I\cap(x_{t}))=d$. If $\operatorname{depth}_{S}I/J\geq d+1$ then $\operatorname{depth}_{S}I/J=d+1$. ###### Proof. In the following exact sequence $0\rightarrow(I:x_{t})/(J:x_{t})\xrightarrow{x_{t}}I/J\rightarrow I/(J+I\cap(x_{t}))\rightarrow 0$ the first term has depth $d+1$ by the Depth Lemma. Now it is enough to apply the above lemma. Let $w_{ij}$ be the least common multiple of $f_{i}$ and $f_{j}$ and set $W$ to be the set of all $w_{ij}\in B$. ###### Lemma 3. If $d=1$ and $B\subset E\cup W$ then $\operatorname{depth}_{S}I/J=1$. ###### Proof. First suppose that $E=\emptyset$, let us say $I=(x_{1},\ldots,x_{r})$. Set $S^{\prime}=K[x_{1},\ldots,x_{r}]$, $I^{\prime}=I\cap S^{\prime}$, $J^{\prime}=J\cap S^{\prime}$. By hypothesis $B\subset S^{\prime}$ and it follows that $(x_{r+1},\ldots,x_{n})I\subset J$ and so $\operatorname{depth}_{S}I=\operatorname{depth}_{S^{\prime}}I^{\prime}=1$. But $\operatorname{depth}_{S}J\geq 2$, if $J\not=0$, and so $\operatorname{depth}_{S}I/J=1$ by the Depth Lemma. Now, suppose that $E\not=\emptyset$. In the following exact sequence $0\rightarrow(x_{1},\ldots,x_{r})/J\cap(x_{1},\ldots,x_{r})\rightarrow I/J\rightarrow I/(J,x_{1},\ldots,x_{r})\rightarrow 0$ the first term has depth 1 as above and the last term has depth $\geq d+1$ since it is generated by squarefree monomials of degrees $\geq 2$ from $E$. Again the Depth Lemma gives $\operatorname{depth}_{S}I/J=1$. ###### Lemma 4. Suppose that $I\subset S$ is generated by some squarefree monomials $f_{1},...f_{r}$ of degree $d$. Assume that for all $b\in B$ all divisors of $b$ of degree $d$ are among $\\{f_{1},\ldots,f_{r}\\}$. Then $\operatorname{depth}_{S}I/J=d$. ###### Proof. Apply induction on $d\geq 1$. If $d=1$ then apply the above lemma. Assume $d>1$. We may suppose that $n\in\operatorname{supp}f_{1}$. $(I:x_{n})$ is an extension of a squarefree monomial ideal $I^{\prime}$ of $S^{\prime}=K[x_{1},\ldots,x_{n-1}]$ which is generated in degree $\geq d-1$. Similarly $(J:x_{n})$ is generated by a squarefree monomial ideal $J^{\prime}$ of $S^{\prime}$. Note that the generators of $I^{\prime}$ of degree $d-1$ have the form $f^{\prime}_{i}=f_{i}/x_{n}$ for $f_{i}\in(x_{n})$, and the squarefree monomials $B^{\prime}$ of degrees $d$ from $I^{\prime}\setminus J^{\prime}$ have the form $b^{\prime}=b/x_{n}$ for some $b\in(B\cap(x_{n}))$. Certainly we must consider also the case when $f_{j}\not\in(x_{n})$. If $x_{n}f_{j}\in J$ then $f_{j}\in(J:x_{n})$ is not in $B^{\prime}$. Otherwise, $f_{j}=(x_{n}f_{j})/x_{n}\in B^{\prime}$. Note that all divisors of degree $d-1$ of each $b^{\prime}\in B^{\prime}$ are among $f^{\prime}_{i}$. By induction hypothesis we have $\operatorname{depth}_{S^{\prime}}I^{\prime}/J^{\prime}=d-1$ and so $\operatorname{depth}_{S}(I:x_{n})/(J:x_{n})=d$. Now it is enough to apply Lemma 1. An obstruction to improve Lemma 3 and the above lemma is given by the following example. ###### Example 1. Let $n=5$, $d=2$, $r=5$, $I=(x_{1}x_{2},x_{1}x_{3},x_{2}x_{3},x_{1}x_{4},x_{3}x_{5})$, $J=(x_{1}x_{2}x_{5},x_{1}x_{4}x_{5},x_{2}x_{3}x_{4},x_{3}x_{4}x_{5})$, $B=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{4},x_{1}x_{3}x_{4},x_{1}x_{3}x_{5},x_{2}x_{3}x_{5}\\}$. We have $\operatorname{depth}_{S}I/J=3$ because $\operatorname{depth}_{S}S/J=3$, $\operatorname{depth}_{S}S/I=2$ and with the help of Depth Lemma. Note that each $b\in B$ is the least common multiple of two generators of $I$, but for example $b=x_{1}x_{2}x_{4}$ has $x_{2}x_{4}\not\in I$ as a divisor of degree $2$. Let $C_{2}=C\cap W$ and $C_{3}$ be the set of all $c\in C$ having all divisors from $B\setminus E$ in $W$. In particular each monomial of $C_{3}$ is the least common multiple of three of $f_{i}$. The converse is not true as shows the following example. ###### Example 2. Let $n=4$, $d=2$, $r=3$, $f_{1}=x_{1}x_{2}$, $f_{2}=x_{2}x_{3}$, $f_{3}=x_{3}x_{4}$, $I=(f_{1},f_{2},f_{3})$ and $J=0$. Then $c=x_{1}x_{2}x_{3}x_{4}$ is the least common multiple of $f_{1},f_{2},f_{3}$ but has a divisor $b=x_{1}x_{2}x_{4}\in B$ which is not the least common multiple of two $f_{i}$. Next theorem is our key result, its proof is based on [17, Theorem 2.1] and will be given in the last section. The main reason that this proof works for $r\leq 3$ but not for $r=4$ is that in the first case $\|C_{3}\|\leq 1$ but in the second one we may have $\|C_{3}\|=4$, which makes the things harder. However, for $r\geq 5$ will appear a new problem since we may have $B\subset W$ and $s\geq 2r$ (for example when $r=5$, $d=2$ we may have $s=10=2r$). We remind that by Theorem 3 we had to check Stanley’s Conjecture only when $s\geq 2r$. ###### Theorem 6. Conjecture 1 holds for $r\leq 3$, the case $r=1$ being given in Theorem 4. ###### Example 3. Let $n=5$, $f_{1}=x_{1}x_{2}$, $f_{2}=x_{1}x_{3}$, $f_{3}=x_{1}x_{4}$, $a=x_{2}x_{3}x_{5}$, $E=\\{a\\}$, $I=(f_{1},f_{2},f_{3},a)$, $J=(x_{4}a)$. We have $w_{12}=f_{1}x_{3}$, $w_{13}=f_{1}x_{4}$, $w_{23}=f_{2}x_{4}$. Set $c=w_{12}x_{4}$, $c_{1}=w_{12}x_{5}$, $c_{2}=w_{23}x_{5}$, $c_{3}=w_{13}x_{5}$. Then $C=\\{c,c_{1},c_{2},c_{3}\\}$ and $B\setminus E=B\cap(\cup_{i}[f_{i},c_{i}])$. Thus $s=7$, $q=4$, $r=3$. It is easy to see that $\operatorname{sdepth}_{S}I/J=3$. Indeed, note that $c_{1}$ is the only $c^{\prime}\in C$ which is multiple of $a$. Suppose that there exists a partition $P$ on $P_{I/J}$ with sdepth $4$. Then we have necessarily in $P$ the interval $[a,c_{1}]$. If $P$ contains the interval $[f_{1},c]$ then it must contain also the intervals $[f_{2},c_{2}]$ and so $[f_{3},c_{3}]$, but then $w_{13}\in[f_{1},c]\cap[f_{3},c_{3}]$, that is the union is not disjoint. If $P$ contains the interval $[f_{1},c_{3}]$ then $P$ contains either $[f_{3},c]$, $[f_{2},c_{2}]$, or $[f_{2},c]$, $[f_{3},c_{2}]$, in both cases the intersection of these two intervals contains $w_{23}$, which is false. By Theorem 6 we get $\operatorname{depth}_{S}I/J\leq 3$, this inequality being in fact an equality. ## 2\. A special case of $r=4$ ###### Theorem 7. Suppose that $I\subset S$ is minimally generated by some squarefree monomials $\\{f_{1},\ldots,f_{r}\\}$ of degrees $d$ such that there exists $c\in C$ with $\operatorname{supp}c\not\subset\cup_{i\in[r]}\operatorname{supp}f_{i}$. If Conjecture 1 holds for $r^{\prime}<r$ and $\operatorname{sdepth}_{S}I/J=d+1$, then $\operatorname{depth}_{S}I/J\leq d+1$. ###### Proof. By [17, Lemma 1.1] we may assume that $C\subset(W)$. By hypothesis, choose $t\in\operatorname{supp}c$ such that $t\not\in\cup_{i\in[r]}\operatorname{supp}f_{i}$. We may suppose that $B\cap(x_{t})=\\{x_{t}f_{1},\ldots x_{t}f_{e}\\}$ for some $1\leq e\leq r$. Set $I_{t}=I\cap(x_{t})$, $J_{t}=J\cap(x_{t})$ and $U_{t}=I_{t}/J_{t}$. Then $B_{t}$ generates $I_{t}$. First assume that $\operatorname{sdepth}_{S}U_{t}\leq d+1$. It follows that $\operatorname{depth}_{S}U_{t}\leq d+1$ by [12, Theorem 4.3]. But $U_{t}\cong(I:x_{t})/(J:x_{t})$ and so $\operatorname{depth}_{S}U_{t}\geq\operatorname{depth}_{S}I/J$ by Lemma 1, which is enough. Now assume that $U_{t}$ has sdepth $\geq d+2$. Let $P_{U_{t}}$ be a partition on $U_{t}$ with sdepth $d+2$ and let $[b_{i},c_{i}]$ be the disjoint intervals starting with $b_{i}=x_{t}f_{i}$, $i\in[e]$. We may suppose that $c_{i}\in C$ for $i\in[e]$. We have $c_{i}=x_{t}w_{ik_{i}}$ for some $1\leq k_{i}\leq r$, $k_{i}\not=i$ because $C\subset(W)$. Note that $x_{t}f_{k_{i}}\in B$ and so $k_{i}\leq e$. We consider the intervals $[f_{i},c_{i}]$. These intervals contain $x_{t}f_{i}$ and $w_{ik_{i}}$. If $w_{ik_{i}}=w_{jk_{j}}$ for $i\not=j$ then we get $c_{i}=c_{j}$ which is false. Thus these intervals are disjoint. Let $I_{e}$ be the ideal generated by $f_{j}$ for $e<j\leq r$ and $B\setminus(\cup_{i=1}^{e}[f_{i},c_{i}])$. Set $J_{e}=I_{e}\cap J$ and $U_{e}=I_{e}/J_{e}$. Note that $c_{i}\not\in I_{e}$ for any $i\in[e]$. In the following exact sequence $0\rightarrow I_{e}/J_{e}\rightarrow I/J\rightarrow I/J+I_{e}\rightarrow 0$ the last term has a partition of sdepth $d+2$ given by the intervals $[f_{i},c_{i}]$ for $1\leq i\leq e$. It follows that $I_{e}\not=J_{e}$ because $\operatorname{sdepth}_{S}I/J=d+1$. Then $\operatorname{sdepth}_{S}I_{e}/J_{e}\leq d+1$ using [18, Lemma 2.2] and so $\operatorname{depth}_{S}I_{e}/J_{e}\leq d+1$ by Conjecture 1 applied for $r^{\prime}<r$. But the last term of the above sequence has depth $>d$ because $x_{t}$ does not annihilate $f_{i}$ for $i\in[e]$. With the Depth Lemma we get $\operatorname{depth}_{S}I/J\leq d+1$. ###### Example 4. Let $n=5$, $r=4$, $f_{1}=x_{2}x_{3}$, $f_{2}=x_{1}x_{2}$, $f_{3}=x_{3}x_{4}$, $f_{4}=x_{3}x_{5}$ and $J=(x_{1}x_{2}x_{4}x_{5})$. We have $w_{12}=x_{1}x_{2}x_{3}$, $w_{13}=x_{2}x_{3}x_{4}$, $w_{14}=x_{2}x_{3}x_{5}$, $w_{34}=x_{3}x_{4}x_{5}$, $w_{23}=x_{1}x_{2}x_{3}x_{4}$, $w_{24}=x_{1}x_{2}x_{3}x_{5}$, $C_{2}=\\{w_{23},w_{24}\\}$, $C=C_{2}\cup\\{x_{1}w_{34},x_{2}x_{3}x_{4}x_{5}\\}$, ${\tilde{I}}_{1}=\\{x_{1}f_{3},x_{1}f_{4},f_{2}\\}\supset J$, ${\tilde{I}}_{4}=\\{f_{3},x_{4}f_{2}\\}\supset J$ and $B\cap(x_{1})=\\{x_{1}f_{3},x_{1}f_{4},x_{4}f_{2},x_{5}f_{2},w_{12}\\}$, $B\cap(x_{4})=\\{w_{13},w_{14},w_{34},x_{4}f_{2},x_{1}f_{3}\\}$. Note that $\operatorname{sdepth}_{S}U_{1}\leq d+1=3$, $\operatorname{sdepth}_{S}U_{4}\leq 3$ because $\|B\cap(x_{1})\|=\|B\cap(x_{4})\|=5>\|C\cap(x_{1})\|+1=\|C\cap(x_{4})\|+1=4$. Thus $\operatorname{depth}_{S}U_{1}=\operatorname{depth}_{S}U_{4}\leq 3$ and so we get $\operatorname{depth}_{S}I/J\leq 3$ using two different $t$. ###### Theorem 8. Suppose that $I\subset S$ is minimally generated by four squarefree monomials $\\{f_{1},\ldots,f_{4}\\}$ of degrees $d$ such that there exists $c\in C$ such that $\operatorname{supp}c\not\subset\cup_{i\in[4]}\operatorname{supp}f_{i}$. If $\operatorname{sdepth}_{S}I/J=d+1$ then $\operatorname{depth}_{S}I/J\leq d+1$. ###### Proof. Apply Theorem 7, since Conjecture 1 holds for $r<4$ by Theorem 6. ## 3\. Proof of Theorem 6 Suppose that $E\not=\emptyset$ and $s\leq q+r$. For $b=f_{1}x_{i}\in B$ set $I_{b}=(f_{2},\ldots,f_{r},B\setminus\\{b\\})$, $J_{b}=J\cap I_{b}$. If $\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$ then let $P_{b}$ be a partition on $I_{b}/J_{b}$ with sdepth $d+2$. We may choose $P_{b}$ such that each interval starting with a squarefree monomial of degree $d$, $d+1$ ends with a monomial of $C$. In $P_{b}$ we have some intervals $[f_{k},f_{k}x_{i_{k}}x_{j_{k}}]$, $1<k\leq r$ and for all $b^{\prime}\in B\setminus\\{b,f_{2}x_{i_{2}},f_{2}x_{j_{2}},\ldots,f_{r}x_{i_{r}},f_{r}x_{j_{r}}\\}]$ an interval $[b^{\prime},c_{b^{\prime}}]$. We define $h:[[\\{f_{2},\ldots,f_{r}\\}\cup B]\setminus\\{b,f_{2}x_{i_{2}},f_{2}x_{j_{2}},\ldots,f_{r}x_{i_{r}},f_{r}x_{j_{r}}\\}]\rightarrow C$ by $f_{k}\rightarrow f_{k}x_{i_{k}}x_{j_{k}}$ and $b^{\prime}\rightarrow c_{b^{\prime}}$. Then $h$ is an injection and $\|\operatorname{Im}\ h\|=s-r\leq q$ (if $s=r+q$ then $h$ is a bijection). ###### Lemma 5. Suppose that the following conditions hold: 1. (1) $r=2$, $4\leq s\leq q+2$, 2. (2) $C\subset((f_{1})\cap(f_{2}))\cup((E)\cap(f_{1},f_{2}))\cup(\cup_{a,a^{\prime}\in E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$, 3. (3) $\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$ for a $b\in(B\cap(f_{1}))\setminus(f_{2})$. Then either $\operatorname{sdepth}_{S}I/J\geq d+2$, or there exists a nonzero ideal $I^{\prime}\subsetneq I$ generated by a subset of $\\{f_{1},f_{2}\\}\cup B$ such that $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for $J^{\prime}=J\cap I^{\prime}$ and $\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$. ###### Proof. Since $\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$ we consider $h$ as above for a partition $P_{b}$ with sdepth $d+2$ of $I_{b}/J_{b}$. We have an interval $[f_{2},c^{\prime}_{2}]$ in $P_{b}$. Suppose that $B\cap[f_{2},c^{\prime}_{2}]=\\{u,u^{\prime}\\}$. A sequence $a_{1},\ldots,a_{k}$ is called a path from $a_{1}$ to $a_{k}$ if the following statements hold: (i) $a_{l}\in B\setminus\\{b,u,u^{\prime}\\}$, $l\in[k]$, (ii) $a_{l}\not=a_{j}$ for $1\leq l<j\leq k$, (iii) $a_{l+1}\|h(a_{l})$ and $h(a_{l})\not\in(b)$ for all $1\leq l<k$. This path is weak if $h(a_{j})\in(u,u^{\prime})$ for some $j\in[k]$. It is bad if $h(a_{k})\in(b)$ and it is maximal if either $h(a_{k})\in(b)$, or all divisors from $B$ of $h(a_{k})$ are in $\\{u,u^{\prime},a_{1},\ldots,a_{k}\\}$. If $a=a_{1}$ we say that the above path starts with $a$. By hypothesis $s\geq 4$ and so there exists $a_{1}\in B\setminus\\{b,u,u^{\prime}\\}$. Set $c_{1}=h(a_{1})$. If $c_{1}\in(b)$ then the path $\\{a_{1}\\}$ is maximal and bad. We construct below, as an example, a path with $k>1$. By recurrence choose if possible $a_{p+1}$ to be a divisor from $B$ of $c_{p}$, which is not in $\\{b,u,u^{\prime},a_{1},\ldots,a_{p}\\}$ and set $c_{p}=h(a_{p})$, $p\geq 1$. This construction ends at step $p=e$ if all divisors from $B$ of $c_{e}$ are in $\\{b,u,u^{\prime},a_{1},\ldots,a_{e}\\}$. If $c_{i}\not\in(b)$ for $1\leq i<e$ then $\\{a_{1},\ldots,a_{e}\\}$ is a maximal path. If one $c_{p}\in(u,u^{\prime})$ then the constructed path is weak. If $c_{e}\in(b)$ then this path is bad. We have three cases: 1) there exist no weak path and no bad path starting with $a_{1}$, 2) there exists a weak path starting with $a_{1}$ but no bad path starts with $a_{1}$, 3) there exists a bad path starting with $a_{1}$. In the first case, set $T_{1}=\\{b^{\prime}\in B:\mbox{there\ \ exists\ \ a\ \ path}\ \ a_{1},\ldots,a_{k}\ \ \mbox{with}\ \ a_{k}=b^{\prime}\\}$, $G_{1}=B\setminus T_{1}$, and $I^{\prime}_{1}=(f_{1},G_{1})$, $I^{\prime}_{2}=(f_{2},G_{1})$, $I^{\prime}_{12}=(f_{1},f_{2},G_{1})$, $I^{\prime\prime}=(G_{1})$, $J^{\prime}_{1}=I^{\prime}_{1}\cap J$, $J^{\prime}_{2}=I^{\prime}_{2}\cap J$, $J^{\prime}_{12}=I^{\prime}_{12}\cap J$, $J^{\prime\prime}=I^{\prime\prime}\cap J$. Note that $I^{\prime\prime}\not=0$ because $b\in I^{\prime\prime}$ and all divisors from $B$ of a monomial $c\in U_{1}=h(T_{1})$ belong to $T_{1}$. Consider the following exact sequence $0\rightarrow I^{\prime}_{12}/J^{\prime}_{12}\rightarrow I/J\rightarrow I/(J,I^{\prime}_{12})\rightarrow 0.$ If $U_{1}\cap(f_{1},f_{2})=\emptyset$ then the last term has depth $\geq d+1$ and sdepth $\geq d+2$ using the restriction of $P_{b}$ to $(T_{1})\setminus(J,I^{\prime}_{12})$ since $h(b^{\prime})\notin I^{\prime}_{12}$, for all $b^{\prime}\in T_{1}$. When the first term has sdepth $\geq d+2$ then by [18, Lemma 2.2] the middle term has sdepth $\geq d+2$. Otherwise, the first term has sdepth $\leq d+1$ and we may take $I^{\prime}=I^{\prime}_{12}$. If $U_{1}\cap(f_{1})=\emptyset$, but $b_{2}\in T_{1}\cap(f_{2})$, then in the following exact sequence $0\rightarrow I^{\prime}_{1}/J^{\prime}_{1}\rightarrow I/J\rightarrow I/(J,I^{\prime}_{1})\rightarrow 0$ the last term has sdepth $\geq d+2$ since $h(b^{\prime})\notin I^{\prime}_{1}$, for all $b^{\prime}\in T_{1}$ and we may substitute the interval $[b_{2},h(b_{2})]$ from the restriction of $P_{b}$ by $[f_{2},h(b_{2})]$, the second monomial from $[f_{2},h(b_{2})]\cap B$ being also in $T_{1}$. As above we get either $\operatorname{sdepth}_{S}I/J\geq d+2$, or $\operatorname{sdepth}_{S}I^{\prime}_{1}/J^{\prime}_{1}\leq d+1$, $\operatorname{depth}_{S}I/(J,I^{\prime}_{1})\geq d+1$. Similarly, we do when $U_{1}\cap(f_{2})=\emptyset$ but $U_{1}\cap(f_{1})\not=\emptyset$. Now, suppose that $b_{1}\in T_{1}\cap(f_{1})$ and $b_{2}\in T_{1}\cap(f_{2})$. We claim to choose $b_{1}\not=b_{2}$ and such that one from $h(b_{1}),h(b_{2})$ is not in $(w_{12})$, let us say $h(b_{1})\not\in(w_{12})$. Indeed, if $w_{12}\not\in B$ and $h(b_{1}),h(b_{2})\in(w_{12})$ then necessarily $h(b_{1})=h(b_{2})$ and it follows $b_{1}=b_{2}=w_{12}$ which is false. Suppose that $w_{12}\in B$ and $h(b_{2})=x_{j}w_{12}$. Then choose $b_{1}=x_{j}f_{1}\in T_{1}$. If $h(b_{1})\in(w_{12})$ then we get $h(b_{1})=h(b_{2})$ and so $b_{1}=b_{2}=w_{12}$ which is impossible. In the following exact sequence $0\rightarrow I^{\prime\prime}/J^{\prime\prime}\rightarrow I/J\rightarrow I/(J,I^{\prime\prime})\rightarrow 0$ the last term has sdepth $\geq d+2$ since we may replace the intervals $[b_{1},h(b_{1})]$, $[b_{2},h(b_{2})]$ of the restriction of $P_{b}$ to $(T_{1})\setminus(J,I^{\prime\prime})$ with the disjoint intervals $[f_{1},h(b_{1})]$, $[f_{2},h(b_{2})]$. Also the last term has depth $\geq d+1$ because in the exact sequence $0\rightarrow(f_{2})/(J,I^{\prime\prime})\cap(f_{2})\rightarrow I/(J,I^{\prime\prime})\rightarrow I/(J,I^{\prime\prime},f_{2})\rightarrow 0$ the end terms have depth $\geq d+1$ since $h(b_{1})\not\in(f_{2})$, otherwise $h(b_{1})\in(w_{12})$, which is false. As above we get either $\operatorname{sdepth}_{S}I/J\geq d+2$, or $\operatorname{sdepth}_{S}I^{\prime\prime}/J^{\prime\prime}\leq d+1$, $\operatorname{depth}_{S}I/(J,I^{\prime\prime})\geq d+1$. In the second case, let $a_{1},\ldots,a_{t_{1}}$ be a weak path and set $c_{j}=h(a_{j})$ for $j\in[t_{1}]$. We may suppose that $c_{t_{1}}\in(u)$, otherwise take a shorter path. Denote $T_{1}$, $U_{1}$ as in the first case, which we keep it fix even we will change a little $h$. Suppose that $a_{t_{1}}\in(f_{2})$. Then change in $P_{b}$ the intervals $[a_{t_{1}},c_{t_{1}}]$, $[f_{2},c^{\prime}_{2}]$ by $[f_{2},c_{t_{1}}]$, $[u^{\prime},c^{\prime}_{2}]$. Thus the new $c^{\prime}_{2}$ is among $\\{c_{1},\ldots,c_{t_{1}}\\}\subset U_{1}$, though the old $c^{\prime}_{2}\not\in U_{1}$. Also the new $u^{\prime}$ is in $T_{1}$. However, if the old $u^{\prime}$ is not a divisor of a $c$ from $U_{1}$, then the proof goes as in the first case using $T^{\prime}_{1}=T_{1}\cup\\{u\\}$, $G^{\prime}_{1}=B\setminus T^{\prime}_{1}$ with $I^{\prime}=I^{\prime}_{2}$, or $I^{\prime}=I^{\prime}_{12}$. Otherwise, $T^{\prime}_{1}$ should be completed because $u^{\prime}$ is not now in $[f_{2},c^{\prime}_{2}]$ and we may consider some paths starting with $u^{\prime}$. Note that there exists a path from $a_{1}$ to $u^{\prime}$ since $u^{\prime}$ is a divisor of a monomial from $U_{1}$. It follows that there exist no bad path starting with $u^{\prime}$. Take ${\tilde{T}_{1}}=T^{\prime}_{1}\cup\\{b^{\prime}\in B:\mbox{there\ \ exists\ \ a\ \ path\ \ from}\ \ u^{\prime}\ \ \mbox{to}\ \ b^{\prime}\\}$ and the proof goes as above with $\tilde{T}_{1}$ instead $T^{\prime}_{1}$, that is with $I^{\prime}$ generated by a subset of $\\{f_{1},f_{2}\\}\cup{\tilde{G}}_{1}$ for ${\tilde{G}}_{1}=B\setminus{\tilde{T}}_{1}$. Now suppose that $a_{t_{1}}\not\in(f_{2})$ but there exists $1\leq v<t_{1}$ such that $a_{v}\in(f_{2})$ and $a_{v}\|c_{t_{1}}$. Then we may replace in $P_{b}$ the intervals $[a_{p},c_{p}],v\leq p\leq t_{1}$ with the intervals $[a_{v},c_{t_{1}}],[a_{p+1},c_{p}],v\leq p<t_{1}$. The old $c_{t_{1}}$ becomes the new $c_{v}$, that is we reduce to the case when $u$ divides $c_{v}$ and $a_{v}\in(f_{2})$, subcase solved above. Remains to study the subcase when there exist no $a_{v}\in(f_{2})$, $1\leq v\leq t_{1}$ with $a_{v}\|c_{t_{1}}$. Then there exists an $a_{t_{1}+1}\in B\cap(f_{2})$, $a_{t_{1}+1}\not=u$ such that $a_{t_{1}+1}\|c_{t_{1}}$. Clearly, $a_{t_{1}+1}\not=u^{\prime}$ because otherwise $c^{\prime}_{2}=c_{t_{1}}$. We have two subcases: $1^{\prime})$ there exists a path $a_{t_{1}+1},\ldots,a_{l}$ such that $h(a_{l})\in(a_{v^{\prime}})$ for some $1\leq v^{\prime}\leq t_{1}$, $2^{\prime})$ for any path $a_{t_{1}+1},\ldots,a_{p}$, any $h(a_{j})$, $t_{1}<j\leq p$ does not belong to $(a_{1},\ldots,a_{t_{1}})$. In the first subcase, we replace in $P_{b}$ the intervals $[a_{j},c_{j}],v^{\prime}\leq j\leq l$ with the intervals $[a_{v^{\prime}},c_{l}],[a_{j+1},c_{j}],v^{\prime}\leq j<l$. The new $h(a_{t_{1}+1})$ is the old $c_{t_{1}}$ and we may proceed as above. In the second case, we set $T_{2}=\\{b^{\prime}\in B:\mbox{there\ \ exists\ \ a\ \ path\ \ from}\ \ a_{t_{1}+1}\ \ \mbox{to}\ \ b^{\prime}\\}.$ Note that any path starting from $a_{t_{1}+1}$ can be completed to a path from $a_{1}$ by adding the monomials $a_{1},\ldots,a_{t_{1}}$. Thus there exists no bad path starting with $a_{t_{1}+1}$, otherwise we can get one starting from $a_{1}$, which is false. If there exists no weak path starting with $a_{t_{1}+1}$ then we proceed as in the first case with $T_{2}$ instead $T_{1}$. If there exists a weak path starting with $a_{t_{1}+1}$ then we proceed as above in case 2) with $T^{\prime}_{2}$, or ${\tilde{T}}_{2}$ instead $T^{\prime}_{1}$, or ${\tilde{T}}_{1}$, except in the subcase $2^{\prime})$ when we will define similarly a $T_{3}$ given by the paths starting with a certain $a_{t_{2}+1}$. Note that the whole set $\\{a_{1},\ldots,a_{t_{2}}\\}$ has different monomials. After several such steps we must arrive in the case $p=t_{m}$ when $\\{a_{1},\ldots,a_{t_{m}}\\}$ has different monomials and the subcase $2^{\prime})$ does not appear. We end this case using $T_{m}$, or $T^{\prime}_{m}$, or ${\tilde{T}}_{m}$ instead $T_{1}$, or $T^{\prime}_{1}$, or ${\tilde{T}}_{1}$. In the third case, let $a_{1},\ldots,a_{t_{1}}$ be a bad path starting with $a_{1}$. Set $c_{j}=h(a_{j})$, $j\in[t_{1}]$. Then $c_{t_{1}}=bx_{l_{1}}$ and let us say $b=f_{1}x_{i}$. If $a_{t_{1}}\in(f_{1})$ then changing in $P_{b}$ the interval $[a_{t_{1}},c_{t_{1}}]$ by $[f_{1},c_{t_{1}}]$ we get a partition on $I/J$ with sdepth $d+2$. Thus we may assume that $a_{t_{1}}\not\in(f_{1})$. If $f_{1}x_{l_{1}}\in\\{a_{1},\ldots,a_{t_{1}-1}\\}$, let us say $fx_{l_{1}}=a_{v}$, $1\leq v<t_{1}$ then we may replace in $P_{b}$ the intervals $[a_{p},c_{p}],v\leq p\leq t_{1}$ with the intervals $[a_{v},c_{t_{1}}],[a_{p+1},c_{p}],v\leq p<t_{1}$. Now we see that we have in $P_{b}$ the interval $[f_{1}x_{l_{1}},f_{1}x_{i}x_{l_{1}}]$ and switching it with the interval $[f_{1},f_{1}x_{i}x_{l_{1}}]$ we get a partition with sdepth $\geq d+2$ for $I/J$. Thus we may assume that $f_{1}x_{l_{1}}\notin\\{a_{1},...,a_{t_{1}}\\}$. Now set $a_{t_{1}+1}=fx_{l_{1}}$. Let $a_{t_{1}+1},\ldots,a_{k}$ be a path starting with $a_{t_{1}+1}$ and set $c_{j}=h(a_{j})$, $t_{1}<j\leq k$. If $a_{p}=a_{v}$ for $v\leq t_{1}$, $p>t_{1}$ then change in $P_{b}$ the intervals $[a_{j},c_{j}],v\leq j\leq p$ with the intervals $[a_{v},c_{p}],[a_{j+1},c_{j}],v\leq j<p$. We have in $P_{b}$ an interval $[f_{1}x_{l_{1}},f_{1}x_{i}x_{l_{1}}]$ and switching it to $[f_{1},f_{1}x_{i}x_{l_{1}}]$ we get a partition with sdepth $\geq d+2$ for $I/J$. Thus we may suppose that in fact $a_{p}\not\in\\{b,a_{1},\ldots,a_{p-1}\\}$ for any $p>t_{1}$ (with respect to any path starting with $a_{t_{1}+1}$). We have three subcases: $1^{\prime\prime})$ there exist no weak path and no bad path starting with $a_{t_{1}+1}$, $2^{\prime\prime})$ there exists a weak path starting with $a_{t_{1}+1}$ but no bad path starts with $a_{t_{1}+1}$, $3^{\prime\prime})$ there exists a bad path starting with $a_{t_{1}+1}$. Set $T_{2}=\\{b^{\prime}\in B:\mbox{there\ exists\ a\ path}\ \ a_{t_{1}+1},\ldots,a_{k}\ \ \mbox{with}\ \ a_{k}=b^{\prime}\\}$, $G_{2}=B\setminus T_{2}$, $U_{2}=h(T_{2})$ in the first subcase, and see that $I^{\prime}$ generated by a subset of $\\{f_{1},f_{2}\\}\cup G_{2}$ chosen as above works. In the second subcase, let $a_{t_{1}+1},\ldots,a_{k}$ be a weak path and set $c_{j}=h(a_{j})$ for $t_{1}<j\leq k$. We may suppose that $c_{k}\in(u)$. Changing $P_{b}$ we may suppose that the new $c^{\prime}_{2}$ is in $U_{2}$ as above. If the old $u^{\prime}$ was not a divisor of a $c\in U_{2}$ then the proof goes as in the first case with $T^{\prime}_{2}=T_{2}\cup\\{u\\}$, $I^{\prime}=I^{\prime}_{2}$. Otherwise, $T^{\prime}_{2}$ should be completed to a ${\tilde{T}_{2}}$ similar to ${\tilde{T}_{1}}$. The proof goes as above with $\tilde{T}_{2}$ instead $T^{\prime}_{2}$. In the third subcase, let $a_{t_{1}+1},\ldots,a_{t_{2}}$ be a bad path starting with $a_{t_{1}+1}$ and set $c_{j}=h(a_{j})$ for $j>t_{1}$. We saw that the whole set $\\{a_{1},\ldots,a_{t_{2}}\\}$ has different monomials. As above $c_{t_{2}}=bx_{l_{2}}$ and we may reduce to the case when $f_{1}x_{l_{2}}\not\in\\{a_{1},\ldots,a_{t_{1}}\\}$. Set $a_{t_{2}+1}=f_{1}x_{l_{2}}$ and again we consider three subcases, which we treat as above. Anyway after several such steps we must arrive in the case $p=t_{m}$ when $b\|c_{t_{m}}$ and again a certain $f_{1}x_{l_{m}}$ is not among $\\{a_{1},\ldots,a_{t_{m}}\\}$ and there exist no bad path starting with $a_{t_{m}+1}=f_{1}x_{l_{m}}$. This follows since we may reduce to the case when the set $\\{a_{1},\ldots,a_{t_{m}}\\}$ has different monomials and so the procedures should stop for some $m$. Finally, using $T_{m}=\\{b^{\prime}\in B:\mbox{there\ exists\ a\ path}\ \ a_{t_{m}+1},\ldots,a_{k}\ \ \mbox{with}\ \ a_{k}=b^{\prime}\\}$ (resp. $T^{\prime}_{m}$, or ${\tilde{T}}_{m}$) as $T_{1}$ (resp. $T^{\prime}_{1}$, or ${\tilde{T}}_{1}$) above we are done. ###### Lemma 6. Suppose that the following conditions hold: 1. (1) $r=3$, $6\leq s\leq q+3$, 2. (2) $C\subset(\cup_{i,j\in[3],i\not=j}(f_{i})\cap(f_{j}))\cup((E)\cap(f_{1},f_{2},f_{3}))\cup(\cup_{a,a^{\prime}\in E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$, 3. (3) There exists $b\in(B\cap(f_{1}))\setminus(f_{2},f_{3})$ such that $\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$. Then either $\operatorname{sdepth}_{S}I/J\geq d+2$, or there exists a nonzero ideal $I^{\prime}\subsetneq I$ generated by a subset of $\\{f_{1},f_{2},f_{3}\\}\cup B$ such that $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for $J^{\prime}=J\cap I^{\prime}$ and $\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$. ###### Proof. We follow the proof of Lemma 5. Since $\operatorname{sdepth}_{S}I_{b}/J_{b}\geq d+2$ we consider $h$ as above for a partition $P_{b}$ with sdepth $d+2$ of $I_{b}/J_{b}$. We have two intervals $[f_{2},c^{\prime}_{2}]$, $[f_{3},c^{\prime}_{3}]$ in $P_{b}$. Suppose that $B\cap[f_{i},c^{\prime}_{i}]=\\{u_{i},u^{\prime}_{i}\\}$, $1<i\leq 3$ . As in Lemma 5 we define a path $a_{1},\ldots,a_{k}$ from $a_{1}$ to $a_{k}$ and a bad path. The above path is weak if $h(a_{j})\in(u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3})$ for some $j\in[k]$. It is maximal if either $h(a_{k})\in(b)$, or all divisors from $B$ of $h(a_{k})$ are in $\\{b,u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3},a_{1},\ldots,a_{k}\\}$. By hypothesis $s\geq 6$ and there exists $a_{1}\in B\setminus\\{b,u_{2},u_{2}^{\prime},u_{3},u^{\prime}_{3}\\}$. Set $c_{1}=h(a_{1})$. If $c_{1}\in(b)$ then the path $\\{a_{1}\\}$ is maximal and bad. We construct below a path with $k>1$. By recurrence choose if possible $a_{p+1}$ to be a divisor from $B$ of $c_{p}$, which is not in $\\{b,u_{2},u^{\prime}_{2},u_{3},u_{3}^{\prime},a_{1},\ldots,a_{p}\\}$ and set $c_{p}=h(a_{p})$, $p\geq 1$. This construction ends at step $p=e$ if all divisors from $B$ of $c_{e}$ are in $\\{b,u_{2},u^{\prime}_{2},u_{3},u_{3}^{\prime},a_{1},\ldots,a_{e}\\}$. If $c_{j}\not\in(b)$ for $1\leq j<e$ then $\\{a_{1},\ldots,a_{e}\\}$ is a maximal path. If one $c_{p}\in(u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3})$ then the constructed path is weak. If $c_{e}\in(b)$ then this path is bad. We may reduce to the situation when $P_{b}$ satisfies the following property: $()$ For all $1\leq i<j\leq 3$ if $w_{ij}\in B\setminus\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$, $1\leq i<j\leq 3$ then $h(w_{ij})\not\in(u_{i},u^{\prime}_{i},u_{j},u^{\prime}_{j})$ if $i>1$. Indeed, suppose that $w_{ij}\in B\setminus\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$ and $h(w_{ij})\in(u_{j})$. Then $h(w_{ij})=x_{l}w_{ij}$ for some $l\not\in\operatorname{supp}w_{ij}$ and we must have let us say $u_{j}=x_{l}f_{j}$. Changing in $P_{b}$ the intervals $[f_{j},c^{\prime}_{j}]$, $[w_{ij},h(w_{ij})]$ with $[f_{j},h(w_{ij})]$, $[u^{\prime}_{j},c^{\prime}_{j}]$ we see that we may assume $u_{j}=w_{ij}$. Suppose that $()$ holds. We have three cases: 1) there exist no weak path and no bad path starting with $a_{1}$, 2) there exists a weak path starting with $a_{1}$ but no bad path starts with $a_{1}$, 3) there exists a bad path starting with $a_{1}$. In the first case, set $T_{1}=\\{b^{\prime}\in B:\mbox{there\ \ exists\ \ a\ \ path}\ \ a_{1},\ldots,a_{t}\ \ \mbox{with}\ \ a_{t}=b^{\prime}\\}$, $G_{1}=B\setminus T_{1}$, and for $k=(k_{1},\ldots,k_{m})$, $1\leq k_{1}<\ldots k_{m}\leq 3$, $0\leq m\leq 3$ set $I^{\prime}_{k}=(f_{k_{1}},\ldots,f_{k_{m}},G_{1})$, $J^{\prime}_{k}=I^{\prime}_{k}\cap J$, and $I^{\prime}_{0}=(G_{1})$, $J^{\prime}_{0}=I^{\prime}_{0}\cap J$ for $m=0$. Note that all divisors from $B$ of a monomial $c\in U_{1}=h(T_{1})$ belong to $T_{1}$, and $I^{\prime}_{0}\not=0$ because $b\in I^{\prime}_{0}$. Consider the following exact sequence $0\rightarrow I^{\prime}_{k}/J^{\prime}_{k}\rightarrow I/J\rightarrow I/(J,I^{\prime}_{k})\rightarrow 0.$ If $U_{1}\cap(f_{1},f_{2},f_{3})=\emptyset$ then the last term of the above exact sequence given for $k=(1,2,3)$ has depth $\geq d+1$ and sdepth $\geq d+2$ using the restriction of $P_{b}$ to $(T_{1})\setminus(J,I^{\prime}_{k})$ since $h(b^{\prime})\notin I^{\prime}_{k}$ , for all $b^{\prime}\in T_{1}$. When the first term has sdepth $\geq d+2$ then by [18, Lemma 2.2] the middle term has sdepth $\geq d+2$ which is enough. If $U_{1}\cap(f_{1},f_{2})=\emptyset$, but there exists $b_{3}\in T_{1}\cap(f_{3})$, then set $k=(1,2)$. In the following exact sequence $0\rightarrow I^{\prime}_{k}/J^{\prime}_{k}\rightarrow I/J\rightarrow I/(J,I^{\prime}_{k})\rightarrow 0$ the last term has sdepth $\geq d+2$ since $h(b^{\prime})\notin I^{\prime}_{k}$, for all $b^{\prime}\in T_{1}$ and we may substitute the interval $[b_{3},h(b_{3})]$ from the restriction of $P_{b}$ by $[f_{3},h(b_{3})]$, the second monomial from $[f_{3},h(b_{3})]\cap B$ being also in $T_{1}$. As above we get either $\operatorname{sdepth}_{S}I/J\geq d+2$, or $\operatorname{sdepth}_{S}I^{\prime}_{1}/J^{\prime}_{1}\leq d+1$, $\operatorname{depth}_{S}I/(J,I^{\prime}_{1})\geq d+1$. Now, we omit other subcases considering only the worst subcase $m=0$. Let $b_{1}\in T_{1}\cap(f_{1})$, $b_{2}\in T_{1}\cap(f_{2})$ and $b_{3}\in T_{1}\cap(f_{3})$. For $1\leq l<j\leq 3$ we claim that we may choose $b_{l}\not=b_{j}$ and such that one from $h(b_{l}),h(b_{j})$ is not in $(w_{lj})$. Indeed, if $w_{lj}\not\in B$ and $h(b_{l}),h(b_{j})\in(w_{lj})$ then necessarily $h(b_{l})=h(b_{j})$ and it follows $b_{l}=b_{j}=w_{lj}$, which is false. Suppose that $w_{lj}\in B$ and $h(b_{j})=x_{p}w_{lj}$. Then choose $b_{l}=x_{p}f_{l}\in T_{1}$. If $h(b_{l})\in(w_{lj})$ then we get $h(b_{l})=h(b_{j})$ and so $b_{l}=b_{j}=w_{lj}$, which is impossible. Therefore, we may choose $b_{j}$ such that $h(b_{1})\not\in(w_{12})$, $h(b_{2})\not\in(w_{23})$. Note that it is possible that $f_{1}\|c$ for some $c\in h(T_{1})$ even $b\not\|c$ for any $c\in U_{1}$. If $h(b_{1})\in(w_{13})$ then we may also choose $h(b_{3})\not\in(w_{13})$. In the case when $h(b_{1})\not\in(w_{13})$, choose any $b_{3}\in T_{1}\cap(f_{3})$ different from the others $b_{j}$. We conclude that the possible intervals $[f_{j},h(b_{j})]$, $j\in[3]$ are disjoint. Next we change the intervals $[b_{j},h(b_{j})]$, $j\in[3]$ from the restriction of $P_{b}$ to $(T_{1})\setminus(J,I^{\prime}_{0})$ by $[f_{j},h(b_{j})]$, the second monomial from $[f_{j},h(b_{j})]\cap B$ being also in $T_{1}$. We claim that $I/(J,I^{\prime}_{0})$ has depth $\geq d+1$. Indeed, in the following exact sequence $0\rightarrow(f_{2})/(f_{2})\cap(J,I^{\prime}_{0},f_{3})\rightarrow I/(J,I^{\prime}_{0},f_{3})\rightarrow I/(J,I^{\prime}_{0},f_{2},f_{3})\rightarrow 0$ the first term has depth $\geq d+1$ because $h(b_{2})\not\in(f_{2})\cap(f_{3})$. If $h(b_{1})\not\in(f_{3})$ then $h(b_{1})\not\in(f_{2},f_{3})\cap(f_{1})$ and so the last term has depth $\geq d+1$. If $h(b_{1})\in(w_{13})$ then we may find a $b^{\prime}\in(B\cap(f_{1}))\setminus(f_{3})$ dividing $h(b_{1})$. It follows that $b^{\prime}\in T_{1}$ and $b^{\prime}\not\in(f_{2},f_{3})\cap(f_{1})$, which implies that the last term has again depth $\geq d+1$. Thus $\operatorname{depth}_{S}I/(J,I^{\prime}_{0},f_{3})\geq d+1$ by the Depth Lemma. Our claim follows from the exact sequence $0\rightarrow(f_{3})/(f_{3})\cap(J,I^{\prime}_{0})\rightarrow I/(J,I^{\prime}_{0})\rightarrow I/(J,I^{\prime}_{0},f_{3})\rightarrow 0$ because the first term has depth $\geq d+1$. Therefore, as above we get either $\operatorname{sdepth}_{S}I/J\geq d+2$, or $\operatorname{sdepth}_{S}I^{\prime}_{0}/J^{\prime}_{0}\leq d+1$, $\operatorname{depth}_{S}I/(J,I^{\prime}_{0})\geq d+1$. In the second case, let $a_{1},\ldots,a_{t_{1}}$ be a weak path and set $c_{j}=h(a_{j})$ for $j\in[t_{1}]$. We may suppose that $c_{t_{1}}\in(u_{2})$, otherwise take a shorter path. Denote $T_{1}$, $U_{1}$ as in the first case. First consider the subcase when $U_{1}\cap(f_{3})=\emptyset$. Suppose that $a_{t_{1}}\in(f_{2})$. Then change in $P_{b}$ the intervals $[a_{t_{1}},c_{t_{1}}]$, $[f_{2},c^{\prime}_{2}]$ by $[f_{2},c_{t_{1}}]$, $[u^{\prime}_{2},c^{\prime}_{2}]$. Thus the new $c^{\prime}_{2}$ is among $\\{c_{1},\ldots,c_{t_{1}}\\}\subset U_{1}$, though the old $c^{\prime}_{2}\not\in U_{1}$. If the old $u^{\prime}_{2}$ is not a divisor of any $c\in U_{1}$ then the proof goes as in the first case with $T^{\prime}_{1}=T_{1}\cup\\{u_{2}\\}$. If the old $u^{\prime}_{2}$ is a divisor of a monomial from $U_{1}$ then $T^{\prime}_{1}$ should be completed because the old $u^{\prime}_{2}$ is not now in $[f_{2},c^{\prime}_{2}]$. Note that there exists a path from $a_{1}$ to $u^{\prime}_{2}$ since $u^{\prime}_{2}$ is a divisor of a monomial from $U_{1}$. It follows that there exist no bad path starting with $u^{\prime}_{2}$. It is worth to mention that the old $c^{\prime}_{2}$ is now in $U_{1}$ and we should consider all pathes starting with divisors of $c^{\prime}_{2}$ from $B$. Take ${\tilde{T}_{1}}=T^{\prime}_{1}\cup\\{b^{\prime}\in B:\mbox{there\ \ exists\ \ a\ \ path\ \ from}\ \ u^{\prime}_{2}\ \ \mbox{to}\ \ b^{\prime}\\}$ and the proof goes as above with $\tilde{T}_{1}$ instead $T^{\prime}_{1}$, that is with $I^{\prime}$ generated by a subset of $\\{f_{1},f_{2},f_{3}\\}\cup{\tilde{G}}_{1}$, where ${\tilde{G}}_{1}=B\setminus{\tilde{T}}_{1}$. Now suppose that $a_{t_{1}}\not\in(f_{2})$ but there exists $1\leq v<t_{1}$ such that $a_{v}\in(f_{2})$ and $a_{v}\|c_{t_{1}}$. Then we may replace in $P_{b}$ the intervals $[a_{p},c_{p}],v\leq p\leq t_{1}$ with the intervals $[a_{v},c_{t_{1}}],[a_{p+1},c_{p}],v\leq p<t_{1}$. The old $c_{t_{1}}$ becomes the new $c_{v}$, that is we reduce to the case when $u_{2}$ divides $c_{v}$ and $a_{v}\in(f_{2})$, subcase solved above. Remains to study the subcase when there exist no $a_{v}\in(f_{2})$, $1\leq v\leq t_{1}$ with $a_{v}\|c_{t_{1}}$. Then there exists an $a_{t_{1}+1}\in B\cap(f_{2})$, $a_{t_{1}+1}\not=u_{2}$ such that $a_{t_{1}+1}\|c_{t_{1}}$. Clearly, $a_{t_{1}+1}\not=u^{\prime}_{2}$ because otherwise $c^{\prime}_{2}=c_{t_{1}}$. We have two subcases: $1^{\prime})$ there exists a path $a_{t_{1}+1},\ldots,a_{l}$ such that $h(a_{l})\in(a_{v^{\prime}})$ for some $1\leq v^{\prime}\leq t_{1}$, $2^{\prime})$ for any path $a_{t_{1}+1},\ldots,a_{p}$, any $h(a_{j})$, $t_{1}<j\leq p$ does not belong to $(a_{1},\ldots,a_{t_{1}})$. In the first subcase, we replace in $P_{b}$ the intervals $[a_{j},c_{j}],v^{\prime}\leq j\leq l$ with the intervals $[a_{v^{\prime}},c_{l}],[a_{j+1},c_{j}],v^{\prime}\leq j<l$. The new $h(a_{t_{1}+1})$ is the old $c_{t_{1}}$ and we may proceed as above. In the second subcase we set $T_{2}=\\{b^{\prime}\in B:\mbox{there\ \ exists\ \ a\ \ path\ \ from}\ \ a_{t_{1}+1}\ \ \mbox{to}\ \ b^{\prime}\\}.$ Note that any path starting from $a_{t_{1}+1}$ can be completed to a path from $a_{1}$ by adding the monomials $a_{1},\ldots,a_{t_{1}}$. Thus there exists no bad path starting with $a_{t_{1}+1}$, otherwise we can get one starting from $a_{1}$, which is false. If there exists no weak path starting with $a_{t_{1}+1}$ then we proceed as in the first case with $T_{2}$ instead $T_{1}$. If there exists a weak path starting with $a_{t_{1}+1}$ then we proceed as above in case 2) with $T^{\prime}_{2}$, or ${\tilde{T}}_{2}$ instead $T^{\prime}_{1}$, or ${\tilde{T}}_{1}$, except in the subcase $2^{\prime})$ when we will define similarly a $T_{3}$ given by the paths starting with a certain $a_{t_{2}+1}$. Note that the whole set $\\{a_{1},\ldots,a_{t_{2}}\\}$ has different monomials. After several such steps we must arrive in the case $p=t_{m}$ when $\\{a_{1},\ldots,a_{t_{m}}\\}$ has different monomials and the subcase $2^{\prime})$ does not appear. We end this case using $T_{m}$, or $T^{\prime}_{m}$, or ${\tilde{T}}_{m}$ instead $T_{1}$, $T^{\prime}_{1}$, or ${\tilde{T}}_{1}$. We should mention that if there exists $b_{1}\in T_{m}$ (or in $T^{\prime}_{m}$, ${\tilde{T}}_{m}$) such that $h(b_{1})\in(f_{1})$ then changing $P_{b}$ as in case 1) we may suppose that $h(b_{1})\not\in(w_{12})$ and $b_{1}\in T_{m}\cap(f_{1})$. Thus we may consider the interval $[f_{1},h(b_{1})]$ disjoint of $[f_{2},c^{\prime}_{2}]$. Consider the subcase when there exist $b_{j}\in T_{1}$, $j=2,3$ such that $h(b_{2})\in(u_{2})$ and $h(b_{3})\in(f_{3})$ but $h(T_{1})\cap(u_{3},u^{\prime}_{3})=\emptyset$. As above we may suppose that after several procedures we changed $P_{b}$ such that $b_{j}\in(f_{j})$ and the new $c^{\prime}_{2}$ is the old $h(b_{2})$. If $h(b_{2})\not\in C_{3}\cup C_{2}$ then we may suppose that $h(b_{2})\not\in(w_{12})$. As in the first case we may change $b_{3}$ such that $h(b_{3})\not\in(w_{23})$. Indeed, the only problem could be if the old $h(b_{3})\in\\{u_{3},u^{\prime}_{3}\\}$, which is not the case. We have no obstruction to change as usual $b_{1}$ such that $h(b_{1})\not\in(w_{13})$ and so note that the interval $[f_{2},h(b_{2})]$ (resp. $[f_{3},h(b_{3})]$, or $[f_{1},h(b_{1})]$) has at most $w_{23}$ (resp. $w_{13}$, or $w_{12}$) from $W$. Thus the intervals $[f_{j},h(b_{j})]$, $j\in[3]$ are disjoint. If $h(b_{2})\in C_{3}$ then either $b_{2}=w_{23}$, or $w_{12}$. But $b_{2}\not=w_{23}$ because otherwise $h(w_{23})\in(u_{2})$ contradicting $()$. Similarly, $b_{2}\not=w_{12}$. If $h(b_{2})=w_{12}$ (resp. $h(b_{2})=w_{23}$) then $b_{2}\not=w_{23}$ (resp. $b_{2}\not=w_{12}$) because otherwise we get a contradiction with $()$. Thus $w_{12}$ (resp. $w_{23}$) is the only monomial of $W$ which belongs to $[f_{2},h(b_{2})]$. Choosing $b_{3}$ such that $h(b_{3})\not\in(w_{23})$ (resp. $h(b_{3})\not\in(w_{12})$) and $b_{1}$ such that $h(b_{1})\not\in(w_{12})$ (resp. $h(b_{1})\not\in(w_{13})$) we get disjoint the corresponding intervals. Now consider the subcase when there exist $b_{j}\in T_{1}$, $j=2,3$ such that $h(b_{2})\in(u_{2})$ and $h(b_{3})\in(u_{3})$. If $h(b_{2})\not\in(f_{3})$ and $h(b_{3})\not\in(f_{2})$ then as above we may assume that with a different $P_{b}$, if necessary, we may reduce to the subcase when $b_{j}\in(f_{j})$, $j=2,3$. In general this is not simple because $h(b_{2})$ as in Example 5 can have no divisors from $B\cap(f_{2})$, which are not in $\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$ and there exist no other $c\in U_{1}$ multiple of $u_{2}$. In such situation we are force to remain on the old $c^{\prime}_{2}$ taking $T^{\prime}_{1}=T_{1}\cup\\{u_{2},u^{\prime}_{2}\\}$ and $U^{\prime}_{1}=U_{1}\cup\\{c^{\prime}_{2}\\}$. If there exists a bad path starting on a divisor from $B\setminus\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$ of $c^{\prime}_{2}$ then we go to case 3). Otherwise, we should consider also the pathes starting with the divisors of $c^{\prime}_{2}$ from $B\setminus\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$ completing $T^{\prime}_{1}$ to ${\tilde{T}}_{1}$. Note that because of $2^{\prime})$ we may speak now about $T_{m}$ instead $T_{1}$. Changing in $P_{b}$ the intervals $[f_{j},c^{\prime}_{j}]$, $[b_{j},h(b_{j})]$, $j=2,3$ with $[f_{j},h(b_{j})]$, $[u^{\prime}_{j},c^{\prime}_{j}]$, $j=2,3$ we may assume the new $c^{\prime}_{2},c^{\prime}_{3}$ are in $U_{m}=h(T_{m})$ for some $m$ and the proof goes as above. If let us say $h(b_{2})\in(f_{3})$ then we must be carefully since it is possible that the new intervals $[f_{j},c^{\prime}_{j}]$ could be not disjoint. A nice subcase is for example when $h(b_{2})$ is a least common multiple of $u_{2},u_{3}$, which we study below. If $w_{23}\in B$ then we we may suppose $u_{2}=w_{23}$. Indeed, if $w_{23}\not\in\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$ and $u_{2}=x_{p}f_{2}$ for some $p\not\in\operatorname{supp}w_{23}$ then $h(b_{2})=x_{p}w_{23}$. Since $b_{2}\not=u_{2}$ and $b_{2}\in(f_{2})$ it follows that $b_{2}=w_{23}$. But this contradicts the property $()$. Suppose that $a_{t_{1}}=b_{2}$. Then note that $h(b_{2})=c_{t_{1}}=x_{p}w_{23}$ for some $p$ and it follows that $a_{t_{1}}=b_{2}=x_{p}f_{2}$ since $a_{t_{1}}\not=w_{23}=u_{2}$. Changing in $P_{b}$ the intervals $[f_{2},c^{\prime}_{2}]$, $[b_{2},h(b_{2})]$ with $[f_{2},h(b_{2})]$, $[u^{\prime}_{2},c^{\prime}_{2}]$, we may assume the new $c^{\prime}_{2}$ is in $U_{m}$. We claim that $w_{23}$ is the only monomial from $B\cap W$ which is in $[f_{2},c^{\prime}_{2}]$. Indeed, $w_{12}$ could be another monomial from $B\cap W$ which is present in the new $[f_{2},c^{\prime}_{2}]$. This could be true only if $a_{t_{1}}=w_{12}$. Thus $h(w_{12})=c_{t_{1}}\in(u_{2})$ which is not possible again by $()$. The same procedure we use to include a new $c^{\prime}_{3}$ in $U_{m}$. Since $u_{2}=w_{23}$ cannot be among $u_{3},u^{\prime}_{3}$ we see that only $w_{13}$ could be among them. Suppose that $u_{3}=w_{13}$. Clearly the new $[f_{3},c^{\prime}_{3}]$ cannot contain $w_{23}$. Choose as in the first case $b_{1}\in(f_{1})$ such that $h(b_{1})\not\in(w_{13})$ and the new intervals $[f_{j},c^{\prime}_{j}]$, $j\in[3]$ are disjoint. If $w_{13}\not\in\\{u_{3},u^{\prime}_{3}\\}$ then we might have only $b_{3}=w_{13}$ and we may repeat the argument. A problem could appear when the new $[f_{j},c^{\prime}_{j}]$, $j=2,3$ contain $w_{12}$, $w_{13}$ because then we may not find $b_{1}$ as before. Note that this problem could appear only when $w_{12},w_{13}\in\\{u_{2},u^{\prime}_{2},u_{3},u^{\prime}_{3}\\}$ because of $()$. We will change the new $c^{\prime}_{2}$ such that will not belong to $(f_{1})$. Changing $P_{b}$ we may suppose that $b_{j}\in(f_{j})$, $j=2,3$ (again this change is not so simple as we saw above). We have $h(b_{2})=c_{t_{1}}=x_{p}w_{12}$ for some $p$ and it follows that $a_{t_{1}}=b_{2}=x_{p}f_{2}$ since $a_{t_{1}}\not=w_{12}=u_{2}$. Suppose that $t_{1}>1$. Thus $a_{t_{1}}\|c_{t_{1}-1}$ and we see that $c_{t_{1}-1}$ is not in $(f_{1})$ because otherwise we get $c_{t_{1}-1}=x_{p}w_{12}=c_{t_{1}}$, which is false. If $a_{t_{1}-1}\in(f_{2})$ then changing in $P_{b}$ the intervals $[a_{t_{1}},c_{t_{1}}]$, $[a_{t_{1}-1},c_{t_{1}-1}]$, $[f_{2},c^{\prime}_{2}]$ by $[f_{2},c_{t_{1}-1}]$, $[u_{2},c_{t_{1}}]$, $[u^{\prime}_{2},c^{\prime}_{2}]$ we see that the new $c^{\prime}_{2}$ is not in $(f_{1})$ and belongs to $U_{m}$. If $w_{12}\in C$ then we get $h(b_{2})=w_{12}$ and the above argument works again, $c_{t_{1}-1}$ being the new $c^{\prime}_{2}$. When $a_{t_{1}-1}\not\in(f_{2})$ but $u^{\prime}_{2}\|c_{t_{1}-1}$ we reduce the problem to the subcase when the path $\\{a_{1},\ldots,a_{t_{1}-1}\\}$ goes from $a_{1}$ to $u^{\prime}_{2}$ and now $u^{\prime}_{2}\not\in(f_{1})$. As above we may change $P_{b}$ such that the new $b_{2}=a_{t_{1}-1}\in(f_{2})$ and the new $c^{\prime}_{2}$, that is the old $c_{t_{1}-1}$ is not in $(f_{1})$. If $a_{t_{1}-1}\not\in(f_{2})$, $u^{\prime}_{2}\not\|c_{t_{1}-1}$ but there exists ${\tilde{a}}\in B\cap(f_{2})$ a divisor of $c_{t_{1}-1}$ then ${\tilde{a}}\not=u_{2}$ because otherwise we get $c_{t_{1}-1}=c_{t_{1}}$. Now we repeat the first part of the case 2). If ${\tilde{a}}=a_{v}$ for some $1\leq v<t_{1}-1$ then changing in $P_{b}$ the intervals $[a_{p},c_{p}]$, $v\leq p<t_{1}$ by $[a_{v},c_{t_{1}-1}]$, $[a_{p+1},c_{p}]$, $v\leq p<t_{1}-1$ we see that the new $c_{v}$ (resp. $c_{v+1}$) is the old $c_{t_{1}-1}$ (resp. $c_{t_{1}}$)). Now changing the intervals $[a_{v},c_{v}]$, $[a_{v+1},c_{v+1}]$, $[f_{2},c^{\prime}_{2}]$ by $[f_{2},c_{v}]$, $[w_{12},c_{v+1}]$, $[u^{\prime}_{2},c^{\prime}_{2}]$ we see that the new $c^{\prime}_{2}\not\in(f_{1})$ and belongs to $U_{m}$. If ${\tilde{a}}\not\in\\{a_{1},\ldots,a_{t_{1}}\\}$ then we are in one of the above subcases $1^{\prime})$, $2^{\prime})$ solved already. We may use this argument to change $c^{\prime}_{j}$, $j=2,3$ such that it is not in $(f_{1})$ anymore, but as long as $h(b_{j})\not=c_{1}$, that is the corresponding $t_{1}>1$. However, we may have $h(b_{j})=c_{1}$ only for one $j>1$, because if for instance $h(b_{3})=c_{1}$ then $c_{1}\in C_{2}\cup C_{3}$. If $c_{1}\in C_{3}$ then we see that $w_{23}\in B$ and $a_{1}=w_{23}$. But this contradicts $()$ because $h(w_{23})\in(u_{2})$. If $c^{\prime}_{2}\in C_{2}$ then $c^{\prime}_{2}=w_{23}$ and either $a_{1}\in(f_{2})$, or $a_{1}\in(f_{3})$, that is $a_{1}$ cannot be $b_{2}$ and $b_{3}$ in the same time. Thus at least one of the new $c^{\prime}_{j}$, $j=2,3$ could be taken $\not\in(f_{1})$. If let us say only $c^{\prime}_{3}\in(f_{1})$ then choose $b_{1}\in T_{1}\cap(f_{1})$ such that $h(b_{1})\not\in(w_{13})$ as before. The interval $[f_{1},h(b_{1})]$ is disjoint from the other new constructed intervals, which is enough as we saw in case 1). In the third case, let $a_{1},\ldots,a_{t_{1}}$ be a bad path starting with $a_{1}$. Set $c_{j}=h(a_{j})$, $j\in[t_{1}]$. Then $c_{t_{1}}=bx_{l_{1}}$ and let us say $b=f_{1}x_{i}$. If $a_{t_{1}}\in(f_{1})$ then changing in $P_{b}$ the interval $[a_{t_{1}},c_{t_{1}}]$ by $[f_{1},c_{t_{1}}]$ we get a partition on $I/J$ with sdepth $d+2$. Thus we may assume that $a_{t_{1}}\not\in(f_{1})$. If $f_{1}x_{l_{1}}\in\\{a_{1},\ldots,a_{t_{1}-1}\\}$, let us say $fx_{l_{1}}=a_{v}$, $1\leq v<t_{1}$ then we may replace in $P_{b}$ the intervals $[a_{p},c_{p}],v\leq p\leq t_{1}$ with the intervals $[a_{v},c_{t_{1}}],[a_{p+1},c_{p}],v\leq p<t_{1}$. Now we see that we have in $P_{b}$ the interval $[f_{1}x_{l_{1}},f_{1}x_{i}x_{l_{1}}]$ and switching it with the interval $[f_{1},f_{1}x_{i}x_{l_{1}}]$ we get a partition with sdepth $\geq d+2$ for $I/J$. Thus we may assume that $f_{1}x_{l_{1}}\notin\\{a_{1},...,a_{t_{1}}\\}$. Now set $a_{t_{1}+1}=fx_{l_{1}}$. Let $a_{t_{1}+1},\ldots,a_{k}$ be a path starting with $a_{t_{1}+1}$ and set $c_{j}=h(a_{j})$, $t_{1}<j\leq k$. If $a_{p}=a_{v}$ for $v\leq t_{1}$, $p>t_{1}$ then change in $P_{b}$ the intervals $[a_{j},c_{j}],v\leq j\leq p$ with the intervals $[a_{v},c_{p}],[a_{j+1},c_{j}],v\leq j<p$. We have in $P_{b}$ an interval $[f_{1}x_{l_{1}},f_{1}x_{i}x_{l_{1}}]$ and switching it to $[f_{1},f_{1}x_{i}x_{l_{1}}]$ we get a partition with sdepth $\geq d+2$ for $I/J$. Thus we may suppose that in fact $a_{p}\not\in\\{b,a_{1},\ldots,a_{p-1}\\}$ for any $p>t_{1}$ (with respect to any path starting with $a_{t_{1}+1}$). We have three subcases: $1^{\prime\prime})$ there exist no weak path and no bad path starting with $a_{t_{1}+1}$, $2^{\prime\prime})$ there exists a weak path starting with $a_{t_{1}+1}$ but no bad path starts with $a_{t_{1}+1}$, $3^{\prime\prime})$ there exists a bad path starting with $a_{t_{1}+1}$. Set $T_{2}=\\{b^{\prime}\in B:\mbox{there\ exists\ a\ path}\ \ a_{t_{1}+1},\ldots,a_{k}\ \ \mbox{with}\ \ a_{k}=b^{\prime}\\}$. We treat the subcases $1^{\prime\prime})$, $2^{\prime\prime})$ as the cases 1), 2) and find $I^{\prime}$ generated by a subset of $\\{f_{1},f_{2},f_{3}\\}\cup G_{2}$, or $\\{f_{1},f_{2},f_{3}\\}\cup G^{\prime}_{2}$, or $\\{f_{1},f_{2},f_{3}\\}\cup{\tilde{G}}_{2}$, where $G_{2}$, $G^{\prime}_{2}$, ${\tilde{G}}_{2}$, are obtained from $T_{2}$ and as above $T^{\prime}_{2}$, or ${\tilde{T}}_{2}$. In the subcase $3^{\prime\prime})$, let $a_{t_{1}+1},\ldots,a_{t_{2}}$ be a bad path starting with $a_{t_{1}+1}$ and set $c_{j}=h(a_{j})$ for $j>t_{1}$. We saw that the whole set $\\{a_{1},\ldots,a_{t_{2}}\\}$ has different monomials. As above $c_{t_{2}}=bx_{l_{2}}$ and we may reduce to the case when $f_{1}x_{l_{2}}\not\in\\{a_{1},\ldots,a_{t_{1}}\\}$. Set $a_{t_{2}+1}=f_{1}x_{l_{2}}$ and again we consider three subcases, which we treat as above. Anyway after several such steps we must arrive in the case $p=t_{m}$ when either we may proceed as in the subcases $1^{\prime\prime})$, $2^{\prime\prime})$, or $b\|c_{t_{m}}$ and again a certain $f_{1}x_{l_{m}}$ is not among $\\{a_{1},\ldots,a_{t_{m}}\\}$ and taking $a_{t_{m}+1}=f_{1}x_{l_{m}}$ there exist no bad path starting with $a_{t_{m}+1}$. This follows since we may reduce to the subcase when the set $\\{a_{1},\ldots,a_{t_{m}}\\}$ has different monomials and so the procedures should stop for some $m$. Finally, using $T_{m}=\\{b^{\prime}\in B:\mbox{there\ exists\ a\ path}\ \ a_{t_{m}+1},\ldots,a_{k}\ \ \mbox{with}\ \ a_{k}=b^{\prime}\\}$ (resp. $T^{\prime}_{m}$, or ${\tilde{T}}_{m}$) as $T_{1}$ (resp. $T^{\prime}_{1}$, or ${\tilde{T}}_{1}$) above we are done. Proof of Theorem 6. By Theorems 2, 3 we may suppose that $2r\leq s\leq q+r$ and we may assume that $E$ contains only monomials of degrees $d+1$ by [15, Lemma 1.6]. Apply induction on $\|E\|$. If $E=\emptyset$ we may apply Theorem 4. Suppose that $\|E\|>0$ and $B\cap(f_{1},\ldots,f_{r})\not=\emptyset$, $r=2,3$, otherwise we get $\operatorname{depth}_{S}I/J\leq d+1$ using [15, Lemma 1.5] applied to any $f_{i}$. We may choose $b\in B\cap(f_{1},f_{2},f_{3})$ which is not in $W$ if $r=2,3$ and $\|B\cap(f_{1},\ldots,f_{r})\|>3\geq\|B\cap W\|$. However, $\|B\cap(f_{1},\ldots,f_{r})\|<2r$ gives $\operatorname{depth}_{S}(f_{1},\ldots,f_{r})/J\cap(f_{1},\ldots,f_{r})\leq d+1$ by Theorem 3 and it follows that $\operatorname{depth}_{S}I/J\leq d+1$ using the Depth Lemma applied to the exact sequence $0\rightarrow(f_{1},\ldots,f_{r})/J\cap(f_{1},\ldots,f_{r})\rightarrow I/J\rightarrow(E)/(J,f_{1},\ldots,f_{r})\cap(E)\rightarrow 0.$ Thus if $r=2,3$ we may suppose to find $b\in B\cap(f_{1},\ldots,f_{r})\setminus W$. Renumbering $f_{i}$ we may suppose that $b\in(f_{1})\setminus(f_{2},\ldots,f_{3})$. Apply induction on $r\leq 3$. Using Theorem 4 and induction hypothesis on $\|E\|$ and $r$ apply [17, Lemma 1.1]. Thus we may suppose that $C\subset((f_{1})\cap(f_{2}))\cup((E)\cap(f_{1},f_{2}))\cup(\cup_{a,a^{\prime}\in E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$, if $r=2$, or $C\subset(\cup_{i,j\in[3],i\not=j}(f_{i})\cap(f_{j}))\cup((E)\cap(f_{1},f_{2},f_{3}))\cup(\cup_{a,a^{\prime}\in E,a\not=a^{\prime}}(a)\cap(a^{\prime}))$ if $r=3$. Set $I^{\prime}_{b}=(f_{2},\ldots,f_{r},B\setminus\\{b\\})$, $J^{\prime}_{b}=I^{\prime}_{b}\cap J$. Clearly $b\not\in I^{\prime}_{b}$ and so in the following exact sequence $0\rightarrow I^{\prime}_{b}/J^{\prime}_{b}\rightarrow I/J\rightarrow I/(J,I^{\prime}_{b})\rightarrow 0$ the last term has depth $\geq d+1$. If the first term has sdepth $\leq d+1$ then it has depth $\leq d+1$ by induction hypothesis on $r$, case $r=1$ being done in Theorem 4. Thus we may suppose that $\operatorname{sdepth}_{S}I^{\prime}_{b}/J^{\prime}_{b}\geq d+2$ and we may apply Lemmas 5, 6. Then we get either $\operatorname{sdepth}_{S}I/J\geq d+2$ contradicting our assumption, or there exists a nonzero ideal $I^{\prime}\subsetneq I$ generated by a subset $G$ of $B$, or by $G$ and a subset of $\\{f_{1},f_{2},f_{3}\\}$ such that $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq d+1$ for $J^{\prime}=J\cap I^{\prime}$ and $\operatorname{depth}_{S}I/(J,I^{\prime})\geq d+1$. In the last case we see that $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq d+1$ by induction hypothesis on $r$, $\|E\|$, or by Theorem 4 and so $\operatorname{depth}_{S}I/J\leq d+1$ by the Depth Lemma applied to the following exact sequence $0\rightarrow I^{\prime}/J^{\prime}\rightarrow I/J\rightarrow I/(J,I^{\prime})\rightarrow 0.\ \ \ \ \ \ \ \Box$ The following bad example it is useful to illustrate somehow our proof. ###### Example 5. Let $n=6$, $r=3$, $d=1$, $f_{i}=x_{i}$ for $i\in[3]$, $E=\\{x_{4}x_{5},x_{5}x_{6}\\}$, $I=(x_{1},x_{2},x_{3},E)$ and $J=(x_{2}x_{4},x_{3}x_{4},x_{1}x_{2}x_{6},x_{1}x_{3}x_{6},x_{1}x_{4}x_{6},x_{1}x_{5}x_{6},x_{2}x_{3}x_{6},x_{2}x_{5}x_{6},$ $x_{3}x_{5}x_{6})$. Then $B=\\{x_{1}x_{2},x_{1}x_{3},x_{2}x_{3},x_{1}x_{4},x_{1}x_{5},x_{2}x_{5},x_{3}x_{5}\\}\cup E$ and $C=\\{x_{1}x_{2}x_{3},x_{1}x_{2}x_{5},x_{2}x_{3}x_{5},x_{1}x_{3}x_{5},x_{1}x_{4}x_{5},x_{4}x_{5}x_{6}\\}.$ Take $b=x_{1}x_{4}$ and $I_{b}=(x_{2},x_{3},x_{1}x_{2},x_{1}x_{3},x_{1}x_{5},E)$, $J_{b}=I_{b}\cap J$. There exists a partition $P_{b}$ with sdepth $3$ on $I_{b}/J_{b}$ given by the intervals $[x_{2},x_{1}x_{2}x_{3}]$, $[x_{3},x_{1}x_{3}x_{5}]$, $[x_{1}x_{5},x_{1}x_{2}x_{5}]$, $[x_{2}x_{5},x_{2}x_{3}x_{5}]$, $[x_{4}x_{5},x_{1}x_{4}x_{5}]$, $[x_{5}x_{6},x_{4}x_{5}x_{6}]$. We have $c^{\prime}_{2}=x_{1}x_{2}x_{3}$, $c^{\prime}_{3}=x_{1}x_{3}x_{5}$ and $u_{2}=x_{2}x_{3}$, $u^{\prime}_{2}=x_{1}x_{2}$, $u_{3}=x_{3}x_{5}$, $u^{\prime}_{3}=x_{1}x_{3}$. Clearly, $u_{2}=w_{23}$. Take $a_{1}=x_{1}x_{5}$, $c_{1}=x_{1}x_{2}x_{5}$, $a_{2}=x_{2}x_{5}$, $c_{2}=x_{2}x_{3}x_{5}$. The path $\\{a_{1},a_{2}\\}$ is maximal weak because the divisors from $B$ of $c_{2}$ are $a_{2},u_{2},u_{3}$. Then $T_{1}=\\{a_{1},a_{2}\\}$ and we change in $P_{b}$ as in the proof the intervals $[x_{2},c^{\prime}_{2}]$, $[a_{2},c_{2}]$ by $[x_{2},c_{2}]$, $[u^{\prime}_{2},c^{\prime}_{2}]$. Thus the new $c^{\prime}_{2}$ is the old $c_{2}$. Now note that this new $c^{\prime}_{2}$ is a multiple of $u_{3}$ and it is the only monomial from $h(T_{1})$, which is a such multiple. Thus we had to take $u_{3}$ in the new $T^{\prime}_{1}$, and $u^{\prime}_{3}$ as well and certainly $c^{\prime}_{3}$ is added to $h(T_{1})$. Clearly, all divisors from $B$ of $c^{\prime}_{3}$ are in $T^{\prime}_{1}=T_{1}\cup\\{u_{3},u^{\prime}_{3}\\}$. But the former $u^{\prime}_{2}$ divides $c_{1}$ and so should be added to $T^{\prime}_{1}$. Thus we have $I^{\prime}=(b,E)$, $J^{\prime}=J\cap I^{\prime}$ and $I/(J,I^{\prime})$ has a partition of sdepth $3$ given by the intervals $[x_{2},c_{2}]$, $[x_{3},c^{\prime}_{3}]$, $[x_{1},x_{1}x_{2}x_{5}]$. If $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\geq 3$ then we get $\operatorname{sdepth}_{S}I/J\geq 3$, which is false. Otherwise, $\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}\leq 2$ and we get $\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq 2$ by [12, Theorem 4.3] and so $\operatorname{depth}_{S}I/J\leq 2$ using the Depth Lemma. ## References * [1] W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised edition. Cambridge University Press (1998). * [2] W. Bruns, C. Krattenthaler, J. Uliczka, Stanley decompositions and Hilbert depth in the Koszul complex, J. 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Popescu, Stanley conjecture on intersections of four monomial prime ideals, Communications in Alg., 41 (2013), 4351-4362, arXiv:AC/1009.5646. * [11] D. Popescu, Graph and depth of a square free monomial ideal, Proceedings of AMS,140 (2012), 3813-3822, arXiv:1104.5596v1, 2011. * [12] D. Popescu, Depth of factors of square free monomial ideals, to appear in Proceedings AMS, arXiv:AC/1110.1963. * [13] D. Popescu, Upper bounds of depth of monomial ideals, J. Commutative Algebra, 5, 2013, 323-327, arXiv:AC/1206.3977. * [14] D. Popescu, Depth of a monomial ideal, in Proceedings of the Seventh Congress of Romanian Mathematicians, 2011. * [15] D. Popescu, A. Zarojanu, Depth of some square free monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, 56(104) (2013),117-124. * [16] D. Popescu, A. Zarojanu, Depth of some special monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, 56(104) (2013), 365-368, arxiv:AC/1301.5171v1. * [17] D. Popescu, A. Zarojanu, Three generated, squarefree, monomial ideals , to appear in Bull. Math. Soc. Sci. Math. Roumanie, 58(106), no 3, (2015), arXiv:AC/1307.8292v6 * [18] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010),773-784. * [19] Y.H. Shen, Lexsegment ideals of Hilbert depth 1 (2012), arXiv:AC/1208.1822v1. * [20] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193. * [21] J. Uliczka, Remarks on Hilbert series of graded modules over polynomial rings, Manuscripta Math., 132 (2010), 159-168. * [22] A. Zarojanu, Stanley Conjecture on intersection of three monomial primary ideals, Bull. Math. Soc. Sc. Math. Roumanie, 55(103),(2012), 335-338.	arxiv-papers	2013-09-19T14:10:31	2024-09-04T02:49:51.158408	{ "license": "Public Domain", "authors": "Adrian Popescu and Dorin Popescu", "submitter": "Adrian Popescu", "url": "https://arxiv.org/abs/1309.4986" }
1309.5000	# Optimally focused cold atom systems obtained using density-density correlations Andika Putra Joint Quantum Institute, University of Maryland, and National Institute of Standards and Technology, College Park, Maryland, 20742, USA Daniel L. Campbell Joint Quantum Institute, University of Maryland, and National Institute of Standards and Technology, College Park, Maryland, 20742, USA Ryan M. Price Joint Quantum Institute, University of Maryland, and National Institute of Standards and Technology, College Park, Maryland, 20742, USA Subhadeep De Joint Quantum Institute, University of Maryland, and National Institute of Standards and Technology, College Park, Maryland, 20742, USA CSIR-National Physical Laboratory, New Delhi 110012, India I. B. Spielman Joint Quantum Institute, University of Maryland, and National Institute of Standards and Technology, College Park, Maryland, 20742, USA ###### Abstract Resonant absorption imaging is a common technique for detecting the two- dimensional column density of ultracold atom systems. In many cases, the system’s thickness along the imaging direction greatly exceeds the imaging system’s depth of field, making the identification of the optimally focused configuration difficult. Here we describe a systematic technique for bringing Bose-Einstein condensates (BEC) and other cold-atom systems into an optimal focus even when the ratio of the thickness to the depth of field is large: a factor of 8 in this demonstration with a BEC. This technique relies on defocus-induced artifacts in the Fourier-transformed density-density correlation function (the power spectral density, PSD). The spatial frequency at which these artifacts first appear in the PSD is maximized on focus; the focusing process therefore both identifies and maximizes the range of spatial frequencies over which the PSD is uncontaminated by finite-thickness effects. ## I Introduction Since the most important technique for obtaining properties of ultracold atoms is direct imaging, a well-designed and well-aligned imaging system is crucial for obtaining high quality data which is valid at all length scales. While large scale properties such as the system’s width or peak density can be obtained with little effort, significant care must be taken for experiments requiring very good spatial resolution Bakr et al. (2010); Sherson et al. (2010), or those studying correlations Folling et al. (2005); Hung et al. (2011). It is difficult to bring objects extended along the imaging axis, such as degenerate Fermi gases Granade et al. (2002); Regal et al. (2003), 3D Mott insulators Greiner et al. (2002), and Bose-Einstein condensates (BECs) Anderson et al. (1995); Davis et al. (1995), into focus particularly after time-of-flight (TOF) expansion because their spatial thickness often exceeds the imaging system’s depth of field. Even for such objects, a high degree of accuracy in focusing is required to minimize imaging artifacts. Understanding and minimizing these artifacts is particularly important when studying density-density correlations, where the artifacts can be confused with the correlation signal under study Hung et al. (2011); Choi et al. (2012); Seo et al. (2013); Langen (2013); De et al. (2012). Here we describe a fairly generic technique for focusing on these extended objects which is far more precise than simply optimizing the “sharpness” of imaged atom clouds. Absorption imaging is a ubiquitous approach for measuring the density distribution of ultracold atom systems Ketterle et al. (1999). A probe beam illuminates the atomic system, and the resulting shadow is imaged onto a scientific camera, typically a charge-coupled device (CCD) or complementary metal-oxide-semiconductor (CMOS) detector. Ideally, the fraction of light absorbed would be directly related to the two-dimensional column density $\rho_{\rm 2D}(x,y)=\int{\rm d}z\;\rho(x,y,z)$ of the atoms along the imaging direction ${\bf e}_{z}$, where $\rho(x,y,z)$ is the density of atoms. If the thickness $\delta z$ along ${\bf e}_{z}$ exceeds the imaging system’s depth of field, then some of the atomic distribution must necessarily be out of focus, invalidating any simple relationship between absorption and column density. Given this, it is a challenge to obtain the optimal focal plane of the extended system that minimizes the artifacts resulting from this defocus, e.g., at the center of a distribution symmetric along ${\bf e}_{z}$. Typically a system is brought into a focus by minimizing the size or apparent diffraction effects from a compact object such as a trapped BEC; in many cases, no such compact reference at the desired image plane is available. In this paper, we present a technique for determining the optimal focus of absorption-imaged extended objects. Using this technique, we identify the focal plane within an accuracy of $2\,\mathrm{\mu m}$ for a $\delta z=150\,\mathrm{\mu m}$ thick object. Specifically, given an object with density-density correlations Hung et al. (2011) with a spatial correlation length $\ell$, we show that observations of correlations in the optical absorption as a function of camera position allow us to bring the object into focus to within a fraction of the depth of field associated with $\ell$, even without knowing the details of the correlation function. This optimal focus is the camera position where the imaged auto-correlation function (ACF) most accurately reflects the atomic density-density correlations, minimizing both defocus-induced artifacts and the resolution limiting effect of the system’s finite thickness Langen (2013); De et al. (2012). In this paper, we review the basic theoretical formulation required to understand light propagating through an absorbing dielectric medium. We then consider several example images created by different idealized objects, in each case noting how to determine their optimal focus. Lastly, we experimentally apply this technique to images of BECs after TOF. ## II Theory Monochromatic light of free-space wavelength $\lambda$ and wavenumber $k_{0}=2\pi/\lambda$ propagating through an object with complex relative permittivity $\varepsilon(\mathbf{r})=\epsilon/\epsilon_{0}$ and relative susceptibility $\chi({\bf r})=\varepsilon(\mathbf{r})-1$, where $\epsilon$ is the permittivity, and $\epsilon_{0}$ is the electric constant, is described by the vectorial wave equation for the electric field $\mathbf{E(r)}$: $\nabla^{2}\mathbf{E(r)}+{k_{0}^{2}}\varepsilon(\mathbf{r})\mathbf{E(r)}=-\nabla\left[\mathbf{E(r)}\cdot\nabla\ln\varepsilon(\mathbf{r})\right].$ (1) In a medium where $\varepsilon(\mathbf{r})$ is slowly varying, the right-hand side (rhs) of Eq. (1) can be neglected, reducing Eq. (1) to separate scalar wave equations $\nabla^{2}E\mathbf{(r)}+k_{0}^{2}\varepsilon(\mathbf{r})E\mathbf{(r)}=0,$ for each vector component of $\mathbf{E(r)}$, e.g., we might have $\mathbf{E(r)}=E\mathbf{(r)}\mathbf{e}_{x}$ for light linearly polarized along $\mathbf{e}_{x}$. ### II.1 Wavefield propagation Here, we cast the above scalar wave equation into the form $\displaystyle-\frac{\partial^{2}E\mathbf{(r)}}{\partial z^{2}}$ $\displaystyle=$ $\displaystyle\left[\nabla^{2}_{\perp}+k_{0}^{2}\right]E\mathbf{(r)}+k_{0}^{2}\chi({\bf r})E(\mathbf{r}),$ (2) suitable for light predominantly traveling along ${\bf e}_{z}$. For a known field configuration at $E({\bf r})$ (such as the probe laser before it interacts with the atoms), Eq. (2) has the formal solution $E\left({\bf r}+\Delta z{\bf e}_{z}\right)=\exp\left[\pm i\Delta z\sqrt{\nabla^{2}_{\perp}+k_{0}^{2}+k_{0}^{2}\chi({\bf r})}\right]\\!E\left({\bf r}\right),$ (3) describing the field propagated a distance $\Delta z$ along ${\bf e}_{z}$. [plus minus sign] Wave propagation in free space [i.e., $\chi({\bf r})=0$ in Eq.(2)] is solved exactly in the angular spectrum representation Novotny and Hecht (2006) $\displaystyle E_{\rm fs}({\bf r}\\!+\\!\Delta z{\bf e}_{z})$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle\mathbf{P}(\Delta z)E({\bf r})$ $\displaystyle\\!\\!=\\!\\!\\!$ $\displaystyle\int\\!\mathrm{d}^{2}\mathbf{k}_{\rm 2D}\\!\left[\tilde{{\bf P}}(\mathbf{k}_{\rm 2D},\Delta z)\tilde{E}(\mathbf{k}_{\rm 2D},z)\right]\\!e^{i{\bf k}_{\rm 2D}\cdot\mathbf{r}_{\rm 2D}},$ for a forward going wave, with the 2D position ${\bf r}_{\rm 2D}=\left(x,y\right)$ and wavevector $\mathbf{k}_{\rm 2D}=\left(k_{x},k_{y}\right)$; the Fourier-transformed wavefield $\tilde{E}({\bf k}_{\rm 2D},z)=\int{\rm d}^{2}{\bf r}_{\rm 2D}\>\exp(-i{\bf k}_{\rm 2D}\cdot\mathbf{r}_{\rm 2D})E(\mathbf{r})$; and the transfer function for propagating a distance $\Delta z$ in free space $\tilde{{\bf P}}\left({\bf k}_{\rm 2D},\Delta z\right)=\exp\left[i\Delta z\left(k_{0}^{2}-k^{2}_{\rm 2D}\right)^{1/2}\right].$ The transfer function behaves differently in two regions of spatial frequencies: for $k^{2}_{\rm 2D}<k_{0}^{2}$, $\tilde{\bf P}$ is oscillatory (propagating regime), and for $k^{2}_{\rm 2D}>k_{0}^{2}$, it is exponentially decaying (evanescent regime). Meanwhile, considering only $\chi({\bf r})$ [neglecting the first term in the rhs of Eq. (2)], the absorption and refraction of light traveling a distance $\Delta z$ is described by $\displaystyle E_{\rm BL}({\bf r}\\!+\\!\Delta z{\bf e}_{z})$ $\displaystyle=$ $\displaystyle\mathbf{Q}(\Delta z)E({\bf r})$ $\displaystyle=$ $\displaystyle\exp\left[ik_{0}\int_{z}^{z+\Delta z}\mathrm{d}z\sqrt{\chi({\bf r})}\right]E({\bf r}).$ Unlike the usual Beer-Lambert (BL, discussed in Sect. II.2), this expression alone does not reflect a good approximation to beam propagation for systems of any significant thickness. ### II.2 Beer-Lambert law and the paraxial approximation To better understand the independent influence of the beam’s propagation and its interaction with matter, we apply the paraxial approximation to Eq. (2), allowing us to draw an analogue between the paraxial wave equation and the Schrödinger equation, which can be solved numerically using a split-step Fourier method (SSFM) J. A. Fleck, Jr. (1995). To understand the difference between Eq. (II.1) and the usual BL law, we again turn to Eq. (2), now assuming that the electric field can be written as $E({\bf r})=\exp\left(ik_{0}z\right)E^{\prime}({\bf r})$, where $E^{\prime}({\bf r})$ is a slowly varying envelope along ${\bf e}_{z}$. Inserting this form into Eq. (2) gives the paraxial wave equation $\displaystyle-2ik_{0}\frac{\partial E^{\prime}\mathbf{(r)}}{\partial z}$ $\displaystyle=$ $\displaystyle\nabla^{2}_{\perp}E^{\prime}\mathbf{(r)}+k_{0}^{2}\chi({\bf r})E^{\prime}(\mathbf{r}),$ (6) where the assumed weak $z$ dependence of $E^{\prime}({\bf r})$ allowed us to drop $\partial^{2}_{z}E^{\prime}({\bf r})$. Like above, the spatial evolution of an initial $E^{\prime}({\bf r})$ can be partitioned into a spectral part $\tilde{\bf P}^{\prime}({\bf k}_{\rm 2D},\Delta z)$ and a coordinate part ${\bf Q}^{\prime}(\Delta z)$, with $\displaystyle\tilde{\bf P}^{\prime}({\bf k}_{\rm 2D},\Delta z)$ $\displaystyle=\exp\left(-i\frac{k_{\rm 2D}^{2}}{2k_{0}}\Delta z\right)$ (7) $\displaystyle{\bf Q}^{\prime}(\Delta z)$ $\displaystyle=\exp\left[i\frac{k_{0}}{2}\int_{z}^{z+\Delta z}\chi({\bf r})\mathrm{d}z\right].$ (8) For the paraxial approximation to be valid, the condition $\left\|\chi({\bf r})\right\|\ll 1$ must also hold: otherwise the ${\bf Q}^{\prime}(\Delta z)$ evolution would lead $E^{\prime}({\bf r})$ to depend strongly on $z$. We numerically evolve the paraxial wave equation [Eq. (6)] along ${\bf e}_{z}$ using a split-step Fourier method (SSFM) Korpel et al. (1986); Feit and J. A. Fleck (1978), where the operators in the rhs of Eq. (6) are split into two: one operator represents wave propagation in a uniform medium using Eq. (7) and the other operator takes into account the effect of refractive index variation using Eq. (8). In the SSFM, we alternately apply the two evolution operators with steps of size $\Delta z$. For each step, the complex amplitude $E^{\prime}(\mathbf{r})$ is propagated first by $\mathbf{P}^{\prime}(\Delta z/2)$, then by $\mathbf{Q}^{\prime}(\Delta z)$, and then again by $\mathbf{P}^{\prime}(\Delta z/2)$. The resulting symmetrized split evolution $E^{\prime}\left({\bf r}+\Delta z{\bf e}_{z}\right)=\mathbf{P^{\prime}}\left(\Delta z/2\right)\mathbf{Q^{\prime}}(\Delta z)\mathbf{P^{\prime}}\left(\Delta z/2\right)E^{\prime}\left({\bf r}\right),$ has its first correction at order $\Delta z^{3}$. The paraxial equations allow us to introduce the depth of field $d_{\rm dof}=\frac{2k_{0}}{k_{\rm max}^{2}}=\frac{l_{\rm min}^{2}}{\pi\lambda},$ (9) where $k_{\rm max}^{2}$ is the largest $k_{\rm 2D}$ of interest and $l_{\rm min}=2\pi/k_{\rm max}$ is the corresponding minimum length scale [these might be specified by: the maximum significant wavevector in $\chi({\bf k}_{\rm 2D},z)$; the resolution of the physical imaging system; or at most by $k_{0}$]. We obtain the BL law by assuming that the system is thin along ${\bf e}_{z}$, i.e., both $\delta z\ll d_{\rm dof}$, and $\tilde{\bf P}^{\prime}({\bf k}_{\rm 2D},\Delta z)$ may be neglected. For purely absorbing materials where $\chi({\bf r})\propto i\sigma_{0}\rho({\bf r})$, this gives the usual BL law $\displaystyle I({\bf r}+\delta z{\bf e}_{z})$ $\displaystyle=$ $\displaystyle\exp\left[-\sigma_{0}\int_{z}^{z+\delta z}\rho({\bf r})\mathrm{d}z\right]I({\bf r})$ (10) describing the attenuation of the free space optical intensity $I({\bf r})=c\epsilon_{0}\left\|E({\bf r})\right\|^{2}/2$ by absorbers of density $\rho({\bf r})$ and scattering cross-section $\sigma_{0}$. This BL result can also be obtained without the paraxial approximation by first neglecting the $\nabla^{2}_{\perp}$ term in Eq. (2) (valid when $k_{\rm max}\delta z\ll 1$: a more strict requirement than in the paraxial approximation where we had $\delta z\ll d_{\rm dof}$) and again assuming $\left\|\chi({\bf r})\right\|\ll 1$, a small relative susceptibility 111Moreover, the gradient term $\nabla\ln\varepsilon(\mathbf{r})$ in Eq. (1) cannot be safely neglected for systems where $\left\|\chi({\bf r})\right\|$ is large or sufficiently rapidly varying, although this would generally imply a breakdown of the paraxial approximation as well.. In experiment, the BL law is generally applied by comparing the intensities $I({\bf r}_{\rm 2D})$ and $I_{0}({\bf r}_{\rm 2D})$ measured with and without atoms present, respectively. This relates the optical depth ${\rm OD}({\bf r}_{\rm 2D})\equiv-\ln\frac{I({\bf r}_{\rm 2D})}{I_{0}({\bf r}_{\rm 2D})}=\sigma_{0}\rho_{\rm 2D}({\bf r}_{2D})$ to the 2D column density. In cold atom experiments, this column density is the primary observable in experiment. ### II.3 Absorption Imaging Figure 1: Dependence of BEC images on image plane position. (a) Intensity corrected optical depth measured $d=-54\,\mathrm{\mu}{\rm m}$, 0 $\mu{\rm m}$, and $54\,\mathrm{\mu}{\rm m}$ from the optimal focus: images (right) and line cuts at $y=0$ (left). (b) The peak optical depth depends only weakly on $d$; is not maximized at $d=0$; and has no structure on the $2\,\mathrm{\mu m}$ scale. Here we consider systems of ultracold atoms illuminated by laser light on a cycling transition, where the atom-light interaction is described by a complex relative susceptibility $\chi({\bf r})=\frac{\sigma_{0}}{k_{0}}\left[\frac{i-2\delta/\Gamma}{1+{I}/{I_{\rm sat}}+\left({2\delta}/{\Gamma}\right)^{2}}\right]\rho(\mathbf{r}).$ $\rho(\mathbf{r})$ is the atomic density; $\delta$ is the laser’s detuning from atomic resonance; $\sigma_{0}=6\pi/k_{0}^{2}$ is the resonant scattering cross-section; $\Gamma$ is the atomic linewidth; and $I_{\rm sat}$ is the saturation intensity Ketterle et al. (1999). The standard BL law is valid for dilute ($\rho\ll k_{0}^{3}$, see Ref. 222A dilute regime, with density $\rho\ll k_{0}^{3}$, can be achieved by letting the atomic cloud to ballistically expand.), spatially thin systems ($k_{0}\delta z\ll 1$), illuminated by low intensity ($I_{0}\ll I_{\rm sat}$) probe beams. The $I_{0}\ll I_{\rm sat}$ requirement can be lifted by introducing the intensity-corrected optical depth ${\rm{OD_{cor}}}({\bf r}_{2D})\equiv-\ln\frac{I({\bf r}_{2D})}{I_{0}({\bf r}_{2D})}+\frac{I_{0}({\bf r}_{2D})-I({\bf r}_{2D})}{I_{\rm sat}},$ (11) which is related to the column density $\rho_{\rm 2D}({\bf r}_{2D})=\frac{{\rm{OD_{cor}}}({\bf r}_{2D})}{\sigma_{0}},$ (12) of dilute ($\rho\ll k_{0}^{3}$), spatially thin systems ($k_{0}\delta z\ll 1$). Due to the limited dynamic range of the camera’s pixels Reinaudi et al. (2007) and the presence of background light, it is technically difficult to reliably detect uncorrected optical depths, larger than $\approx 4$. Thus, we deliberately select $I_{0}>I_{\rm sat}$, saturating the transition with $I_{0}$ such that ${\rm OD_{cor}}<3$. In addition, the spatial thickness of many cold atom systems exceed the depth of field leaving parts of its distribution along imaging direction inevitably out of focus, thereby invalidating Eq. (12). Even for dilute clouds (after sufficient TOF), images taken an equal distance above and below the focal plane can differ. This lack of symmetry makes a straightforward determination of the optimal focus difficult (lensing effects from even slightly off- resonance imaging beams and aberrations in the imaging system can complicate the situation further.) To illustrate this difficulty, we consider images of BECs with the focal plane displaced a distance $d=-54\,\mathrm{\mu m}$, 0 ${\rm\mu m}$, and $54\,\mathrm{\mu m}$ from the BECs’ center (see Fig. 1). Because the BEC is thick compared to the depth of field, Eq. (12) does not hold; in addition lensing effects cause the cloud’s peak $\rm{OD_{cor}}$ to behave asymmetrically when the focus is behind or in front of the cloud. In these images, there are no sharp features that identify the optimal focus at the micron level. Owing to the weak dependence of large-scale parameters such as peak-height or width on defocus, such precise focusing is not required in many experiments. As we see below, experiments that study correlations within such images are extremely sensitive to defocus and new methods are required. Our technique brings images such as these into focus, identifying an optimal focal plane at the $\approx 2\,\mathrm{\mu m}$ level. ### II.4 Modeling To obtain a basic understanding of our approach, we first consider the defocused image of a $1\,\mathrm{\mu m}$ thick absorbing medium, inhomogenous ${\bf e}_{x}$-${\bf e}_{y}$ plane, bounded above and below by vacuum, with, $\chi({\bf r})=ig(x,y)$ for $z\in(-0.5\,\mathrm{\mu m},0.5\,\mathrm{\mu m})$, where $g(x,y)\geq 0$ is a Poisson distributed random variable. Like atoms illuminated on resonance, this medium has a purely imaginary susceptibility. The illuminating light is modeled by a plane wave with wavelength $\lambda=780\,\mathrm{nm}$ suitable for imaging our 87Rb Bose-Einstein condensates 333In our SSFM simulation of light traversing this medium, we used a $\Delta z=1\,\mathrm{\mu m}$ step size.. While this object has no visible structure, by virtue of its spectrally flat density-density correlation function, it can be brought into focus. The imaged intensity pattern $I(x,y)$ from this $1\,\mathrm{\mu m}$ layer appears random at various distances from focus, but its correlations become oscillatory. To reveal this information, we turn to its spatial power spectral density: the magnitude squared of $I(x,y)$’s Fourier transform 444The Wiener- Khincin theorem states that the spectral decomposition of the autocorrelation function is equal to the power spectral density.. The PSD is circularly symmetric in the spatial frequency $\mathbf{k}_{\rm 2D}=(k_{x},k_{y})$ plane. Fig. 2a shows the PSD in this $k=\left\|\mathbf{k}_{\rm 2D}\right\|$ “radial” direction as a function of distance from focus $d$. This PSD has a fringe pattern; the wavevector of the first minimum exceeds the maximum imaged wavevector only near the image’s focus at $d=0$ ${\mu m}$. The physical origin of this structure can be understood by turning to the paraxial wave equations [Eqs. (7) and (8)], and by first studying a single absorber at ${\bf r}=0$ illuminated by a plane wave $E_{0}^{\prime}({\bf r}_{\rm 2D},0^{-})=E_{0}$. Equation (8) shows that a thin absorber simply changes the amplitude of the field, leaving its phase untouched, and for simplicity, we assume this absorber has a Gaussian profile in the ${\bf e}_{x}$-${\bf e}_{y}$ plane with width $w_{0}$. Thus the electric field just following the absorber is changed by $\delta E^{\prime}({\bf r}_{\rm 2D},0^{+})=-\delta E\exp\left[-r^{2}/w_{0}^{2}\right]$, with $r^{2}=x^{2}+y^{2}$. The propagation of such a gaussian mode by a distance $d$ along ${\bf e}_{z}$ can be solved exactly in the paraxial approximation, and in the spectral basis this is $\displaystyle\delta\tilde{E}^{\prime}({\bf k}_{\rm 2D},d)$ $\displaystyle=-\pi w^{2}_{0}\delta E\exp\left[-\frac{w_{0}^{2}k_{\rm 2D}^{2}}{4}\left(1+\frac{2i}{w_{0}^{2}k_{0}}d\right)\right].$ The total field from an absorber located at a different location ${\bf r}_{0}$ in the ${\bf e}_{x}$-${\bf e}_{y}$ plane simply acquires an overall phase factor $\exp\left[-i{\bf k}_{\rm 2D}\cdot{\bf r}_{0}\right]$. We now compute the experimentally relevant optical depth by taking the reverse Fourier transform of the full electric field, computing the intensity, then the optical depth, and taking the Fourier transform to obtain (retaining terms of order $\delta E/E_{0}$) $\displaystyle\widetilde{\rm OD}$ $\displaystyle=\frac{2\pi w_{0}^{2}\delta E}{E_{0}}\exp\left(-\frac{w_{0}^{2}k_{\rm 2D}^{2}}{4}\right)\cos\left(\frac{k_{\rm 2D}^{2}d}{2k_{0}}\right),$ (13) with the same overall phase factor depending on the initial position. Averaging over $N$ randomly placed absorbers therefore gives an overall signal scaling as $\sqrt{N}$ with a random overall phase. Taking the magnitude squared gives the PSD $\displaystyle{\rm PSD}_{\rm thin}$ $\displaystyle={\rm PSD}_{0}\times\cos^{2}\left(\frac{k_{\rm 2D}^{2}d}{2k_{0}}\right),$ with $\displaystyle{\rm PSD}_{0}$ $\displaystyle=N\left(\frac{2\pi w_{0}^{2}\delta E}{E_{0}}\right)^{2}\exp\left(-\frac{w_{0}^{2}k_{\rm 2D}^{2}}{2}\right).$ This quantity has zeros located at $k_{\rm zero}[n]=\sqrt{2\pi(n+1/2)k_{0}/d}$ for integer $n$. In our numerical simulation, the minima follow the functional form $k_{\rm zero}[n]=A[n]\|d\|^{-1/2}$ as shown by the dotted lines in Fig. 2, with $A[0]\approx 5.08$ and $A[1]\approx 8.76$ for the first and second zeros: the expected values for $A[n]$. Thus, for this thin apparently structureless system, fringes in the PSD allow us to identify the focal plane. Figure 2: (a) Spatial PSD of the intensity produced by $1\,\mathrm{\mu m}$ thick layer of randomly distributed scatterers showing that fringes diverge in focus. (b) PSD produced by a $100\,\mathrm{\mu m}$ thick sheet of random columnar scatterers. (c) PSD produced by a $100\,\mathrm{\mu m}$ thick sheet of random scatterers. The dotted lines are functional forms of the lowest curved-fringes, and in each case $d$ is measured from the objects’ center. To demonstrate the technique of finding optimal focus of an extended object, we now consider a second disordered scattering potential with a columnar structure, now $100\,\mathrm{\mu m}$ thick, i.e., $\chi({\bf r})=ig(x,y)$ for $z\in(-50\,\mathrm{\mu m},50\,\mathrm{\mu m})$, where again $g(x,y)\geq 0$ is a Poisson distributed random variable. This object’s PSD is plotted as a function of distance $d$ from its center in Fig. 2b; in addition to the same fringe pattern as for the $1\,\mathrm{\mu m}$ thick case, the PSD now vanishes at specific spatial frequencies independent of $d$. To model this, we note that the absorbers can now be located at a distance $z$ from the symmetry plane, so in Eq. (13), we replace $d\rightarrow d-z$ and integrate $z$ from $-\delta z/2$ to $\delta z/2$, which ultimately gives the PSD $\displaystyle{\rm PSD}_{\rm col}$ $\displaystyle={\rm PSD}_{0}\times\cos^{2}\left(\frac{k_{\rm 2D}^{2}d}{2k_{0}}\right){\rm sinc}^{2}\left(\frac{k_{\rm 2D}^{2}\delta z}{4k_{0}}\right).$ This predicts the appearance of additional zeros located at $k^{\prime}_{\rm zero}[m]=\sqrt{4\pi mk_{0}/\delta z}$ for non-zero integer $m$ (this is an artifact of the box-like density distribution of atoms, and would be greatly softened in real systems where the density drops smoothly to zero). In our example, the lowest order horizontal fringes is located at $k^{\prime}_{\rm zero}[1]=1.00\,\mathrm{\mu m^{-1}}$. Here again, we easily determine the optimal focus, $d=0\ {\mu m}$, from the diverging curved-fringes. Next, we consider a scattering potential fully disordered in 3D, again with a $100\,\mathrm{\mu m}$ thickness, i.e., $\chi({\bf r})=ig(x,y,z)$ for $z\in(-50\,\mathrm{\mu m},50\,\mathrm{\mu m})$, where $g(x,y,z)\geq 0$ is a Poisson distributed random variable. In this case, the independent random scatterers along imaging direction causes the PSD to rapidly loose structure with increasing $k_{\rm 2D}$ (see Fig. 2c). Here too, our random scatter model can be applied, giving $\displaystyle{\rm PSD}_{\rm rnd}$ $\displaystyle={\rm PSD}_{0}\times\left[\frac{\cos\left(k_{\rm 2D}^{2}d/k_{0}\right){\rm sinc}\left(k_{\rm 2D}^{2}\delta z/2k_{0}\right)+1}{2}\right].$ This reduces to our earlier result when $\delta z\rightarrow 0$ for a thin system and shows that, while the same fringes exist, they are rapidly attenuated for larger spatial frequencies, where the signal approaches a constant background value. However, in principle the curved-fringes still allow the optimal focus to be identified. ## III Optimal focusing of Elongated Bose-Einstein Condensates Using on our model, we now consider absorption imaged BECs and implement the technique presented in previous section to find the optimal focus. Figure 3: (a) Absorption imaged elongated BEC with density fluctuations. (b) 1D PSD of column density along weakly trap direction ${\bf e}_{x}$ as a function of $t_{\rm TOF}$. (c) Values of $k$ where the 1D PSD is minimum. The two lowest such $k$-fringes are depicted. Symbols denote the fringe locations extracted from (b) plotted along with Lorentzian fits (dotted lines), determining the optimal focus. The solid curves depict theoretical functional forms for the two lowest order fringes. In (b) and (c) the dashed line marks $k=k^{\prime}_{\rm zero}[1]=0.82\,\mathrm{\mu m^{-1}}$ for our condensate thickness of $150\,\mathrm{\mu m}$; the dotted line marks $k=k^{\prime}_{\rm zero}[1]/\sqrt{2}$, below which the ACF of the focused images reliably reflects the ACF $n_{\rm 2D}({\bf r}_{\rm 2D})$. We prepared $N=7\times 10^{5}$ atom 87Rb Bose-Einstein condensates in the $\left\|5\mbox{S}_{1/2},F=1,m_{F}=0\right\rangle$ electronic ground state in a crossed-dipole trap with frequencies $\omega_{x,y,z}=2\pi\times(3.1,135,135)\,\mathrm{Hz}$. In situ, the BECs were javelin shaped owing to the extremely anisotropic confining potential. After a 17 ms to 21 ms TOF, we repumped into the $f=2$ manifold, and resonantly imaged on the $\left\|5\mbox{S}_{1/2},f=2,m_{F}=2\right\rangle$ to $\left\|5\mbox{P}_{3/2},f=3,m_{F}=3\right\rangle$ cycling transition with a $\lambda\approx 780.2\,\mathrm{nm}$ probe laser. The imaging system consisted of a CCD camera and two pairs of lenses functioning as a compound microscope, magnifying the intensity pattern at the object by a factor of $\approx 6$ at the image plane. The first pair of objective lenses, with effective focal length (efl) $\hat{f_{1}}=53.6\,\mathrm{mm}$, collimated the light diffracted by the cloud and were separated by a distance $D=\hat{f_{1}}+\hat{f_{2}}$ from a second pair of lenses with a $\hat{f_{2}}=325\,\mathrm{mm}$ efl. The resulting $0.23$ numerical aperture implies that a $10.6\,\mathrm{\mu m}$ diffraction-limited spot on our CCD sensor is larger than its $5.6\,\mathrm{\mu m}$ pixel size. The associated $1.7\,\mathrm{\mu m}$ spot-size on the cloud gives a $d_{\rm dof}=18.6\,\mathrm{\mu m}$ depth of field in our imaging system Inoue and Oldenbourg (1995). Instead of varying the distance from focus by physically moving imaging lenses or the CCD, we changed the time during which the BEC fell along ${\bf e}_{z}$ and obtained absorption images with TOF times $t_{\rm TOF}$ from $17.0\,\mathrm{ms}$ to $21.0\,\mathrm{ms}$. At these TOFs, the condensates’ radii were $R_{y,z}\approx 75(5)\,\mathrm{\mu m}$ and $R_{x}\approx 210(10)\,\mathrm{\mu m}$. Initially, the cloud was elongated in the harmonic trap with aspect ratio $43$ to 1. The initial $43:1$ aspect ratio was reduced to $2.65:1$ after TOF, and the transverse size of the cloud exceeded the imaging depth of field by a factor of $8$. Figure 3b shows the 1D PSD of the atoms’ corrected optical depths along ${\bf e}_{z}$, which is directly related to the absorption intensity through Eq. (11). The fluctuations in the BEC’s density distribution behave like the randomly modulated $\chi({\bf r})$ in our example systems, creating a recurring fringe pattern in the PSD spectrum as obtained in Fig. 3c. The fringes are quite pronounced for quasi one-dimensional BECs, where initial phase fluctuations map into pancake-shaped density fluctuations arrayed along the initially long axis after TOF Dettmer et al. (2001). Despite the decreased contrast at high spatial frequencies due to the BEC extent along ${\bf e}_{z}$, we clearly observe fringes curving as a function of $t_{\rm TOF}$ in Fig. 3c. This allows us to determine the optimal focus of the system. From the above experimental data, we fit the two lowest order fringes to $k_{m}\left[(d-z_{0})^{2}/\delta z^{2}+1\right]^{-1/4}$, a peaked function with the expected $d^{-1/2}$ behavior away from $z_{0}$. The fits give an optimal focus location of $z_{0}[0]=1836(2)\,\mathrm{\mu m}$ using the zeroth order fringe or of $z_{0}[1]=1837(2)\,\mathrm{\mu m}$ using the first order fringe. These values correspond to a TOF of $19.36(1)\,\mathrm{ms}$. We are thus able to determine the optimal focus within $\approx 2\,\mathrm{\mu m}$ or equivalently $\approx 10\,\mathrm{\mu s}$ in TOF. Comparing the experimental data to the theoretical forms, we notice that the fringes are slightly asymmetrical with their locations slightly below theoretical ones for larger TOF. Based on our simulations, this likely results from the $z$ dependent magnification of our imaging system, which changes by about 10% as the atoms fall from 1420 $\mu{\rm m}$ to 2150 $\mu{\rm m}$ (17 ms to 21 ms TOF). ## IV Summary We presented a systematic method to bring clouds of ultracold atoms, particularly initially elongated BECs, into an optimal focus. The density fluctuations in the BECs after TOF acted like random scatterers, creating diffraction pattern which changed predictably as a function of distance from the optimal focus. Using TOF absorption imaging, we demonstrated this method, pinpointing the optimal focus of the BEC to within 2 $\mu$m for a $150\,\mathrm{\mu m}$ thick BEC. This robust technique is easily implemented, requires no hardware changes, and uses a minimum of computation. ###### Acknowledgements. We thank F. E. Becerra, A. Hu, and W. D. Phillips for a careful reading of the manuscript. We acknowledge the financial support from the NSF through the Physics Frontier Center at JQI, and the ARO with funds from both the Atomtronics MURI and DARPA’s OLE Program. ## References * Bakr et al. (2010) W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, Science 329, 547-550 (2010). * Sherson et al. (2010) J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. 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Lett. 32, 3143 (2007). * Inoue and Oldenbourg (1995) S. Inoue and R. Oldenbourg, _Handbook of Optics_ (McGraw-Hill, 1995), 2nd ed. * Dettmer et al. (2001) S. Dettmer, D. Hellweg, P. Ryytty, J. J. Arlt, W. Ertmer, K. Sengstock, D. S. Petrov, G. V. Shlyapnikov, H. Kreutzmann, L. Santos, et al., Phys. Rev. Lett. 87, 160406 (2001).	arxiv-papers	2013-09-19T14:37:27	2024-09-04T02:49:51.169761	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Andika Putra and Daniel L. Campbell and Ryan M. Price and Subhadeep De\n and I. B. Spielman", "submitter": "Andika Putra", "url": "https://arxiv.org/abs/1309.5000" }
1309.5049	∎ 11institutetext: R. Liao 22institutetext: B. Bellalta 33institutetext: T.C. Minh 44institutetext: J. Barcelo 55institutetext: M. Oliver 66institutetext: Department of Information and Communication Technologies Universitat Pompeu Fabra, Barcelona, 08018, Spain 66email: {ruizhi.liao, boris.bellalta, trang.cao, jaume.barcelo, miquel.oliver}@upf.edu # Uni-MUMAC: A Unified Down/Up-link MU-MIMO MAC Protocol for IEEE 802.11ac WLANs Ruizhi Liao Boris Bellalta Trang Cao Minh Jaume Barcelo Miquel Oliver ###### Abstract Due to the dominance of the downlink traffic in Wireless Local Area Networks (WLANs), a large number of previous research efforts have been put to enhance the transmission from the Access Point (AP) to stations (STAs). The downlink Multi-User Multiple-Input Multiple-Output (MU-MIMO) technique, supported by the latest IEEE amendment-802.11ac, is considered as one of the key enhancements leading WLANs to the Gigabit era. However, as cloud uploading services, Peer-to-Peer (P$2$P) and telepresence applications get popular, the need for a higher uplink capacity becomes inevitable. In this paper, a unified down/up-link Medium Access Control (MAC) protocol called Uni-MUMAC is proposed to enhance the performance of IEEE 802.11ac WLANs by exploring the multi-user spatial multiplexing technique. Specifically, in the downlink, we implement an IEEE 802.11ac-compliant MU-MIMO transmission scheme to allow the AP to simultaneously send frames to a group of STAs. In the uplink, we extend the traditional one round channel access contention to two rounds, which coordinate multiple STAs to transmit frames to the AP simultaneously. $2$-nd round Contention Window ($CW_{\text{2nd}}$), a parameter that makes the length of the $2$-nd contention round elastic according to the traffic condition, is introduced. Uni-MUMAC is evaluated through simulations in saturated and non-saturated conditions when both downlink and uplink traffic are present in the system. We also propose an analytic saturation model to validate the simulation results. By properly setting $CW_{\text{2nd}}$ and other parameters, Uni-MUMAC is compared to a prominent multi-user transmission scheme in the literature. The results exhibit that Uni-MUMAC not only performs well in the downlink-dominant scenario, but it is also able to balance both the downlink and uplink throughput in the emerging uplink bandwidth-hungry scenario. ###### Keywords: MAC MU-MIMO down/up-link IEEE 802.11ac WLANs ## 1 Introduction IEEE 802.11 Wireless Local Area Networks (WLANs) is becoming an indispensable part of our life, at homes and working places. Due to the problems, such as frame collisions and protocol overheads, the throughput of WLANs is significantly lower than the raw data rate of what the Physical (PHY) layer can achieve ciscoAC . The evolution of Internet traffic is going to exacerbate this low-throughput problem. The Internet traffic shifts from web browsings and file transfers to a wide variety of applications, many of which integrate content-rich files provided by users DBLP:conf/icumt/KihlOLA10 ; DBLP:journals/telsys/WamserPSHT11 . This shift, mainly driven by the bandwidth-hungry multimedia applications (e.g., web HDTV, video sharing and wireless display), demands a performance increase in both downlink and uplink of WLANs ciscoWhiteP . Spatial multiplexing is one of the current trends (the spatial diversity and the frame aggregation are among others) aiming at improving the performance of wireless systems. IEEE 802.11n 5307322 supports spatial multiplexing in the point-to-point communication mode (i.e., Single-user MIMO or SU-MIMO). The point-to-multipoint communication mode, for example, the transmission from the Access Point (AP) to multiple stations (STAs) (i.e., downlink Multi-user MIMO or MU-MIMO), is supported by the latest IEEE amendment-802.11ac 6359961 . However, the uplink MU-MIMO enhancement, which is crucial to mitigate collisions and to satisfy the performance requirements in the uploading- intensive scenario, has not been considered by any IEEE standard. In this paper, we propose a unified down/up-link MU-MIMO Medium Access Control (MAC) protocol called Uni-MUMAC, which coordinates distributed STAs to exploit the spatial multiplexing gain to improve the performance of IEEE 802.11ac WLANs. The main contributions are summarized as follows. 1) Two separate MU- MIMO MAC protocols, one for the downlink transmission 5733223 and the other one for the uplink transmission 6314214 , are integrated into a unified MU- MIMO MAC protocol. Compared to 5733223 and 6314214 , where only one-way traffic is considered (i.e., the downlink or the uplink), the presence of both downlink and uplink transmissions has been taken into account. 2) A special focus is placed at finding the most suitable value of the $2$-nd round Contention Window ($CW_{\text{2nd}}$) to obtain the highest system throughput, and the impact of the optimized uplink transmission on the downlink is discussed. With the optimized $CW_{\text{2nd}}$ and other properly configured parameters (e.g., the number of aggregated frames and the queue length of the AP), Uni-MUMAC is then extensively evaluated through simulations in the downlink-dominant and the down/up-link balanced traffic scenarios in IEEE 802.11ac based WLANs. 3) An analytic model is developed to validate the simulation results, and a prominent proposal in the literature is implemented to compare with our scheme. The rest of the paper is organized as follows. First, Section 2 explores some of the key MU-MIMO MAC proposals in the literature. Then, Section 3 introduces the modified frame structure and detailed Uni-MUMAC operating procedures. After that, Section 4 gives the considered scenarios to evaluate Uni-MUMAC, the saturation throughput model, simulation results and observations. Finally, Section 5 concludes the paper and discusses the future research challenges. ## 2 Related Work Most previous work has put efforts on adjusting MAC parameters or extending MAC functions to improve the performance of WLANs. In the downlink, the spatial multiplexing technique has recently gained much attention. To support it, many proposals in the literature adopt the following MAC procedure. The AP firstly sends out a modified Request to Send (RTS) containing a group of targeted STAs, then those listed STAs estimate the channel, add the estimated Channel State Information (CSI) into the extended Clear to Send (CTS) and send it back. As soon as the AP receives all successful CTSs, it precodes the outgoing signals and sends multiple data frames simultaneously. Cai et al. in conf/globecom/CaiSZSMW08 propose a distributed MU-MIMO MAC protocol that modifies RTS and CTS frames to estimate the channel, based on which, the AP is able to concurrently transmit frames to multiple STAs. Kartsakli et al. in DBLP:conf/icc/KartsakliZAV09 consider an infrastructured WLAN and propose four multi-user scheduling schemes to simultaneously transmit frames to STAs. The results show that the proposal achieves notable gains compared to that of the single user case. Gong et al. in DBLP:conf/globecom/GongPSWM10 propose a modified Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) protocol with three different ACK- replying mechanisms. The authors claim that the proposed protocol can provide a considerable performance improvement against the beamforming based approach when Signal-to-noise Ratio (SNR) is high. Zhu et al. in DBLP:conf/ccnc/ZhuBKAN12 investigate the required MAC modifications to support downlink MU-MIMO transmissions by focusing on the fairness issue. The proposed Transmit Opportunity (TXOP) sharing scheme not only obtains a higher throughput but is also more fair than the conventional mechanism. Cha et al. in 6214021 compare the performance of a downlink MU-MIMO scheme with a Space Time Block Coding (STBC) based frame aggregation scheme. The results show that the former produces a higher throughput than the latter if transmitted frames are of similar length. The uplink enhancement is getting more attention as the popularity of P$2$P and cloud applications increases. In general, there are two broad categories of uplink MU-MIMO MAC enhancements, namely, the un-coordinated access and the coordinated access. The former utilizes the MAC random mechanism to decide which STAs are allowed for data transmissions, while the latter employs the AP to schedule STAs’ uplink access. Some of the un-coordinated uplink access schemes are sampled as follows. In DBLP:conf/wcnc/JinJHS08 , Jin et al. evaluate the performance of uplink MU- MIMO transmissions in the IEEE 802.11 basic access mode, where the simultaneous uplink transmissions are on the random access basis and the channel coefficients of each STA are assumed to be known by the AP. In 4025043 , Zheng et al. present a Distributed Coordination Function (DCF) enhancement called Two-Round RTS Contention (TRRC) to take advantage of the spatial domain. The proposed scheme allows STAs to contend for the channel after a successful RTS is detected. In kutse , Tan et al. present a distributed MAC scheme called Carrier Counting Multiple Access (CCMA), where a beacon that contains the uplink access threshold is announced by the AP periodically. Based on the threshold, STAs count the number of ongoing transmissions by monitoring preambles, and then decide to contend for the channel or stay idle. In BabichC10 , Babich et al. investigate the theoretical model of asynchronous frame transmissions, where a STA is allowed to transmit even if other STAs are already transmitting. Some of the coordinated uplink access schemes are overviewed as follows. In ttanc , Tandai et al. propose a synchronized uplink transmission scheme coordinated by the AP. On receiving requests from STAs, the AP broadcasts a pilot-Requesting CTS (pR-CTS) to schedule STAs’ pilot transmissions for estimating the channel. After obtaining the CSI, the AP sends a Notifying-CTS (N-CTS) to inform the selected STAs to transmit frames in parallel. In SZhou , Zhou et al. propose a two-round channel contention mechanism, which divides the MAC procedure into two parts, namely, the random access and the data transmission. The random access terminates when the AP receives a predefined number of successful RTSs, and then the data transmission follows. In journals/twc/Zhang10 , Zhang et al. further extends the two contention rounds to multiple rounds, which enable more STAs to be involved in parallel uplink transmissions. The proposed protocol can fall-back to the single-round mode automatically on condition that the traffic is low and the single-round scheme can provide higher throughput. In 6302115 , Jung et al. present an asynchronous uplink Multi-Packet Reception (MPR) scheme, where an additional feedback channel is assumed to be employed by the AP to acknowledge the successful frame receptions along with other ongoing transmissions. Only a few work has combined the downlink and the uplink transmissions together. In DBLP:conf/macom/ShenLSDWZ12 , Shen et al. propose a High Throughput MIMO (HT-MIMO) MAC protocol, which utilizes frequency signatures to differentiate simultaneously-received control messages. The proposal works in the Point Coordination Function (PCF) mode, hence both downlink and uplink transmissions can be only initiated by the AP. In 4917548 , Jin et al. focus on the unbalanced throughput problem between downlink and uplink, where a Contention Window (CW) adjustment scheme and a random piggyback scheme are proposed to increase the downlink throughput ratio. In DBLP:conf/wcnc/LiAL10 , Li et al. propose a multi-user transmission MAC scheme, which supports the Multi-Packet Transmission (MPT) in the downlink and multiple control frame receptions (e.g., CTSs or ACKs) in the uplink, while simultaneous data transmissions from multiple STAs are not considered. Due to the simplicity, the MAC scheme of DBLP:conf/wcnc/LiAL10 is implemented to compare with our proposal. ## 3 Uni-MUMAC Operations Uni-MUMAC is based on the IEEE 802.11 Enhanced Distributed Channel Access (EDCA), which relies on the CSMA/CA mechanism to share the wireless channel. EDCA can operate in either the basic access mode or the optional RTS/CTS handshaking one. In this paper, Uni-MUMAC adopts and extends the RTS/CTS scheme for the following reasons: 1) The AP can notify the uplink contending STAs about the number of available antennas by a modified control frame; 2) The AP can estimate the CSI from the RTS/CTS exchanging process; 3) The distributed STAs can be synchronized from the exchanging process to transmit to the AP in parallel. ### 3.1 Frame Structure #### 3.1.1 PHY Frame Structure The PHY frame structure of IEEE 802.11ac is shown in Figure 1, where VHT PLCP, PPDU and MPDU stand for Very High Throughput Physical Layer Convergence Protocol, PLCP Protocol Data Unit and MAC Protocol Data Unit, respectively. As shown from the frame structure, PPDU consists of the PHY preamble and MPDUs. IEEE 802.11ac specifies that all MPDUs must be transmitted in the format of Aggregated-MPDU (A-MPDU), where aggregated MPDUs are separated by MPDU delimiters. Before being delivered to the PHY layer, a service field and a tail field are appended to the A-MPDU. The PHY preamble is formed by $3$ legacy fields for the backward compatibility (i.e., L-STF, L-LTF and L-SIG) and some newly introduced VHT fields 6359961 6140087 . Figure 1: PHY frame format of IEEE 802.11ac IEEE 802.11ac introduces these VHT fields to assist WLANs in obtaining the high performance. A Group Identifier (Group-ID) field is added in VHT Signal Field-A (VHT-SIG-A), which is used to inform the targeted STAs about the followed MU-MIMO transmission, the order and the position of each STA’s corresponding stream. A complete Group-ID table is created and disseminated by the AP, and will be recomputed as STAs associate or de-associate to the AP. Since the number of STAs’ combinations can exceed the available number of Group-ID in a large basic service set, and the down/up-link channel may be different, thus, we assume a single Group-ID can reference to multiple transmission sets along with other PHY preamble features that could be used to resolve the intended STAs aboul2013managing . In other words, there will be always at least one proper Group-ID entry that can be mapped to the intended transmission set. VHT Long Training Field (VHT-LTF) can contain an orthogonal training sequence that is known by both the transmitter and the receiver to estimate the MIMO channel. The number of VHT-LTF fields should not be less than the number of transmitted spatial streams to precisely estimate the channel. The legacy and VHT-SIG-A fields adopt the low rate modulation scheme to make the preamble understandable to all STAs, while the rest VHT fields and A-MPDU are transmitted using the VHT modulation scheme. In this paper, a single modulation and coding scheme (MCS), i.e., $16$-QAM with $1/2$, is utilized for all frames to simplify the simulation, although the extension to various MCS for different frames and STAs is straightforward. Here, we only introduce the PHY features that are closely related to the proposed protocol. The readers please refer to 6359961 for details of other PHY features. #### 3.1.2 MAC Frame Structure The control frames of Uni-MUMAC are shown in Figures 2 and 3. In the downlink, the control frames are MU-RTS, MU-CTS and MU-ACK. MU-RTS keeps the standard RTS frame structure, because the AP can utilize the Group-ID field of the PHY frame to notify targeted receivers. MU-CTS and MU-ACK add a transmitter address field to the original CTS and ACK frames in order to facilitate the AP to differentiate multiple responding STAs. Note that MU-CTS and MU-ACK coincidentally have the same frame structure as the standard RTS frame after adding a transmitter address field to the original CTS and ACK frames. Figure 2: Frame structure of standard RTS In the uplink, all frame modifications are limited to the AP side to reduce STAs’ computing consumption. These modified frames are Ant-CTS (CTS with antenna information), G-CTS (Group CTS) and G-ACK (Group ACK), as shown in Figure 3. An antenna information field is added to Ant-CTS, which is broadcast by the AP to announce the number of available antennas (after one antenna is occupied in the first contention round) and the start of the $2$-nd contention round. G-CTS and G-ACK have the identical frame structure, where the receiver address field is removed and replaced by the Group-ID field in the IEEE 802.11ac PHY frame, while a transmitter address field is added to indicate the AP address. The G-CTS frame is used to inform STAs the start of the data transmission, and G-ACK is used to indicate the successful reception of data frames. (a) Ant-CTS (b) G-CTS & G-ACK Figure 3: Modified frames for uplink transmissions ### 3.2 Successful Downlink Transmissions Figure 4 shows a successful Uni-MUMAC downlink transmission. Initially, the channel is assumed busy (B). After the channel has been idle for an Arbitration Inter Frame Space (AIFS), a random backoff (BO) drawn from CW starts to count down and is frozen as soon as the channel is detected as busy. Suppose the AP first wins the channel contention and sends a MU-RTS. Then, the STAs who are included in Group-ID reply with MU-CTSs sequentially as the indicated order. Those STAs who are not included in the MU-RTS will set the Network Allocation Vector (NAV) to defer their transmissions. After a MU-CTS is received, the AP will measure the channel through the training sequence included in the PHY preamble, and then uses the estimated CSI to precode the simultaneously-transmitted frames. As being precoded, the frames destined to different STAs will not interfere with each other. Finally, STAs send MU-ACKs at the same time to acknowledge the successful reception of data frames. Figure 4: A successful Uni-MUMAC downlink transmission Note that, the uplink channel is assumed to be the same as the downlink one in this paper. In other words, the implicit CSI feedback, namely, the AP estimates the channel using the training sequence included in the MU-CTS, is adopted. The reason is that the explicit CSI feedback will need more computing capability at STAs and require an extra field with substantial volume in the MU-CTS to include the measured CSI, which may not be suitable for STAs in some capacity or power constraint scenarios. ### 3.3 Successful Uplink Transmissions In the uplink, a standard RTS is sent to the AP by the STA that won the $1$-st round channel contention. Instead of replying a CTS, an Ant-CTS is broadcast by the AP with two functions: 1) to notify the STA about the successful reception of the RTS, and 2) to inform other STAs that the number of available antennas and the start of the $2$-nd contention round. The STAs who have frames to send will compete for the available spatial streams in the $2$-nd contention round. A new random $BO$ ($BO_{\text{2nd}}$) drawn from $[0,CW_{\text{2nd}}-1]$ starts to count down, and a RTS will be sent if $BO_{\text{2nd}}$ of a STA reaches $0$. The number of available antennas of the AP decreases by one each time an uplink RTS is successfully received. The $2$-nd contention round finishes as: 1) all available antennas of the AP are occupied or 2) a predefined duration of the $2$-nd contention round elapses in case there are not enough contending STAs (the maximum duration of the $2$-nd contention round is set to $CW_{\text{2nd}}$ slots). As soon as the $2$-nd contention round finishes, a G-CTS is sent by the AP to indicate the readiness for receiving multiple frames in parallel. The G-CTS frame includes the STAs who have successfully sent RTSs during both $1$-st and $2$-nd contention rounds. When the G-CTS is received by the targeted STAs, they are synchronized to send data frames to the AP. Finally, the AP acknowledges the received data frames with G-ACK. An example of a successful uplink transmission is shown in Figure 5, in which illustrating case, the AP has $3$ antennas, STA $2$ picks $BO_{\text{2nd}}=0$ and STA $3$ picks $BO_{\text{2nd}}=1$ from $[0,CW_{\text{2nd}}-1]$, respectively. Figure 5: A successful Uni-MUMAC uplink transmission It is important to point out that the RTSs sent by STAs in the $2$-nd contention round could collide with G-CTS sent by the AP. For example, in the case that the RTS sent by a STA who claims the AP’s last available antenna is not heard by some STAs (hidden terminals), which therefore believe that the AP still has available antennas. Then, after a Short Inter Frame Space (SIFS) interval, the G-CTS sent by the AP and RTSs sent by the hidden STAs would collide. To avoid this unexpected scenario, STAs are forced to wait for a Multi-User SIFS interval in the $2$-nd contention round. MU-SIFS is an interval longer than SIFS but shorter than AIFS, which not only prioritizes the AP to send the G-CTS, but also avoids STAs to misunderstand MU-SIFS as an idle channel. ### 3.4 Frame Collisions Collisions will occur in both $1$-st and $2$-nd contention rounds if more than one STA choose the same random backoff value. On sending a RTS, EDCA specifies that the STA has to set a timer according to Equation (1) to receive the expected CTS, where $T_{\text{CTS}}$ represents the transmission duration of a CTS frame. If CTS is not received before the timer expires, the STAs who previously sent RTSs assume that collisions occurred. These RTS-sending STAs will compete for the channel access after the expiration of the timer. For the RTS-receiving STAs, none of RTSs can be decoded correctly. Therefore, after the collision time, the receiving STAs will wait for an Extended Inter Frame Space (EIFS, as shown in Equation (2)) interval to compete for the channel access together with those RTS-sending STAs. As shown in Figure 6 (Ant-CTS and MU-CTSs with dotted lines mean these frames would be transmitted if there were no collisions), collisions in the $1$-st contention round include two cases: 1) collisions among STAs; 2) collisions between STAs and the AP. Since STAs can not differentiate these two cases, the collision time has to be set according to the duration of the longer frame, which is $T_{\text{MU-RTS}}$. In addition, the $\text{CTS}_{\text{timer}}$ and the EIFS interval also have to be extended according to $\text{MU- CTS}_{\text{timer}}$ (as shown in Equation (3), where $N$ is the number of AP’s antennas) and Multi-User EIFS (MU-EIFS, as shown in Equation (4)), to take the scenario that the AP is involved in collisions into account. Figure 6: Collisions in the $1$-st contention round $\displaystyle\text{CTS}_{\text{timer}}=\text{SIFS}+T_{\text{CTS}}$ (1) $\displaystyle\text{EIFS}=\text{SIFS}+T_{\text{CTS}}+\text{AIFS}$ (2) $\displaystyle\text{MU-CTS}_{\text{timer}}=N\cdot(\text{SIFS}+T_{\text{MU- CTS}})$ (3) $\displaystyle\text{MU-EIFS}=N\cdot(\text{SIFS}+T_{\text{MU- CTS}})+\text{AIFS}$ (4) If collisions occur in the $2$-nd contention round, the colliding STAs will not be indicated as the receivers in the Group-ID field of G-CTS. Therefore, only the STAs that have successfully sent RTSs in both contention rounds are allowed to transmit frames to the AP at the same time (as illustrated in Figure 7). Figure 7: RTS collisions in the $2$-nd contention round ### 3.5 Other Considerations In IEEE 802.11 EDCA, a STA renews its $BO$ if the channel contention was successful. For the STAs who did not win the contention, the frozen $BO$ is used for the next contention round. In this paper, $BO$ of the $1$-st contention round is renewed after collisions in the $1$-st round or if the STA is the initiator of the two-round process. Although both STA $1$ and STA $2$ participate in the transmission as shown in Figure 7, STA $1$ is considered to be the initiator. In other words, STA $1$ will have a new random $BO$ in the followed $1$-st contention round, while STA $2$ will use the frozen $BO$. It is more straightforward regarding the $BO_{\text{2nd}}$ renewal policy. Each STA draws a fresh $BO_{\text{2nd}}$ from $CW_{\text{2nd}}$ as soon as a new $2$-nd contention round starts. G-CTS is sent out by the AP when the number of available antennas reaches zero or the duration of the $2$-nd contention round drains. As soon as the Ant-CTS is sent, the AP sets the G-CTS timer to account for up to $CW_{\text{2nd}}$ slots (as shown in Equation (5)). $\displaystyle\text{G-CTS}_{\text{timer}}=CW_{\text{2nd}}\cdot(\text{MU- SIFS}+T_{\text{RTS}})$ (5) ## 4 Performance Evaluation Uni-MUMAC is evaluated using an analytic model and simulations. The analytic model is adapted from Bianchi’s saturation throughput model bianchi2000performance to support MU-MIMO transmissions in both downlink and uplink. The simulation is implemented in `C++` using the Component Oriented Simulation Toolkit (COST) library GilChen and the SENSE simulator chen2005sense . A single-hop WLAN implementing Uni-MUMAC is considered as shown in Figure 8. It consists of one AP and $M$ STAs with an error-free channel. The AP employs an array of $N$ antennas, while each STA has only one antenna. The data frame has a fixed length of $L$ bits. The parameters used to evaluate Uni-MUMAC are listed in Table 1. Figure 8: Down/Up-link Uni-MUMAC transmissions ### 4.1 Saturation Throughput Analysis Table 1: System parameters Parameters \| Values ---\|--- Channel Bandwidth \| $40$ MHz Modulation & Coding Scheme \| $16$-QAM with $1/2$ Guard Interval \| $0.8$ $\mu$s Queue Length of STA & AP \| $Q_{\text{sta}}=50$, $Q_{\text{ap}}=M^{2}$ Frame Length ($L$) \| $8000$ bits MAC Header ($L_{\text{MAC}}$) \| $272$ bits MPDU Delimiter ($L_{\text{delimiter}}$) \| $32$ bits Service Bits ($L_{\text{service}}$) \| $16$ bits Tail Bits ($L_{\text{tail}}$) \| $6$ bits RTS/MU-RTS/MU-CTS/MU-ACK \| $160$ bits Ant-CTS \| $120$ bits G-CTS/G-ACK \| $112$ bits Idle Slot ($\sigma$) \| $9$ $\mu$s SIFS, MU-SIFS and AIFS \| $16$, $20$ and $34$ $\mu$s $CW$ \| $32$ AP Antennas ($N$) \| $1$, $2$, $4$ No. of Iteration ($N_{\text{iteration}}$) \| $100000$ Let $\tau=\frac{2}{(CW+1)}$ be the transmission probability of a node in a random slot, where $CW$ is the size of the $1$-st round contention window. Then, the probability that the channel is idle is: $p_{\text{i}}=(1-\tau)^{M+1}.$ (6) The probability that the channel sees a successful transmission slot, $p_{\text{s}}$, is given by: $\displaystyle p_{\text{s}}=\binom{M+1}{1}\tau(1-\tau)^{M}=(M+1)\tau(1-\tau)^{M},$ (7) which accounts for that a single node (either the AP or a STA) successfully wins the $1$-st round channel contention. By deducting $p_{\text{i}}$ and $p_{\text{s}}$, the probability that the channel observes a collision slot, $p_{\text{c}}$, is obtained: $\displaystyle p_{\text{c}}=1-p_{\text{i}}-p_{\text{s}}.$ (8) In the saturated condition, a successful downlink transmission always contains $N$ (the number of AP antennas) data streams. Therefore, the number of bits of a successful downlink transmission ($N_{\text{b,down}}$) is: $\displaystyle N_{\text{b,down}}=\alpha\cdot N\cdot N_{\text{f}}\cdot L\cdot p_{\text{s}},$ (9) where $\alpha=\frac{1}{M+1}$ is the probability that a transmission is from the AP, and $N_{\text{f}}$ is the number of aggregated frames in an A-MPDU. The calculation of the successfully received number of bits of uplink ($N_{\text{b,up}}$) has to account for successful transmissions of both $1$-st and $2$-nd contention rounds: $\displaystyle N_{\text{b,up}}=(1-\alpha)\cdot N_{\text{f}}\cdot L\cdot p_{\text{s}}\cdot\sum_{\text{x}=1}^{N}p_{\text{x\\_ant}}\cdot\text{x},$ (10) where $p_{\text{x\\_ant}}$ is the probability that x (x $\in[1,N]$) antennas of the AP have been used for the uplink transmission. In other words, one antenna has been obtained by a STA in the $1$-st contention round, and x-1 antennas have been successfully obtained by STAs in the $2$-nd contention round. The duration of a successful downlink transmission, $T_{\text{s,down}}$, is: $\displaystyle T_{\text{s,down}}=\text{AIFS}+T_{\text{MU- RTS}}+N\cdot(T_{\text{MU-CTS}}+\text{SIFS})+T_{\text{A-MPDU}}+T_{\text{MU- ACK}}+2\cdot\text{SIFS}.$ (11) An example to calculate the duration of a MU-RTS frame and a data frame using the system parameters of Table 1 is given in Equation (12). $T_{\text{PHY}}(N)=36+N\cdot 4$ $\mu$s are the duration of PHY header (the number of the VHT-LTF fields is proportional to the number of AP antennas $N$); $L_{\text{service}}$, $L_{\text{tail}}$ and $L_{\text{delimiter}}$ are the length of the service field, the tail field and the MPDU delimiter; $L_{\text{DBPS}}$ and $T_{\text{symbol}}$ are the number of data bits in a symbol and the symbol duration; $N_{\text{f}}$ is the number of aggregated frames in an A-MPDU; $L_{\text{MU-RTS}}$ and $L_{\text{MAC}}$ are the length of MU-RTS and the MAC header respectively. More detailed calculation of the frame duration can be found in 6287486 . $\displaystyle\begin{cases}T_{\text{MU- RTS}}=T_{\text{PHY}}(N)+\Big{\lceil}\frac{L_{\text{service}}+L_{\text{MU- RTS}}+L_{\text{tail}}}{L_{\text{DBPS}}}\Big{\rceil}T_{\text{symbol}}\\\ T_{\text{A-MPDU}}=T_{\text{PHY}}(N)+\Big{\lceil}\frac{L_{\text{service}}+N_{\text{f}}\cdot(L_{\text{MAC}}+L+L_{\text{delimiter}})+L_{\text{tail}}}{L_{\text{DBPS}}}\Big{\rceil}T_{\text{symbol}}\end{cases}$ (12) The duration of a successful uplink transmission, $T_{\text{s,up}}$, is: $\displaystyle T_{\text{s,up}}=\text{AIFS}+T_{\text{RTS}}+T_{\text{Ant- CTS}}+E_{\text{2nd- slots}}+T_{\text{G-CTS}}+T_{\text{A-MPDU}}+T_{\text{G-ACK}}+4\cdot\text{SIFS},$ (13) where $E_{\text{2nd-slots}}$ stands for the average duration of the $2$-nd contention round. $\displaystyle E_{\text{2nd-slots}}=(T_{\text{RTS}}+\text{MU- SIFS})\cdot\sum_{\text{k}=1}^{CW_{\text{2nd}}}p_{\text{k\\_Slot}}\cdot\text{k},$ (14) where $p_{\text{k\\_Slot}}$ is the probability that there are k (k $\in[1,CW_{\text{2nd}}]$) slots in the $2$-nd contention round. As a STA can not differentiate if collisions of the $1$-st round are caused by the AP or other STAs, the collision time has to be set according to the duration of the longer frame: $\displaystyle T_{\text{c}}=\text{AIFS}+T_{\text{MU-RTS}}+N\cdot(T_{\text{MU- CTS}}+\text{SIFS}).$ (15) The average duration of a channel slot is: $\displaystyle T_{\text{average}}=\alpha\cdot p_{\text{s}}\cdot T_{\text{s,down}}+(1-\alpha)\cdot p_{\text{s}}\cdot T_{\text{s,up}}+p_{\text{c}}\cdot T_{\text{c}}+p_{\text{i}}\cdot\sigma.$ (16) Equation (17) gives a simple example to calculate $p_{\text{2\\_ant}}$, in which case, the AP has $2$ antennas and $CW_{\text{2nd}}=2$: $\displaystyle p_{\text{2\\_ant}}=\binom{M-1}{1}\frac{1}{CW_{\text{2nd}}}\left(1-\frac{1}{CW_{\text{2nd}}}\right)^{M-2}+\binom{M-1}{1}\frac{1}{CW_{\text{2nd}}}\left(1-\frac{1}{CW_{\text{2nd}}}\right)^{M-2}\cdot p_{\text{1\\_fail}}.$ (17) The first part of Equation (17) stands for that only one STA is successful in the $1$-st slot. The second part represents that only one STA is successful in the $2$-nd slot, which is conditioned on that the $1$-st slot fails ($p_{\text{1\\_fail}}$, no STAs or more than one STA chooses the $1$-st slot). Note that the similar condition is not required for the first part, because the $2$-nd round contention finishes as soon as a STA wins the $1$-st slot regardless the choices of other STAs of other slots. As $CW_{\text{2nd}}$ increases, the closed form of $p_{\text{2\\_ant}}$ becomes infeasible due to various combination of conditions for a STA to succeed in different slots. Therefore, we utilize a semi-analytic algorithm to calculate $p_{\text{x\\_ant}}$ and $p_{\text{k\\_Slot}}$, the pseudo code of which is shown in Algorithm 1. Input: $N$, $M$, $CW_{\text{2nd}}$, $N_{\text{iteration}}$ Output: $p_{\text{x\\_ant}}$, $p_{\text{k\\_Slot}}$ for _$i\leq N_{\text{iteration}}$_ do for _$\textbf{STA\\_id}=1:M$_ do $rand\\_value(STA\\_id)=\lfloor rand()CW_{\text{2nd}}\rfloor$ $\ignorespaces\triangleright$ $rand\\_value$: random value chosen by each STA $CW2nd\\_Array(rand\\_value(STA\\_id))\texttt{++}$ $\ignorespaces\triangleright$ $CW2nd\\_Array$: no. of times a $CW_{\text{2nd}}$ value is chosen end for $count=1$ $1st\\_success=\textbf{false}$ for _$\textbf{length}=1:CW_{\text{2nd}}$_ do if _$CW2nd\\_Array(length)\texttt{==}1$_ then $num\\_sta\\_success(count)\texttt{++}$ $\ignorespaces\triangleright$ $num\\_sta\\_success$: no. of STAs won the $2$-nd round contention if _$1st\\_success\texttt{==}\textbf{true}$_ then $num\\_sta\\_success(count\texttt{-}1)\texttt{--}$ end if if _$count\texttt{==}1$_ then $1st\\_success=\textbf{true}$ end if if _$count\texttt{==}N\texttt{-}1$_ then break end if $count\texttt{++}$ end if if _$length\texttt{==}CW_{\text{2nd}}$_ then $slot\\_id=CW_{\text{2nd}}$ end if end for $num_{\text{slot\\_success}}(slot\\_id)\texttt{++}$ $\ignorespaces\triangleright$ $num_{\text{slot\\_success}}$: no. of slots used in the $2$-nd round contention end for for _$\text{x}=1:N\texttt{-}1$_ do $p_{\text{(x+1)\\_ant}}=num\\_sta\\_success(\text{x})/N_{\text{iteration}}$ end for $p_{\text{1\\_ant}}=1-\sum_{x=2}^{N}p_{\text{x\\_ant}}$ for _$k=1:CW_{\text{2nd}}$_ do $p_{\text{k\\_Slot}}=num_{\text{slot\\_success}}(\text{k})/N_{\text{iteration}}$ end for Algorithm 1 Algorithm for $p_{\text{x\\_ant}}$ and $p_{\text{k\\_Slot}}$ Finally, the collision probability of a node, $\displaystyle P_{\text{collision}}=1-(1-\tau)^{M},$ (18) and down/up-link throughput are derived: $\displaystyle\begin{cases}S_{\text{down}}=\frac{N_{\text{b,down}}}{T_{\text{average}}}\\\ S_{\text{up}}=\frac{N_{\text{b,up}}}{T_{\text{average}}}.\end{cases}$ (19) The transmission probability $\tau$, equations (18) and (19) form a non-linear system, which can be resolved by an iterative numerical technique kumar2005new . ### 4.2 System Performance against $CW_{\text{2nd}}$ In this sub-section, the performance of Uni-MUMAC is evaluated by increasing $CW_{\text{2nd}}$, with the goal to find a suitable $CW_{\text{2nd}}$ value that maximizes the system performance. Two traffic conditions are considered: 1) the saturated one, as shown in Figure 9, and 2) the non-saturated one, as shown in Figure 10. The saturated condition means that both the AP and STAs always have frames to transmit. Obviously, there is no $2$-nd round channel access when the AP has $1$ antenna, which is why the results keep constant as $N=1$. Note that the plots include both analysis and simulation results in the saturated condition, while the plots include only simulation results of the non-saturated condition. As shown in Figure 9, when the WLAN is saturated (i.e., both downlink and uplink are saturated), $CW_{\text{2nd}}$ has very small impact on the downlink throughput (AP’s throughput). However, for the uplink, the importance of choosing an appropriate $CW_{\text{2nd}}$ is observed. For example, the uplink throughput (STAs’ throughput) approaches its maximum when $CW_{\text{2nd}}\in[8,12]$ as $M=8$ (Figure 9(a)) and when $CW_{\text{2nd}}\in[12,16]$ as $M=15$ (Figure 9(b)). (a) $M=8$ (b) $M=15$ Figure 9: Saturated throughput against $CW_{\text{2nd}}$ In the non-saturated condition, we set the traffic load for each STA and the AP to $1.4$ Mbps and $11.2$ Mbps, respectively. In Figure 10(a), the downlink throughput ($N=2$ and $4$) obtains the highest value when $CW_{\text{2nd}}\in[4,8]$, and then decreases as $CW_{\text{2nd}}$ keeps increasing. The reason is that the continuous increase of $CW_{\text{2nd}}$ leads to longer uplink transmissions that harm the downlink ones. Figure 10(b) shows that the average delay increases as $CW_{\text{2nd}}$ increases. Note that, the average delay remains at a relatively low level when the system is in the non-saturated condition, for example, the average delay of STAs when $CW_{\text{2nd}}\in[4,34]$ and the average delay of the AP when $N=4$ and $CW_{\text{2nd}}\in[4,8]$. However, the average delay of the AP ($N=4$) increases sharply as the downlink traffic approaches saturation. (a) $M=8$, STA $1.4$ Mbps, AP $11.2$ Mbps (b) $M=8$, STA $1.4$ Mbps, AP $11.2$ Mbps Figure 10: Non-saturated throughput & Average delay It is also observed that the downlink throughput, as the network becomes saturated, is much lower than the uplink one. The reasons are as follows. First, the AP bottle-neck effect. It is due to the fact that the AP manages all traffic to and from STAs in a WLAN, while it has the same probability to access the channel as the STAs due to the random backoff mechanism of CSMA/CA. In addition, the inherently high traffic load at the AP results in that the downlink is saturated in most of the time. Thirdly, a favorable value of $CW_{\text{2nd}}$ for the uplink does not mean the same benefit to the downlink. For example, as shown in the Figure 9, the uplink obtains the highest throughput when $CW_{\text{2nd}}$ is set approximately to $M$ ($CW_{\text{2nd}}\approx M$), while the downlink transmission prefers a value of $CW_{\text{2nd}}$ as small as possible. In order to mitigate the AP bottle-neck effect and compensate the downlink disadvantage when STAs choose a big $CW_{\text{2nd}}$, we set the maximum number of frames that the AP can aggregate in an A-MPDU to $M$ ($N_{\text{f}}\leq M$), while keeping the number of frames aggregated by each STA to $1$ in the following simulations. Also, the queue length of the AP is set to quadratically increase with the number of STAs ($Q_{\text{ap}}=M^{2}$) to statistically guarantee that there are enough frames destined to different STAs 6287486 . In Figures 11 and 12, the performance of Uni-MUMAC is evaluated in the same condition as done in Figures 9 and 10 except that the network adopts the new frame aggregation scheme (AP’s $N_{\text{f}}\leq M$, STA’s $N_{\text{f}}=1$) and the new queue length ($Q_{\text{ap}}=M^{2}$, $Q_{\text{sta}}=50$). The results show that Uni-MUMAC manages to avoid the extremely low downlink throughput when the system is saturated (Figure 11) and keeps the downlink transmission always in the non-saturation area (Figure 12(a)), which is not achieved in Figure 10(a). The average delay of the AP (Figure 12(b)) is much lower compared to that of the AP in Figure 10(b), which is because the system remains in the non-saturated condition by employing the frame aggregation scheme. (a) $M=8$ (b) $M=15$ Figure 11: Saturated throughput when AP aggregates frames The results from Figure 11 also show that the system can roughly obtain the maximum performance when $CW_{\text{2nd}}\in[M-4,M+4]$. For example, in the case that the AP has $4$ antennas, the system throughput (AP+STA) reaches its maximum when $CW_{\text{2nd}}\in[6,8]$ as $M=8$ and $CW_{\text{2nd}}\in[12,16]$ as $M=15$, respectively. Therefore, the optimum value of $CW_{\text{2nd}}$ is fixed to $M$ in the following simulations. (a) $M=8$, STA $1.4$ Mbps, AP $11.2$ Mbps (b) $M=8$, STA $1.4$ Mbps, AP $11.2$ Mbps Figure 12: Non-saturated throughput & Average delay when AP aggregates frames ### 4.3 System Performance against $M$ In this sub-section, the performance of Uni-MUMAC is evaluated against the number of STAs in the downlink-dominant and the down/up-link balanced traffic scenarios, where $M$ is increased from $1$ to $35$, the maximum number of frames aggregated at the AP is set to $M$ and the $2$-nd round Contention Window is also set to $M$. The two traffic scenarios are specified as follows. 1. 1. Downlink-dominant: This is the traditional WLAN traffic scenario, where the AP manages a much heavier traffic load compared to that of STAs. Therefore, the traffic load of the AP is set to be $4$ times higher than that of each STA. For instance, if the traffic load of a STA is $0.8$ Mbps and there are $5$ STAs, the traffic load of the AP will be $4\cdot 0.8\cdot 5=16$ Mbps. 2. 2. Down/up-link balanced: This is one of WLAN traffic types that not only includes P$2$P applications, which have already been around for some years, but also includes those emerging content-rich file sharing and video calling applications. Therefore, the traffic load of the AP is set to be the same as that of each STA. In this case, if there are $5$ STAs, and each STA has $0.8$ Mbps traffic load, the traffic load of the AP will be $0.8\cdot 5=4$ Mbps. The multi-user MAC scheme (LI-MAC) proposed by Li et al. in DBLP:conf/wcnc/LiAL10 is implemented and used as a reference (named as AP/STA-LI in the legend) to compare with Uni-MUMAC. For fair comparison, LI- MAC and Uni-MUMAC adopt the same configuration parameters (as shown in Table 1). The key features of LI-MAC and Uni-MUMAC are illustrated in Table 2. Table 2: Key features of LI-MAC and Uni-MUMAC MAC Schemes Modification Downlink Uplink LI-MAC DBLP:conf/wcnc/LiAL10 MU- RTS/CTS Multi-packet + Parallel-control-frame TX One-round single-packet TX Uni-MUMAC MU-RTS/CTS/ACK/SIFS G-CTS/ACK, Ant-CTS Multi-packet + Sequential- control-frame TX Two-round Multi-packet TX Figure 13(a) shows the throughput by increasing the number of STAs in the downlink-dominant traffic scenario. It is with clear advantage to employ a higher number of antennas at the AP. The downlink throughput is much higher than the uplink one before the system gets saturated. The reasons for that are twofold: 1) the AP traffic load is inherently higher than that of STAs, and 2) the AP adopts the frame aggregation scheme. As the system becomes saturated, the throughput of both downlink and uplink decreases as $M$ increases. As shown in Figure 13(a), the uplink throughput of LI-MAC ($N=4$) is the same as that of Uni-MUMAC ($N=1$), which is because LI-MAC adopts the baseline DCF in the uplink. As the uplink throughput approaches saturation ($M=15$), the downlink throughput of LI-MAC starts to decrease. The downlink throughput of Uni-MUMAC can achieve higher gains when the network is not saturated, which is because the proposed $2$-nd round transmission increases the uplink transmission efficiency, and therefore decreases the number of AP’s channel contenders. However, as the number of STAs further increases, where both up/down-link saturate, LI-MAC outperforms Uni-MUMAC, which is because Uni- MUMAC suffers a high collision rate in the 2-nd round that prolongs the 2-nd round duration. However, it is important to point out that neither LI-MAC or Uni-MUMAC is able to work sustainably in the saturated condition. (a) Downlink-dominant: STA $0.8$ Mbps, AP $3.2$ Mbps (b) P$2$P Scenario: STA $0.8$ Mbps, AP $0.8$ Mbps Figure 13: Throughput against $M$ Figure 13(b) shows the throughput against $M$ in the down/up-link balanced traffic scenario. As expected, Uni-MUMAC achieves the balanced downlink and uplink throughput. This is because the AP and STAs are set to have the same traffic load, and more importantly, the frame aggregation scheme (AP’s $N_{\text{f}}\leq M$, STA’s $N_{\text{f}}=1$) counteracts the STAs’ collective advantage on the channel access. Comparing with Uni-MUMAC, the downlink throughput of LI-MAC achieves better performance when the uplink is saturated, which is because the duration of collisions in the uplink of LI-MAC is much shorter than that of Uni-MUMAC. However, the drawback is that LI-MAC has a big throughput gap between the AP and STAs, which does not satisfy the traffic requirements of the considered scenario. Figure 14 shows the average delay against $M$. Both downlink and uplink delays increase with $M$, and grow significantly as the downlink or the uplink traffic approaches the saturation. After the system gets saturated, the average delay becomes steady. It is worth pointing out that the average delay of STAs is higher than that of the AP when $M$ becomes bigger. The reason for that is that the transmission duration of the AP gets longer as $M$ increases (due to the frame aggregation scheme), which makes STAs waiting longer to access the channel. (a) Downlink-dominant: STA $0.8$ Mbps, AP $3.2$ Mbps (b) P$2$P Scenario: STA $0.8$ Mbps, AP $0.8$ Mbps Figure 14: Average delay against $M$ Figure 15 shows the $1$-st round collision probability increases with $M$ and converges when the system becomes saturated, which confirms the down/up-link saturation trend as discussed in Figures 13 and 14. It is interesting to note that the collision probability of STAs is higher than that of the AP when the system is non-saturated. The reason for that is a STA transmits less frequently than the AP in the non-saturated condition, which results in a lower conditional collision probability for the AP. It can be clearly explained by Equation 20, where $p_{\text{ap}}$ and $\tau_{\text{ap}}$ ($p_{\text{sta}}$ and $\tau_{\text{sta}}$) are the $1$-st round collision probability and the transmission probability of the AP (or a STA) in the non- saturated condition: (a) Downlink-dominant: STA $0.8$ Mbps, AP $3.2$ Mbps (b) P$2$P Scenario: STA $0.8$ Mbps, AP $0.8$ Mbps Figure 15: $1$-st round collision probability against $M$ $\displaystyle\begin{cases}p_{\text{ap}}=1-(1-\tau_{\text{sta}})^{M}\\\ p_{\text{sta}}=1-(1-\tau_{\text{sta}})^{M-1}\cdot(1-\tau_{\text{ap}}).\end{cases}$ (20) Figure 16 shows the $2$-nd round collision probability against $M$. It is clear that the $2$-nd round collision probability is higher when the system traffic load is higher. In the low number of STAs area, the $2$-nd round collision probability when the AP has $2$ antennas is sometimes lower than that when the AP has $4$ antennas. The reason is that, a higher number of antennas at the AP usually means a longer duration of the $2$-nd contention round, which increases the chances of collisions in the $2$-nd round. For example, in a case that the AP employs $2$ antennas, the $2$-nd contention round finishes as soon as a STA successfully wins the still-available antenna of the AP; while in a case that the AP employs more than $2$ antennas, the $2$-nd contention round continues, therefore increasing the $2$-nd round collision probability. Figure 16: $2$-nd round collision probability against $M$ ## 5 Conclusions & Future Research Challenges In this paper, a unified MU-MIMO MAC protocol called Uni-MUMAC, which supports both MU-MIMO downlink and uplink transmissions for IEEE 802.11ac WLANs, is proposed. We evaluate it through an analytic model and simulations. A prominent MAC scheme from the literature is implemented and compared with Uni- MUMAC. By analyzing the simulation results, we observe that the $2$-nd round Contention Window $CW_{\text{2nd}}$, which is tuned to optimize the uplink transmission, is however not bringing the same benefit to the downlink one. An adaptive frame aggregation scheme and a queue scheme are applied at the AP to offset this disadvantage. By properly setting the aforementioned parameters, the results show that a WLAN implementing Uni-MUMAC is able to avoid the AP bottle-neck problem and performs very well in both the traditional downlink- dominant and emerging down/up-link balanced traffic scenarios. The results also show that a higher system capacity can be achieved by employing more antennas at the AP. Uni-MUMAC gives us insight about the interaction of down/up-link transmissions and how different parameters that control the system can be tuned to achieve the maximum performance. Based on the study of this paper, we considered the following aspects as the future research challenges or next steps for Uni- MUMAC. 1. 1. Adaptive Scheduling Scheme: As discussed in the paper, a parameter that optimizes the uplink could be unfavorable to the downlink. Therefore, an adaptive scheduling algorithm that takes several key parameters into account and compensates those STAs whose interests are harmed would play a significant role on obtaining the maximum performance while maintaining the fairness. As implied from the results, these parameters include: the size of A-MPDU, the queue length, the spatial-stream/frame allocation, the number of nodes/antennas, and other key parameters that control down/up-link transmissions. 2. 2. Traffic Differentiation: Another future research challenge is to provide new traffic differentiation capability in the uplink in addition to the one defined in IEEE 802.11e amendment 11e . The new traffic differentiation should be able to limit the number of STAs that can participate in the $2$-nd contention round to reduce $2$-nd round collisions. A possible solution could be to create a table at the AP with information about the priority of each traffic flow and the queue length of each STA, and then to utilize this table to control the $2$-nd contention round. 3. 3. Multi-hop Mesh Networks: There are more challenges that need to be considered in designing MAC to operate in multi-hop wireless networks. First, the hidden- node problem. It is still an open challenge to find mechanisms that efficiently solve the collisions caused by hidden nodes. A collision-free scheme proposed in barcelo2012distributed or the handshake based coordinated access could be a starting point to combat the hidden-node collisions in wireless mesh networks. Secondly, due to the heterogeneity of mesh nodes (e.g., different number of antennas at nodes), MAC protocols for wireless mesh networks need to be designed with the capability of swiftly switching among MU-MIMO, SU-MIMO, multi-packet and single-packet transmission schemes. Thirdly, MAC and routing protocols need to be jointly designed. There could be multiple destinations involved in a MU-MIMO transmission, and some destinations could be out of the one-hop transmitting range, in which case, routing strategies should be able to forward multiple packets to different nodes in parallel. ###### Acknowledgements. This work has been supported by the Spanish Government and the Catalan Government under projects TEC2012-32354 (Plan Nacional I+D), CSD2008-00010 (Consolider-Ingenio Program) and SGR2009#00617. ## References (1) Cisco, “802.11ac: The Fifth Generation of Wi-Fi,” in Cisco White Paper, pp. 1–25, 2012. * (2) M. Kihl, P. Ödling, C. Lagerstedt, and A. 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Oliver, “On the distributed construction of a collision-free schedule in multi-hop packet radio networks,” Springer Telecommunication Systems, pp. 1–14, 2013.	arxiv-papers	2013-09-19T16:49:16	2024-09-04T02:49:51.179424	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ruizhi Liao, Boris Bellalta, Trang Cao Minh, Jaume Barcelo, Miquel\n Oliver", "submitter": "Ruizhi Liao", "url": "https://arxiv.org/abs/1309.5049" }
1309.5064	11institutetext: US Naval Research Laboratory-Code 6792, Nonlinear System Dynamics Section, Plasma Physics Division, Washington, DC 20375 Bloomberg School of Public Health-Johns Hopkins University, 615 N. Wolfe Street, Baltimore, MD 21205 # Capturing pattern bi-stability dynamics in delay-coupled swarms L. Mier-y-Teran-Romero and I. B. Schwartz 1122111122 ###### Abstract Swarms of large numbers of agents appear in many biological and engineering fields. Dynamic bi-stability of co-existing spatio-temporal patterns has been observed in many models of large population swarms. However, many reduced models for analysis, such as mean-field (MF), do not capture the bifurcation structure of bi-stable behavior. Here, we develop a new model for the dynamics of a large population swarm with delayed coupling. The additional physics predicts how individual particle dynamics affects the motion of the entire swarm. Specifically, (_1_) we correct the center of mass propulsion physics accounting for the particles’ velocity distribution; (_2_) we show that the model we develop is able to capture the pattern bi-stability displayed by the full swarm model. ###### pacs: 05.45.-a ###### pacs: 89.75.Kd ###### pacs: 87.23.Cc Nonlinear dynamics Pattern formation in complex systems Population dynamics ## 1 Introduction Recently, much attention has been given to the study of interacting multi- agent, particle or swarming systems in various natural [1, 2, 3, 4] and engineering [5, 6] fields. These multi-agent swarms can self-organize and form complex spatio-temporal patterns even when the coupling between agents is weak. Many of these investigations have been motivated by a multitude of biological systems such as schooling fish, swarming locusts, flocking birds, bacterial colonies, ant movement, etc. [7, 8, 9, 10, 11, 12, 13, 14], and have also been applied to the design of systems of autonomous, communicating robots or agents [15, 16, 17] and mobile sensor networks [18, 19]. The excellent overviews [20, 21] discuss the diverse biological contexts where swarming occurs, as well as different modeling approaches. We note that in spite of all these investigations, understanding how swarming patterns self-organize, as well as predicting their stability are still very much open problems. A number of different mathematical modeling techniques have been applied to investigate aggregating particle systems. One possibility is to treat the system at a single-individual level, using ordinary differential equations (ODEs) or delay differential equations (DDEs) to describe their motion in space and time [22, 23, 24, 25]. Another possibility that is applicable with sufficiently dense numbers of particles involves the use of continuum models with averaged velocity and agent density fields that are governed by partial differential equations (PDEs) [9, 26, 27, 11]. Moreoever, a number of researchers have incorporated random noise effects into their models that are able to produce transitions from one pattern to another [28, 29]. The study of these systems has been enriched by tools from statistical physics [30] since both first and second order phase transitions have been found in the formation of coherent states [22, 31]. An important aspect of understanding self-organizing swarms patterns is that of delay in the coupling between individual agents. Time delay appears in many systems for several reasons: 1) finite time information transfer; 2) time required to acquire measurement information; 3) computation time required for generating the control instructions; and 4) actuation time required for the instructions to be applied. In general, time delay reflects an important property inherited in all swarms due to actuation, control, communication, and computation [5, 32]. The occurrence of time delays in interacting particle systems and in dynamical systems in general has been shown to have profound dynamical consequences, such as destabilization and synchronization [33, 34]. Since time delays in the engineering of autonomous robot-systems are almost unavoidable, incorporating them into the mathematical models is particularly important. Initially, such studies focused on the case of one or a few discrete time delays. More recently, however, the complex situation of several and random time delays has been researched [35, 36, 37]. Another important case is that of distributed time delays, when the dynamics of the system depends on a continuous interval in its past instead of on a discrete instant [38, 39, 40]. When examining self-organizing patterns in swarms, different attractors emerge depending on initial conditions, and/or the the addition of external noise. Co-existing bi- and multi-stable swarming patterns have been observed in a multitude of models [41, 42, 43]. Because of the existence of co-existing patterns, time delayed swarming systems display transitions between different spatio-temporal patterns if there is an adequate balance between the strength of the attractive coupling, the duration of time delay and the external noise intensity, [29, 44, 45]. Often, mean-field approximations to swarm dynamics cannot capture the possibility of different patterns for the same parameter values [44]; this failure is a consequence of the mean field missing the details of the particle distribution about the swarm center of mass. To redress the bi-stable formation problem, we develop a new reduced model based on a higher-order approximation that is able to predict bi-stable patterns in globally coupled swarm models with delayed interactions. ## 2 The Reduced Swarm Model Derivation We consider the dynamics of a two dimensional system of $N$ particles being acted on by the influence of self-propulsion and mutual attraction. Let $\mathbf{f}(\mathbf{r,\dot{r}})$, denote a self propelling force, and an interaction potential function between particles be given by $U(\mathbf{r}_{i}-\mathbf{r}_{j})$. In our description, the attraction between particles does not occur instantaneously, but rather in a time delayed fashion due to finite communication speeds and processing times. We describe the general motion of the particles by the following dimensionless equations: $\displaystyle\ddot{\mathbf{r}}_{i}(t)\mathbf{=\mathbf{f}}(\mathbf{\mathbf{r},\dot{r}}_{i})-\frac{a}{N}\sum_{j=1,j\neq i}^{N}\nabla_{\mathbf{r}_{i}}U(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau))$ (1) for $i=1,2\ldots,N$. Here, $\mathbf{r}_{i}(t)$ and $\dot{\mathbf{r}}_{i}(t)$ denote the two-dimensional position and velocity of particle $i$ at time $t$, respectively. We assume $\mathbf{f}(\mathbf{r},\dot{\mathbf{r}})=(1-\dot{\mathbf{r}}_{i}^{2})\dot{\mathbf{r}}_{i}$ to describe the self-propulsion of agent $i$, where $\dot{\mathbf{r}}_{i}^{2}=\dot{\mathbf{r}}_{i}\cdot\dot{\mathbf{r}}_{i}$. We denote the parameter $a$ as the coupling constant and measures the strength of attraction between agents. At time $t$, agent $i$ is attracted to the position of agent $j$ at the past time $t-\tau$. If we assume the form of our model is based on the normal form for particles near a supercritical bifurcation corresponding to the onset of coherent motion [46], then the leading term of the potential function may be considered to be quadratic; i.e., $\nabla_{\mathbf{r}_{i}}U(\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau))=\mathbf{r}_{i}(t)-\mathbf{r}_{j}(t-\tau)$. Specific mathematical models of this kind have been extensively used to study the dynamics of swarm patterns [47, 28, 48, 29, 46, 44]. Certainly, the choice of potential function has a fundamental impact on the type of long-term patterns that the system may acquire as well as on determining their dimensionality and characteristic spatio-temporal length scales [49]; many of these scales may be explicitly obtained for the patterns arising from Eqs. (1) [44]. Numerous potential functions appropriate to different biological and engineering situations have been carefully investigated [3, 2]; most of them possess one or more minima where attraction and repulsion are balanced. Our potential may be thought of as a first, quadratic approximation to the minima of more complicated potential landscapes. We obtain a reduced description for the swarm dynamics in Eqn. (1) by defining the center of mass of the swarm (CM), $\mathbf{R}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i}(t)$, and the three tensors $\displaystyle\mathbf{C_{rr}}=\frac{1}{N}\sum_{i=1}^{N}\delta\mathbf{r}_{i}\delta\mathbf{r}_{i},\qquad\mathbf{C_{vv}}=\frac{1}{N}\sum_{i=1}^{N}\delta\dot{\mathbf{r}}_{i}\delta\dot{\mathbf{r}}_{i},$ (2) $\displaystyle\mathbf{C_{rv}}=\frac{1}{N}\sum_{i=1}^{N}\delta\mathbf{r}_{i}\delta\dot{\mathbf{r}}_{i},$ where $\delta\mathbf{r}_{i}=\mathbf{r}_{i}-\mathbf{R}$. The above tensors represent all of the second moments of the particles’ position and velocity relative to the center of mass. Here, $\mathbf{a}\mathbf{b}$ is the exterior product of the vectors $\mathbf{a}$ and $\mathbf{b}$ and has matrix components $(\mathbf{a}\mathbf{b})_{ij}=a_{i}b_{j}$. Note that $\mathbf{C_{rr}}$ and $\mathbf{C_{vv}}$ are symmetric tensors with non-negative diagonal elements, whereas $\mathbf{C_{rv}}$ has neither of these properties. The dynamical equation for the center of mass is obtained from the relation $\ddot{\mathbf{R}}(t)=\frac{1}{N}\sum_{i=1}^{N}\ddot{\mathbf{r}}_{i}(t)$, while the equations for the tensors are found by taking time derivatives of $\delta\mathbf{r}_{i}$ and recalling $\sum_{i}\delta\mathbf{r}_{i}=0$. In our derivation, we drop all possible third order moments (that take the form of third order tensors) and justify it as follows. The swarm equations are rotationally-invariant in space and so the time-asymptotic patterns that arise tend to have particle position and velocity distributions that are symmetric with respect to the CM. For large numbers of particles organized in such symmetric patterns, these third order moments are composed of mutually cancelling terms since they are of odd power in either in the position or the velocity relative to the CM. Finally, we close the system of equations by approximating fourth order moments of the form $\frac{1}{N}\sum_{i=1}^{N}\delta\dot{\mathbf{r}}^{2}_{i}\mathbf{a}_{i}\mathbf{b}_{i}$ (where $\mathbf{a}_{i}$ and $\mathbf{b}_{i}$ are either $\delta{\mathbf{r}}_{i}$ or $\delta\dot{\mathbf{r}}_{i}$) by $\displaystyle\frac{1}{N}\sum_{i=1}^{N}\left\langle\delta\dot{\mathbf{r}}^{2}_{i}\right\rangle\mathbf{a}_{i}\mathbf{b}_{i}=\textrm{tr}(\mathbf{C_{vv}})\ \frac{1}{N}\sum_{i=1}^{N}\mathbf{a}_{i}\mathbf{b}_{i},$ (3) and dropping all higher order moments. Here, $\textrm{tr}(\mathbf{C_{vv}})$ denotes the trace of $\mathbf{C_{vv}}$. Our mean-field approximation including up to second moments (MF2M) finally takes the form: $\displaystyle\ddot{\mathbf{R}}=$ $\displaystyle P(t)\dot{\mathbf{R}}-2\dot{\mathbf{R}}\cdot\mathbf{C_{vv}}-a\left(\mathbf{R}(t)-\mathbf{R}(t-\tau)\right),$ (4a) $\displaystyle\ddot{\mathbf{C}}_{\mathbf{rr}}=$ $\displaystyle 2\mathbf{C}_{\mathbf{vv}}-2\left(\dot{\mathbf{R}}\cdot{\mathbf{C}}_{\mathbf{vr}}\dot{\mathbf{R}}+\dot{\mathbf{R}}{\mathbf{C}}_{\mathbf{rv}}\cdot\dot{\mathbf{R}}\right)$ $\displaystyle\qquad+P(t)\dot{\mathbf{C}}_{\mathbf{rr}}-2a{\mathbf{C}}_{\mathbf{rr}},$ (4b) $\displaystyle\dot{\mathbf{C}}_{\mathbf{rv}}=$ $\displaystyle\mathbf{C}_{\mathbf{vv}}-2\dot{\mathbf{R}}\cdot{\mathbf{C}}_{\mathbf{vr}}\dot{\mathbf{R}}+P(t){\mathbf{C}}_{\mathbf{rv}}-a{\mathbf{C}}_{\mathbf{rr}},$ (4c) $\displaystyle\dot{\mathbf{C}}_{\mathbf{vv}}=$ $\displaystyle-2\left(\dot{\mathbf{R}}\cdot\mathbf{C_{vv}}\dot{\mathbf{R}}+\dot{\mathbf{R}}\mathbf{C_{vv}}\cdot\dot{\mathbf{R}}\right)+2P(t)\mathbf{C_{vv}}$ $\displaystyle\qquad-a\dot{\mathbf{C}}_{\mathbf{rr}},$ (4d) where we let $P(t)=\left(1-\dot{\mathbf{R}}^{2}-\textrm{tr}(\mathbf{C_{vv}})\right)$. Note that no evolution equation is needed for the tensor $\mathbf{C}_{\mathbf{vr}}$ since $\mathbf{C}_{\mathbf{vr}}=\mathbf{C}_{\mathbf{rv}}^{\top}$. Interestingly, no second moment tensor appears in a time-delayed form. This is because the acceleration due to time delay that each particle and the center of mass undergo is the same, up to $\mathcal{O}(1/N)$. Thus, relative to the center of mass (as the second moment tensors are themselves measured) the particles undergo no time-delayed acceleration, to the order mentioned. Also, note that (_i_) Eqns. (4) reduce to the first-order mean-field approximation of [44] when all second-moment tensors are neglected; and that (_ii_) the linearization of Eqns. (4) about the trivial state decouples the CM and the second moment tensors. The standard linear stability calculation shows that second moment tensor equations render the trivial solution of Eqns. (4) unstable for all parameter values. The instability of the trivial solution agrees with what intuition tells us about Eqn. (1): the slightest difference in position or velocity among the particles will accelerate them via self- propulsive and attractive forces and make the CM and second-moment tensors depart from the stationary solution. This effect is not captured by the simple MF model since it does not account at all for how the particles are distributed about the center of mass. Figure 1: As in [44], Hopf (blue) and pitchfork (red) bifurcation curves in $a$ and $\tau$ space that delimit different regions of long-term dynamical behavior as prediceted by the first-order MF approximation: uniformly travelling state (A), ring state (B) and rotating state (C). (Color online) ## 3 Model analysis and physical interpretation The velocity second moment introduces two corrections into the propulsion of the CM equation (4a) in a physically meaningful way. To see this, first note that while the self-propulsive force of each individual particle always lies along its velocity vector, the cumulative self-propulsion of all particles is not necessarily directed along the CM velocity vector. The first tensor correction accounts for how the particle dispersal slows down the CM propulsion along its velocity and appears in the term $P(t)\dot{\mathbf{R}}=\left(1-\dot{\mathbf{R}}^{2}-\textrm{tr}(\mathbf{C_{vv}})\right)\dot{\mathbf{R}}$. For example, consider that all particles move coherently with the exact same velocity vectors; since their self-propelling forces are also coherent the CM self-propulsion term $\left(1-\dot{\mathbf{R}}^{2}-\textrm{tr}(\mathbf{C_{vv}})\right)\dot{\mathbf{R}}$ is maximal, in the sense that $\textrm{tr}(\mathbf{C_{vv}})=0$. Otherwise, when the particles are becoming dispersed $\textrm{tr}(\mathbf{C_{vv}})>0$, their individual self-propulsion does not add up coherently and this makes the self-propulsion of the CM weaker. The second tensor correction is the term $-2\dot{\mathbf{R}}\cdot\mathbf{C_{vv}}$ and, since in contrast to the term $P(t)\dot{\mathbf{R}}$ it is not necessarily directed along the velocity vector $\dot{\mathbf{R}}$, it represents a correction for the fact that the CM propulsion may have a component orthogonal to $\dot{\mathbf{R}}$ because of the dispersal of particles. An important consequence of (_i_) and (_ii_) is that all of the bifurcations found previously for the MF approximation [44] are inherited by the MF2M system in Eqns. (4). These bifurcation boundaries delimit the parameter regions where the MF approximation predicts different spatio-temporal patterns to be adopted in the long-time limit (Fig. 1). (A) A uniformly travelling state is composed of particles collapsed together and moving at constant speed in a given direction. (B) A ring state is formed by particles distributed almost uniformly along a circle, some of them moving clockwise and others counterclockwise, while the center of mass is at rest at the center (see Fig. 4a below). (C) A rotating state, in which all particles collapse to a point and move in a circular orbit (see Fig. 4b). The spatio-temporal patterns (A), (B) and (C) are captured by our MF2M approximation and manifest themselves as follows. The uniformly travelling/rotating states have trivial components for the second moment tensors (indicating the collapse of all particles to the CM) but uniform motion/periodic oscillations for the position of the CM. In contrast, the ring state has a stationary position for the CM but periodic oscillations for all second moment tensor components. The periodic oscillations of the tensor components in the ring state are due to the fact that particles are not distributed quite uniformly along the ring in either position or velocity space and their spread about the CM (second moment tensors) in both spaces has periodic variations. Figure 2: Period of sustained oscillations for the first component of the CM vector (left column) and the tensor component $C_{\mathbf{vv}}(1,1)$ (right column) for different values of the time delay $\tau$ and for $a=2$. Blue circles correspond to the full swarm described by Eqn. (1) and red crosses to our MF2M approximation. Initial conditions favor convergence to the rotating/ring state along the top/bottom row. The swarm converges to the ring state for $\tau\lesssim 1.5$ ($\tau\lesssim 2.1$) along the top (bottom) row and to the rotating state for higher $\tau$’s. In the ring state, the period of oscillation of $C_{\mathbf{vv}}(1,1)$ is approximately $\pi/\sqrt{a}=2.22$, as shown analytically. The agreement with other tensor components is similar. See text for more details. (Color online) We employ numerical simulations using a fourth order Runge-Kutta algorithm with quadratic interpolation for the delayed terms to see how our MS2M system captures the convergence to different spatio-temporal patterns. The coupling parameter is fixed at $a=2$ and the time delay is varied so as to cross the boundary between regions B and C of Fig. 1. We consider two types of initial conditions over the interval $[-\tau,0]$: in the first, the particles are distributed at random in a box with sides of length 0.05 and velocities uniformly distributed in [0.5 0.55] for the $x$ component and in [-0.025 0.025] for the $y$ component. In the second initial condition we distribute $N=300$ particles at random in the unit box with velocity components randomly distributed between -0.5 and 0.5. The first initial condition favors convergence to the rotating state while the second one favors convergence to the ring state. We note that although the initial conditions for the position and velocity vectors appear physically inconsistent, this is of no consequence since no time-delayed velocity terms appear in the equations. Starting from $\tau=0.5$, the full swarm and our MF2M systems first converge asymptotically to the ring state (Fig. 2). As the time delay increases, the convergence is to the rotating state instead. However, we find the hysterisis loop characteristic of bi-stable behavior: the transition occurs at different values of $\tau$ for the two different initial conditions. Remarkably, our MF2M system is not only able to accurately predict the periods of oscillation for the ring/rotating states of the full swarm equations, but it also identifies the value of the time delay at which the pattern transition occurs for each of the two initial conditions. Figure 3: Oscillation amplitude of the first component of the CM vector (left column) and the tensor component $C_{\mathbf{vv}}(1,1)$ (right column), after the decay of transients, for different values of the time delay $\tau$ and for $a=2$. Blue circles correspond to the full swarm described by Eqn. (1) and red crosses to our MF2M approximation. Initial conditions favor convergence to the rotating/ring state along the top/bottom row. The swarm converges to the ring state for $\tau\lesssim 1.5$ ($\tau\lesssim 2.1$) along the top (bottom) row and to the rotating state for higher $\tau$’s. The agreement with other tensor components is similar. (Color online) As noted above, the second moment tensors undergo periodic oscillations when the particles adopt the ring state. The period of these tensor oscillations may be approximated directly from Eqns. (4). In this spatio-temporal pattern, the CM is fixed and individual particles move at unit speed [44]. Substituting $\mathbf{R}=\dot{\mathbf{R}}=0$ and $\textrm{tr}(\mathbf{C_{vv}})=1$ in Eqns. (4) yields a linear system with oscillatory solutions with period $\pi/\sqrt{a}$. On the other hand, the period of oscillation of the center of mass position in the rotating state is determined by a complicated non-linear equation that may be obtained by resorting to polar coordinates and agrees perfectly with numerical simulations [44]. Thus, while the simple MF approximation fails to capture the bi-stable behavior displayed by the full swarm model, our MF2M approximation is able to do so. Figure 4: Top panel: hysterisis loop of the CM speed for the full model (solid blue) and our MF2M (dashed red) approximation, obtained with a time varying time delay (loop is traversed counter-clockwise in time). A zero/non-zero CM speed indicates convergence to the ring/rotating state. Transient dynamics at the time delay values where pattern switching occurs ($\tau=1.1$, 2.4) have been removed. Bottom panels: time snapshots of the spatial distribution of the swarm particles at the points labeled (a) and (b) of the top panel. In the rotating state of panel (b) all particles have collapsed to a point and move with the same velocity (blue arrow) around a circle; a portion of their time track is also shown (red line). (Color online). An important signature of the spatio-temporal patterns of our system is the amplitude of oscillations, which we also extract from the results of our numerical simulations (Fig. 3). The hysterisis of the system becomes equally evident in these amplitude vs. time delay plots. For the two initial conditions described before, our MF2M model is able to capture the oscillation amplitude of the CM (Fig. 3, left column) with accuracy as well as the time delay value at which the long-time convergence switches from the ring to the rotating state. However, the amplitude of oscillation of the tensor component $C_{\mathbf{vv}xx}$ is not captured as well, particularly for time delay values near the MF bifurcation (Fig. 3, right column). This departure is due to the neglected higher-order moments and finite-particle effects. An additional way to visualize the bistable nature of the ring and rotating state attractors is by forcing the system to undergo hysterisis via a time- dependent time delay. To this end, we start both the full swarm system and our MF2M approximation with a time delay of $\tau=1.1$, increase it slowly up to $\tau=2.4$ and then bring it back down to its starting value. Using the speed of the CM as a proxy for the state of the system, we see the swarm converge to the ring state (CM speed of zero) up until the time delay reaches 2.4; at this point, the swarm switches to the rotating state (non-zero CM speed) and persists in this state until the time delay is back to 1.1 (Fig. 4, top panel). Figs. 4(a) and (b) show snapshots of the swarm in different states, taken at the points indicated in the top panel. ## 4 Discussion and conlusions A more realistic version of the model studied here should modify our choice of potential function to include the local, hard-body repulsion among individuals. Our previous work shows that the patterns and transitions discussed here do not fundamentally change with the addition of small, local repulsive forces between particles. Stronger repulsion certainly can destabilize the coherent structures. Crucially, we note that since repulsion causes the particles to spread out in space, their distribution about the center of mass becomes even more important for determining their group dynamics. Although a systematic study is beyond the scope of this work, we expect our extended MF2M model to be highly useful in frameworks where repulsion between particles cannot be neglected. We also expect our multi-moment approach to extend the MF2M model framework to situations in which the time delay for inter-particle interactions is distributed. This situation is of interest since, analogously to inter- particle repulsion, randomly distributed time delays cause the individual agents to occupy larger portions of space, making their second moments non- negligible [45]. Conversely, in some realistic applications the particle distribution about the center of mass can have an effect on the magnitude of the time delay. In our MF2M framework, this would make the time delay be dependent on the second moment tensors. To summarize, we derived a new MF approximation of a general model swarming system in order to account for higher order moments about the CM. Notably, our extended MF2M model is able to account for the bi-stability of spatio-temporal patterns that are displayed by the original swarming particles. This is in sharp contrast to the MF approximation for this system, which cannot capture this bi-stability and other complex behavior of the time delayed swarming system that we studied. With the inclusion of higher order moments, it is clear to what extent bi-stability depends on the particle distribution of the swarm about its center of mass. Adding the additional physics to the model, although higher dimensional than just a mean field, should allow more general low dimensional approaches to more accurately predict the structure of bi- stability in large population swarms. ###### Acknowledgements. The authors gratefully acknowledge the Office of Naval Research for their support through ONR contract no. N0001412WX20083 and the Naval Research Laboratory 6.1 program contract no. N0001412WX30002. LMR and IBS are supported by Award Number R01GM090204 from the National Institute Of General Medical Sciences. 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1309.5189	# On the similarity of Tensors111P. Yuan’s research is supported by the NSF of China (Grant No. 11271142) and the Guangdong Provincial Natural Science Foundation(Grant No. S2012010009942), L. You’s research is supported by the Zhujiang Technology New Star Foundation of Guangzhou (Grant No. 2011J2200090) and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (Grant No.2012gjhz0007). Pingzhi Yuan222Email address: [email protected]., Lihua You333Corresponding author: [email protected]. (School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China ) ###### Abstract In this paper, we characterize all similarity relations when $m\geq 3$, obtain some interesting properties which are different from the case $m=2$, and show that the results of matrices about the Jordan canonical form cannot be extended to tensors. AMS classification: 15A18; 15A69 Keywords: tensor; product; similarity; hypergraph. ## 1 Introduction Since the work of Qi [5] and Lim [3], the study of tensors and the spectra of tensors (and hypergraphs) and their various applications have attracted much attention and interest. An order $m$ dimension $n$ tensor $\mathbb{A}=(a_{i_{1}i_{2}\ldots i_{m}})_{1\leq i_{j}\leq n\hskip 5.69046pt(j=1,\ldots,m)}$ over the complex field $\mathbb{C}$ is a multidimensional array with all entries $a_{i_{1}i_{2}\ldots i_{m}}\in\mathbb{C}\,(i_{1},\ldots,i_{m}\in[n]=\\{1,\ldots,n\\})$. The majorization matrix $M(\mathbb{A})$ of the tensor $\mathbb{A}$ is defined as $(M(\mathbb{A}))_{ij}=a_{ij\ldots j},(i,j\in[1,n])$ by Pearson [4]. The unit tensor of order $m$ and dimension $n$ is the tensor $\mathbb{I}=(\delta_{i_{1},i_{2},\ldots,i_{m}})$ with $\delta_{i_{1},i_{2},\ldots,i_{m}}=\left\\{\begin{array}[]{cc}1,&{\rm if\hskip 5.69046pt}i_{1}=i_{2}=\ldots=i_{m};\\\ 0,&{\rm otherwise}.\end{array}\right.$ Let $\mathbb{A}$ (and $\mathbb{B}$) be an order $m\geq 2$ (and $k\geq 1$), dimension $n$ tensor, respectively. Recently, Shao [6] defined a general product $\mathbb{A}\mathbb{B}$ to be the following tensor $\mathbb{D}$ of order $(m-1)(k-1)+1$ and dimension $n$: $d_{i\alpha_{1}\ldots\alpha_{m-1}}=\sum\limits_{i_{2},\ldots,i_{m}=1}^{n}a_{ii_{2}\ldots i_{m}}b_{i_{2}\alpha_{1}}\ldots b_{i_{m}\alpha_{m-1}}\quad(i\in[n],\,\alpha_{1},\ldots,\alpha_{m-1}\in[n]^{k-1}).$ The tensor product possesses a very useful property: the associative law ([6], Theorem 1.1). With the general product, the following definition of the similarity relation of two tensors was proposed by Shao [6]. ###### Definition 1.1. ([6], Definition 2.3) Let $\mathbb{A}$ and $\mathbb{B}$ be two order $m$ dimension $n$ tensors. Suppose that there exist two matrices $P$ and $Q$ of order $n$ with $P\mathbb{I}Q=\mathbb{I}$ such that $\mathbb{B}=P\mathbb{A}Q$, then we say that the two tensors are similar. It is easy to see that the similarity relation is an equivalent relation, and similar tensors have the same characteristic polynomials, and thus they have the same spectrum (as a multiset). For example, the permutation similarity and the diagonal similarity (see also [6, 7, 8]) are two special kinds of the similarity of tensors. ###### Definition 1.2. ([6]) Let $\mathbb{A}$ and $\mathbb{B}$ be two order $m$ dimension $n$ tensors. We say that $\mathbb{A}$ and $\mathbb{B}$ are permutational similar, if there exists some permutation matrix $P$ of order $n$ such that $\mathbb{B}=P\mathbb{A}P^{T}$, where $\sigma\in S_{n}$ is a permutation on the set $[n]$ and $P=P_{\sigma}=(p_{ij})$ is the corresponding permutation matrix of $\sigma$ with $p_{ij}=1\Leftrightarrow j=\sigma(i)$. ###### Definition 1.3. ([6], Definition 2.4) Let $\mathbb{A}$ and $\mathbb{B}$ be two order $m$ dimension $n$ tensors. We say that $\mathbb{A}$ and $\mathbb{B}$ are diagonal similar, if there exists some invertible diagonal matrix $D$ of order $n$ such that $\mathbb{B}=D^{-(m-1)}\mathbb{A}Q$. About the matrices $P$ and $Q$ in Definition 1.1, [6] showed the following proposition. ###### Proposition 1.4. ([6], Remark 2.1) If $P$ and $Q$ are two matrices of order $n$ with $P\mathbb{I}Q=\mathbb{I}$, where $\mathbb{I}$ is the order $m$ dimension $n$ unit tensor, then both $P$ and $Q$ are invertible matrices. In this paper, we characterize all similarity relations when $m\geq 3$ in Section 2, and in Section 3, we obtain some interesting properties which are different from the case $m=2$, and show that the results of matrices about the Jordan canonical form cannot be extended to tensors. ## 2 Main results In this section, we will prove Theorem 2.3. We first prove the following two lemmas. ###### Lemma 2.1. Let $P=(p_{ij})$ and $Q$ be two matrices of order $n$ with $P\mathbb{I}Q=\mathbb{I}$, $\mathbb{I}$ the order $m$ dimension $n$ unit tensor. Take $\mathbb{I}Q=\mathbb{A}=(a_{i_{1}i_{2}\ldots i_{m}})$, then we have (i) $a_{i_{1}i_{2}\ldots i_{m}}=0$ when $(i_{2},i_{3},\ldots,i_{m})\neq(i_{2},i_{2},\ldots,i_{2})$; (ii) $PM(\mathbb{A})=I$, where $M(\mathbb{A})$ is the majorization matrix of $\mathbb{A}$ and $I$ is the unit matrix of order $n$. ###### Proof. For the proof of (i), we let $\alpha=i_{2}i_{3}\ldots i_{m}\neq i_{2}i_{2}\ldots i_{2}$. By $P\mathbb{A}=\mathbb{I}$, we have $\delta_{i\alpha}=\sum_{j=1}^{n}p_{ij}a_{j\alpha},\,i=1,2,\ldots,n.$ Since $\alpha\neq i_{2}i_{2}\ldots i_{2}$, then $\delta_{i\alpha}=0$, it follows that $P(a_{1\alpha},a_{2\alpha},\ldots,a_{n\alpha})^{T}=(0,0,\ldots,0)^{T}.$ Observe that $P$ is an invertible matrix by Proposition 1.4, thus $a_{j\alpha}=0,j=1,2,\ldots,n$, and the conclusion of (i) follows. For the proof of (ii), by $P\mathbb{A}=\mathbb{I}$, we have $\delta_{ij\ldots j}=\sum_{u=1}^{n}p_{iu}a_{uj\ldots j}=\sum_{u=1}^{n}p_{iu}(M(\mathbb{A}))_{uj}=(PM(\mathbb{A}))_{ij}.$ Hence $PM(\mathbb{A})=I$, this completes the proof of (ii).∎ ###### Lemma 2.2. Let $\mathbb{I}$ be the order $m\geq 3$ dimension $n$ unit tensor. Suppose that $Q=(q_{ij})$ is a matrix of order $n$ such that $(\mathbb{I}Q)_{i\alpha}=0$ for all $i\in[n]$ and all $\alpha\neq j\ldots j,j\in[n]$, then there is at most one nonzero element in every row of $Q$. ###### Proof. Since $(\mathbb{I}Q)_{i_{1}i_{2}\ldots i_{m}}=\sum_{j_{2},\ldots,j_{m}=1}^{n}\delta_{i_{1}j_{2}\ldots j_{m}}q_{j_{2}i_{2}}\ldots q_{j_{m}i_{m}}=q_{i_{1}i_{2}}q_{i_{1}i_{3}}\ldots q_{i_{1}i_{m}},$ by the assumption, for every $\alpha=i_{2}i_{3}\ldots i_{m}\neq i_{2}i_{2}\ldots i_{2}$, we get $q_{i_{1}i_{2}}q_{i_{1}i_{3}}\ldots q_{i_{1}i_{m}}=0.$ (2.1) If $q_{i_{1}t}\neq 0$ for some $t\in[n]\setminus\\{i_{2}\\}$, then we take $i_{3}=\ldots=i_{m}=t$. By (2.1) we have $q_{i_{1}i_{2}}q_{i_{1}t}^{m-2}=0$, hence $q_{i_{1}i_{2}}=0$. Since the choice of $i_{2}$ is arbitrary, then we have proved that for any $i\in[n]$, there is at most one nonzero element in $\\{q_{i1},q_{i2},\ldots,q_{in}\\}$. This completes the proof of the lemma.∎ ###### Theorem 2.3. Let $\mathbb{I}$ be the unit tensor of order $m\geq 3$ and dimension $n\geq 2$. Suppose that $P$ and $Q$ are two matrices of order $n$ with $P\mathbb{I}Q=\mathbb{I}$, then there exist a permutation matrix $R_{\sigma}$ and an invertible diagonal matrix $D$, such that $Q=DR_{\sigma}$ and $P=R_{\sigma}^{T}D^{1-m}$, where $\sigma\in S_{n}$ is a permutation on the set $[n]$. ###### Proof. Since $\mathbb{I}=P\mathbb{I}Q=P(\mathbb{I}Q)$, combining Lemmas 2.1 and 2.2, we obtain that there is at most one nonzero element in every row of $Q$. Note that $Q$ is an invertible matrix by Proposition 1.4, hence there is precisely one nonzero element, say, $q_{i\sigma(i)}\neq 0,1\leq i\leq n$ in every row of $Q$. Obviously, $\sigma(1),\sigma(2),\ldots,\sigma(n)$ is a permutation of $\\{1,2,\ldots,n\\}$, hence $Q=DR_{\sigma}$ is the product of an invertible diagonal matrix $D=diag(d_{11},\ldots,d_{nn})$, where $d_{ii}=q_{i\sigma(i)}$ and a permutation matrix $R_{\sigma}=(r_{ij})$, where $r_{ij}=1$ if $j=\sigma(i)$ and $r_{ij}=0$ otherwise. Now by the general tensor product, we get $(M(\mathbb{I}Q))_{ij}=(\mathbb{I}Q)_{ij\ldots j}=q_{ij}^{m-1}=\left\\{\begin{array}[]{cc}q_{i\sigma(i)}^{m-1}&{\rm if\hskip 5.69046pt}j=\sigma(i);\\\ 0&{\rm otherwise}.\end{array}\right.$ Hence $M(\mathbb{I}Q)$ is the product of the invertible diagonal matrix $D^{m-1}=diag(d_{11}^{m-1},\ldots,d_{nn}^{m-1})$ and the permutation matrix $R_{\sigma}=(r_{ij})$. By Lemma 2.1 (ii), we have $PM(\mathbb{I}Q)=I$, it follows that $P=R_{\sigma}^{T}D^{1-m}$, is the product of the permutation matrix $R_{\sigma}^{T}$ and the invertible diagonal matrix $D^{1-m}=diag(d_{11}^{1-m},\ldots,d_{nn}^{1-m})$. ∎ ###### Remark 2.4. By the proof of Theorem 2.3, we see that $Q$ and $P$ are closely related with $QP=D^{2-m}$ and $PQ=R_{\sigma}^{T}D^{2-m}R_{\sigma}$ are invertible diagonal matrices. Note that when $m=2$, $P,Q$ are invertible matrices and $PQ=QP=I$. It implies that the case of $m\geq 3$ is completely different from the case $m=2$. Let $\mathbf{P}_{n}$ be the set of all permutation matrices of order $n$, $\mathbb{P}_{n}$ be the set of all matrices which have the same zero patterns with some permutation matrix of order $n$. Clearly, $\mathbf{P}_{n}\subseteq\mathbb{P}_{n}$. For example, $S=\left(\begin{array}[]{cc}0&1\\\ 1&0\\\ \end{array}\right)\in\mathbf{P}_{2},T=\left(\begin{array}[]{cc}0&2\\\ -3&0\\\ \end{array}\right)\in\mathbb{P}_{2}$, where $S$ and $T$ have the same zero pattern. ###### Remark 2.5. Let $\mathbb{I}$ be the unit tensor of order $m\geq 3$ and dimension $n\geq 2$. Suppose that $P$ and $Q$ are two matrices of order $n$ with $P\mathbb{I}Q=\mathbb{I}$, then $P,Q\in\mathbb{P}_{n}$. ###### Remark 2.6. By Definition 1.1, we know that Theorem 2.3 gives a characterization for the similarities of tensors with order $m\geq 3$ dimension $n$, i.e., as for the similarity of tensors, we need only consider the permutation similarity, the diagonal similarity and their compositions. ###### Theorem 2.7. Let $\mathbb{A}$ and $\mathbb{B}$ be two order $m\geq 3$ dimension $n$ tensors. If the tensors $\mathbb{A}$ and $\mathbb{B}$ are similar, then there exists a tensor $\mathbb{C}$ such that $\mathbb{A}$ and $\mathbb{C}$ are diagonal similar, and $\mathbb{B}$ and $\mathbb{C}$ are permutational similar. ###### Proof. By Definition 1.1 and Theoren 2.3, there exist a permutation matrix $R$ and an invertible diagonal matrix $D$ of order $n$ such that $Q=DR$, $P=R^{T}D^{1-m}$ and $\mathbb{B}=P\mathbb{A}Q$. Take $\mathbb{C}=D^{1-m}\mathbb{A}D$, then $\mathbb{B}=R^{T}\mathbb{C}R=R^{T}\mathbb{C}(R^{T})^{T}$. By Definitions 1.2 and 1.3, the results hold. ∎ ## 3 Some applications Let $Z(\mathbb{A})$ be the tensor obtained by replacing all the nonzero entries of $\mathbb{A}$ by one. Then $Z(\mathbb{A})$ is called the zero- nonzero pattern (or simply the zero pattern) of $\mathbb{A}$. Let $a$ be a complex number, we define $Z(a)=1$ if $a\not=0$ and $Z(a)=0$ if $a=0$. ###### Lemma 3.1. Let $\mathbb{A}=(a_{i_{1}i_{2}\ldots i_{m}})_{1\leq i_{j}\leq n\hskip 3.41418pt(j=1,\ldots,m)}$ and $\mathbb{B}=(b_{i_{1}i_{2}\ldots i_{m}})_{1\leq i_{j}\leq n\hskip 2.84544pt(j=1,\ldots,m)}$ be two order $m\geq 3$ dimension $n$ tensors. If the tensors $\mathbb{A}$ and $\mathbb{B}$ are diagonal similar, then $Z(\mathbb{A})$=$Z(\mathbb{B})$. ###### Proof. By Definition 1.3, there exists an invertible diagonal matrix $D=diag(d_{11},\ldots,d_{nn})$ of order $n$ such that $\mathbb{B}=D^{1-m}\mathbb{A}D$. Then $b_{i_{1}i_{2}\ldots i_{m}}=(D^{1-m}\mathbb{A}D)_{i_{1}i_{2}\ldots i_{m}}$ $=\sum\limits_{j_{1},\ldots,j_{m}=1}^{n}(D^{1-m})_{i_{1}j_{1}}a_{j_{1}j_{2}\ldots j_{m}}d_{j_{2}i_{2}}\ldots d_{j_{m}i_{m}}$ $=a_{i_{1}i_{2}\ldots i_{m}}d^{1-m}_{i_{1}i_{1}}d_{i_{2}i_{2}}\ldots d_{i_{m}i_{m}}.$ Therefore $b_{i_{1}i_{2}\ldots i_{m}}\not=0\Leftrightarrow a_{i_{1}i_{2}\ldots i_{m}}\not=0$, and thus $Z(\mathbb{A})$=$Z(\mathbb{B})$. ∎ ###### Theorem 3.2. Let $\mathbb{A}$ and $\mathbb{B}$ be two order $m\geq 3$ dimension $n$ tensors. If the tensors $\mathbb{A}$ and $\mathbb{B}$ are similar, then $Z(\mathbb{A})$ and $Z(\mathbb{B})$ are permutational similar. ###### Proof. By Theorem 2.7, there exists a tensor $\mathbb{C}$ such that $\mathbb{A}$ and $\mathbb{C}$ are diagonal similar, and $\mathbb{B}$ and $\mathbb{C}$ are permutational similar. It is easy that $Z(\mathbb{A})=Z(\mathbb{C})$ by Lemma 3.1. Now we show that $Z(\mathbb{B})$ and $Z(\mathbb{C})$ are permutational similar. Let $R$ be a permutation matrix of order $n$ such that $\mathbb{B}=R\mathbb{C}R^{T}$. Then $(Z(\mathbb{B}))_{i_{1}i_{2}\ldots i_{m}}=(Z(R\mathbb{C}R^{T}))_{i_{1}i_{2}\ldots i_{m}}$ $=Z(\sum\limits_{j_{1},\ldots,j_{m}=1}^{n}r_{i_{1}j_{1}}c_{j_{1}j_{2}\ldots j_{m}}(R^{T})_{j_{2}i_{2}}\ldots(R^{T})_{j_{m}i_{m}})$ $=Z(\sum\limits_{j_{1},\ldots,j_{m}=1}^{n}r_{i_{1}j_{1}}c_{j_{1}j_{2}\ldots j_{m}}r_{i_{2}j_{2}}\ldots r_{i_{m}j_{m}})$ $=Z(c_{\sigma(i_{1})\sigma(i_{2})\ldots\sigma(i_{m})})$ $=\sum\limits_{j_{1},\ldots,j_{m}=1}^{n}r_{i_{1}j_{1}}Z(c_{j_{1}j_{2}\ldots j_{m}})r_{i_{2}j_{2}}\ldots r_{i_{m}j_{m}}.$ So $Z(\mathbb{B})=RZ(\mathbb{C})R^{T}$, and thus $Z(\mathbb{A})$ and $Z(\mathbb{B})$ are permutational similar. ∎ ###### Corollary 3.3. Let $\mathbb{A}$ and $\mathbb{B}$ be two order $m\geq 3$ dimension $n$ tensors, $N(\mathbb{A})$ the number of the nonzero entries of tensor $\mathbb{A}$. If the tensors $\mathbb{A}$ and $\mathbb{B}$ are similar, then $N(\mathbb{A})=N(\mathbb{B})$. ###### Proof. By Theorem 3.2, there exists a permutation matrix $R=R_{\sigma}=(r_{ij})$ of order $n$ such that $Z(\mathbb{B})=RZ(\mathbb{A})R^{T}$, where $\sigma\in S_{n}$ is a permutation on the set $[n]$ and $r_{ij}=1\Leftrightarrow j=\sigma(i)$. Then similar to the proof of Theorem 3.2, we have $(Z(\mathbb{B}))_{i_{1}i_{2}\ldots i_{m}}=(RZ(\mathbb{A})R^{T})_{i_{1}i_{2}\ldots i_{m}}$ $=\sum\limits_{j_{1},\ldots,j_{m}=1}^{n}r_{i_{1}j_{1}}(Z(\mathbb{A}))_{j_{1}j_{2}\ldots j_{m}}(R^{T})_{j_{2}i_{2}}\ldots(R^{T})_{j_{m}i_{m}}$ $=\sum\limits_{j_{1},\ldots,j_{m}=1}^{n}r_{i_{1}j_{1}}(Z(\mathbb{A}))_{j_{1}j_{2}\ldots j_{m}}r_{i_{2}j_{2}}\ldots r_{i_{m}j_{m}}$ $=(Z(\mathbb{A}))_{\sigma(i_{1})\sigma(i_{2})\ldots\sigma(i_{m})}.$ Therefore $(Z(\mathbb{B}))_{i_{1}i_{2}\ldots i_{m}}\not=0\Leftrightarrow(Z(\mathbb{A}))_{\sigma(i_{1})\sigma(i_{2})\ldots\sigma(i_{m})}\not=0$, and thus $N(\mathbb{A})$=$N(\mathbb{B})$. ∎ ###### Remark 3.4. Note that the result of Corollary 3.3 does not hold when $m=2$. For example, let $P=\left(\begin{array}[]{cc}1&1\\\ 1&0\\\ \end{array}\right),$ $Q=\left(\begin{array}[]{cc}1&-1\\\ 1&0\\\ \end{array}\right),$ $A=\left(\begin{array}[]{cc}0&0\\\ 1&0\\\ \end{array}\right),$ $B=PAQ=\left(\begin{array}[]{cc}1&-1\\\ 1&-1\\\ \end{array}\right).$ It is clear $PIQ=QIP=I$, then the matrix $A$ and $B$ are similar, but $N(A)=1\not=N(B)=4.$ ###### Definition 3.5. ([1]) Let $\mathbb{A}=(a_{i_{1}i_{2}\ldots i_{m}})_{1\leq i_{j}\leq n\hskip 3.41418pt(j=1,\ldots,m)}$ be an order $m\geq 3$ dimension $n$ tensor. If $a_{i_{1}i_{2}\ldots i_{m}}\equiv 0$ whenever $\min\\{i_{2},\ldots,i_{m}\\}<i_{1}$, then $\mathbb{A}$ is called an upper triangular tensor. If $a_{i_{1}i_{2}\ldots i_{m}}\equiv 0$ whenever $\max\\{i_{2},\ldots,i_{m}\\}>i_{1}$, then $\mathbb{A}$ is called a low triangular tensor. If $\mathbb{A}$ is either upper or low triangular tensor, then $\mathbb{A}$ is called a triangular tensor. If $a_{i_{1}i_{2}\ldots i_{m}}\equiv 0$ whenever $i_{1}i_{2}\ldots i_{m}\not=i_{1}i_{1}\ldots i_{1}$, then $\mathbb{A}$ is called a diagonal tensor. Clearly, a triangular tensor is both an upper and a low triangular tensor. ###### Corollary 3.6. Let $\mathbb{A}$ and $\mathbb{B}$ be two order $m\geq 3$ dimension $n$ tensors. Suppose that $\mathbb{A}$ and $\mathbb{B}$ are similar, and $\mathbb{A}$ is a diagonal tensor, then $\mathbb{B}$ is a diagonal tensor. ###### Proof. We only need to show $(Z(\mathbb{B}))_{i_{1}i_{2}\ldots i_{m}}=0$ (and thus $b_{i_{1}i_{2}\ldots i_{m}}=0)$ for any $i_{1}i_{2}\ldots i_{m}\not=i_{1}i_{1}\ldots i_{1}$ where $i_{1}\in[n]$. By Theorem 3.2, there exists a permutation matrix $R=R_{\sigma}=(r_{ij})$ of order $n$ such that $Z(\mathbb{B})=RZ(\mathbb{A})R^{T}$, where $\sigma\in S_{n}$ is a permutation on the set $[n]$ and $r_{ij}=1\Leftrightarrow j=\sigma(i)$. Then by the proof of Corollary 3.3, we have $(Z(\mathbb{B}))_{i_{1}i_{2}\ldots i_{m}}\not=0\Leftrightarrow(Z(\mathbb{A}))_{\sigma(i_{1})\sigma(i_{2})\ldots\sigma(i_{m})}\not=0$. Note that $i_{1}i_{2}\ldots i_{m}\not=i_{1}i_{1}\ldots i_{1}$ if and only if $\sigma(i_{1})\sigma(i_{2})\ldots\sigma(i_{m})\not=\sigma(i_{1})\sigma(i_{1})\ldots\sigma(i_{1})$, therefore $(Z(\mathbb{B}))_{i_{1}i_{2}\ldots i_{m}}=0$ when $i_{1}i_{2}\ldots i_{m}\not=i_{1}i_{1}\ldots i_{1}$ since $\mathbb{A}$ is a diagonal tensor. ∎ ###### Remark 3.7. Note that the result of Corollary 3.6 does not hold when $m=2$. For example, all real symmetric matrices are similar to diagonal matrices, but symmetric matrices are not diagonal matrices in general. Another example, if a real matrix $A$ of order $n$ has $n$ distinct eigenvalues, then $A$ is similar to diagonal matrices, but $A$ is not necessarily a diagonal matrix. ###### Definition 3.8. A Jordan block $J_{k}(\lambda)$ is an upper triangular matrix of order $k$ of the form $P=\left(\begin{array}[]{ccccc}\lambda&1&&&\\\ &\lambda&1&&\\\ &&\ddots&\ddots&\\\ &&&\ddots&1\\\ &&&&\lambda\end{array}\right).$ There are $k-1$ terms 1 in the superdiagonal; the scalar $\lambda$ appears $k$ times on the main diagonal. All other entries are zero, and $J_{1}(\lambda)=[\lambda]$. ###### Theorem 3.9. ([2], Jordan canonical form theorem) Let $A$ be a given complex matrix of order $n$. There is a nonsingular matrix $S$ of order $n$ such that $A=SJS^{-1}$, where $J=\left(\begin{array}[]{ccccc}J_{n_{1}}(\lambda_{1})&&&0\\\ &J_{n_{2}}(\lambda_{2})&&\\\ &&\ddots&\\\ 0&&&J_{n_{k}}(\lambda_{k})\end{array}\right)$ is a Jordon matrix with $n_{1}+n_{2}+\ldots+n_{k}=n$, and $J$ is unique up to permutations of the diagonal Jordan blocks, called the Jordan canonical form of $A$. The eigenvalues $\lambda_{i},i=1,\ldots,k$ are not necessarily distinct. If $A$ is a real matrix with only real eigenvalues, then the similarity matrix $S$ can be taken to be real. Clearly, the Jordon matrix $J$ is an upper triangular matrix. ###### Remark 3.10. The results of matrices about the Jordan canonical form cannot be extended to tensors since not all tensors are similar to some upper triangular tensor by Corollary 3.3. ## References * [1] S.L. Hu, Z.H. Huang, C. Ling, L.Q. Qi, On determinants and eigenvalue theory of tensors, Journal of Symbolic Computation, 50(2013), 508-531. * [2] R.A. Horn, C.R. Johnson, Matrix Analysis, Posts and Telecom Press, 2005. * [3] L.H. Lim, Singular values and eigenvalues of tensors, a rational approach, in proceedings 1st IEEE international workshop on computational advances of adaptive processing (2005), 129-132. * [4] K. Pearson, Essentially positive tensors, Int. J. Algebra, 4(2010),421-427. * [5] L. Qi, Eigenvalues of a real supersymmetric tensor, Symbolic Comput. 40(2005), 1302-1324. * [6] J.Y Shao, A general product of tensors with applications, Linear Algebra and its Appl. 439(2013), 2350-2366. * [7] Y Yang and Q. Yang, Further results for Perron-Frobenius theorem for non-negative tensors, SIAM J. Matrix Anal. Appl. 31(2010), 2517-2530. * [8] Y Yang and Q. Yang, Further results for Perron-Frobenius theorem for non-negative tensors II, SIAM J. Matrix Anal. Appl. 31(2011), 1236-1250.	arxiv-papers	2013-09-20T07:11:32	2024-09-04T02:49:51.199986	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Pingzhi Yuan, Lihua You", "submitter": "Lihua You", "url": "https://arxiv.org/abs/1309.5189" }
1309.5231	11institutetext: Institute for Astronomy and Astrophysics, Section Computational Physics, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany 11email: [email protected], [email protected], [email protected] 22institutetext: Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, 10125 Torino, Italy # Radiation hydrodynamics integrated in the code PLUTO Stefan M. Kolb 11 Matthias Stute 11 Wilhelm Kley 11 Andrea Mignone 22 (Received ; accepted 9. September 2013) ###### Abstract Aims. The transport of energy through radiation is very important in many astrophysical phenomena. In dynamical problems the time-dependent equations of radiation hydrodynamics have to be solved. We present a newly developed radiation-hydrodynamics module specifically designed for the versatile MHD code PLUTO . Methods. The solver is based on the flux-limited diffusion approximation in the two-temperature approach. All equations are solved in the co-moving frame in the frequency independent (grey) approximation. The hydrodynamics is solved by the different Godunov schemes implemented in PLUTO , and for the radiation transport we use a fully implicit scheme. The resulting system of linear equations is solved either using the successive over-relaxation (SOR) method (for testing purposes), or matrix solvers that are available in the PETSc library. We state in detail the methodology and describe several test cases in order to verify the correctness of our implementation. The solver works in standard coordinate systems, such as Cartesian, cylindrical and spherical, and also for non-equidistant grids. Results. We have presented a new radiation-hydrodynamics solver coupled to the MHD-code PLUTO that is a modern, versatile and efficient new module for treating complex radiation hydrodynamical problems in astrophysics. As test cases, either purely radiative situations, or full radiation-hydrodynamical setups (including radiative shocks and convection in accretion discs) have been studied successfully. The new module scales very well on parallel computers using MPI. For problems in star or planet formation, we have added the possibility of irradiation by a central source. ###### Key Words.: radiation transport – irradiation – hydrodynamics – accretion disc ## 1 Introduction Radiative effects play a very important role in nearly all astrophysical fluid flows, ranging from planet and star formation to the largest structures in the universe. Coupling the equations of radiation transport to those of (magneto-)hydrodynamics (MHD) has been studied for decades, and comprehensive treatments can be found for example in text books by Mihalas & Mihalas (1984) or Pomraning (1973). The numerical implementation of two-temperature radiation hydrodynamics (in the diffusion approximation) into multi-dimensional MHD/HD- codes has been done already over twenty years ago in various implementations, for example by Eggum et al. (1988), Kley (1989), in the ZEUS-code (Stone et al., 1992), and later by Turner & Stone (2001). In order to study, for example, the dynamics and characteristics of stellar atmospheres together with convection, more accurate solvers for the radiation transport based on the method of short characteristics have been developed, see Davis et al. (2012) and Freytag et al. (2012) for the present status. This can then be coupled to the hydrodynamics using the Variable Eddington Tensor method (Jiang et al., 2012). Another approach is the $\mathrm{M}1$ closure model where the radiative moment equations are closed at a higher level (González et al., 2007; Aubert & Teyssier, 2008). Despite this progress it is still useful and desirable to have a method at hand which solves the interaction of matter and radiation primarily within the bulk part of the matter which may be optically thick. In such type of applications, the method of flux-limited diffusion (FLD, see Levermore & Pomraning (1981)) has its clear merits and is still implemented into existing MHD-codes, for example in NIRVANA (Kley et al., 2009) to study the planet formation process, in RAMSES (Commerçon et al., 2011) for protostellar collapse simulations, and in combination with a multi-frequency irradiation tool into PLUTO (Kuiper et al., 2010) for massive star formation. Since the 3D-MHD code PLUTO (Mignone et al., 2007) is becoming increasingly popular within the computational astrophysics community, we added a publicly available radiation module, which is based on the two-temperature FLD- approximation, as described by Commerçon et al. (2011). PLUTO solves the equations of hydrodynamics and magnetohydrodynamics including the non-ideal effects of viscosity, thermal conduction and resistivity by means of shock- capturing Godunov-type methods. Several Riemann solvers, several time stepping methods and interpolation schemes can be chosen. Additionally, we added a ray- tracing routine that allows for additional irradiation by a point source in the center. Treating the irradiation in a ray-tracing approach, guarantees the long-range character of the radiation better than FLD (Kuiper et al., 2012; Kuiper & Klessen, 2013). The paper is organized as follows. In section 2.1, we briefly introduce the equations of hydrodynamics including radiation transport. Additionally we describe the general idea behind the flux limited diffusion approximation. In section 3, we present the discretization of the equations and the solver of the resulting matrix equation, and present our numerical implementation of irradiation. In section 4, we present six different test cases to show the correctness of the implemented equations: four test cases with an analytical solution (section 4.1 to 4.4) and two others in which our results are compared with those from other codes (section 4.5 and 4.6). We end with a summary and conclusions. ## 2 Radiation hydrodynamics ### 2.1 The equations Even though the PLUTO-environment includes the full MHD-equations and non- ideal effects such as viscosity, we restrict ourselves here to the Euler equations of ideal hydrodynamics. Radiation effects are included in the two- temperature approximation, which implies an additional equation for the radiation energy. In order to follow the transport of radiation, we apply the flux-limited diffusion approximation and treat the exchange of energy and momentum between the gas and the radiation field with additional terms in the gas momentum and energy equations. The system of equations then read: $\displaystyle\frac{\partial}{\partial t}\rho+\nabla\cdot(\rho\@vec{v})$ $\displaystyle=0$ (1) $\displaystyle\frac{\partial}{\partial t}\rho\@vec{v}+\nabla\cdot\left(\rho\@vec{v}\otimes\@vec{v}\right)+\nabla p$ $\displaystyle=+\rho(\@vec{a}_{\mathrm{ext}}+\@vec{a}_{\mathrm{rad}})$ (2) $\displaystyle\frac{\partial}{\partial t}e+\nabla\cdot\left[\left(e+p\right)\@vec{v}\right]$ $\displaystyle=+\rho\@vec{v}\cdot(\@vec{a}_{\mathrm{ext}}+\@vec{a}_{\mathrm{rad}})$ (3) $\displaystyle\phantom{=}\hskip 3.0pt-\kappa_{\mathrm{P}}\rho c(a_{\mathrm{R}}T^{4}-E)$ $\displaystyle\frac{\partial}{\partial t}E+\nabla\cdot\@vec{F}$ $\displaystyle=\kappa_{\mathrm{P}}\rho c\left(a_{\mathrm{R}}T^{4}-E\right)$ (4) The first three equations (1-3) describe the evolution of the gas motion, where $\rho$ is the gas density, $p$ the thermal pressure, $\@vec{v}$ the velocity, $e=\rho\,\epsilon+1/2\,\rho\,v^{2}$ the total energy density (i.e., the sum of internal and kinetic) of the gas without radiation, and $\@vec{a}_{\mathrm{ext}}$ an acceleration caused by external forces (e.g. gravity), not induced by the radiation field (see below). This system of equations is closed by the ideal gas relation $p=(\gamma-1)\,\rho\,\epsilon=\rho\frac{k_{\mathrm{B}}\,T}{\mu\,m_{\mathrm{H}}},$ (5) where $\gamma$ is the ratio of specific heats, $T$ the gas temperature, $k_{\mathrm{B}}$ the Boltzmann constant, $\mu$ the mean molecular weight, and $m_{\mathrm{H}}$ the mass of hydrogen. The specific internal energy can be written as $\epsilon=c_{\mathrm{V}}\,T$, with the specific heat capacity given by $c_{\mathrm{V}}=\frac{k_{\mathrm{B}}}{(\gamma-1)\mu m_{\mathrm{H}}}.$ (6) Here, we assume constant $\gamma$ and $\mu$, which also implies a constant $c_{\mathrm{V}}$. The evolution of the radiation energy density $E$ is given by eq. (4), where $\@vec{F}$ denotes the radiative flux, $\kappa_{\mathrm{P}}$ the Planck mean opacity, $c$ the speed of light and $a_{\mathrm{R}}$ the radiation constant. The fluid is influenced by the radiation in two different ways. First, the radiation may be absorbed or emitted by the fluid leading to variation of its energy density. This variation is given by the expression $\kappa_{\mathrm{P}}\rho c\left(a_{\mathrm{R}}T^{4}-E\right)$, see right hand side of equation (3) and (4). The second effect is that of radiation pressure. We include this term as an additional acceleration to the momentum equation, $\@vec{a}_{\mathrm{rad}}=\frac{\kappa_{\mathrm{R}}}{c}\@vec{F}$. The present implementation does not include the advective transport terms for the radiation energy and radiative pressure work in eqs. (3) and (4). For the relatively low temperature protoplanetary disk application that we consider here these terms are of minor importance. If required, these terms can be treated in our implementation straightforwardly within PLUTO by adding additional source terms. ### 2.2 The flux-limited diffusion approximation The system of equations shown cannot be solved without further assumptions for the radiative flux $\@vec{F}$. Here we use the flux limited diffusion approximation (FLD) where the radiation flux is given by a diffusion approximation $\@vec{F}=-\lambda\,\frac{c}{\kappa_{\mathrm{R}}\,\rho}\,\nabla E\,,$ (7) with the Rosseland mean opacity $\kappa_{\rm R}$. The flux-limiter $\lambda$ describes approximately the transition from very optically thick regions with $\lambda=1/3$ to optically thin regimes, where $\@vec{F}\rightarrow- cE\frac{\nabla E}{\|\nabla E\|}$. This leads to the formal definition of the flux-limiter which is a function of the dimensionless quantity $R=\frac{\|\nabla E\|}{\kappa_{\mathrm{R}}\rho E}\,,$ (8) with the following behaviour: $\lambda(R)=\left\\{\begin{array}[]{ll}\frac{1}{3},&R\to 0\\\ \frac{1}{R},&R\to\infty\end{array}\right.$ (9) Physically sensible flux-limiters thus have to fulfil the equation (9) in the given limits and describe the behaviour between the limits approximately. We have implemented three different flux-limiters: $\displaystyle\lambda(R)$ $\displaystyle=\frac{1}{R}\left(\coth R-\frac{1}{R}\right)$ (10) $\displaystyle\lambda(R)$ $\displaystyle=\left\\{\begin{array}[]{l l}\frac{2}{3+\sqrt{9+12R^{2}}}&\quad 0\leq R\leq\frac{3}{2}\\\ \frac{1}{1+R+\sqrt{1+2R}}&\quad\frac{3}{2}<R\leq\infty\\\ \end{array}\right.$ (13) $\displaystyle\lambda(R)$ $\displaystyle=\left\\{\begin{array}[]{l l}\frac{2}{3+\sqrt{9+10R^{2}}}&\quad 0\leq R\leq 2\\\ \frac{10}{10R+9+\sqrt{180R+81}}&\quad 2<R\leq\infty\\\ \end{array}\right.$ (16) from Levermore & Pomraning (1981), Minerbo (1978), and Kley (1989), respectively. A comparison of them is presented in Kley (1989). In general it is necessary to solve the equations for each frequency which appears in the physical problem. However, here we use the grey approximation in which all radiative quantities including the opacities are integrated over all frequencies. In our treatment scattering is not accounted for directly, but it is included in the effective isotropic absorption and emission coefficients. ## 3 Solving the radiation part ### 3.1 Reformulation of the equations Instead of solving system of equations (1-4) directly as a whole, the problem is split into two steps. In the first step, PLUTO is used to solve the equations of fluid dynamics with the additional force caused by the radiation. This corresponds to the equations (1) to (3) with the additional acceleration, $\@vec{a}_{\mathrm{rad}}$, but without the interaction term between the matter and radiation (last term in eq. 3). By using PLUTO for solving the non- radiative part of the equations, we are not limited to the Euler equations, but are able to use the full capabilities of PLUTO for solving the equations of hydrodynamics or magnetohydrodynamics, including the effects of viscosity and magnetic resistivity. In a second, additional step we solve the radiation energy equation (4) and for the corresponding heating-cooling term in the internal energy of the fluid: $\left.\begin{array}[]{lll}\displaystyle{\frac{\partial}{\partial t}E-\nabla\cdot\left(\frac{c\lambda}{\kappa_{\mathrm{R}}\rho}\nabla E\right)}&=&\phantom{-}\kappa_{\mathrm{P}}\rho c\left(a_{\mathrm{R}}T^{4}-E\right)\\\ \displaystyle{\frac{\partial}{\partial t}\rho\epsilon}&=&-\kappa_{\mathrm{P}}\rho c\left(a_{\mathrm{R}}T^{4}-E\right)\end{array}\quad\right\\}$ (17) In order to obtain the radiation energy density, we solve the system of coupled equations (17). Within one time step PLUTO advances the hydrodynamical quantities, i.e. the density $\rho$, the velocity $\@vec{v}$ and a temporary pressure $p$ from time $t^{n}$ to the time $t^{n+1}$, where the time step, $\Delta t=t^{n+1}-t^{n}$, is determined by PLUTO using the CFL conditions, presently without including the radiation pressure. These depend on the used time stepping method in PLUTO , for more information see Mignone et al. (2007) and the userguide of PLUTO . The physical process of radiation transport takes place on time scales much shorter than the one in hydrodynamics. In order to use the same time step for hydrodynamics and the radiation transport, we apply an implicit scheme to handle the radiation diffusion and the coupling between matter and radiation described by equation (17). Because of the coupling of the equations, the method will update $T$ and $E$ simultaneously, which leads formally to a nonlinear set of coupled equations. As outlined below, the system is solved for the radiation energy density $E$. From the new values for $E$, we compute the new fluid temperature (see eq. 21 below) and update the fluid pressure by using the ideal gas relation from equation (5). This is then used within PLUTO to calculate a new total gas energy $e$. ### 3.2 Discretization In order to discretize the equations (17), we apply a finite volume method. For that purpose we integrate over the volume of a grid cell and transform the divergence into a surface integral. Furthermore, we replace the gradient of $E$ by finite differences, and apply an implicit scheme. The discretization scheme has been implemented in 3D for Cartesian, cylindrical and spherical polar coordinates including all the necessary geometry terms for the divergence and gradient. Since the density has been updated already in the hydrodynamical part of the solver, we can replace $\frac{\partial\rho\epsilon}{\partial t}$ with $\rho\,c_{\mathrm{V}}\,\frac{\partial T}{\partial t}$, which is valid for a constant heat capacity. Then the resulting discretized equations for the radiative part can be written as $\displaystyle\frac{E_{i,j,k}^{n+1}-E_{i,j,k}^{n}}{\Delta t}$ (18) $\displaystyle=$ $\displaystyle G_{x1}^{r}K_{i+\frac{1}{2},j,k}^{n}\frac{E_{i+1,j,k}^{n+1}-E_{i,j,k}^{n+1}}{\Delta{x_{1}}_{i+\frac{1}{2}}}-G_{x1}^{l}K_{i-\frac{1}{2},j,k}^{n}\frac{E_{i,j,k}^{n+1}-E_{i-1,j,k}^{n+1}}{\Delta{x_{1}}_{i-\frac{1}{2}}}$ $\displaystyle+$ $\displaystyle G_{x2}^{r}K_{i,j+\frac{1}{2},k}^{n}\frac{E_{i,j+1,k}^{n+1}-E_{i,j,k}^{n+1}}{\Delta{x_{2}}_{j+\frac{1}{2}}}-G_{x2}^{l}K_{i,j-\frac{1}{2},k}^{n}\frac{E_{i,j,k}^{n+1}-E_{i,j-1,k}^{n+1}}{\Delta{x_{2}}_{j-\frac{1}{2}}}$ $\displaystyle+$ $\displaystyle G_{x3}^{r}K_{i,j,k+\frac{1}{2}}^{n}\frac{E_{i,j,k+1}^{n+1}-E_{i,j,k}^{n+1}}{\Delta{x_{3}}_{k+\frac{1}{2}}}-G_{x3}^{l}K_{i,j,k-\frac{1}{2}}^{n}\frac{E_{i,j,k}^{n+1}-E_{i,j,k-1}^{n+1}}{\Delta{x_{3}}_{k-\frac{1}{2}}}$ $\displaystyle+$ $\displaystyle{\kappa_{\mathrm{P}}^{n}}_{i,j,k}\,\rho_{i,j,k}^{n}c\left(a_{\mathrm{R}}(T_{i,j,k}^{n+1})^{4}-E_{i,j,k}^{n+1}\right)\,,$ and for the thermal energy (or temperature, respectively) $\frac{T_{i,j,k}^{n+1}-T_{i,j,k}^{n}}{\Delta t}=-\frac{{\kappa_{\mathrm{P}}}_{i,j,k}^{n}\,c}{c_{\mathrm{V}}}\,\left(a_{\mathrm{R}}\left(T_{i,j,k}^{n+1}\right)^{4}-E_{i,j,k}^{n+1}\right)\,.$ (19) Here, the superscript $n$ refers to the values of all variables after the most recent update from the hydrodynamical step. In order to simplify the notation for the separate radiation module, we assume the update takes place from time $n$ to $n+1$. The subscripts $i,j,k$ refer to the 3 spatial directions of the computational grid, where all variables are located at the cell centers. Half- integer indices refer to cell interfaces. The physical sizes (proper length) of each cell in the 3 spatial directions $m$ ($m=1,2,3$) are given by $\Delta{x_{m}}$, where we additionally allow for non-equidistant grids. The effective radiative diffusion coefficient (defined at cell centers) is given by $K_{i,j,k}^{n}=\frac{c\lambda(R_{i,j,k})}{{\kappa_{\mathrm{R}}}_{i,j,k}^{n}\,\rho_{i,j,k}^{n}}\,,$ where $R_{i,j,k}$ is calculated from eq. (8) by central differencing. Values at cell interfaces are obtained by linear interpolation. The factors $G^{l,r}_{xm}$ are geometrical terms defined, respectively, as the left and right surface areas divided by the cell volume in the direction given by $m=1,2,3$. In the recent work by Bitsch et al. (2013b) the difference equations have been written out in more detail for Cartesian, equidistant grids. The required opacities are evaluated using the values of $\rho$ and $T$ after the hydrodynamical update at time $t^{n}$. As mentioned before, equations (17) constitute a set of coupled nonlinear equations. The non-linear term $(T_{i,j,k}^{n+1})^{4}$ that appears in equation (19) is linearised using the method outlined in Commerçon et al. (2011) $(T_{i,j,k}^{n+1})^{4}=(T_{i,j,k}^{n})^{4}\left(1+\frac{T_{i,j,k}^{n+1}-T_{i,j,k}^{n}}{T_{i,j,k}^{n}}\right)^{4}\approx 4(T_{i,j,k}^{n})^{3}T_{i,j,k}^{n+1}-3(T_{i,j,k}^{n})^{4}\,.$ (20) Using this approximation, we obtain an equation for computing the new temperature in terms of the new radiation energy density, $E_{i,j,k}^{n+1}$, and the old temperatures, $T_{i,j,k}^{n}$ $T_{i,j,k}^{n+1}=\frac{{\kappa_{\mathrm{P}}}_{i,j,k}^{n}c\left(3a_{\mathrm{R}}(T_{i,j,k}^{n})^{4}+E_{i,j,k}^{n+1}\right)\Delta t+c_{\mathrm{V}}T_{i,j,k}^{n}}{c_{\mathrm{V}}+4{\kappa_{\mathrm{P}}}_{i,j,k}^{n}ca_{\mathrm{R}}(T_{i,j,k}^{n})^{3}\Delta t}\,.$ (21) The expression can be substituted into eq. (18) to obtain a linear system of equations for the new radiation energies $E_{i,j,k}^{n+1}$, that can be solved using standard matrix solvers, see section 3.4. The new temperature can then be calculated from eq. (21). We implemented several boundary conditions for the radiation energy density including periodic, symmetric and fixed value. ### 3.3 Irradiation In order to couple possible irradiation to the radiation transport equations, a new source term, $S$, has to be added to the right hand side of the thermal energy equation in system (17) $\frac{\partial\rho\epsilon}{\partial t}=-\kappa_{\mathrm{P}}\rho c(a_{\mathrm{R}}T^{4}-E)+S\,.$ (22) This results in an additional term, $S_{i,j,k}/(\rho_{i,j,k}c_{\rm V})$, in Eq. 19, correspondingly in equation (21), and in a modification of the right hand side of the resulting matrix equation for $E_{i,j,k}^{n+1}$. For the present implementation, we assume that the irradiating source is located at the centre of a spherical coordinate system. Therefore it is straightforward to compute the optical depth $\tau_{i,j,k}$ even for simulations using parallel computers. Assuming that a ray of light travels along the radial direction from the origin to the grid cell $i,j,k$ under consideration, the optical depth from the inner radius $r_{0}$ to the $i$th grid cell with radius $r_{i}$ can be simply expressed as the integral along the radial coordinate, $\tau_{i,j,k}=\int_{r_{0}}^{r_{i}}\kappa_{\star}\rho(r)\,dr\approx\sum_{n=0}^{i}\kappa_{\star}{{}_{n,j,k}}\,\rho_{n,j,k}\Delta r_{n}$ (23) where $\Delta r_{n}$ is the radial length of the $n$th grid cell, and $\kappa_{\star}$ the opacity used for irradiation. For the sake of readability, we write $\tau_{i}$ instead of $\tau_{i,j,k}$ in the following. We use $\kappa_{\star}=\kappa_{\rm P}$ in the test case with irradiation presented in section 4.3. Additionally $\kappa_{\star}$ can be defined by the user as well as the other opacities. Re-emission of the photons which were absorbed in the cell volume is handled in our treatment by the heating-cooling term see equation (17). The luminosity of the source is given by $L_{\star}=4\pi R_{\star}^{2}\sigma T_{\star}^{4}\,,$ (24) where $\sigma$ denotes the Stefan-Boltzmann constant, $T_{\star}$ is the temperature of the star and $R_{\star}$ its radius. In order to compute the amount of irradiated energy which is absorbed by a specific grid cell we have to know the surface area $A$ of a grid cell oriented perpendicular to the radiation from the star and the flux $f$ at the radius $r$. This surface area $A$ is given by the expression $A_{i,j,k}=\int\limits_{\theta_{j}}^{\theta_{j+1}}\int\limits_{\phi_{k}}^{\phi_{k+1}}\,dA=r_{i}^{2}(\phi_{k+1}-\phi_{k})(\cos\theta_{j}-\cos\theta_{j+1})\,,$ (25) where $\theta$ is the azimuthal and $\phi$ the polar angle in the spherical coordinate system. Without absorption the flux $f$ is given by the expression $f=\frac{L_{\star}}{4\pi r^{2}}=\sigma T_{\star}^{4}\left(\frac{R_{\star}}{r}\right)^{2}\,.$ (26) The amount of energy per time which arrives at the surface of the grid cell $(i,j,k)$ is $H_{i,j,k}=A_{i,j,k}f=(\phi_{k+1}-\phi_{k})(\cos\theta_{j}-\cos\theta_{j+1})\sigma T_{\star}^{4}R_{\star}^{2}\,,$ (27) again without absorption. If the irradiated energy is partly absorbed, the remaining amount of energy per time is then $H_{i,j,k}e^{-\tau_{i,j,k}}$. Using these results, we can compute the energy density per time, $S$, which is absorbed by one grid cell ($i,j,k$) $\displaystyle S_{i,j,k}$ $\displaystyle=\frac{H_{i,j,k}e^{-\tau_{i}}-H_{i,j,k}e^{-\tau_{i+1}}}{V_{i,j,k}}=\frac{H_{i,j,k}\left(e^{-\tau_{i}}-e^{-\tau_{i+1}}\right)}{V_{i,j,k}}$ $\displaystyle=\frac{3\sigma T_{\star}^{4}R^{2}_{\star}\left(e^{-\tau_{i}}-e^{-\tau_{i+1}}\right)}{(r_{i+1}^{3}-r_{i}^{3})}\,,$ (28) with the volume of a grid cell $\displaystyle V_{i,j,k}$ $\displaystyle=\int\limits_{r_{i}}^{r_{i+1}}\int\limits_{\theta_{j}}^{\theta_{j+1}}\int\limits_{\phi_{k}}^{\phi_{k+1}}r^{2}sin\theta\,dr\,d\theta\,d\phi$ $\displaystyle=\frac{1}{3}(r_{i+1}^{3}-r_{i}^{3})(\cos\theta_{j}-\cos\theta_{j+1})(\phi_{k+1}-\phi_{k})\,.$ (29) The absorbed energy density per time, $S_{i,j,k}$, is computed for each grid cell before solving the matrix equation. A similar treatment of irradiation has been described recently by Bitsch et al. (2013b), for a multi-frequency implementation see Kuiper et al. (2010). ### 3.4 The matrix solver We implemented two different solvers for the matrix equation. The first one uses the method of successive over-relaxation (SOR), and as a faster and more flexible solver we use the PETSc111For more information visit the website http://www.mcs.anl.gov/petsc or have a look at Balay et al. (2012). library. From the PETSc library we use the Krylov subspace iterative method and a preconditioner to solve the matrix equation. For all test cases described we used gmres (Generalized Minimal Residual) as iterative methode and bjacobi (Block Jacobi) as preconditioner. Beside others the convergence of the SOR algorithm and the PETSc library can be estimated using the following criteria $\left\\|\@vec{r}^{(k)}\right\\|<\max(\epsilon_{\mathrm{r}}\cdot\left\\|\@vec{b}\right\\|,\epsilon_{\mathrm{a}})$ (30) where $\@vec{b}$ is the right hand side of the matrix equation $A\@vec{x}=\@vec{b}$, $\@vec{r}^{(k)}=\@vec{b}-A\@vec{x}^{(k)}$ is the residual vector for the k-th iteration of the solver and $\@vec{x}$ is the solution vector (here the radiation energy density). As norm we used here the $L_{2}$ norm. The quantities $\epsilon_{\mathrm{r}}$ and $\epsilon_{\mathrm{a}}$ are the relative and absolute tolerance, respectively, and are problem dependent, with a common value of $10^{-50}$ for $\epsilon_{\mathrm{a}}$. For the test cases in section 4 we use relative tolerances $\epsilon_{\mathrm{r}}$ between $10^{-5}$ and $10^{-8}$. The criterion (30) is the default one used by the PETSc library. For more information about the convergence test in PETSc the reader should refer to section 4.3.2. of Balay et al. (2012). The solver performance in a parallel environment is described in section 4.6.4. ## 4 Test cases In order to verify the implemented method, we simulated several test problems and compared the results with either corresponding analytical solutions or calculations done with different numerical codes. Most of the tests correspond to one-dimensional problems. In order to model those, we have used quasi one- dimensional domains, with a very long cuboid that has the height $h$, width $w$ and a length $l$. The length $l$ is much larger than the width or height, and for simplicity we use $w=h$. We performed some of the tests in all three implemented coordinate systems (Cartesian, cylindrical and spherical) and in three different alignments of the cuboid along each coordinate direction. This is done to check whether the geometry factors are correct. In the case of a non-Cartesian coordinate system we placed the cuboid at large distances $r$ from the origin such that the domain approximately describes a Cartesian setup. We use for all test cases the solver based on the PETSc library with the default iterative solver gmres and the pre-conditioner bjacobi. ### 4.1 Linear diffusion test Figure 1: Linear diffusion test at the time $t=4.2\cdot 10^{-12}\,\mathrm{s}$. The simulated (read dots) and the analytical (black line) solution is plotted. We also plot the solution with the absolute value of the relative error (blue dashed line) which belongs to the axis on the right. Figure 2: Time evolution for the linear diffusion test from time $t=0\,\mathrm{s}$ to $4.2\cdot 10^{-12}\,\mathrm{s}$ at three different positions at $x=0\,\mathrm{cm}$ (black lines), at $x=0.5\,\mathrm{cm}$ (blue lines) and at $x=1.0\,\mathrm{cm}$ (red lines). The dotted lines for each position belong to the simulated solution, the solid lines to the analytical solution and the dashed lines show the relative error which belong to the axis on the right. The following test is adapted from Commerçon et al. (2011). The initial profile of the radiation energy density is set to a delta function which is then evolved in time and compared to the analytical one dimensional solution. We perform this test in all implemented coordinate systems (Cartesian, cylindrical and spherical coordinates) as described above, which results in nine different simulations. The used domain is quasi one-dimensional and the equations of hydrodynamics are not solved in this test. Only the radiation diffusion equation $\frac{\partial}{\partial t}E=\nabla\cdot\left(\frac{c\lambda}{\kappa_{\mathrm{R}}\rho}\nabla E\right)$ (31) is solved which we obtain from equations (17) by setting $\kappa_{\mathrm{P}}=0$. An analytical solution to equation (31) can be calculated in the one dimensional case with a constant flux-limiter $\lambda=\frac{1}{3}$ and a constant product of the Rosseland opacity and density, here we set $\kappa_{\mathrm{R}}\rho=1\,\mathrm{cm^{-1}}$. The equation to solve is then given by $\frac{\partial}{\partial t}E(x,t)=\frac{c}{3}\frac{\partial^{2}}{\partial x^{2}}E(x,t)$ (32) with solution $E(x,t)=\frac{\tilde{E}_{0}}{\sqrt{\frac{4}{3}c\pi t}}e^{-\frac{3x^{2}}{4ct}}\,,$ (33) where $\tilde{E}_{0}$ is the integral over the initial profile of the energy density, $E(x,t=0)$. Note that in the quasi one-dimensional case (using a stretched 3D domain) $\tilde{E}_{0}$ has the units $\mathrm{erg\,cm^{-2}}$. #### 4.1.1 Setup The domain is a cuboid with a length of $4\,\mathrm{cm}$ and a width and height of $0.04\,\mathrm{cm}$. We used here $301\times 3\times 3$ grid cells. The initial profile of the radiation energy density in the quasi one- dimensional case is set by $E_{i}=\begin{cases}1\,\mathrm{\frac{erg}{cm^{3}}},&i=1,2,\dots,N\text{ with }i\neq\frac{N}{2}\\\\[5.0pt] \hskip 10.0pt\frac{\tilde{E}_{0}}{\Delta x},&i=\frac{N}{2}\\\ \end{cases}$ (34) where $\Delta x$ is the length of a grid cell. For numerical reasons, we have set $E_{i}$ for $i\neq\frac{N}{2}$ to the value $1\,\mathrm{erg\,cm^{-3}}$ instead of $0\,\mathrm{erg\,cm^{-3}}$. This choice is not problematic, since $\tilde{E}_{0}/\Delta x\gg 1\,\mathrm{erg\,cm^{-3}}$ for our chosen value of $\tilde{E}_{0}=10^{5}\,\mathrm{erg\,cm^{-2}}$. The initial values for pressure and density are $p=1\,\mathrm{g\,cm^{-1}\,s^{-2}}$ and $\rho=1\,\mathrm{g\,cm^{-3}}$. Furthermore we use $\kappa_{\mathrm{R}}=1\,\mathrm{cm^{2}\,g^{-1}}$ for the Rosseland opacity. All boundary conditions are set to periodic except for the boundary conditions at the beginning and end of the quasi one-dimensional domain, which are set to outflow. For the matrix solver we used a relative tolerance of $\epsilon_{r}=10^{-8}$. The simulation starts at $t=0\,\mathrm{s}$ with an constant time step of $\Delta t=1\cdot 10^{-14}\,\mathrm{s}$ and stops at $t=4.2\cdot 10^{-12}\,\mathrm{s}$. #### 4.1.2 Results The numerical solution $E_{\mathrm{n}}$ and the analytical solution $E_{\mathrm{a}}$ from equation (33) are plotted in the figures 1 and 2 together with the absolute value of the relative error. In figure 1 the radiation energy density is plotted against the position at the time $t=4.2\cdot 10^{-12}\,\mathrm{s}$. The relative error in the relevant range from $-1\,\mathrm{cm}$ to $1\,\mathrm{cm}$ is always below one percent. In figure 2 the time evolution from $t=0\,\mathrm{s}$ to $4.2\cdot 10^{-12}\,\mathrm{s}$ is shown for the positions $x=\\{0,0.5,1.0\\}\,\mathrm{cm}$ coded in the colors black, blue and red, respectively. The results shown in this figure depend strongly on the position. For the position $x=0\,\mathrm{c}m$ the error is, for all times later than $t=4\cdot 10^{-13}\,\mathrm{s}$, below one percent and decreases with time. For the other positions, the behaviour is different. The relative error rises and after a while it decreases. This behaviour can be explained by looking at figure 1. The error is higher at the diffusion front. This region moves with time and causes the effect for the other positions. The test shows that the time evolution of the radiation energy density is reproduced correctly. As described, this test was performed in different coordinate systems and orientations, with the same results. ### 4.2 Coupling test Figure 3: Coupling test from $t=10^{-20}\,\mathrm{s}$ to $t=10^{-4}\,\mathrm{s}$ with three different initial gas energy densities. The reference solution (black lines) and the simulated results for the initial energy density $e_{0}=10^{10}\,\mathrm{erg\,cm^{-3}}$ (red dots), $e_{0}=10^{6}\,\mathrm{erg\,cm^{-3}}$ (blue dots) and $e_{0}=10^{2}\,\mathrm{erg\,cm^{-3}}$ (green dots) are plotted. The purpose of this test from Turner & Stone (2001) is to check the coupling between radiation and the fluid. For this purpose we simulate a stationary fluid which is initially out of thermal equilibrium. In this simulation the radiation energy density is the dominant energy which is constant over the whole simulation. The system of equations (17) decouples in this case and, in addition, it is not necessary to solve the matrix equation for $E$. By setting $\sigma_{\mathrm{P}}=\kappa_{\mathrm{P}}\rho$ and $T=\frac{p}{\rho}\frac{\mu m_{\mathrm{H}}}{k_{\mathrm{B}}}$ from eq. (5) with the assumption that $\sigma_{\mathrm{P}}$ and $\rho$ are constant, we can rewrite the thermal energy equation of the system (17) as $\frac{\,\mathrm{d}e}{\,\mathrm{d}t}=\underbrace{c\sigma_{\mathrm{P}}E}_{C_{\mathrm{1}}}-\underbrace{c\sigma_{\mathrm{P}}a_{\mathrm{R}}\left(\frac{\gamma-1}{\rho}\frac{\mu m_{\mathrm{H}}}{k_{\mathrm{B}}}\right)^{4}}_{C_{\mathrm{2}}}e^{4}\,.$ (35) With the used approximations, the coefficients $C_{\mathrm{1}}$ and $C_{\mathrm{2}}$ are constant. The solution to Eq. (35) can be calculated analytically in terms of an algebraic equation which would have to be solved iteratively. Hence, we integrate Eq. (35) numerically using a Runge Kutta solver of $4$-th order scheme with adaptive step size. In the following we refer to this solution as the reference solution. Information on the expected behaviour of the solution can be obtained directly from the differential equation. It is clear that in the final equilibrium state (with $\frac{\,\mathrm{d}e}{\,\mathrm{d}t}=0$) the gas temperature has to be equal to the radiation temperature $T=\sqrt[4]{\frac{E}{a_{\mathrm{R}}}}$, thus the final gas energy density will be $e_{\mathrm{final}}=\left(\frac{C_{\mathrm{1}}}{C_{\mathrm{2}}}\right)^{\frac{1}{4}}\,.$ (36) If the initial gas energy density $e_{0}$ is much lower than $e_{\mathrm{final}}$, we can neglect the second term in eq. (35) at the beginning, thus $e(t)=C_{\mathrm{1}}\,t+e_{0}$. The corresponding coupling time can be estimated to $\tau=\frac{e_{\mathrm{final}}-e_{0}}{C_{\mathrm{1}}}\,.$ (37) On the other hand, if $e_{0}\gg e_{\mathrm{final}}$, we can neglect the first term in eq. (35) and derive $e(t)\propto\left(C_{\mathrm{2}}\,t\right)^{-\frac{1}{3}}\quad\mathrm{and}\quad\tau=\frac{1}{e_{\mathrm{final}}^{3}\,C_{\mathrm{2}}}\,.$ (38) #### 4.2.1 Setup The computational domain is identical to that of the linear diffusion test in section 4.1. For the grid we use a resolution of $25\times 3\times 3$ grid cells. As before we do not solve the equations of hydrodynamics and the boundary conditions are quite simple. All boundaries are set to periodic boundary conditions. The constants we used are set to: radiation energy density $E=10^{12}\,\mathrm{erg\,cm^{-3}}$, density $\rho=10^{-7}\,\mathrm{g\,cm^{-3}}$, opacity $\sigma_{\mathrm{P}}=4\cdot 10^{-8}\,\mathrm{cm^{-1}}$, mean molecular weight $\mu=0.6$ and the ratio of specific heats $\gamma=5/3$. The simulations starts at $t=0\,\mathrm{s}$ with an initial time step of $\Delta t=10^{-20}\,\mathrm{s}$ and evolves until $t=10^{-4}\,\mathrm{s}$. After each step the time step is increased by 1% in order to speed-up the computation. The simulation is done with three different initial gas energy densities, $e_{0}=10^{10}\,\mathrm{erg\,cm^{-3}}$, $e_{0}=10^{6}\,\mathrm{erg\,cm^{-3}}$ and $e_{0}=10^{2}\,\mathrm{erg\,cm^{-3}}$. #### 4.2.2 Results Figure 3 shows the numerical gas energy density and the reference solution plotted against time for the three different initial values of $e$. The agreement of both results is excellent for all initial values. From the figure we see that in the limit of small and large initial $e_{0}$, we find exactly the behaviour as predicted by the estimates for eq. (35). The analytic estimates for the coupling time $\tau$ from equation (37) agrees very well with our results. The estimate for $e_{0}=10^{2}\,\mathrm{erg\,cm^{-3}}$ is $\tau=5.88\cdot 10^{-8}\,\mathrm{s}$ and for $e_{0}=10^{6}\,\mathrm{erg\,cm^{-3}}$ we calculated $\tau=5.78\cdot 10^{-8}\,\mathrm{s}$. In the case that $e_{\mathrm{final}}<e_{0}$ the estimate in eq. (38) is approximate $\tau=5.88\cdot 10^{-8}\,\mathrm{s}$. We have to mention here that this test verifies primarily the correctness of equation (21). As in the linear diffusion test, this test was performed in three different coordinate systems in different orientations, with the same results. ### 4.3 Coupling test with irradiation Figure 4: Coupling test with enabled irradiation from $t=10^{-7}\,\mathrm{s}$ to $t=10^{3}\,\mathrm{s}$ at three different distances $d$ from the inner boundary of the domain. The reference solution (black lines) and the simulated results for the energy density $e=10^{2}\,\mathrm{erg\,cm^{-3}}$ are plotted at distance $d=0\,\mathrm{cm}$ (red dots), $d=3\cdot 10^{4}\,\mathrm{cm}$ (blue dots) and $d=3\cdot 10^{5}\,\mathrm{cm}$ (green dots). Figure 5: Radial dependency of the gas energy density for the coupling test with enabled irradiation. The gas energy density is plotted at different times, $t=1.12\cdot 10^{-4}\,\mathrm{s}$ (red dots), $t=1.62\cdot 10^{-2}\,\mathrm{s}$ (blue dots), $t=1.43\,\mathrm{s}$ (green dots), $t=1.72\cdot 10^{1}\,\mathrm{s}$ (yellow dots) and $t=9.18\cdot 10^{2}\,\mathrm{s}$ (magenta dots). The dots represent the numerical solution and the solid black lines the reference solution. The position $d$ is again measured relative to the inner boundary of the quasi one-dimensional domain. This test is in its basic setup the same as that from section 4.2, but with irradiation enabled, i.e. equation (22) is solved instead of the second equation in (17). As described in section 3.3, irradiation is limited to spherical coordinates, which we use for this test. With the same assumptions as in section 4.2 , i.e., that $\sigma_{\mathrm{P}}$ and $\rho$ are constant and with the definitions for $\sigma_{\mathrm{P}}$, $e$, $p$ as well as for $T$, it is possible to rewrite $S$ from equation (28) to $S(r)=\frac{3\sigma T_{\star}^{4}R^{2}_{\star}e^{-\sigma_{\mathrm{P}}\left(r-r_{\mathrm{0}}\right)}\left(1-e^{-\sigma_{\mathrm{P}}\Delta r}\right)}{\left(r+\Delta r\right)^{3}-r^{3}}\,,$ (39) and obtain for equation (22) $\frac{\,\mathrm{d}e}{\,\mathrm{d}t}=\underbrace{S(r)+c\sigma_{\mathrm{P}}E}_{C_{\mathrm{1}}(r)}-\underbrace{c\sigma_{\mathrm{P}}a_{\mathrm{R}}\left(\frac{\gamma-1}{\rho}\frac{\mu m_{\mathrm{H}}}{k_{\mathrm{B}}}\right)^{4}}_{C_{\mathrm{2}}}e^{4}\,.$ (40) The reference solution is computed in the same way as before although it now depends on the distance $r$ from the star. The quasi one-dimensional domain starts at $r=9000\cdot 10^{5}\,\mathrm{cm}$ and ends at $r=9003\cdot 10^{5}\,\mathrm{cm}$ and we use $300\times 3\times 3$ grid cells. The domain size in $\theta$ and $\phi$ direction was chosen in a way such that the grid cells are nearly quadratic. For the simulation we use a constant radiation energy density of $E=10^{-2}\,\mathrm{erg\,cm^{-3}}$, a density of $\rho=10^{-5}\,\mathrm{g\,cm^{-3}}$ , a Rosseland opacity of $\kappa_{\rm R}=10\,\mathrm{cm^{2}\,g^{-1}}$ and a Planck opacity of $\kappa_{\mathrm{P}}=\kappa_{\mathrm{R}}$ which corresponds to $\sigma_{\mathrm{P}}=10^{-4}\,\mathrm{cm^{-1}}$. The opacity for the irradiation $\kappa_{\star}$ is set to $\kappa_{\mathrm{P}}$. For the star, the temperature was set to $T_{\star}=6000\,\mathrm{K}$ and the radius to $R_{\star}=8.1\cdot 10^{8}\,\mathrm{cm}$. Additionally we make the assumption that there is no absorption in the region between the surface of the star and the inner boundary of the computation domain. Figure 4 shows the gas energy density plotted against time with an initial gas energy of $e=10^{2}\,\mathrm{erg\,cm^{-3}}$ at three different positions $d=0\,\mathrm{cm}$, $d=3\cdot 10^{4}\,\mathrm{cm}$ and $d=3\cdot 10^{5}\,\mathrm{cm}$ where $d$ is measured relative to the inner boundary of the quasi one-dimensional domain. In figure 5 the radial dependency of the gas energy density is plotted for the same simulation at five different times. As expected, the results show that the gas energy density at a time later than $t=10^{2}\,\mathrm{s}$ becomes constant and depends on the distance from the star. The simulated and reference solution show a excellent agreement. ### 4.4 A steady state test Figure 6: Comparison between the numerical (red dots) and the analytical (black line) solution of the steady state test after $t=1200\,\mathrm{s}$. In addition the absolute value of the relative error (blue dashed line) is plotted related to the axis on the right side. Note this axis is logarithmic. The original version of this test was published in Flaig (2011). We consider a one-dimensional stationary setup with a given density stratification. In the steady state, the time derivatives in the equations (17) vanish and the system is reduced to the following equation for the radiation energy density $0=\nabla\cdot\,\left(\frac{c\lambda}{\kappa_{\mathrm{R}}\rho}\nabla E\right)\,.$ (41) A further reduction is obtained when we rewrite this equation in one dimension along the z-axis in Cartesian coordinates. The equation is then much simpler and can be written as $\frac{\,\mathrm{d}}{\,\mathrm{d}z}\left(\frac{c\lambda}{\kappa_{\mathrm{R}}\rho}\frac{\,\mathrm{d}}{\,\mathrm{d}z}E\right)=0\,.$ (42) In general the expression $\frac{c\lambda}{\kappa_{\mathrm{R}}\rho}$ is not known analytically for realistic opacities. In order to circumvent this problem, we define the effective optical depth $\tau_{\mathrm{eff}}=\int\,\mathrm{d}\tau_{\mathrm{eff}}=\int_{z_{b}}^{z_{a}}\kappa_{\mathrm{eff}}\rho\,\mathrm{d}z$ where $z_{a}$ and $z_{b}$ are the lower and upper boundaries of the quasi one- dimensional domain, respectively, and $\kappa_{\mathrm{eff}}$ is the effective opacity given by $\kappa_{\mathrm{eff}}=\frac{1}{3}\frac{\kappa_{\mathrm{R}}}{\lambda}$. By using $\,\mathrm{d}\tau_{\mathrm{eff}}=\kappa_{\mathrm{eff}}\rho\,\mathrm{d}z$, equation (42) can be rewritten as: $\frac{\,\mathrm{d}}{\,\mathrm{d}z}\frac{\,\mathrm{d}}{\,\mathrm{d}\tau_{\mathrm{eff}}}E=0\,.$ (43) The solution of this equation is then given by $E=\left(E(\tau_{\mathrm{eff}}=1)-E(\tau_{\mathrm{eff}}=0)\right)\tau_{\mathrm{eff}}+E(\tau_{\mathrm{eff}}=0)$ (44) where $E(\tau_{\mathrm{eff}}=0)$ and $E(\tau_{\mathrm{eff}}=1)$ are the radiation energy density at the position where the effective optical depth has the values zero or one, respectively. Thus, in the static case the radiation energy has a linear dependence on the optical depth $\tau_{\mathrm{eff}}$ for all opacity laws. #### 4.4.1 Initial setup The domain was chosen to have an arbitrary length of $300\,\mathrm{cm}$ and a width and height of $3\,\mathrm{cm}$ and $300\times 3\times 3$ grid cells were used. This test is performed without solving the hydrodynamical equations, instead we solved equations (17) for a fixed density and opacity law, and evolved the solution, until a stationary state has been reached. For the radiation boundary conditions, we used boundary conditions with fixed values of $E$ at the lower and upper boundary of the domain. At the lower boundary we have chosen $E=a_{\mathrm{R}}T^{4}$ with a temperature of $T=2000\,\mathrm{K}$. Because the stratification is optically thin at the upper boundary we want to allow the radiation to escape freely from the domain. For this reason we simply set the temperature to a very small value at the upper boundary, here $T=10\,\mathrm{K}$. All other boundary conditions have been set to periodic. The density stratification is given by $\rho(z)=\rho_{0}e^{\frac{1}{2}\left(\frac{z-z_{a}}{0.46\cdot\left(z_{b}-z_{a}\right)}\right)^{2}}\,.$ (45) The initial temperature profile can be chosen randomly in principle, but in order to speed up the computation we used a linear temperature profile starting at $z_{a}$ with $T=2000\,\mathrm{K}$ and ending at $z_{b}$ with $T=10\,\mathrm{K}$. From this temperature profile we assigned pressure values using equation (5). The radiation energy density $E$ inside the domain is also set using the gas temperature profile and $E=a_{\mathrm{R}}T^{4}$. The ratio of specific heats and the mean molecular weight are set to $\gamma=1.43$ and $\mu=0.6$, respectively. As flux-limiter we have chosen equation (13), for the Rosseland mean opacity $\kappa_{\rm R}$ we use data from Lin & Papaloizou (1985), and the Planck mean opacity is set to $\kappa_{\rm P}=\kappa_{\rm R}$. The initial time step is $\Delta t=0.3\,\mathrm{s}$ and it is increased slightly with time in order to speed-up the computation and to keep the number of iterations done by the matrix solver nearly constant. This simulation was preformed with a relative tolerance of $\epsilon_{r}=10^{-6}$ for the matrix solver. #### 4.4.2 Results A steady state is reached approximately after $t=1200\,\mathrm{s}$. In figure 6, we plot the radiation energy density against the effective optical depth $\tau_{\text{eff}}$ from our numerical solution (red dots) together with the analytical solution from equation (44). The parameters $E(\tau_{\mathrm{eff}}=1)-E(\tau_{\mathrm{eff}}=0)$ and $E(\tau_{\mathrm{eff}}=0)$ have been obtained by fitting equation (44) to the numerical solution. We have to note here that $E(\tau_{\mathrm{eff}}=0)$ is determined by interpolation between ghost cells and active cells near the upper boundary $z_{\mathrm{b}}$. Hence, the radiation temperature in the active region can be much larger than $10\,\mathrm{K}$, a value was specifically chosen to be very small. We also plot the absolute value of the relative error $\left\|(E_{n}-E_{a})/E_{a}\right\|$. The results from the simulation agree very well with the analytical prediction. As we can see from figure 6, the largest deviation from the analytical solution is at small values of $\tau_{\mathrm{eff}}$ with an relative error around one percent. As the linear diffusion and the coupling test, this test was performed in all three coordinate systems and in different orientations, with the same results. ### 4.5 Radiation shock (a) Subcritical shock (b) Supercritical shock Figure 7: Sub- and supercritical shock test. In both cases we plot the radiation temperature (blue line) and the gas temperature (red line) against $s=z-v\cdot t$ where $z$ is the position along the quasi one-dimensional domain and $v$ the piston velocity. The subcritical shock 7(a) is shown at time $t=3.8\cdot 10^{4}\,\mathrm{s}$ and the supercritical shock 7(b) at $t=7.5\cdot 10^{3}\,\mathrm{s}$. In this section we extend the previous tests and solve now the full equations of hydrodynamics and radiation transport simultaneously, testing the complete new module within PLUTO environment. #### 4.5.1 Initial setup Following a set-up from Ensman (1994), a shock is generated in a quasi one- dimensional domain. This test case is more complex than the ones before, and it is not possible to derive an analytical solution. Instead we compare our results with the simulations of Commerçon et al. (2011). The computational domain is chosen to have a length of $7\cdot 10^{10}\,\mathrm{cm}$ and a width and height of $3.418\cdot 10^{7}\,\mathrm{cm}$ with $2048\times 4\times 4$ grid cells. The initial density and temperature are set to $\rho=7.78\cdot 10^{-10}\,\mathrm{g\,cm^{-3}}$ and $T=10\,\mathrm{K}$. The initial radiation energy density is set by the equation $E=a_{\mathrm{R}}T^{4}$. For the flux- limiter we employ the Minerbo-formulation according to eq. (13), and for the opacity we use $\kappa_{\mathrm{R}}\cdot\rho=\kappa_{\mathrm{P}}\cdot\rho=3.1\cdot 10^{-10}\,\mathrm{cm^{-1}}$. Furthermore the ratio of specific heats is set to $\gamma=7/5$ and the mean molecular weight to $\mu=1$, in analogy to Commerçon et al. (2011). The time step is computed through the CFL condition of PLUTO for wich we assume a value of $0.4$. For the solver we took the not so accurate but robust tvdlf which uses a simple Lax-Friedrichs scheme. For generating the radiative shock, the following boundary conditions are used: in the direction of the shock propagation, we employ a reflective boundary condition at the lower boundary and a zero-gradient at the upper boundary of the domain. The remaining boundaries are set to periodic. For the relative tolerance used by the matrix solver we have chosen a value of $\epsilon_{r}=10^{-5}$. The shock is generated by applying an initial velocity $v$ to the gas. The velocity is directed towards the reflecting boundary condition which acts as a wall. The shock propagates then from the wall back into the domain. Depending on the velocity, the shock is sub- or supercritical, i.e., the temperature behind the shock front is larger or equal than the temperature upstream (in front of the shock front), respectively. In this test we simulate both cases: the subcritical shock with a velocity of $v=6\cdot 10^{5}\,\mathrm{cm\,s^{-1}}$ and the supercritical shock with $v=20\cdot 10^{5}\,\mathrm{cm\,s^{-1}}$. #### 4.5.2 Results For a better comparison with the results of simulations, where the material is at rest and a moving piston causes the shock, we introduce the quantity $s$. This quantity is given by the relation $s=z-v\cdot t$ where $z$ is the position along the quasi one-dimensional domain. Note that this quantity is called $z$ in Commerçon et al. (2011). Fig. 7 shows the radiation temperature (blue line) and the gas temperature (red line) against the previously defined quantity $s$ for both the subcritical (at $t=3.8\cdot 10^{4}\,\mathrm{s}$) and supercritical case (at $t=7.5\cdot 10^{3}\,\mathrm{s}$). In the supercritical case the pre- and post-shock gas temperature are equal, as expected. In the subcritical case these temperatures can be estimated analytically (Ensman, 1994; Mihalas & Mihalas, 1984; Commerçon et al., 2011). In table 1, the analytical estimates and the numerical values from our simulations and the results from Commerçon et al. (2011) are shown together. Here $T_{2}$ is the post-shock temperature, $T_{-}$ the pre-shock temperature and $T_{+}$ the spike temperature. In the equations, $R_{\rm G}=\frac{k_{\mathrm{B}}}{\mu m_{\mathrm{H}}}$ is the perfect gas constant, $\sigma_{\mathrm{SB}}=\frac{ca_{\mathrm{R}}}{4}$ the Stefan-Boltzmann constant, and $u$ is the velocity of the shock relative to the upstream material (or vice versa) in our case $u=7.19\cdot 10^{5}\,\mathrm{cm\,s^{-1}}$. \| analytical estimate \| numerical solution \| Commerçon et al. ---\|---\|---\|--- $T_{2}\approx\frac{2(\gamma-1)u^{2}}{R_{\rm G}(\gamma+1)^{2}}$ \| $\sim\phantom{1}865\,\mathrm{K}$ \| $\phantom{1}816.6\,\mathrm{K}$ \| $\phantom{1}825\,\mathrm{K}$ $T_{-}\approx\frac{\gamma-1}{\rho uR_{\rm G}}\frac{2\sigma_{\mathrm{SB}}T_{2}^{4}}{\sqrt{3}}$ \| $\sim\phantom{1}315\,\mathrm{K}$ \| $\phantom{1}331.9\,\mathrm{K}$ \| $\phantom{1}275\,\mathrm{K}$ $T_{+}\approx T_{2}+\frac{3-\gamma}{\gamma+1}T_{-}$ \| $\sim 1075\,\mathrm{K}$ \| $1147.1\,\mathrm{K}$ \| $1038\,\mathrm{K}$ Table 1: Comparison of the results from the radiation shock test with analytical estimates and the results from Commerçon et al. (2011) for the pre- shock $T_{-}$ and post-shock $T_{2}$ gas temperature as well as the spike temperature $T_{+}$. The results agree in general with the analytical estimates and the results from Commerçon et al. (2011). The analytical estimate for the post-shock temperature is higher than the numerical results with both codes. We have to note here that the analytical estimate depends on $u$ and differs therefore from the values given in Commerçon et al. (2011). The pre-shock and spike temperatures agree reasonably well with the analytical estimates in our simulations but are higher than the results from Commerçon et al. (2011). The differences of our numerical solution to the analytical estimates might due to the fact that we ignored the advective terms in the radiation energy density in eq. (4) that may play a role in this dynamic situation. Additionally, it is noteworthy that the position of the shock front is very well reproduced. This test was performed in Cartesian coordinates. ### 4.6 Accretion disc The goal of this last test is to compare the results of different codes on a more complex two-dimensional physical problem that involves the onset of convective motions. For this purpose we model a section of an internally heated, viscous accretion disc in spherical coordinates $(r,\theta,\phi)$ where $r$ is the distance to the centre of the coordinate system, $\theta$ the polar angle measured from the $z$-axis in cylindrical coordinates and $\phi$ the azimuth angle. The setup follows the standard disc model used in Kley et al. (2009). The tests proceed in two steps. In a first setup we reduce the complexity of the problem and consider a static problem, i.e., without solving the equations of hydrodynamics. This will demonstrate that the equilibrium between viscous heating and radiative cooling is treated correctly in our implementation. In the second setup we consider the full hydrodynamic problem and study the onset of convection in discs. #### 4.6.1 The initial setup For both, the static and the dynamical case we use the same initial setup. The radial extent ranges from $r_{\text{min}}=0.4$ to $r_{\text{max}}=2.5$, where all lengths are given in units of the semi-major axis of Jupiter $a_{\text{jup}}=5.2\,\mathrm{AU}$. In the vertical direction the domain extends from $\theta_{\text{min}}=83\,\mathrm{{}^{\circ}}$ to $\theta_{\text{max}}=90\,\mathrm{{}^{\circ}}$ and in $\phi$ direction from $\phi_{\text{min}}=0\,\mathrm{{}^{\circ}}$ to $\phi_{\text{max}}=360\,\mathrm{{}^{\circ}}$. In the three coordinate directions $(r,\theta,\phi)$ we use $256\times 32\times 4$ grid cells. The disc aspect ratio $h$ is set to $h=\frac{H}{s}=0.05\,\mathrm{}$ where $s=r\sin\theta$ describes the (radial) distance from the $z$-axis in cylindrical coordinates, and $H$ is the disc’s vertical scale height. The viscosity $\nu$ is set to a value of $\nu=10^{15}\,\mathrm{cm^{2}\,s^{-1}}$, and the mean molecular weight to $\mu=2.3\,\mathrm{}$. For the ratio of specific heats we have used different values, as specified below. The density stratification can be obtained from vertical hydrostatic equilibrium, assuming a temperature that is constant on cylinders, $T=T(s)$. It follows (Masset et al., 2006) $\rho(r,\theta)=\rho_{0}\cdot s^{-1.5}\exp\left(\frac{\sin{\theta}-1}{h^{2}}\right)$ (46) where the quantity $\rho_{0}$ was chosen such that the total mass of the disc is $M_{\text{disc}}=0.01\cdot M_{\star}$, where $M_{\star}$ is the mass of the central star of the system which is set to the mass of the sun, $M_{\star}=M_{\odot}$. The mass within the computational domain is then $1/2M_{\text{disc}}$ because we only compute the upper half of the disc. The radial variation leads to a surface density profile of $\Sigma\propto r^{-1/2}$, which is the equilibrium profile for constant viscosity, and vanishing mass flux through the disc. The pressure $p$ is set by the isothermal relation $p=\rho c_{\mathrm{s}}^{2}$, with the speed of sound $c_{\mathrm{s}}=H\Omega_{\mathrm{K}}$ and the Keplerian angular velocity $\Omega_{\mathrm{K}}=\sqrt{\frac{GM_{\star}}{s^{3}}}\,,$ with the gravitational constant $G$. The temperature can be computed through equation (5) and results in $T=\frac{\mu m_{\mathrm{H}}}{k_{\mathrm{B}}}\frac{p}{\rho}$. The initial velocities are set to zero except for the angular velocity $v_{\phi}$ which is set to $v_{\phi}=\sqrt{\frac{\left(1-2h^{2}\right)GM_{\star}}{s}}\,.$ For the Rosseland mean opacity $\kappa_{\rm R}$ we use data from Lin & Papaloizou (1985), and the Planck mean opacity is set to $\kappa_{\rm P}=\kappa_{\mathrm{R}}$. The displayed simulations have been performed in the rotating frame in which the coordinate system rotates with the constant angular velocity of $\Omega_{\mathrm{K}}$ at $a_{\text{jup}}$, but for non- rotating systems identical results are obtained. As before the radiation energy density is initialised to $E=a_{\mathrm{R}}T^{4}$. For density, pressure and radial velocity we apply reflective radial boundary conditions and the angular velocity is set to the Keplerian values. In the azimuthal direction periodic boundary conditions are used for all variables. In the vertical direction we apply an equatorial symmetry and reflective boundary condition for $\theta_{min}$. The radiation boundary conditions are set to reflective for the $r$ direction (both lower and upper), in $\theta$-direction we use a fixed value of $E=a_{\mathrm{R}}T^{4}$ with $T=5\,\mathrm{K}$ at $\theta_{min}$ (which denotes the disc surface), and a symmetric boundary condition holds at the disc’s midplane $\theta_{max}$. For the $\phi$-direction we use periodic boundary conditions. In both cases we used for the matrix solver a relative tolerance of $\epsilon_{r}=10^{-8}$. In the simulation with hydrodynamics we use the Riemann-solver hllc222Harten, Lax, Van Leer approximate Riemann Solver with the contact discontinuity. (a) $t=10\text{ orbits}$ (b) $t=100\text{ orbits}$ Figure 8: Radial mid-plane temperature profile in the simulations with PLUTO (red dots) and with the code RH2D (black line) after $t=10\,\mathrm{orbits}$ 8(a) and $t=100\,\mathrm{orbits}$ 8(b), together with the absolute value of the relative error (blue dashed line) which belongs to the log axis on the right. #### 4.6.2 The static case In this test case only the radiative equations are solved without the hydrodynamics. In order to account for the viscous heating in this case, we add an additional dissipation contribution, $D$, to the right hand side of the internal energy equation in (17). We consider standard viscous heating, and include only the main contribution due to the approximately Keplerian shear flow. At the individual grid points the dissipation is then given by $D_{i,j,k}=r_{i}^{2}\rho_{i,j,k}\nu\left(\frac{\partial\Omega_{i,j,k}}{\partial r_{i}}\right)^{2}\,,$ (47) where $\nu$ is the constant viscosity and $\Omega_{i,j,k}$ the angular velocity at the individual grid points. In summary we solve the same equations as in the case with irradiation, when we substitute $S_{i,j,k}$ with $D_{i,j,k}$. (a) $\gamma=5/3$ (b) $\gamma=1.1$ Figure 9: Radial mid-plane temperature profile in the simulation with PLUTO (red line), RH2D (black line) and NIRVANA (blue line) in the quasi-equilibrium state after 100 orbits in the case with $\gamma=5/3$ without convection 8(a) and in the strongly convective case with $\gamma=1.1$ 8(b). Additionally we added the results of a simulation performed with PLUTO where we use a logarithmic grid in $r$-direction (green line). In the steady state, the time derivatives in the equations (17) vanish and the system is reduced to the following equation for the radiation energy density $\nabla\cdot\,\left(\frac{c\lambda}{\kappa_{\mathrm{R}}\rho}\nabla E\right)=D.$ (48) In optically thick regions, $E=a_{\rm R}T^{4}$ and eq. (48) determines the temperature stratification within the disc. Figure 10: Vertical slice of the disc temperature at $t=100\,\mathrm{orbits}$ in the dynamical case with $\gamma=1.1$ showing convection cells. Also plotted in the inset is the enlarged region from $r=0.4\,a_{\text{jup}}$ to $0.6\,a_{\text{jup}}$ with the velocity field in the $r-\theta$ plane (black arrows). The simulation starts at $t=0\,\mathrm{orbits}$ and is evolved until $t=100\,\mathrm{orbits}$ are reached, where one orbit corresponds to the Keplerian orbital period at the distance of $a_{\text{jup}}$ which is given here by $3.732\cdot 10^{8}\,\mathrm{s}$. The initial and overall time step was chosen as $\Delta t=10^{-3}\,\mathrm{orbits}=3.732\cdot 10^{5}\,\mathrm{s}$. The results for the static case are shown in figure 8 using here a value of $\gamma=7/5$ for the adiabatic index. The plots show the radial temperature profile of the accretion disc in the mid-plane for the simulations after 10 orbits (top panel) and after 100 orbits (bottom panel). We display results of two different simulations, one done with the code PLUTO (red dots) using the described methods, and the second (black lines) run with the code RH2D (Kley, 1989). The result from both codes are nearly identical. Even after $100\,\mathrm{orbits}$ the absolute value of the relative error is always less than $2\%$. The test shows that the time-scale of the radiative evolution, as well as the equilibrium state is captured correctly. We note that the code RH2D uses the one-temperature approach of radiation transport in this case. #### 4.6.3 The dynamical case The final equilibrium of the described static case does not depend on the magnitude of $\gamma$, because the viscous heating is independent of it, see eq. (48). The situation is different, however, for the dynamical cases, where the hydrodynamical evolution of the flow is taken into account. Since the time scale of the radiative transport depends on $\gamma$ (through eq. 6), one might expect the possibility of convective instability, see for example the recent work by Bitsch et al. (2013a). This is indeed the case for small enough values of $\gamma$. In order to demonstrate the correctness of our implementation also for the full dynamical problem, we modelled two discs, one with $\gamma=5/3$ which clearly shows no convection, and the other with $\gamma=1.1$ which shows strong convection. The initial setup was identical to that described before, but now we solve the equations of viscous hydrodynamics with radiation transport, but without irradiation and explicit dissipation. Please note that for viscous flows the energy generation due to viscous dissipation is automatically included in the total energy equation. The equations (1) to (3) are solved by PLUTO , and the system of equations (17) are solved as described in section 3. Since this setup is very dynamical and requires a more complex interplay of hydrodynamics and radiative transport, we use an additional third code, NIRVANA, for comparison. The NIRVANA code has been used in Kley et al. (2009) and Bitsch et al. (2013a) on very similar setups. The results of the two cases are shown in Fig. 9. In the top panel (a) we display the result for the $\gamma=5/3$ case which is not convective. Here, the agreement between the codes is excellent with the maximum deviation in the percentage range. In the lower panel (b) we display the results for the $\gamma=1.1$ case. Here the radiative transport time-scale is enhanced which leads to a strongly convective situation, which can be seen in the raggedness of the curves. In this simulation we doubled the spatial resolution, compared with the $\gamma=5/3$ case, such that the convection cells are reasonably well resolved, see figure 10. The agreement between the three different codes is very good, despite of the very different solution methods for the hydrodynamics equations: PLUTO uses the total energy equation with a Riemann- solver while RH2D and NIRVANA use a second-order upwind scheme and the thermal energy equation. Additionally, the latter two codes use the full dissipation function and the one-temperature approach. #### 4.6.4 Parallel scaling Figure 11: Parallel scaling benchmark results for the static accretion disc test case. We plot here the number of processor cores against $\frac{t_{64}}{t_{\mathrm{N}}}$ where $t_{N}$ is the runtime used on $N$ processors accordingly for $t_{64}$. The used run-times with full hydrodynamics and radiation transport for $64$, $128$, $256$, $512$ and $1024$ cpu cores (red crosses) are shown together with the ideal case (black dashed line). In order to test the parallel scaling of our new implementation, we used the same setup as in section 4.6.3 and increased the number of grid cells to $1024\times 64\times 256$. The computations were only run until $t=5\,\mathrm{orbits}$, and we used the solver PETSc. So we were able to run the test on $64$ up to $1024$ processor cores within a reasonable time. The simulations were run on clusters of the BWGrid which are equipped with Intel Xeon E5440 cpus and have a low latency InfiniBand network. In figure 11 we show the results of the simulations performed with full hydrodynamics and radiation transport. The run-time increases nearly by a factor of two when doubling the number of cores. With this setup, solving the hydrodynamics equations needs between $40\%$ and $50\%$ of the computation time and the radiation transport the remaining $60\%$ to $50\%$, however, these numbers are strongly problem-dependent. Therefore even up to 1024 cores, we see good agreement with ideal scaling. According to Amdahl’s law the full code, including the original part of PLUTO and our implementation of the radiation transport, is well parallelised. ## 5 Summary and conclusions We described the implementation of a new radiation module to the PLUTO code. The module solves for the flux-limited diffusion approximation in the two- temperature approach. For discretisation the finite volume method is used, and the resulting difference equations couple the updates of the temperature and radiation energy density. Due to possibly severe time step limitations, the set of equations is solved implicitly. For treating the non-linearity of the temperature in the matter-radiation coupling term, we utilize the method of Commerçon et al. (2011). The accuracy of the implementation has been verified using different physical and numerical setups. The first set of tests deals with purely radiative problems that include the purely diffusive evolution towards an equilibrium, and special setups to test the coupling terms between radiative and thermal energy. A newly developed setup checks for the correct inclusion of the irradiation from a central source in a spherical coordinate system. In the second test suite we study the full simultaneous evolution of hydrodynamics and radiation. First, sub- and super-critical radiative shock simulations are performed and their outcomes agree very well with published results of identical setups. Finally, we study the onset of convection in internally heated viscous discs, and find very good agreement between 3 different, independent hydrodynamical codes. This last test also allowed us to test the correct implementation in a spherical coordinate system and a non- equidistant logarithmic grid. Our numerical performance tests indicate excellent parallel scaling, up to at least 1024 processors. The current version of the radiation module comes with routines for the Rosseland mean opacity from Lin & Papaloizou (1985) and Bell & Lin (1994). Additionally it is possible to use the Rosseland and Planck mean opacities from Semenov et al. (2003). The described radiation module can be easily used within the PLUTO -environment. It can be found on the webpage 333http://www.tat.physik.uni- tuebingen.de/~pluto/pluto_radiation/ as a patch for the version 4.0 of PLUTO . ###### Acknowledgements. We gratefully thank the bwGRiD project444bwGRiD (http://www.bw-grid.de), member of the German D-Grid initiative, funded by the Ministry for Education and Research (Bundesministerium für Bildung und Forschung) and the Ministry for Science, Research and Arts Baden-Württemberg (Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg). for the computational resources. We gratefully acknowledges support through the German Research Foundation (DFG) through grant KL 650/11 within the Collaborative Research Group FOR 759: The formation of Planets: The Critical First Growth Phase. We thank Rolf Kuiper for many stimulating discussions, either physical or technical. ## References * Aubert & Teyssier (2008) Aubert, D. & Teyssier, R. 2008, MNRAS, 387, 295 * Balay et al. (2012) Balay, S., Brown, J., , et al. 2012, PETSc Users Manual, Tech. Rep. ANL-95/11 - Revision 3.3, Argonne National Laboratory * Bell & Lin (1994) Bell, K. R. & Lin, D. N. C. 1994, ApJ, 427, 987 * Bitsch et al. 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1309.5260	# Quasi Regular Semilattice and Association Schemes in Singular Linear Space Zhang Baohuan1 Yue Mengtian2 Li Zengti1 1.Math. and Inf. College, Langfang Teachers’ College, Langfang, 065000, China 2.Department of Scientific Research, Langfang Teachers’ College, Langfang, 065000, China [email protected] Abstract Let $\mathbb{F}_{q}^{n+l}$ denote the $(n+l)$-dimensional singular linear space over a finite field $\mathbb{F}_{q}$. For a fixed integer $m\leq\min\\{n,l\\}$, denote by $\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$ the set of all subspaces of type $(t,t_{1})$, where $t_{1}\leq t\leq m$. Partially ordered by ordinary inclusion, one family of quasi regular semilattices is obtained. Moreover, we obtain a association schemes and discuss the bound of a $M$-clique. AMS classification : 20G40, 05B35 Key words: Quasi regular semilattice; Association scheme; $M$-clique ## 1 Introduction It is well known that lattice is an important part of poset’s theory. its theory play an important role in many branches of mathematics, such as computer logical design. The results on the lattices generated by transitive sets of subspaces under finite classical groups may be found in Huo, Liu and Wan [8, 9, 10]. In [4], Guo discussed the lattices associated with finite vector spaces and finite affine spaces. In [12], P. Delsarte discussed the regular semilattices in finite vector spaces. In this paper, we obtain a new quasi regular semilattice and a new association scheme in the singular linear space. More over we discuss the bounds of a $M$-clique. The rest of this paper is organized as followed. In section 2, we discuss some definitions and terminologies about lattices, regular semilattices, association schemes and $M$-cliques. In section 3, we construct a family of quasi regular semilattice, and then compute its parameters. In section 4, we obtain a new association. In section 5, we discuss the bound of a $M$-clique. ## 2 Preliminaries Let $(P,\leq)$ be a poset. We write $a<b$ whenever $a\leq b$ and $a\neq b$. If $P$ has the minimum (respectively maximum) element, then we denote it by 0 (respectively $\upharpoonleft$), and say that $P$ is a poset with 0 (respectively $\upharpoonleft$). A poset $P$ is said to be a semilattice if $a\wedge b:=\rm{inf}\\{a,b\\}$ exist for any two elements $a,b\in P$. Let P be a finite poset with 0. If there is a function $r$ from $P$ to set of all the nonnegative integers such that * (1) r(0)=0, * (2) $r(b)=r(a)+1$, if $a\lessdot b$. Then $r$ is said to be the rank function on $P$. Note that the rank function on $P$ is unique if it exists. Let $P$ be a semilattice, and let $P=X_{0}\cup X_{1}\cup\cdots\cup X_{m}$, where $X_{i}=\\{x\in P\|r(x)=i\\},i=0,1,\cdots,m.$ The semilattice $(P,\leq)$ is called regular if the following three properties hold: * (i) Given $y\in X_{m},z\in X_{r}$ with $z\leq y$, the number of points $u\in X_{s}$ such that $z\leq u\leq y$ is a constant $\mu(r,s)$. * (ii) Given $u\in X_{s}$ , the number of points $z\in X_{r}$, such that $z\leq u$ is a constant $\nu(r,s)$. * (iii) Given $a\in X_{r},,y\in X_{m}$, with $a\wedge y\in X_{j}$, the number of pairs $(b,z)\in X_{s}\times X_{m}$ such that $b\leq z,b\leq y,a\leq z$ is a constant $\pi(j,r,s).$ In this paper, we define the concept of quasi regular semilattice as follows. The semilattice $(P,\leq)$ is called quasi regular if the following three properties hold: * (i) Given $y\in X^{m^{\prime}}_{m},z\in X^{r^{\prime}}_{r}$ with $z\leq y$, the number of points $u\in X^{s^{\prime}}_{s}$ such that $z\leq u\leq y$ is a constant $\mu(r(r^{\prime}),s(s^{\prime});m^{\prime})$. * (ii) Given $u\in X^{s^{\prime}}_{s}$ , the number of points $z\in X^{r^{\prime}}_{r}$, such that $z\leq u$ is a constant $\nu(r(r^{\prime}),s(s^{\prime}))$. * (iii) Given $a\in X^{r^{\prime}}_{r},y\in X^{m_{1}}_{m}$, with $a\wedge y\in X^{j^{\prime}}_{j}$, the number of pairs $(b,z)\in X^{s^{\prime}}_{s}\times X^{m^{\prime}}_{m}$ such that $b\leq z,b\leq y,a\leq z$ is a constant $\pi(j(j^{\prime}),r(r^{\prime}),s(s^{\prime});m^{\prime}).$ Here $X_{i}=X_{i}^{0}\cup X_{i}^{1}\cup\cdots\cup X_{i}^{i}$, and $X_{i}^{j}\cap X_{i}^{k}=\emptyset$ for $j\neq k.$ Let $X$ be a finite set of vertices. A $d$-class association scheme on $X$ consists of a set of $d+1$ symmetric relations $R_{0},R_{1},\cdots,R_{d}$ on $V$, with identity relation $R_{0}=\\{(x,x)\|x\in V\\},$ such that any two vertices are in precisely one relation. Denoted by $(X,\\{R_{i}\\}_{0\leq i\leq d})$. Furthermore, there are intersection numbers $p^{k}_{ij}$ such that for any $(x,y)\in R_{k}$, the number of vertices $z$ such that $(x,z)\in R_{i}$ and $(z,y)\in R_{j}$ equals $p^{k}_{ij}$. The nontrivial relations can be considered as graphs, which in our case are undirected. One immediately sees that the respective graphs are regular with degree $v_{i}=p_{ii}^{0}$. For the corresponding adjacency matrices $A_{i}$ the axioms of the scheme are equivalent to $\sum_{i=0}^{d}A_{i}=J,\ A_{0}=I,\ A_{i}=A_{i}^{T},\ A_{i}A_{j}=\sum_{k=0}^{d}p_{ij}^{k}A_{k}.$ It follows that the adjacency matrices generate a $(d+1)$-dimensional commutative algebra $\mathfrak{A}$ of symmetric matrices. This algebra was first studied by Bose and Mesner [19] and is called the Bose-Mesner algebra of the scheme. A nonzero vector $\alpha\in\mathbb{R}^{\|X\|}$ is said to be $\theta$-positive , if $C_{i}(\alpha)=v_{i}^{-1}<\alpha,A_{i}\alpha>\geq 0$ holds for every $i$. Let $M$ be a subset of $[0,n]$, containing $0$, Any positive vector $\alpha$ is called an $M$-clique if it satisfies $C_{j}(\alpha)=v_{j}^{-1}<\alpha,A_{j}\alpha>=0,\ for\ all\ j\not\in M$ Let $x_{0}$ be a point of $X$ and $s$ be an integer in $[0,n]$, A given $\theta$-positive vector $\alpha\in\mathbb{R}^{\|X\|}$ is called a unicoloured vector of center $x_{0}$ and colour $s$, if the following condition holds: $\alpha(x)=0,\ unless\ (x,x_{0})\in R_{s}$ Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a prime power. For two non-negative integers $n$ and $l$, $\mathbb{F}_{q}^{n+l}$ denotes the $(n+l)$-dimensional row vector space over $\mathbb{F}_{q}$. The set of all $(n+l)\times(n+l)$ nonsingular matrices over $\mathbb{F}_{q}$ of the form $\left(\begin{array}[]{cc}T_{11}&T_{12}\\\ 0&T_{22}\end{array}\right),$ where $T_{11}$ and $T_{22}$ are nonsingular $n\times n$ and $l\times l$ matrices, respectively, forms a group under matrix multiplication, called the singular general linear group of degree $n+l$ over $\mathbb{F}_{q}$ and denoted by $GL_{n+l,n}(\mathbb{F}_{q})$. Let $P$ be an $m$-dimensional subspace of $\mathbb{F}_{q}^{n+l}$, denote also by $P$ a $m\times(n+l)$ matrix of rank $m$ whose rows span the subspace $P$ and call the matrix $P$ a matrix representation of the subspace $P$ . There is an action of $GL_{n+l,n}(\mathbb{F}_{q})$ on $\mathbb{F}_{q}^{n+l}$ defined as follows $\begin{array}[]{rrr}\mathbb{F}_{q}^{n+l}\times GL_{n+l,n}(\mathbb{F}_{q})&\longrightarrow&\mathbb{F}_{q}^{n+l},\\\ ((x_{1},\cdots,x_{n},x_{n+1},\cdots,x_{n+l}),T)&\longmapsto&(x_{1},\cdots,x_{n},x_{n+1},\cdots,x_{n+l})T.\end{array}$ The above action induces an action on the set of subspaces of $\mathbb{F}_{q}^{n+l}$; i.e., a subspace $P$ is carried by $T\in GL_{n+l,n}(\mathbb{F}_{q})$ to the subspace $PT$. The vector space $\mathbb{F}_{q}^{n+l}$ together with the above group action is called the $(n+l)$-dimensional singular linear space over $\mathbb{F}_{q}$. For $1\leq i\leq n+l$, let $e_{i}$ be the row vector in $\mathbb{F}_{q}^{n+l}$ whose $i$-th coordinate is $1$ and all other coordinates are $0$. Denote by $E$ the $l$-dimensional subspace of $\mathbb{F}_{q}^{n+l}$ generated by $e_{n+1},e_{n+2},\cdots,e_{n+l}$. A $m$-dimensional subspace $P$ of $\mathbb{F}_{q}^{n+l}$ is called a subspace of type $(m,k)$ if $\dim(P\cap E)=k$. For a fixed subspace $P$ of type $(m_{1},k_{1})$ in $\mathbb{F}_{q}^{n+l}$, let $\mathcal{M}^{\prime}(m_{1},k_{1};m,k;n+l,n)$ denote the set of all the subspaces of type $(m,k)$ containing $P$. $\mathcal{M}^{\prime}(m_{1},k_{1};m,k;n+l,n)$ is non-empty if and only if $0\leq k_{1}\leq k\leq l$ and $0\leq m_{1}-k_{1}\leq m-k\leq n$. Let $N^{\prime}(m_{1},k_{1};m,k;n+l,n)=\|\mathcal{M}^{\prime}(m_{1},k_{1};m,k;n+l,n)\|.$ The formula is given by Kaishun Wang, Jun Guo, Fenggao Li (see [7]) as follows. $N^{\prime}(m_{1},k_{1};m,k;n+l,n)=q^{(l-k)(m-k- m_{1}+k_{1})}\left[n-(m_{1}-k_{1})\atop(m-k)-(m_{1}-k_{1})\right]_{q}\left[l-k_{1}\atop k-k_{1}\right]_{q}.$ ###### Lemma 2.1 Let $0\leq k_{1}\leq k_{2}\leq k\leq l,0\leq m_{1}-k_{1}\leq m_{2}-k_{2}\leq m-k\leq n.$ For $i=1,2,3$, assume that $U_{i}$ be a set of all the subspaces of type $(m_{i},k_{i})$ in $\mathbb{F}_{q}^{n+l}$, and $U$ be a set of all the subspaces of type $(m,k)$ in $\mathbb{F}_{q}^{n+l}$ such that $U_{1}\subseteq U_{2}\subseteq U$ and $U_{3}\subseteq U$. if $U_{3}\cap U_{2}=U_{1}$, then the number of $U_{3}$ is $q^{(\delta_{3}-\delta_{1})(\delta_{2}-\delta_{1}+k-k_{3})+(k_{3}-k_{1})(k_{2}-k_{1})}\left[\delta-\delta_{2}\atop\delta_{3}-\delta_{1}\right]_{q}\left[k-k_{2}\atop k_{3}-k_{1}\right]_{q},$ where $\delta=m-k,\delta_{i}=m_{i}-k_{i},i=1,2,3$. Proof. Let $\delta_{1}=m_{1}-k_{1},\delta_{2}=m_{2}-k_{2},\delta_{3}=m_{3}-k_{3},\delta=m-k$. By the transitivity of $GL_{n+l,n}(\mathbb{F}_{q})$ on the set of subspaces of the same type, we may assume that $\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \ \ \delta_{1}&n-\delta_{1}&k_{1}&l-k_{1}\end{array}\\\ U_{1}=\left(\begin{array}[]{llll}I&0&0&0\\\ 0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ \delta_{1}\\\ k_{1}\end{array}\end{array}$ $\begin{array}[]{l}\begin{array}[]{llllll}\ \ \ \ \ \ \ \ \ \delta_{1}&\delta_{2}-\delta_{1}&n-\delta_{2}&k_{1}&k_{2}-k_{1}&l-k_{2}\end{array}\\\ U_{2}=\left(\begin{array}[]{llllll}I&0&0&0&0&0\\\ 0&0&0&I&0&0\\\ 0&I&0&0&0&0\\\ 0&0&0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ \delta_{1}\\\ k_{1}\\\ \delta_{2}-\delta_{1}\\\ k_{2}-k_{1}\end{array}\end{array}$ and $\begin{array}[]{l}\begin{array}[]{llllllll}\ \ \ \ \ \ \ \ \ \ \delta_{1}&\delta_{2}-\delta_{1}&\delta-\delta_{2}&n-\delta&k_{1}&k_{2}-k_{1}&k-k_{2}&l-k\end{array}\\\ U=\left(\begin{array}[]{llllllll}I&0&0&0&0&0&0&0\\\ 0&0&0&0&I&0&0&0\\\ 0&I&0&0&0&0&0&0\\\ 0&0&0&0&0&I&0&0\\\ 0&0&I&0&0&0&0&0\\\ 0&0&0&0&0&0&I&0\\\ \end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ \delta_{1}\\\ k_{1}\\\ \delta_{2}-\delta_{1}\\\ k_{2}-k_{1}\\\ \delta-\delta_{2}\\\ k-k_{2}\end{array}\end{array}$ Since $U_{3}\cap U_{2}=U_{1},$ we have $\begin{array}[]{l}\begin{array}[]{llllllll}\ \ \ \ \ \ \ \ \ \delta_{1}&\delta_{2}-\delta_{1}&\delta-\delta_{2}&n-\delta&k_{1}&k_{2}-k_{1}&k-k_{2}&l-k\end{array}\\\ U_{3}=\left(\begin{array}[]{llllllll}I&0&0&0&0&0&0&0\\\ 0&0&0&0&I&0&0&0\\\ 0&u_{32}&u_{33}&0&0&u_{36}&u_{37}&0\\\ 0&0&0&0&0&u_{46}&u_{47}&0\\\ \end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ \delta_{1}\\\ k_{1}\\\ \delta_{3}-\delta_{1}\\\ k_{3}-k_{1}\\\ \end{array}\end{array}$ where rank $u_{47}=k_{3}-k_{1}$. Note that there are $\left[k-k_{2}\atop k_{3}-k_{1}\right]_{q}$ choices for $u_{47}$. By the transitivity of $GL_{n+l,n}(\mathbb{F}_{q})$ on the set of subspaces of the same type, the number of $U_{3}$’s does not depend on the particular choice of $u_{47}$. Pick $u_{47}=(I^{(k_{3}-k_{1})},0)$. Then $U_{3}$ has a matrix representation $\begin{array}[]{l}\begin{array}[]{lllllllll}\ \ \delta_{1}&\delta_{2}-\delta_{1}&\delta-\delta_{2}&n-\delta&k_{1}&k_{2}-k_{1}&k_{3}-k_{1}&k-k_{2}-k_{3}+k_{1}&l-k\end{array}\\\ \left(\begin{array}[]{lllllllll}I&0&0&0&0&0&0&0&0\\\ 0&0&0&0&I&0&0&0&0\\\ 0&u_{32}&u_{33}&0&0&u_{36}&0&u^{\prime}_{37}&0\\\ 0&0&0&0&0&u_{46}&I&0&0\\\ \end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ \delta_{1}\\\ k_{1}\\\ \delta_{3}-\delta_{1}\\\ k_{3}-k_{1}\\\ \end{array}\end{array}$ where rank $u_{33}=\delta_{3}-\delta_{1}$. Therefore the number of $U_{3}$ is equal to $q^{(\delta_{3}-\delta_{1})(\delta_{2}-\delta_{1}+k-k_{3})+(k_{3}-k_{1})(k_{2}-k_{1})}\left[\delta-\delta_{2}\atop\delta_{3}-\delta_{1}\right]_{q}\left[k-k_{2}\atop k_{3}-k_{1}\right]_{q}.$ $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ ## 3 The quasi regular semilattice For a fixed integer $m\leq\min\\{n,l\\}$, denote by $\mathcal{L}^{m}(\mathbb{F}_{q}^{n+l})$ the set of all subspaces of type $(t,t_{1})$, where $t_{1}\leq t\leq m$. If we partially order $\mathcal{L}^{m}(\mathbb{F}_{q}^{n+l})$ by the ordinary inclusion, then $\mathcal{L}^{m}(\mathbb{F}_{q}^{n+l})$ is a semilattice, denoted by $\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$. For any $A\in\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$, The rank function of $\mathcal{L}_{o}(\mathbb{F}_{q}^{n+l})$ is defined as follows $r(A)=\dim(A).$ Let $X_{i}=\\{B\in\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})\|r(B)=i\\},$ and $X_{i}^{j}=\\{B\in X_{i}\|\dim(B\cap E)=j\\},j=0,1,\cdots,i,$ where $E=\langle e_{n+1},e_{n+2},\cdots,e_{n+l}\rangle\in\mathbb{F}_{q}^{n+l}.$ We will prove that $\mathcal{L}^{m}_{o}(\mathbb{F}_{q}^{n+l})$ is a quasi regular semilattice and compute its parameters. ###### Lemma 3.1 Let $A\in X^{r_{1}}_{r}$, $C\in X^{m_{1}}_{m}$ and $A\leq C$. Then the number of $B\in X^{s_{1}}_{s}$ such that $A\leq B\leq C$ is equal to $\mu(r(r_{1}),s(s_{1});m_{1})=q^{(s-s_{1}-r+r_{1})(m_{1}-s_{1})}\left[\begin{array}[]{l}m-r+r_{1}-m_{1}\\\ s-s_{1}-r+r_{1}\end{array}\right]_{q}\left[\begin{array}[]{l}m_{1}-r_{1}\\\ s_{1}-r_{1}\end{array}\right]_{q}$ Proof. By the transitivity of $GL_{n+l}(\mathbb{F}_{q})$ on the set of subspaces of the same type, we may assume that $\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \ r-r_{1}&n-r+r_{1}&r_{1}&l-r_{1}\end{array}\\\ A=\left(\begin{array}[]{llll}I&0&0&0\\\ 0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ r-r_{1}\\\ r_{1}\end{array}\end{array}$ and $\begin{array}[]{l}\begin{array}[]{llllll}\ \ \ \ \ \ \ \ r-r_{1}&m-m_{1}-r+r_{1}&n-m+m_{1}&r_{1}&m_{1}-r_{1}&l-m_{1}\end{array}\\\ C=\left(\begin{array}[]{llllll}I&0&0&0&0&0\\\ 0&0&0&I&0&0\\\ 0&I&0&0&0&0\\\ 0&0&0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ r-r_{1}\\\ r_{1}\\\ m-m_{1}-r+r_{1}\\\ m_{1}-r_{1}\end{array}.\end{array}$ Since $A\leq B\leq C,$ we have $\begin{array}[]{l}\begin{array}[]{llllll}\ \ \ \ \ \ \ \ r-r_{1}&m-m_{1}-r+r_{1}&n-m+m_{1}&r_{1}&m_{1}-r_{1}&l-m_{1}\end{array}\\\ B=\left(\begin{array}[]{llllll}I&0&0&0&0&0\\\ 0&0&0&I&0&0\\\ 0&u_{32}&0&0&u_{35}&0\\\ 0&0&0&0&u_{45}&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ r-r_{1}\\\ r_{1}\\\ s-s_{1}-r+r_{1}\\\ s_{1}-r_{1}\end{array}\end{array},$ where rank $u_{45}=(s_{1}-r_{1})$. Note that there are $\left[m_{1}-r_{1}\atop s_{1}-r_{1}\right]_{q}$ choices for $u_{45}$. By the transitivity of $GL_{n+l}(\mathbb{F}_{q})$ on the set of subspaces of the same type, the number of $B$’s does not depend on the particular choice of $u_{45}$. Pick $u_{45}=(I^{(s_{1}-r_{1})},0)$. Then $B$ has a matrix representation $\begin{array}[]{l}\begin{array}[]{lllllll}\ \ \delta_{2}&\delta_{1}-\delta_{2}&n-\delta_{1}&r_{1}&s_{1}-r_{1}&m_{1}-s_{1}&l-m_{1}\end{array}\\\ \left(\begin{array}[]{lllllll}I&0&0&0&0&0&0\\\ 0&0&0&I&0&0&0\\\ 0&u_{32}&0&0&0&u^{\prime}_{35}&0\\\ 0&0&0&0&I&0&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ \delta_{2}\\\ r_{1}\\\ \delta_{3}-\delta_{2}\\\ s_{1}-r_{1}\end{array}.\end{array}$ where $\delta_{1}=m-m_{1},\delta_{2}=r-r_{1},\delta_{3}=s-s_{1}$. Therefore the number of $B$ is equal to $q^{(s-s_{1}-r+r_{1})(m_{1}-s_{1})}\left[\begin{array}[]{l}m-r+r_{1}-m_{1}\\\ s-s_{1}-r+r_{1}\end{array}\right]_{q}\left[\begin{array}[]{l}m_{1}-r_{1}\\\ s_{1}-r_{1}\end{array}\right]_{q}.$ $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ ###### Lemma 3.2 If $B\in X^{s_{1}}_{s}$, then the number of $A\in X^{r_{1}}_{r}$ such that $A\leq B$ is equal to $\nu(r(r_{1}),s(s_{1}))=q^{(r-r_{1})(s_{1}-r_{1})}\left[s-s_{1}\atop r-r_{1}\right]_{q}\left[s_{1}\atop r_{1}\right]_{q}.$ Proof. By the transitivity of $G_{n+l,l}(\mathbb{F}_{q})$ on the set of subspaces of the same type, we may assume that $\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \ s-s_{1}&n-s+s_{1}&s_{1}&l-s_{1}\end{array}\\\ B=\left(\begin{array}[]{llll}I&0&0&0\\\ 0&0&I&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ s-s_{1}\\\ s_{1}\end{array}\end{array}.$ Since $A\leq B,$ we have $\begin{array}[]{l}\begin{array}[]{llll}\ \ \ \ \ \ \ \ s-s_{1}&n-s+s_{1}&s_{1}&l-s_{1}\end{array}\\\ A=\left(\begin{array}[]{llll}u_{11}&0&u_{13}&0\\\ 0&0&u_{23}&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ r-r_{1}\\\ r_{1}\end{array}\end{array},$ where rank $u_{23}=r_{1}$. Note that there are $\left[s_{1}\atop r_{1}\right]_{q}$ choices for $u_{23}$. By the transitivity of $GL_{n+l}(\mathbb{F}_{q})$ on the set of subspaces of the same type, the number of $A$’s does not depend on the particular choice of $u_{23}$. Pick $u_{23}=(I^{(r_{1})}\ 0)$. Then $A$ has a matrix representation $\begin{array}[]{l}\begin{array}[]{lllll}s-s_{1}&n-s+s_{1}&r_{1}&s_{1}-r_{1}&l-s_{1}\end{array}\\\ \left(\begin{array}[]{lllll}u_{11}&0&0&u^{\prime}_{13}&0\\\ 0&0&I&0&0\end{array}\right)\end{array}\begin{array}[]{l}\begin{array}[]{l}\\\ r-r_{1}\\\ r_{1}\end{array}\end{array}.$ Therefore the number of subspace $A$ is equal to $\nu(r(r_{1}),s(s_{1}))=q^{(r-r_{1})(s_{1}-r_{1})}\left[s-s_{1}\atop r-r_{1}\right]_{q}\left[s_{1}\atop r_{1}\right]_{q}.$ $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ ###### Lemma 3.3 Let $A\in X^{r_{1}}_{r}$ and $B\in X^{m_{2}}_{m}$. Assume that $A\wedge B\in X^{j_{1}}_{j}$, and $(C,D)\in X_{s}^{s_{1}}\times X_{m}^{m_{1}}$ . If $C\leq D,C\leq B,A\leq D$, then the number of $(C,D)$ is equal to $\begin{array}[]{ll}&\pi(j(j_{1}),r(r_{1}),s(s_{1});m_{1})\\\ =&\sum_{0\leq i\leq j,0\leq i_{1}\leq\min\\{i,j_{1}\\}}q^{(s-s_{1}-i+i_{1})(j-j_{1}-i+i_{1}+m_{1}-s_{1})+(s_{1}-i_{1})(j_{1}-i_{1})}\\\ &\times\left[(m-m_{1})-(j-j_{1})\atop(s-s_{1})-(i-i_{1})\right]_{q}\left[m_{1}-j_{1}\atop s_{1}-i_{1}\right]_{q}N^{\prime}(r+s-i,r_{1}+s_{1}-i_{1};m,m_{1};n+l,n).\end{array}$ Proof. Since $A\leq D,C\leq B$, we have $A\wedge C\leq A\wedge B\leq D.$ Since $C\leq D,$ and $C\wedge(A\wedge B)=A\wedge(B\wedge C)=A\wedge C\in X^{i_{1}}_{i},0\leq i\leq j,0\leq i_{1}\leq\min\\{i,j_{1}\\}.$ For $i\in[0,j]$ and $i_{1}\in[0,\min\\{i,j_{1}\\}]$, by Lemma 2.1, the number of $C$ is equal to $q^{(s-s_{1}-i+i_{1})(j-j_{1}-i+i_{1}+m_{1}-s_{1})+(s_{1}-i_{1})(j_{1}-i_{1})}\left[(m-m_{1})-(j-j_{1})\atop(s-s_{1})-(i-i_{1})\right]_{q}\left[m_{1}-j_{1}\atop s_{1}-i_{1}\right]_{q}.$ Since $C\leq D,A\leq D$, $C+A\leq D$. It follows from $A\wedge C\in X_{i}^{i_{1}}$ that $A+C$ is a subspace of type $(r+s-i,r_{1}+s_{1}-i_{1})$. Therefore, the number of $D$ is equal to $N^{\prime}(r+s-i,r_{1}+s_{1}-i_{1};m,m_{1};n+l,n)$. Hence the desired result follows. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ By Lemma 3.1,Lemma 3.2 and Lemma 3.3, we obtain the following theorem. ###### Theorem 3.4 Semilattice $\mathcal{L}_{0}^{m}(\mathbb{F}_{q}^{n+l})$ is a quasi regular semilattice. Its parameters are given by the formulas $\mu(r(r_{1}),s(s_{1});m_{1})=q^{(s-s_{1}-r+r_{1})(m_{1}-s_{1})}\left[\begin{array}[]{l}m-r+r_{1}-m_{1}\\\ s-s_{1}-r+r_{1}\end{array}\right]_{q}\left[\begin{array}[]{l}m_{1}-r_{1}\\\ s_{1}-r_{1}\end{array}\right]_{q},$ $\nu(r(r_{1}),s(s_{1}))=q^{(r-r_{1})(s_{1}-r_{1})}\left[s-s_{1}\atop r-r_{1}\right]_{q}\left[s_{1}\atop r_{1}\right]_{q},$ and $\begin{array}[]{ll}&\pi(j(j_{1}),r(r_{1}),s(s_{1});m_{1})\\\ =&\sum_{0\leq i\leq j,0\leq i_{1}\leq i}q^{(s-s_{1}-i+i_{1})(j-j_{1}-i+i_{1}+m_{1}-s_{1})+(s_{1}-i_{1})(j_{1}-i_{1})}\\\ &\times\left[(m-m_{1})-(j-j_{1})\atop(s-s_{1})-(i-i_{1})\right]_{q}\left[m_{1}-j_{1}\atop s_{1}-i_{1}\right]_{q}N^{\prime}(r+s-i,r_{1}+s_{1}-i_{1};m,m_{1};n+l,n).\end{array}$ ## 4 Association schemes Given $d$ and $d_{0}$ with $0\leq d_{0}\leq d\leq\min\\{n,l\\}$. For $h=\min\\{k,d_{0}\\},$ we define $\widetilde{X_{k}}=\left\\{\begin{array}[]{ll}X_{k}^{0}\cup\cdots\cup X_{k}^{h},&if\ 0\leq k\leq d-1,\\\ X_{d}^{d_{0}},&if\ k=d,\end{array}\right.$ and $R_{k}=\\{(B,C)\in X_{d}^{d_{0}}\times X_{d}^{d_{0}}\|B\wedge C\in\widetilde{X_{d-k}}\\}.$ Then the $R=\\{R_{0},R_{1},\cdots,R_{d}\\}$ is a partition of $X_{d}^{d_{0}}\times X_{d}^{d_{0}}$. For any $B,C\in X_{d}^{d_{0}}$, let $A_{k(k^{\prime})}(B,C)=\left\\{\begin{array}[]{ll}1,&if\ B\wedge C\in X^{k^{\prime}}_{d-k},\\\ 0,&otherwise.\end{array}\right.$ Then, for $h=\min\\{d-k,d_{0}\\}$, we define $A_{k}=\left\\{\begin{array}[]{ll}A_{k(0)}+A_{k(1)}+\cdots+A_{k(h)},&if\ 1\leq k\leq d,\\\ A_{0(d_{0})},&if\ k=0.\end{array}\right.$ Let $D_{i,k}(B,C)=\left\\{\begin{array}[]{ll}1,&if\ B\leq C,\\\ 0,&otherwise.\end{array}\right.$ The $D_{i,k}$ is a Riemann matrix indexed by $\widetilde{X_{i}}\times\widetilde{X_{k}}$. For convenience, we write $D_{i}=D_{i,d}$. Let $\mathfrak{A}=\langle A_{0},A_{2},\cdots,A_{d}\rangle$ denote the $(d+1)$-dimensional real vector space generated by the $A_{0},A_{2},\cdots,A_{d}$. Let $C_{i}=D^{T}_{i}D_{i}$, for $i=0,1,\cdots,d.$ We have the lemma as follows. ###### Lemma 4.1 The matrices $C_{0},C_{1},\cdots,C_{d}$ generate $\mathfrak{A}$, with $C_{t}=\sum_{h=0}^{\min\\{t,d_{0}\\}}\nu(t(h),d(d_{0}))A_{0}+\sum_{k=1}^{d}\sum_{\lambda=0}^{\min\\{d-k,d_{0}\\}}\sum_{h=0}^{\min\\{t,d_{0}\\}}\nu(t(h),(d-k)(\lambda))A_{k},$ (1) where $t=0,1,\cdots,d.$ Proof. For any $(A,B)\in X_{d}^{d_{0}}\times X_{d}^{d_{0}}$, the number of $F\in\widetilde{X_{t}}$ such that $F\leq A\wedge B$ is the $(A,B)$-entry in two members of (1). It implies that the system (1) holds. On the other hand, its matrix has rank $d+1$. Hence, the result is obtained. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ ###### Lemma 4.2 Let $A\in X^{d_{1}}_{d},B\in X^{s_{1}}_{s}$, then the number of $D\in X^{r_{1}}_{r}$ such that $D\leq A$ and $B\wedge D\in X^{j_{1}}_{j}$ is a constant $\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})$. Proof. For a given $k$ and $k_{1}$, with $0\leq k\leq h=\min\\{r,s\\}$ and $0\leq k_{1}\leq k$. We assume that $(C,D)\in(X_{k}^{k_{1}},X_{r}^{r_{1}})$ and $C\leq B,C\leq D\leq A.$ Counting the number of the pairs $(C,D)$ in two different ways, we obtain $\sum_{j=0}^{h}\sum_{j_{1}=0}^{j}\nu(k(k_{1}),j(j_{1}))\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})=\nu(k(k_{1}),s(s_{1}))\mu(k(k_{1}),r(r_{1});d_{1}).$ (2) This yields a system $(\ref{el2})$ of linear equations in unknown $\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})$ with fixed $r(r_{1}),s(s_{1})$. Since the matrix $[\nu(k(k_{1}),j(j_{1}))]$ of the system $(\ref{el2})$ is a nonsingular, these equations uniquely determine the $\psi(j(j_{1}),r(r_{1}),s(s_{1});d_{1})$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ ###### Theorem 4.3 The system $(X_{d}^{d_{0}},R)$ is an association scheme. Proof. We establish the matrix relation as follows $C_{r}C_{s}=\sum_{k=0}^{d}\sum_{\lambda=0}^{h_{1}}\left\\{\sum_{j=0}^{r}\sum_{j^{\prime}=0}^{h_{2}}\left[\sum_{r^{\prime}=0}^{h_{3}}\left(\psi(j(j^{\prime}),r(r^{\prime}),(d-k)(\lambda);d_{0})\sum_{s^{\prime}=0}^{h_{4}}\pi(j(j^{\prime}),r(r^{\prime}),s(s^{\prime});d_{0})\right)\right]\right\\}A_{k}.$ (3) Here $h_{1}=\min\\{d-k,d_{0}\\},h_{2}=\min\\{j,d_{0}\\},h_{3}=\min\\{r,d_{0}\\},h_{4}=\min\\{s,d_{0}\\}.$ In fact, By the definition, for any $D,F\in X_{d}^{d_{0}}$, the $(D,F)$-entry of $C_{r}C_{s}$ is the number of triples $(A,B,C)\in\widetilde{X_{r}}\times\widetilde{X_{s}}\times\widetilde{X_{d}}$ such that $A\leq D\wedge C$ and $B\leq F\wedge C$ hold. Let $D\wedge F\in X_{d-k}^{\lambda}$. We assume that $A\wedge F\in X_{j}^{j^{\prime}}$ with fixed $j$ and $j^{\prime}$, by Lemma 4.2, there are $\sum_{r^{\prime}=0}^{\min\\{r,d_{0}\\}}\psi(j(j^{\prime}),r(r^{\prime}),(d-k)(\lambda);d_{0})$ choices for $A\leq D$. For each $A$, by Lemma 3.3, there are $\sum_{s^{\prime}=0}^{\min\\{s,d_{0}\\}}\pi(j(j^{\prime}),r(r^{\prime}),s(s^{\prime});d_{0})$ choices for pairs $(B,C)$ with $B\leq F\wedge C,A\leq C$. Hence, (3) holds. By Lemma 4.1, $\mathfrak{A}$ is a Bose-Mesner algebra. It implies the assertion. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ ## 5 $M$-clique In this section, we always assume that $l=0$, $X=X_{d}^{0}$. By Theorem 4.3, we obtain a association scheme $\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$ in $\mathbb{F}_{q}^{n}$. Next, we will discuss the bound of $M$-clique on association scheme $(X,\\{R_{i}\\}_{0\leq i\leq d})$. We associate the following linear programming problem, in the real variables $\xi_{k}(k\in K=\\{k\in[0,d]\|m_{k}>1\\}):$(see [5]) $\left\\{\begin{array}[]{l}{\rm min}(f^{\prime})=1+\sum_{k\in K}(1-b_{k})\xi_{k},\\\ \hbox{subject to}\ \xi_{k}\geq 0,\ \hbox{all}\ k\in K,\\\ \hbox{and to}\ \sum_{k\in K}(P_{i}(k)-b_{k}v_{i})\xi_{k}\leq v_{i},\ \hbox{all}\ i\in M\backslash\\{0\\}.\end{array}\right.$ (4) Here $P_{i}(k)$ is eigenvalue of a association scheme, $v_{i}$ is degree of a association scheme, $b_{k}=(v_{s}^{-1}P_{s}(k))^{2},k\in[0,d]$. ###### Proposition 5.1 ([5]) The problem (4) admits a solution $\min(f^{\prime})$, and any unicoloured $M$-clique $\alpha$ satisfies $\hat{c}_{0}(\alpha)\leq\min(f^{\prime})$. By Proposition 5.1, we obtain the following theorem. ###### Theorem 5.2 Let $\alpha\in\mathbb{R}^{\|X\|}$ be a unicoloured $M$-clique on association scheme $\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$ with $M=\\{0,1\\}$ and colour $s=1$. then $\hat{c}_{0}(\alpha)=<\alpha,\alpha>^{-1}<\alpha,1>^{2}\leq\frac{q^{d+1+n}-3q^{2d+1}+5q^{d+1}-2q^{d}+2q^{n+d}-3q^{n+1}}{q^{n+d}-q^{2d+1}+2q^{d+1}-q^{d}-q^{n+1}}.$ Proof. By the ([12]), the eigenvalues of an association scheme of $\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$ are given by the formula $P_{i}(r)=\sum_{j=0}^{i}(-1)^{i-j}\left[d-j\atop i-j\right]_{q}\left[d-r\atop j\right]_{q}\left[n-d+j-r\atop j\right]_{q}q^{rj+{i-j\choose 2}}.$ Pick $\xi_{k}=\xi\delta_{k,1}$. It is known that the degree of the $\mathfrak{X}=(X,\\{R_{i}\\}_{0\leq i\leq d})$ are given by the formula $v_{i}=q^{i^{2}}\left[n-d\atop i\right]_{q}\left[d\atop i\right]_{q}.$ By the above formulas, we obtain $v_{1}=\frac{q(q^{n-d}-1)(q^{d}-1)}{(q-1)^{2}},$ $P_{1}(1)=-\frac{q^{d}-1}{q-1}+\frac{q(q^{d-1}-1)(q^{n-d}-1)}{(q-1)^{2}},$ For $s=1,i=1$, we have $(P_{1}(1)-b_{1}v_{1})\xi\leq v_{1}$, $(P_{1}(1)-(v_{1}^{-1}P_{1}(1))^{2}v_{1})\xi\leq v_{1}$. It implies that $(v_{1}P_{1}(1)-P_{1}(1)^{2})\xi\leq v_{1}^{2}$. Since $v_{1}-P_{1}(1)=\frac{q^{n}-1}{q-1}>0.$ We have $\xi\leq\frac{v_{1}^{2}}{p_{1}(1)(v_{1}-p_{1}(1)}.$ and $1-b_{1}=1-\frac{p_{1}(1)^{2}}{v_{1}^{2}}=\frac{v_{1}^{2}-p_{1}(1)^{2}}{v_{1}^{2}}>0.$ It follows that $\begin{array}[]{ll}\min(f^{\prime})&=1+(1-b_{1})\xi\\\ &\leq 1+\frac{v_{1}+p_{1}(1)}{p_{1}(1)}\\\ &=\frac{q^{d+1+n}-3q^{2d+1}+5q^{d+1}-2q^{d}+2q^{n+d}-3q^{n+1}}{q^{n+d}-q^{2d+1}+2q^{d+1}-q^{d}-q^{n+1}}.\end{array}$ By Proposition 5.1, we have $\hat{c}_{0}(\alpha)\leq\frac{q^{d+1+n}-3q^{2d+1}+5q^{d+1}-2q^{d}+2q^{n+d}-3q^{n+1}}{q^{n+d}-q^{2d+1}+2q^{d+1}-q^{d}-q^{n+1}}.$ $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$ ## Acknowledgement This research is supported by NSF of Hebei Province (A2013408009), NSF of Hebei Education Department(ZH2012082), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No.20121303110005) and the foundation of Langfang Teachers’ College (LSBS201205). ## References * [1] * [2] M. 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Guo, A construction of distance-regular graphs from subspaces in d-bounded distance-regular graphs, ARS Combinatoria 98 (2011), 135-148.	arxiv-papers	2013-09-20T13:09:48	2024-09-04T02:49:51.219744	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Zhang Baohuan, Yue Mengtian, Li Zengti", "submitter": "Li Zengti", "url": "https://arxiv.org/abs/1309.5260" }
1309.5275	# An open dataset for research on audio field recording archives: freefield1010 Dan Stowell and Mark D. Plumbley Centre for Digital Music, Queen Mary University of London [email protected] ###### Abstract We introduce a free and open dataset of 7690 audio clips sampled from the field-recording tag in the Freesound audio archive. The dataset is designed for use in research related to data mining in audio archives of field recordings / soundscapes. Audio is standardised, and audio and metadata are Creative Commons licensed. We describe the data preparation process, characterise the dataset descriptively, and illustrate its use through an auto-tagging experiment. ## 1 Introduction Digital sound archives hold vast resources of material, including speech, music, and naturalistic and ethnographic field recordings (Ranft, 2004). However, there are still many challenges in organising and searching in these archives. In recent decades, research fields such as automatic speech recognition (ASR) and music information retrieval (MIR) have developed automatic methods for labelling and transcribing specific types of sound. Yet even if we put speech and music to one side, we still have a large and valuable range of recorded sound, and there has been relatively little work in organising and searching this non-speech-non-music audio. Such research may come under the umbrella of “computational auditory scene analysis” (CASA) (Wang and Brown, 2006). Tasks have begun to be addressed in recent years such as automatically labelling the type of audio scene, or automatically detecting and labelling the events within the audio scene (Giannoulis et al., 2013, and citations therein). Important for research development is the existence of standard datasets that can be independently reused by researchers. This has motivated many community efforts in ASR and in MIR. In our own recent work we created a set of “audio scene” recordings as part of the IEEE AASP “D-CASE” challenge, consisting of 30-second binaural recordings made by three recordists in locations around London (Giannoulis et al., 2013). Such focussed datasets are valuable for developing algorithms for the specific tasks considered, but they are quite different from most audio archives, being tightly calibrated in their production and of moderate size. In order for datasets to be relevant to applications in sound archives, they need to be large enough that they (a) reflect the diversity of content in audio archives and (b) give some indication of scalability issues for analysis and for visualisation/navigation. However, large datasets are expensive and time-consuming to record from scratch, while most existing archives cannot be freely redistributed due to the associated copyright and licensing terms. It is illustrative that the creators of a recent large pop-music dataset (the Million Song Dataset) worked around this issue by distributing not the audio, but some pre-computed features derived from the audio (Bertin-Mahieux et al., 2011). This enabled them to distribute a large open dataset of relevance to pop music archives, but as a consequence it restricted the types of analysis possible: researchers are constrained to using the specific pre-computed features they provided. For our purposes, a notable initiative is the Freesound archive,111http://freesound.org/ which hosts extensive holdings of crowdsourced audio recordings, reusable under Creative Commons and public domain licences. Established in 2005, it holds more than 160,000 sounds from thousands of users around the world. It contains a wide range of sound types, including field recordings, recordings from contact mics and hydrophones, and synthetic sounds. Freesound is already usable for various research purposes, having open licensing conditions and an easy-to-use application programming interface (API). However, it is a large crowdsourced and continually-updated collection, not a fixed dataset. Files are continually added and removed, and metadata changed; it is too large to easily redistribute among researchers; various licences are used, not all mutually compatible; and files are in various formats (e.g. WAV/AIFF/ MP3/Ogg, number of channels, sample-rate, duration) which can be inconvenient for those developing algorithms. Also, a crowdsourced archive such as Freesound is typically more heterogeneous than traditional archives such as the British Library Sound Archive, whose curation involves manual attention to file formats and metadata (Ranft, 2004). We therefore chose to compile a free and open dataset of a wide range of sounds, standardised and curated from a fraction of the extensive holdings of the Freesound archive. This dataset is intended to be of use to researchers developing methods for working with field recordings in audio archives. In the following we describe how we designed and prepared the data. We then describe how anyone can access the data and work with it, and illustrate with an automatic recognition experiment to infer the presence of tags. ## 2 Dataset In order to maximise the potential usefulness of the dataset, before preparation we considered the following design criteria: Content: Firstly we aimed to reflect the content that a general audio archive might collect. Given the range of archive policies, from crowdsourced to strictly curated, we opted for a middle way, by using Freesound contributions but only those under the field-recording tag. Initial inspection of the various tags in Freesound determined that this tag was mostly free of interpretation issues, unlike for example ambience which Freesound users often use for ambient field recordings but also for synthesised atmospheric soundtrack sounds. On the other hand, we decided not to manually curate the collection to a specific definition of “field recording”, in part because such curation is difficult to apply to material from unknown third parties, and also because audio archive collections are rarely so narrowly construed. Licensing: Freesound contains material under various licences. Most common are the Creative Commons CC-BY licence and the “CC0” public domain dedication, though there is a small proportion of older material under other licences such as Creative Commons “sampling” licences which are not compatible with CC-BY. We wished to be able to apply a single open licence to the overall dataset, so we restricted ourselves to CC-BY and CC0 material, which means the overall dataset can be published under CC-BY. We also needed to respect the attribution requirements in the CC-BY source material, which we implemented by ensuring we stored the author metadata with each file as well as a URL to link back to the original source. Size and duration: We aimed to produce a dataset of manageable storage size, for ease of redistribution, yet suitably diverse. We also wished to produce sound excerpts of a standard and relatively short duration, so that they could be used in listening tests without risking listener fatigue, and so that automatic tests could run efficiently. These motivations led us to settle on a fixed ten-second duration for each excerpt. File formats: We also aimed to use a standardised file format. Freesound allows users to upload sounds of any sample rate, any number of channels, and in various file formats (uncompressed and lossy-compressed). We chose not to use audio coming from lossy compressed file formats, in case of artifacts introduced by the codecs. We also chose to convert all downloaded sounds to standard CD-quality mono WAV files. We considered standardising on 24-bit and/or 96 kHz as recommended in the IASA “TC04” archiving standard (IASA Technical Committee, 2009). However, in our experience 16-bit PCM has been more widely compatible than 24-bit PCM: the latter is not well handled by some older versions of Matlab and some Python audio libraries, although this situation is improving. Also the majority of the original downloaded audio was in 44.1 kHz. We therefore settled on 16-bit 44.1 kHz. In the crowdsourced Freesound archive, amplitude levels are uncontrolled, which may be problematic for listening tests. We therefore chose to amplitude-normalise each excerpt in our dataset. Dataset partitioning: In data mining and machine-learning experiments, it is useful to have a dataset partitioned into separate subsets—e.g. one for training and another for testing (Witten and Frank, 2005, Chapter 5). To facilitate this we chose to partition our data into ten equally-sized subsets. Partitioning can be purely random or can be stratified: for example, if the data was intended for an experiment detecting the presence of geotags, then the partitions could be arranged such that each partition had an equal mix of geotagged and non-geotagged data (Witten and Frank, 2005, Chapter 5). However, our dataset is intended for various purposes and not for a single specific experiment, so we opted for the simple random partitioning. Researchers can choose to use these subsets for comparability with others, or to perform their own partitioning. ### 2.1 Preparation We first obtained specific permission from Freesound to perform our relatively large-scale data download. We created a Python script based on the official freesound-python code, which we used to download the files from Freesound.222https://github.com/danstowell/freesound-python/tree/tagsearch The script was run to download all files matching all of the following criteria: * • Tagged field-recording (which contained 17807 sounds in total) * • Length 10 seconds or greater * • Audio file format WAV * • Published under either the CC-BY licence or CC0 * • Audio with 1 or 2 channels * • Audio sample format one of: pcm16, pcm24, pcm32 Each file was saved along with its metadata in JSON format.333http://tools.ietf.org/html/rfc4627 A small number of files (44) failed to download completely; these were detected by using the sox command- line audio tool (v 14.3.2) to attempt WAV decoding, and deleting the files which reported end-of-file errors or similar. The script was run in July 2013, taking about a week to download 328 GB of material. We then prepared a 10-second standardised excerpt from each audio file, taken from the middle of the audio recording, and used sox to convert it to a standardised file format: WAV, single-channel, sample rate 44.1 kHz, 16-bit PCM, amplitude normalised to $-2$ dB (empirically selected as the maximum gain before clipping). We inspected the excerpts for any further issues, by listening to all 10-second extracts. We found a few (seven) which were pure silence with DC offset. While this may sometimes be valid audio when considered in context, it led to normalisation issues, and could be problematic for some applications; we decided to remove these files. The above procedure resulted in 7690 sound excerpts with accompanying metadata. We placed the exerpts into 10 separate partitions, where the allocation was by pseudorandom shuffle initialised with a fixed seed value for repeatability (given in the Python script referenced above). Each of the 10 partitions has about 128 minutes of audio; the dataset totals over 21 hours of audio. Figure 1: Density plot of all geolocation tags in the dataset. Source code for this plot is in Appendix A. Each file is associated with various metadata (author, date, licence etc.) including a median of 7 tags per file (range 1–68). Around 40% of the files come with geolocation metadata. Figure 1 shows a density plot for all geolocations in the dataset. It indicates a broad geographic spread, although with much the strongest density in Europe (perhaps understandable given that Freesound is a European project), and a relative lack of sounds tagged from Africa and Russia. ### 2.2 Availability The full dataset is available online, hosted on the Internet Archive,444http://archive.org/details/freefield1010 and our institutional repository555http://c4dm.eecs.qmul.ac.uk/rdr/handle/123456789/35, under an overall CC-BY licence. For portability, each of the ten subsets of the data is presented as a single zip file which can fit on a data CD (around 570 MB per subset). We list MD5 checksums of these zip files so that their integrity can be verified. The metadata stored alongside the audio is easy to work with. Appendix A gives a simple Python code example for working with the JSON metadata, in this case the code used to generate Figure 1. ## 3 A classification experiment To demonstrate the dataset in use, we conducted an experiment using a binary classification paradigm using audio content analysis to infer the presence/absence of particular tags (sometimes called auto-tagging (Ellis et al., 2011)). For this we used the simple baseline classifier smacpy presented in Giannoulis et al. (2013), which provides a standard implementation of a common approach based on Gaussian mixture modelling of Mel frequency cepstral coefficients (MFCCs). Freesound uses a free-tagging system, with users able to associate an arbitrary number of tags with an audio file, and also able to create new tags. We thus manually chose a small selection of tags which appear relatively often—birdsong, city, people, nature, train, voice, water—to use for this experiment, creating a binary classification task for each one. We note that we might expect the tags to have a varying directness of connection with the audio content: we would expect recordings tagged voice to feature human voice sounds relatively prominently, whereas recordings tagged city might contain many sounds in ensemble, with perhaps no sonic component always present. We further added two pseudo-tags to the study, based on other metadata attributes: an indicator of whether or not the item comes with geolocation data (__geotagged), and an indicator of whether the item is CC-BY licensed (__ccby), as opposed to the only other licence present, public domain. These pseudo-tags in principle have no direct connection to the audio content, although there may be correlations due to circumstantial effects (for example, geolocation might more often be stored when recording outdoor scenes). We therefore expected only mild if any ability to predict these metadata attributes from audio. Table 1: Prevalence of selected tags in freefield1010. Tag \| Num tagged \| Proportion (%) ---\|---\|--- birdsong \| 198 \| 2.6 city \| 562 \| 7.3 nature \| 905 \| 11.8 people \| 321 \| 4.1 train \| 411 \| 5.3 voice \| 556 \| 7.2 water \| 707 \| 9.2 __geotagged \| 3058 \| 39.8 __ccby \| 6111 \| 79.5 Table 1 provides a summary of the prevalence of the selected tags in the freefield1010dataset. It is important to note that, for the “true” tags especially, each tag is present in only a minority of the items; the ratio of positive to negative instances is highly skewed. This has consequences for how we evaluate automatic classification: rather than using raw accuracy, which fails to account for this skew, we use the area under the curve (AUC) statistic derived from a receiver operating characteristic (ROC) curve (Provost et al., 1998; Fawcett, 2006). Our experiment proceeded as follows: for each of the selected tags and pseudo- tags, we performed a ten-fold cross-validation experiment using the folds defined by the ten subsets of freefield1010. This means that for each fold, we used nine of the ten subsets as training data for the classifier, where the presence/absence of the tag was the binary attribute to be learnt, and then tested the classifier using the audio from the one remaining subset. For each such run, we calculated the numbers of correct and incorrect decisions, and used this to calculate the AUC statistic (as in Fawcett (2006)). Source code for this experiment is available online.666https://github.com/danstowell/smacpy/tree/freefield1010 Figure 2: Results for automatic inference of tag presence/ absence. Plot shows the mean and the 95% confidence interval of the AUC score for each tag (and pseudo-tag) studied, across ten-fold crossvalidation. Results in Figure 2 show that the tags can be automatically inferred from audio with varying degrees of reliability. The best result is for birdsong at 82% AUC, while the weakest result for a true tag is city at 63%. Note that the standard interpretation of an AUC value is that it tells us the probability that the algorithm will rank a random positive instance higher than a random negative instance (Fawcett, 2006); chance performance is always 50% for the AUC statistic. Results for the two pseudo-tags attain around 58% AUC: above chance, but still very weak. It indicates that there is some mild difference in audio content between the positive and negative instances, but much less than for the true tags. The classification performance for true tags appears to show some connection with the “directness” issue raised above: the tags which yield weakest recognition performance (city, people, nature) can be said to have an indirect connection with the audio content. However these results are illustrative only, using a baseline classifier rather than a leading-edge algorithm. The strongest performance (82%) is still much lower than is desirable for a binary classifier deployed in a live system. The 95% confidence intervals (error bars) in Figure 2 are relatively small and well-separated. This illustrates that the dataset is of sufficient size to make inferences about the relative predictability of these tags from the audio content, and also that there is relative consistency among the ten folds of the data. ## 4 Conclusions In this paper we have described the preparation of a free and open audio dataset, designed primarily for use in research on data mining of audio archives. The dataset is derived from a subset of the Freesound archive, but fixed and standardised so as to facilitate reproducible research. In an auto- tagging experiment we demonstrated that the dataset can be used to probe issues such as the differential predictability of tags from audio. We hope that the dataset will prove useful to others. ## Acknowledgments We wish to thank the Freesound developers and maintainers at the Music Technology Group in Universitat Pompeu Fabra, for running the excellent Freesound archive and for allowing us API access to the data. DS & MP are supported by an EPSRC Leadership Fellowship EP/G007144/1. ## References * Bertin-Mahieux et al. [2011] T. Bertin-Mahieux, D. P. W. Ellis, B. Whitman, and P. Lamere. The Million Song Dataset. In _Proceedings of the 12th International Conference on Music Information Retrieval (ISMIR-11)_ , pages 591–596, Miami, FL, USA, Oct 2011. * Ellis et al. [2011] K. Ellis, E. Coviello, and G. Lanckriet. Semantic annotation and retrieval of music using a bag of systems representation. In _Proceedings of the 12th International Conference on Music Information Retrieval (ISMIR-11)_ , pages 723–728, Miami, FL, USA, Oct 2011. * Fawcett [2006] T. Fawcett. An introduction to ROC analysis. _Pattern recognition letters_ , 27(8):861–874, 2006. doi: 10.1016/j.patrec.2005.10.010. * Giannoulis et al. [2013] D. Giannoulis, D. Stowell, E. Benetos, M. Rossignol, M. Lagrange, and M. D. Plumbley. A database and challenge for acoustic scene classification and event detection. In _Proceedings of the European Signal Processing Conference (EUSIPCO 2013)_ , 2013. * IASA Technical Committee [2009] IASA Technical Committee. _Guidelines on the Production and Preservation of Digital Audio Objects_. Number IASA-TC04. International Association of Sound and Audiovisual Archives, Aarhus, Denmark, second edition, 2009. URL http://www.iasa-web.org/tc04/audio-preservation. * Provost et al. [1998] F. J. Provost, T. Fawcett, and R. Kohavi. The case against accuracy estimation for comparing induction algorithms. In _Proceedings of the International Conference on Machine Learning (ICML)_ , pages 445–453, 1998. * Ranft [2004] R. Ranft. Natural sound archives: Past, present and future. _Anais da Academia Brasileira de Ciências_ , 76(2):456–460, 2004. doi: 10.1590/S0001-37652004000200041. * Wang and Brown [2006] D. L. Wang and G. J. Brown, editors. _Computational Auditory Scene Analysis: Principles, Algorithms, and Applications_. IEEE Press, New York, 2006. * Witten and Frank [2005] I. H. Witten and E. Frank. _Data Mining: Practical Machine Learning Tools and Techniques_. Morgan Kaufmann, San Francisco, CA, USA, 2nd edition, 2005. ## Appendix A: Code example The following Python code was used to generate Figure 1 (using Python 2.7). ⬇ import glob, json import matplotlib.pyplot as plt import matplotlib.cm as cm ffpath = ’./freefield1010’ outfile = ’plots/plotgeo.pdf’ lats = [], lons = [] for onepath in glob.iglob(’%s//.json’ % ffpath): jsonfile = open(onepath, ’r’) jsondata = json.load(jsonfile) jsonfile.close() geo = jsondata.get(u’geotag’) if geo != None: lats.append(float(geo[u’lat’])) lons.append(float(geo[u’lon’])) # plot plt.figure() plt.hexbin(lons, lats, gridsize=25, bins=’log’, cmap=cm.binary) plt.xlabel(’Longitude’) plt.ylabel(’Latitude’) cb = plt.colorbar() cb.set_label(’log10(N)’) plt.savefig(outfile, papertype=’A4’, format=’pdf’)	arxiv-papers	2013-09-20T14:12:04	2024-09-04T02:49:51.227314	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Dan Stowell and Mark D. Plumbley", "submitter": "Dan Stowell", "url": "https://arxiv.org/abs/1309.5275" }
1309.5415	# Light-weight Flexible Magnetic Shields For Large-Aperture Photomultiplier Tubes P. DeVore D. Escontrias T. Koblesky C. J. Lin D. W. Liu K. B. Luk J. Ngan J. C. Peng C. Polly J. Roloff H. Steiner S. Wang J. Wong M. Yeh Department of Physics, University of California, Berkeley, CA 94720, U.S.A. Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A. Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. Department of Physics, Chinese University of Hong Kong, Hong Kong, China Chemistry Department, Brookhaven National Laboratory. Upton, NY 11973, U.S.A. (August 20, 2013) ###### Abstract Thin flexible sheets of high-permeability FINEMET® foils encased in thin plastic layers have been used to shield various types of 20-cm-diameter photomultiplier tubes from ambient magnetic fields. In the presence of the Earth’s magnetic field this type of shielding is shown to increase the collection efficiency of photoelectrons and can improve the uniformity of response of these photomultiplier tubes. ###### keywords: Magnetic Shield , Photomultiplier tube , FINEMET ###### PACS: 07.55.Nk , 85.60.Ha , , , , , , , , , , , , , ## 1 Introduction It is well known that the performance of most photomultiplier tubes (PMTs) is susceptible to even small local magnetic field such as that of the Earth, which is typically in the range of 0.4 to 0.5 gauss [1]. The impact of this kind of weak magnetic field can be reduced by shielding the PMT, especially the region near the photocathode, with high-permeability materials such as mu- metal. We describe here a new type of magnetic shield that has been developed at the Lawrence Berkeley National Laboratory for use in the Daya Bay Reactor Neutrino Oscillation Experiment [2]. It utilizes FINEMET, a novel thin flexible high- permeability material, to form a truncated cone around a large-aperture PMT behind the photocathode. In the following sections we describe the detailed characteristics of this shield, and present experimental results showing its impact on the performance of some selected PMTs. In particular we will show that it reduces variations in the PMT response caused by changes in the relative orientation of the PMT and magnetic field, and that it significantly improves the electron collection efficiency of the PMT. These improvements have a direct impact on the physics objectives of an experiment in that they result in a greater uniformity of response of the detector, and thereby yield a more reliable measurement of the energy of an event. ## 2 Effect of Magnetic Field on PMT We have investigated the effect of the magnetic field on the gain and charge collection of some large-aperture PMTs. These results are of particular relevance to experiments where the events of interest produce at most a few photoelectrons (PEs) in a given PMT. The magnetic field also affects multi- photoelectron events, but in this case it is not possible to separate the effects caused by gain variation from those attributable to collection efficiency. In this study we used Electron Tubes 9354KB [3], Hamamatsu R5912 [4], and Photonis XP1806 [5] 20-cm PMTs. ### 2.1 Effect of Magnetic Field on Collected Amount of Charge We determined the effect of the local magnetic field on the response of a PMT by positioning the PMT with its polar axis perpendicular to the field, as shown in Figure 1(a). The PMT was mounted on a holder that could be rotated about the polar axis from outside of the dark box containing the PMT. Light from a pulsing blue LED illuminated the entire photocathode, and its intensity could be controlled as needed in the study. The charge associated with the PMT signal was measured with an analog-to-digital converter (ADC) as a function of the angle between the local magnetic field and a reference on the PMT, denoted by $\phi$. In this study, the polarized key on the socket of the PMT, shown in Figure 1(b), was chosen as the reference direction. When this reference direction was parallel (normal) to the local field, $\phi$ was $0^{\circ}$ ($90^{\circ}$). Figure 1: (a) Orientation of the PMT with respect to the local magnetic field for evaluating response. (b) PMT reference direction: the polarized key between pin 1 and pin 20 indicated by the arrow is used as the PMT reference direction. The $\phi$-dependence of the charge variation at some operating high voltages for the Electron Tubes 9354KB, Hamamatsu R5912, and Photonis XP1806 PMTs is shown in Figures 2, 3 and 4 respectively. The charge variation, $\Delta Q$, is defined as $\Delta Q(\phi)=\left(\frac{Q(\phi)-\overline{Q}}{\overline{Q}}\right)\times 100\%$ (1) where $Q(\phi)$ is the amount of charge collected at angle $\phi$, and $\overline{Q}$ is the average amount of collected charge given by $\overline{Q}=\frac{1}{N}\sum_{i=1}^{N}Q(\phi_{i})$ (2) with $N$ being the number of measurements done at various $\phi$ values. For each type of PMT the result is quite insensitive to the applied voltage. The difference in the observed behaviour between the three PMT models is a consequence of the different designs of the dynode structure and focussing scheme. Figure 5 shows the opening to the first dynode for these three kinds of PMTs. Electron Tubes 9354KB has the smallest opening (most affected) whereas Photonis XP1806 has the largest opening (least affected). Figure 2: Effect of magnetic field on collecting charge for an unshielded Electron Tube 9354KB PMT as a function of the angle between the PMT reference direction and the magnetic field. Figure 3: Effect of magnetic field on collecting charge for an unshielded Hamamatsu R5912 PMT as a function of the angle between the PMT reference direction and the magnetic field. Figure 4: Effect of magnetic field on collecting charge for an unshielded Photonis XP1806 PMT as a function of the angle between the PMT reference direction and the magnetic field. Figure 5: Opening to the first dynode. From left to right: Electron Tubes 9354KB, Hamamatsu R5912, and Photonis XP1809. ### 2.2 Effect of Magnetic Field on Single-photoelectron (SPE) Spectrum A revealing way to study the influence of the magnetic field on the detected charge is to use single-photoelectron events. By making this measurement without a magnetic shield we can directly determine how the shield affects both the gain and the collection efficiency of the PMT. The SPE spectrum can be obtained by reducing the light intensity on the photocathode until only about 10% of the triggers produce a signal. From Poisson statistics about 95% of the observed pulses are produced by SPEs. The spectrum of the charge collected at the anode under this condition is shown in Figure 6. Since the charge of a single electron is known a priori, the calibrated ADC channel of the SPE peak can be used to determine the absolute gain of the PMT. Photoelectrons generated at the photocathode are directed to the first dynode by a set of focussing electrodes. The configuration of the focussing elements and the effective area of the first dynode determines the collection efficiency. The Lorentz force, v $\times$ B, on the photoelectrons in the rather long distance between the photocathode and the first dynode causes some of them to be swept away from the first dynode and thereby reduces the collection efficiency. The number of events in the SPE distribution is proportional to the efficiency of collecting the produced SPE by the first dynode of the PMT. Using a Hamamatsu R5912 without a magnetic shield we have obtained SPE spectra with the PMT reference direction parallel ($\phi=0^{\circ}$) and perpendicular ($\phi=90^{\circ}$) to the local magnetic field. The observed ADC spectrum of each orientation, a convolution of the SPE distribution and the pedestal, is shown in Figure 6. The observed spectrum was fitted to two Gaussian functions, one for the pedestal and the other one for the SPE distribution, and an exponential function for modeling the non-Gaussian component of the pedestal. The gain , peak-to-valley ratio (P/V ratio) and charge resolution of the PMT were obtained from the best fits which are displayed in Figure 6. The results are tabulated in Table 1. It is interesting to note that the change in gain between the two orientations is small. From Figure 3, we see a 15% difference in the charge variation between these two angular positions; that is, the collection efficiency has changed by about 15%. This result suggests that the majority of the secondary electrons produced within the dynodes are not significantly affected by the magnetic field. However, the peak-to-valley ratio gets worse when the magnetic effect is more pronounced. A series of measurements was made to determine how the collection efficiency in the presence of a magnetic field depends on the potential difference between the photocathode and first dynode. The results showed almost no effect on the collection efficiency when the potential difference was within 250 V of its nominal value, typically several hundred volts, for a gain of $10^{7}$. Figure 6: Magnetic effect on the SPE distribution of a Hamamatsu R5912 without shielding. Figure on the left is obtained with the PMT reference direction parallel to the magnetic field ($\phi=0^{\circ}$); figure on the right is determined with the magnetic field perpendicular to the PMT reference direction ($\phi=90^{\circ}$). The red curve in each figure is the best fit to the spectrum. Table 1: SPE Spectra of a Hamamatsu R5912 in the presence of magnetic field. PMT Orientation \| Gain \| P/V Ratio \| Charge Resolution ---\|---\|---\|--- $\varparallel$ magnetic field ($\phi=0^{\circ}$) \| $(1.030\pm 0.003)\times 10^{7}$ \| 2.77 $\pm$ 0.12 \| 0.343 $\pm$ 0.003 $\perp$ magnetic field ($\phi=90^{\circ}$) \| $(1.043\pm 0.004)\times 10^{7}$ \| 2.13 $\pm$ 0.11 \| 0.333 $\pm$ 0.003 ## 3 FINEMET magnetic shield In this section we describe the magnetic shield made of FINEMET, the material we used for reducing the effect of magnetic field on the Electron Tubes 9354KB, Hamamatsu R5912, and Photonis XP1806 PMTs. ### 3.1 FINEMET® FINEMET is a trademark of Hitachi Materials Ltd. for an amorphous metallic alloy ribbon made up of nano-crystalline grains of iron, silicon, boron, and small amounts of copper and niobium [6]. As shown in Table 2, this novel soft magnetic alloy has high saturation magnetic flux density, low core loss, and high permeability over a wide range of frequencies. Its permeability varies by less than $\pm$10% between $-50~{}^{\circ}$C and $150~{}^{\circ}$C, with very small aging effects. We used 0.12-mm-thick FINEMET sheets for fabricating the magnetic shields. This thin flexible material is made of a layer of 18 $\mu$m-thick FINEMET FT-3M tape laminated with 25-$\mu$m-thick hot-melt adhesive between two polyethylene terephthalate (PET) films. It can operate in a temperature range between $-40~{}^{\circ}$C and $80~{}^{\circ}$C. It is interesting to note that one layer of FINEMET can reduce the local magnetic field of $4\times 10^{-5}$ T by about 20 dB (from Figure 4 of reference [7]). Property \| Value ---\|--- Density \| 7300 kg/m3 Resistivity \| 1.2 $\mu\Omega$-m Curie temperature \| 570 ∘C Magnetic flux density (DC H = 800 A/m) \| 1.23 T Maximum relative permeability (DC) \| 70,000 Table 2: Selected physical properties of Hitachi FINEMET FT-3M [7]. In addition to the physical properties provided by the manufacturer and listed in Table 2, we have measured the amount of natural radioactivity of a few samples of FINEMET. Our results were $(18\pm 5)$ ppb for the 238U series, $(15\pm 5)$ ppb for the 232Th series, and $(4\pm 2)$ ppm for K, which translate to decay rates of about 0.2 Bq/kg, 0.06 Bq/kg, and 0.1 Bq/kg for 238U, 232Th, and 40K respectively. We have observed that if it is not stored in a ventilated location progenies of radon can accumulate gradually on the surfaces of the laminated FINEMET foil. Besides radioactivity, we have also investigated the compatibility of FINEMET with mineral oil and 18-M$\Omega$ ultra-pure water, which are commonly used in nuclear and particle-physics experiments. By submerging a piece of FINEMET sample with known dimensions in each liquid at elevated temperature for accelerated aging studies, the absorbance of the liquid was monitored as a function of time. For mineral oil at 40 ∘C, with a ratio of contact surface to liquid volume (S/V) of 0.39 cm-1, its absorbance as a function of wavelength and time is shown in Figure 7. The change in absorbance due to the presence of FINEMET is minute for wavelengths between 300 nm and 650 nm. The absorbance of purified water at 70 ∘C for a FINEMET sample with a S/V of 0.32 cm-1 is summarized in Figure 8. In the vicinity of 350 nm, the rate of change in absorbance levels off in about two weeks after a relatively fast degradation in the beginning. Note that the integrity of FINEMET has not changed in both liquids. Figure 7: Absorbance of mineral oil at $40~{}^{\circ}$C as a function of wavelength for a submerged FINEMET sample. Figure 8: Absorbance of purified water at $70~{}^{\circ}$C as a function of wavelength for a submerged FINEMET sample. ### 3.2 Fabrication of Magnetic Shield As illustrated in Figure 9, a truncated conical magnetic shield for the Hamamatsu R5912 PMTs used in the Daya Bay Experiment was realized by cutting a two-dimensional fan out of a piece of FINEMET foil with a water jet. The conically shaped shield was made by wrapping the fan around the PMT as shown in Figure 10. It was firmly held in place between the equator of the glass bulb and the acrylic potting shell surrounding the base by tucking the tabs into the pre-cut slots. The shield is flexible, so any deformation of the cone can be taken out easily by simple manipulation of the surface. The weight of such a FINEMET shield for a typical 20-cm PMT is about 20 g. Figure 9: Two-dimensional pattern of an unfolded truncated conical FINEMET magnetic shield. Figure 10: Cross section of a Hamamatsu R5912 PMT used in Daya Bay with a truncated conical magnetic shield. The dashed line is the photocathode. ### 3.3 Performance of the FINEMET magnetic shield The effectiveness of the truncated conical FINEMET shield depends on the slant height of the cone. For the Hamamatsu R5912 assembly used in the Daya Bay experiment, the magnetic shield extends from the acrylic sealing shell to the equator of the bulb, where the photocathode ends. We have evaluated the performance of the FINEMET magnetic shields with slant heights of 15.4 cm, 16.7 cm, 18.1 cm, 19.2 cm, and 23.0 cm, where the 23 cm-long shield extends to the plane tangent to the apex of the PMT bulb. The nominal slant height of the shield used in Daya Bay is 15.4 cm, ending right at the contact point of the bulb near the equator. For the Photonis XP1806 PMT assembly we used shields with slant heights of 14.3 cm and 23.6 cm which extended to the equator and the apex of the PMT bulb respectively. The corresponding slant heights are 14.5 cm and 22.5 cm for the Electron Tubes 5394KB PMT. #### 3.3.1 Improvement on the collection efficiency With the polar axis of the PMT normal to the local magnetic field as shown in Figure 1, the amount of collected charge relative to the one without shielding at $\phi=0^{\circ}$ (relative collection efficiency) for the Hamamatsu R5912 PMT as a function of the slant height of the FINEMET shield for different $\phi$ angles is shown in Figure 11. The magnetic shield clearly improves the collection of charge. Relative to the case of without any shielding, a FINEMET shield with a slant height of 15.4 cm (23.0 cm) improves the average collection efficiency by 8% (19%). Furthermore, as shown in Figure 12, the charge variation as a function of the $\phi$ angle is reduced when the slant height increases. For example, without shielding, the peak-to-peak difference in the charge variation is about 23%. A FINEMET shield with a slant height of 23.0 cm shrinks this difference to approximately 5%. Even a shield that covers only up to the equator (15.4 cm) allows us to achieve almost the same performance. This implies that the shield provides a less non-uniform collection efficiency to this level over the entire photocathode. Similarly, the results on collecting charge and minimizing the variation of the collection efficiency for the Photonis XP1806 and Electron Tubes 9354KB PMTs with FINEMET shields are depicted in Figures 13, 14, 15 and 16. For the Photonis XP1806 PMT, on average, the collection efficiency is increased by about 5% (14%) for a shield with a slant height of 14.3 cm (23.6 cm), and the variation in the amount of charge collected is reduced from 10% to 1.3% which is close to our systematic limit. The most dramatic improvement is achieved with the Electron Tubes 9354KB PMT. In this case the average collection efficiency is up by about 25% for a FINEMET shield extending to the equator and 40% for shields reaching to the apex of the PMT bulb. The variation in the collection efficiency is reduced from 46% with no shield to 21% (5%) for a shield with a slant height of 14.5 cm (22.5 cm). Figure 11: Relative collection efficiency as a function of the slant height of truncated conical FINEMET shield for a Hamamatsu R5912 PMT oriented at different $\phi$ angles. Figure 12: Charge variation due to magnetic field for Hamamatsu R5912 PMT with FINEMET shields of different slant heights. Figure 13: Relative collection efficiency as a function of slant height of truncated conical FINEMET shield for a Photonis XP1806 PMT oriented at different $\phi$ angles. Figure 14: Charge variation due to magnetic field for Photonis XP1806 PMT with FINEMET shields of different slant heights. Figure 15: Relative collection efficiency as a function of slant height of truncated conical FINEMET shield for an Electron Tubes 9354KB PMT oriented at different $\phi$ angles. Figure 16: Charge variation due to magnetic field for Electron Tubes 9354KB PMT with FINEMET shields of different slant heights. #### 3.3.2 Improvement on the SPE Spectrum In Table 3 we summarize the results on the SPE spectrum for the Hamamatsu R5912 PMT with FINEMET shields of different sizes. Compared to the results for the case without shielding (Table 1), we see that the shield can improve the gain by 5%, P/V ratio by up to 17% (34%) for $\phi=0^{\circ}$ ($90^{\circ}$), and charge resolution by less than 10%. Furthermore, the variation of the P/V ratio due to different orientation of the PMT relative to the magnetic field is significantly reduced. Consequently the systematic uncertainty in the charge measurement is minimized. Table 3: Characteristics of the SPE Spectrum for a Hamamatsu R5912 PMT with FINEMET magnetic shields of different slant heights. PMT Reference Direction \| Gain \| P/V Ratio \| Charge Resolution ---\|---\|---\|--- Slant height = 15.4 cm $\varparallel$ magnetic field ($\phi=0^{\circ}$) \| $(1.080\pm 0.003)\times 10^{7}$ \| 3.27 $\pm$ 0.18 \| 0.316 $\pm$ 0.002 $\perp$ magnetic field ($\phi=90^{\circ}$) \| $(1.092\pm 0.003)\times 10^{7}$ \| 2.86 $\pm$ 0.14 \| 0.324 $\pm$ 0.003 Slant height = 18.1 cm $\varparallel$ magnetic field ($\phi=0^{\circ}$) \| $(1.091\pm 0.003)\times 10^{7}$ \| 3.17 $\pm$ 0.16 \| 0.311 $\pm$ 0.002 $\perp$ magnetic field ($\phi=90^{\circ}$) \| $(1.088\pm 0.03)\times 10^{7}$ \| 3.03 $\pm$ 0.15 \| 0.318 $\pm$ 0.003 Slant height = 23.0 cm $\varparallel$ magnetic field ($\phi=0^{\circ}$) \| $(1.081\pm 0.003)\times 10^{7}$ \| 3.20 $\pm$ 0.16 \| 0.315 $\pm$ 0.002 $\perp$ magnetic field ($\phi=90^{\circ}$) \| $(1.085\pm 0.003)\times 10^{7}$ \| 2.93 $\pm$ 0.14 \| 0.311 $\pm$ 0.002 #### 3.3.3 Position-dependence of PMT Response We determined the response of a Hammamatsu R5912 PMT at each point of a grid on the photocathode by measuring the PE spectrum with a pulsing LED [8]. The orientation of the PMT is shown in Figure 17. We split the light pulse from the LED into two parts with optical fibers. One part of the LED light was sent to a reference PMT to monitor the intensity of the LED, and the other part to the photocathode of the Hamamatsu PMT. A collimator was used to limit the spot size of the incident light at the photocathode to a circle with a radius of 2 mm. The LED intensity was set to produce at least 20 photoelectrons per pulse on the photocathode. Runs of ten seconds duration at a pulse rate of 100 Hz were used to measure the response of the PMT for every point on the photocathode. An ADC was gated by the pulse generator used to drive the LED. A computer recorded the integrated counts in each ADC channel for each run. The light source was moved under computer control to the next point on the grid until the whole surface of the photocathode was scanned. Figure 17: (a) Orientation of the photocathode of a Hamamatsu R5912 PMT with respect to the local magnetic field. The x-y plane is normal to the polar axis of the PMT. The magnetic field is along the negative-y direction. (b) Position-dependent response of a Hamamatsu R5912 PMT with no magnetic shield. The colored scale is the average number of detected PEs. Figure 18: Position- dependent response of a Hamamatsu R5912 PMT with a FINEMET shield (a) with a slant height of 15.4 cm, (b) with a slant height of 20.5 cm. The colored scale is the average number of detected PEs. In the absence of any magnetic field we expect rather uniform collection efficiency over the photocathode. However, the Earth’s magnetic field in the transverse direction creates a position-dependent response. In this case the field acting on a PMT with no shielding produces large non-uniformity as shown in Figure 17. The lowest collection efficiency occurs in a region centered at x $\approx$ 6 cm and y = 0 cm. In addition, the response of the PMT is almost symmetric about the x-axis. This result is a direct consequence of the fact that the internal structure of the PMT is symmetric about the x-axis, so that with the magnetic field in the negative y-direction the response of the PMT should be symmetric across the x-axis. This symmetry should not exist for the y-axis because the internal structure of the PMT is different across the y-axis, and also because a magnetic field in the negative y-direction will affect the motion of electrons in the x-direction. A FINEMET shield around the PMT makes the response more uniform over the whole photocathode. The results for the shields with slant height of 15.4 cm and 20.5 cm are presented in Figure 18. The 20.5-cm shield has a more uniform response than the 15.4-cm shield, although even with this long shield there are still significant residual non-uniformities. However, both of these shields clearly produce a better performance for the PMT than that obtained with no shield. ## 4 Conclusions We have presented the performance of a truncated conical magnetic shield made of FINEMET. Mechanically, it is thin, light, flexible, relative cheap, and easy to handle. In addition, the laminated FINEMET sheet is compatible with mineral oil and purified water. We have demonstrated that FINEMET is very effective in shielding large-aperture PMTs against magnetic field. As a result, the collection efficiency of the PMT is improved and the response of the PMT is more uniform across the entire photocathode. ## 5 Acknowledgements We would like to express our gratitude to the technical staffs of the Lawrence Berkeley National Laboratory for their excellent support. This work was partially supported by the Director, Office of Science, Office of Basic Energy Sciences, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231and DE-AC02-98CH10886, the U.S. National Science Foundation, the Research Grant Council of the Hong Kong Special Administrative Region, China (Project Nos. CUHK 1/07C and CUHK3/CRF/10). J.N. and J. W. were also partially supported by the Department of Physics, Chinese University of Hong Kong. ## References * [1] Hamamatsu Photonics, ”Photomultiplier Tubes: Basics and Applications”, 3rd Edition, (2007). * [2] X. Guo et. al. (Daya Bay Collaboration), hep-ex/0701029 (2007). * [3] Electron Tubes 9354KB Data Sheet, Electron Tubes Inc., Sept., 2002. * [4] Hamamatsu R5912 Data Sheet, Hamamatsu Co., Sept., 1998. * [5] Photonis Hemispherical PMT Data Sheet, Photonis Co, Sept., 2006. * [6] R. Lebourgeois et al., J. Magn. Magn. Mater. 254-255, 191 (2003). * [7] Catalog number HL-FM16-F, Hitachi Metals, Ltd. (http://www.hitachi-metals.co.jp). * [8] T. Koblesky et. al., Nucl. Instrum. Method A670, 40 (2012).	arxiv-papers	2013-09-20T23:45:35	2024-09-04T02:49:51.241812	{ "license": "Public Domain", "authors": "P. DeVore, D. Escontrias, T. Koblesky, C. J. Lin, D. W. Liu, K. B.\n Luk, J. Ngan, J. C. Peng, C. Polly, J. Roloff, H. Steiner, S. Wang, J. Wong\n and M. Yeh", "submitter": "Kam Biu Luk", "url": "https://arxiv.org/abs/1309.5415" }
1309.5427	# Latent Fisher Discriminant Analysis Gang Chen Department of Computer Science and Engineering SUNY at Buffalo [email protected] ###### Abstract Linear Discriminant Analysis (LDA) is a well-known method for dimensionality reduction and classification. Previous studies have also extended the binary- class case into multi-classes. However, many applications, such as object detection and keyframe extraction cannot provide consistent instance-label pairs, while LDA requires labels on instance level for training. Thus it cannot be directly applied for semi-supervised classification problem. In this paper, we overcome this limitation and propose a latent variable Fisher discriminant analysis model. We relax the instance-level labeling into bag- level, is a kind of semi-supervised (video-level labels of event type are required for semantic frame extraction) and incorporates a data-driven prior over the latent variables. Hence, our method combines the latent variable inference and dimension reduction in an unified bayesian framework. We test our method on MUSK and Corel data sets and yield competitive results compared to the baseline approach. We also demonstrate its capacity on the challenging TRECVID MED11 dataset for semantic keyframe extraction and conduct a human- factors ranking-based experimental evaluation, which clearly demonstrates our proposed method consistently extracts more semantically meaningful keyframes than challenging baselines. ## 1 Introduction Linear Discriminant Analysis (LDA) is a powerful tool for dimensionality reduction and classification that projects high-dimensional data into a low- dimensional space where the data achieves maximum class separability [10, 12]. The basic idea in classical LDA, known as the Fisher Linear Discriminant Analysis (FDA) is to obtain the projection matrix by minimizing the within- class distance and maximizing the between-class distance simultaneously to yield the maximum class discrimination. It has been proved analytically that the optimal transformation is readily computed by solving a generalized eigenvalue problem [12]. In order to deal with multi-class scenarios [10], LDA can be easily extended from binary-case and generally used to find a subspace with $d-1$ dimensions for multi-class problems, where $d$ is the number of classes in the training dataset. Because of its effectiveness and computational efficiency, it has been applied successfully in many applications, such as face recognition [3] and microarray gene expression data analysis. Moreover, LDA was shown to compare favorably with other supervised dimensionality reduction methods through experiments [20]. However, LDA expects instance/label pairs which are surprisingly prohibitive especially for large training data. In the last decades, semi-supervised methods have been proposed to utilize unlabeled data to aid classification or regression tasks under situations with limited labeled data, such as Transductive SVM (TSVM) [21, 14] and Co-Training [4]. Correspondently, it is reasonable to extend the supervised LDA into a semi-supervised method, and many approaches [5, 26, 20] have been proposed. Most of these methods are based on transductive learning. In other words, they still need instance/label pairs. However, many real applications require bag-level labeling [1], such as object detection [11] and event detection [19]. In this paper, we propose a Latent Fisher Discriminant Analysis model (or LFDA in short), which generalizes Fisher LDA model [10]. Our model is inspired by MI-SVM [1] or latent SVM [11] and multiple instance learning problems [8, 17]. On the one hand, recently applications in image and video analysis require a kind of bag-level label. Moreover, using latent variable model for this kind of problem shows great improvement on object detection [11]. On the other hand, the requirement of instance/label pairs in the training data is surprisingly prohibitive especially for large training data. The bag-level labeling methods are a good solution to this problem. MI-SVM or latent SVM is a kind of discriminative model by maximizing a posterior probability. Our model unify the discriminative nature of the Fisher linear discriminant with a data driven Gaussian mixture prior over the training data in the Bayesian framework. By combining these two terms in one model, we infer latent variables and projection matrix in alternative way until convergence. We demonstrate this capability on MUSK and Corel data sets for classification, and on TRECVID MED11 dataset for keyframe extraction on five video events [19]. ## 2 Related Work Linear Discriminant Analysis (LDA) has been a popular method for dimension reduction and classification. It searches a projection matrix that simultaneously maximizes the between-class dissimilarity and minimizes the within-class dissimilarity to increase class separability, typically for classification applications. LDA has attracted an increasing amount of attention in many applications because of its effectiveness and computational efficiency. Belhumeur et al proposed PCA+LDA [3] for face recognition. Chen et al projects the data to the null space of the within-class scatter matrix and then maximizes the between-class scatter in this space [6] to deal with the situation when the size of training data is smaller than the dimensionality of feature space. [22] combines the ideas above, maximizes the between-class scatter matrix in the range space and the null space of the within-class scatter matrix separately and then integrates the two parts together to get the final transformation. [25] is also a two-stage method which can be divided into two steps: first project the data to the range space of the between-class scatter matrix and then apply LDA in this space. To deal with non-linear scenarios, the kernel approach [21] can be applied easily via the so-called kernel trick to extend LDA to its kernel version, called kernel discriminant analysis [2], that can project the data points nonlinearly. Recently, sparsity induced LDA is also proposed [18]. However, many real-world applications only provide labels on bag-level, such as object detection and event detection. LDA, as a classical supervised learning method, requires a training dataset consisting of instance and label pairs, to construct a classifier that can predict outputs/labels for novel inputs. However, directly casting LDA as a semi-supervised method is challenging for multi-class problems. Thus, in the last decades, semi- supervised methods become a a hot topic. One of the main trend is to extend the supervised LDA into a semi-supervised method [5, 26, 20], which attempts to utilize unlabeled data to aid classification or regression tasks under situations with limited labeled data. [5] propose a novel method, called Semi- supervised Discriminant Analysis, which makes use of both labeled and unlabeled samples. The labeled data points are used to maximize the separability between different classes and the unlabeled data points are used to estimate the intrinsic geometric structure of the data. [20] propose a semi-supervised dimensionality reduction method which preserves the global structure of unlabeled samples in addition to separating labeled samples in different classes from each other. Most of these semi-supervised methods model the geometric relationships between all data points in the form of a graph and then propagate the label information from the labeled data points through the graph to the unlabeled data points. Another trend prefers to extent LDA into an unsupervised senarios. For example, Ding et al propose to combine LDA and K-means clustering into the LDA-Km algorithm [9] for adaptive dimension reduction. In this algorithm, K-means clustering is used to generate class labels and LDA is utilized to perform subspace selection. Our solution is a new latent variable model called Latent Fisher Discriminant Analysis (LFDA), which complements existing latent variable models that have been popular in the recent vision literature [11] by making it possible to include the latent variables into Fisher discriminant analysis model. Unlike previous latent SVM [11] or MI-SVM [1] model, we extend it with prior data distribution to maximize a joint probability when inferring latent variables. Hence, our method combines the latent variable inference and dimension reduction in an unified Bayesian framework. ## 3 Latent Fisher discriminant analysis We propose a LFDA model by including latent variables into the Fisher discriminant analysis model. Let ${\mathcal{X}}=\\{{\bf x_{1}},{\bf x_{2}},...,{\bf x_{n}}\\}$ represent $n$ bags, and the corresponding labels ${\mathcal{L}}=\\{l_{1},l_{2},...,l_{n}\\}$. For each ${\bf x_{i}}\in\mathcal{X}$, ${\bf x_{i}}$ can be treated as a bag (or video) in [1], and its label $l_{i}$ is categorical and assumes values in a finite set, e.g. $\\{1,2,...,C\\}$. Let ${\bf x_{i}}\in\mathbb{R}^{d\times n_{i}}$, which means it contains $n_{i}$ instances (or frames), ${\bf x_{i}}=\\{x_{i}^{1},x_{i}^{2},...,x_{i}^{n_{i}}\\}$, and its $j^{th}$ instance is a vector in $\mathbb{R}^{d}$, namely $x_{i}^{j}\in\mathbb{R}^{d}$. Fisher’s linear discriminant analysis pursue a subspace $\mathcal{Y}$ to separate two or more classes. In other words, for any instance $x\in\mathcal{X}$, it searches for a projection $f:x\rightarrow y$, where $y\in\mathbb{R}^{d^{\prime}}$ and $d^{\prime}\leq d$. In general, $d^{\prime}$ is decided by $C$, namely $d^{\prime}=C-1$. Suppose the projection matrix is $\mathcal{P}$, and $y=f(x)=\mathcal{P}x$, then latent Fisher LDA proposes to minimize the following ratio: $\displaystyle\centering({\mathcal{P}^{}})=\operatornamewithlimits{arg\,min}_{\mathcal{P},z}J({\mathcal{P},z})=\operatornamewithlimits{arg\,min}_{\mathcal{P},z}\textrm{trace}\left(\frac{\mathcal{P}^{T}\Sigma_{w}(x,z)\mathcal{P}}{\mathcal{P}^{T}\Sigma_{b}(x,z)\mathcal{P}}+\beta\mathcal{P}^{T}\mathcal{P}\right)\@add@centering$ (1) where $z$ is the latent variable, $\beta$ is an weighting parameter for regularization term. The set $z\in Z(x)$ defines the possible latent values for a sample $x$. In our case, $z\in\\{1,2,...,C\\}$. $\Sigma_{b}(x,z)$ is between class scatter matrix and $\Sigma_{w}(x,z)$ is within class scatter matrix. However, LDA is dependent on a categorical variable (i.e. the class label) for each instance $x$ to compute $\Sigma_{b}$ and $\Sigma_{w}$. In our case, we only know bag-level labels, not on instance-level labels. To minimize $J({\mathcal{P}},z)$, we need to solve $z(x)$ for any given $x$. This problem is a chicken and egg problem, and can be solved by alternating algorithms, such as EM [7]. In other words, solve $\mathcal{P}$ in Eq. (1) with fixed $z$, and vice versa in an alternating strategy. ### 3.1 Updating $z$ Suppose we have found the projection matrix $\mathcal{P}$, and corresponding subspace $\mathcal{Y}=\mathcal{P}\mathcal{X}$, where $\mathcal{Y}=\\{{\bf y_{1}},{\bf y_{2}},...,{\bf y_{n}}\\}$. Instead of inferring latent variables at instance-level in latent SVM, we propose latent variable inference at clustering-level in the projected space $\mathcal{Y}$. That means all elements in the same cluster have same label. Such assumption is reasonable because elements in the same cluster are close to each other. On the other hand, cluster-level inference can speed up the learning process. We extend mixture discriminative analysis model in [13] by incorporating latent variables over all instances for an given class. As in [13], we assume each class $i$ is a $K$ components of Gaussians, $\centering p(x\|\lambda_{i})=\sum_{j=1}^{K}\pi_{i}^{j}g(x\|\mu_{i}^{j},\Sigma_{i}^{j})\@add@centering$ (2) where $x$ is a $d$-dimensional continuous-valued data vector (i.e. measurement or features); $\pi_{i}=\\{\pi_{i}^{j}\\}_{j=1}^{K}$ are the mixture weights, and $g(x\|\mu_{i}^{j},\Sigma_{i}^{j})$, $j\in[1,K]$, are the component Gaussian densities with $\mu_{i}=\\{\mu_{i}^{j}\\}_{j=1}^{K}$ as mean and $\Sigma_{i}=\\{\Sigma_{i}^{j}\\}_{j=1}^{K}$ as covariance. $\lambda_{i}$ is the parameters for class $i$ which we need to estimate, $\lambda_{i}=\\{\pi_{i},\mu_{i},\Sigma_{i}\\}$. Hence, for each class $i\in\\{1,2,\dots,C\\}$, we can estimate $\lambda_{i}$ and get its $K$ subsets $S_{i}=\\{S_{i}^{1},S_{i}^{2},...,S_{i}^{K}\\}$ with EM algorithm. Suppose we have the discriminative weights (or posterior probability) for the $K$ centers in each class, $w_{i}=\\{w_{i}^{1},w_{i}^{2},...,w_{i}^{K}\\}$, which are the posterior probability determined by the latent FDA and will be discussed later. We maximize one of the following two equations: Maximizing a posterior probability: $\displaystyle\centering\mu_{i}^{j}=\operatornamewithlimits{arg\,max}_{\mu_{i}^{j}\in\mu_{i},j\in[1,K]}w_{i}=\operatornamewithlimits{arg\,max}_{\mu_{i}^{j}\in\mu_{i},j\in[1,K]}p(z_{i}\|\mu_{i},\mathcal{P})\@add@centering$ (3a) Maximizing the joint probability with prior: $\displaystyle\centering\mu_{i}^{j}=\operatornamewithlimits{arg\,max}_{\mu_{i}^{j}\in\mu_{i},j\in[1,K]}(\pi_{i}\circ w_{i})\@add@centering$ (3b) where $z_{i}$ is the latent label assignment, $\pi_{i}$ is the prior clustering distributions for $\lambda_{i}$ in class $i$, $w_{i}$ is the posterior (or weight) determined by kNN voting (see further) in the subspace and $\circ$ is the pointwise production or Hadamard product. We treat Eq. (3a) as the latent Fisher discriminant analysis model (LFDA), because it takes the same strategy as the latent SVM model [1, 11]. As for Eq. (3b), we extend LFDA by combining the both factors (representative and discriminative) together, and find the cluster $S_{i}^{j}$ in class $i$ by maximizing Eq. (3b). In a sense, Eq. (3b) considers the prior distribution from the training dataset, thus, we treat it as the joint latent Fisher discriminant analysis model (JLFDA) or LFDA with prior. In the nutshell, we propose a way to formulate discriminative and generative methods under the unified Bayesian framework. We comparatively analyze both of these models (Section 4). Consequently, if we select the cluster $S_{i}^{j}$ with the mean $\mu_{i}^{j}$ which maximizes the above equation for class $i$, we can relabel all samples $x$ positive for class $i$ and the rest negative, subject to $y={\mathcal{P}}x$ and $y\in S_{i}^{j}$. Then, we construct a new training data ${\mathcal{X^{+}}}=\\{{\bf x_{1}^{+}},{\bf x_{2}^{+}},...,{\bf x_{n}^{+}}\\}$, with labels ${\mathcal{L^{+}}}=\\{{\bf z_{1}^{+}},{\bf z_{2}^{+}},...,{\bf z_{n}^{+}}\\}$, where ${\bf x_{i}^{+}}=S_{i}^{j}$ for class $i$ with $n_{i}^{j^{\prime}}$ elements, and its labels ${\bf z_{i}^{+}}=\\{z_{i}^{1},z_{i}^{2},...,z_{i}^{n_{i}^{j^{\prime}}}\\}$ on instance level. Obviously, ${\bf x_{i}^{+}}\subseteq{\bf x_{i}}$ and ${\mathcal{X^{+}}}$ is a subset of ${\mathcal{X}}$. The difference between ${\mathcal{X^{+}}}$ and ${\mathcal{X}}$ lies that every element $x_{i}^{+}\in{\bf x_{i}^{+}}$ has label $z(x_{i}^{+})$ decided by Eq. (LABEL:eq:Eq3sub), while ${\bf x_{i}}\subset\mathcal{X}$ only has bag level label. ### 3.2 Updating projection $\mathcal{P}$ When we have labels for the new training data ${\mathcal{X^{+}}}$, we use the Fisher LDA to minimize $J({\mathcal{P}},z)$. Note that Eq. (1) is invariant to the scale of the vector $\mathcal{P}$. Hence, we can always choose $\mathcal{P}$ such that the denominator is simply $\mathcal{P}^{T}\Sigma_{b}\mathcal{P}={\bf 1}$. For this reason we can transform the problem of minimizing Eq. (1) into the following constrained optimization problem [10, 12, 24]: $\displaystyle\centering{\mathcal{P}^{}}\@add@centering$ $\displaystyle=\operatornamewithlimits{arg\,min}_{{\mathcal{P}}}\textrm{trace}\left(\mathcal{P}^{T}\Sigma_{w}(x,z)\mathcal{P}+\beta\mathcal{P}^{T}\mathcal{P}\right)$ $\displaystyle\quad\textrm{s. t. }{\mathcal{P}^{T}\Sigma_{b}(x,z)\mathcal{P}}={\bf 1}$ (4) where 1 is the identity matrix in $\mathbb{R}^{d^{\prime}\times d^{\prime}}$. The optimal Multi-class LDA consists of the top eigenvectors of $(\Sigma_{w}(x,z)+\beta)^{\dagger}\Sigma_{b}(x,z)$ corresponding to the nonzero eigenvalues [12], here $(\Sigma_{w}(x,z)+\beta)^{\dagger}$ denotes the pseudo-inverse of $\Sigma_{w}(x,z)+\beta$. After we calculated $\mathcal{P}$, we can project ${\mathcal{X^{+}}}$ into subspace ${\mathcal{Y^{+}}}$. Note that in the subspace ${\mathcal{Y^{+}}}$, any $y^{+}\in{\mathcal{Y^{+}}}$ preserves the same labels as in the original space. In other words, ${\mathcal{Y^{+}}}$ has corresponding labels ${\mathcal{L^{+}}}$ at element level, namely $z(y^{+})=z(x^{+})$. In general, multi-class LDA [24] uses kNN to classify new input data. We compute $w_{i}$ using the following kNN strategy: for each sample $x\in{\mathcal{X}}$, we get $y=\mathcal{P}x$ by projecting it into subspace ${\mathcal{Y}}$. Then, for $y\in{\mathcal{Y}}$, we choose its $N$ nearest neighbors from ${\mathcal{Y^{+}}}$, and use their labels to voting each cluster $S_{i}^{j}$ in each class $i$. Then, we compute the following posterior probability: $\displaystyle\centering w_{i}^{j}\@add@centering$ $\displaystyle=p(z_{i}=1\|\mu_{i}^{j})=p(\mu_{i}^{j}\|z_{i}=1)p(z_{i}=1)$ $\displaystyle=p(z_{i}=1)\frac{p(\mu_{i}^{j},z_{i}=1)}{\sum_{i=1}^{C}p(\mu_{i}^{j},z_{i}=1)}$ (5) It counts all $y\in{\mathcal{Y}}$ fall into $N$ nearest neighbor of $\mu_{i}^{j}$ with label $z_{i}$. Note that kNN is widely used as the classifier in the subspace after LDA transformation. Thus, Eq. (3.2) consider all training data to vote the weight for each discriminative cluster $S_{i}^{j}$ in every class $i$. Hence, we can find the most discriminative cluster $S_{i}^{j}$, s.t. $w_{i}^{j}>w_{i}^{k}$, $k\in[1,K],k\neq j$. Algorithm. We summarize the above discussion in pseudo code. To put simply, we update ${\mathcal{P}}$ and $z$ in an alternative manner, and accept the new projection matrix ${\mathcal{P}}$ with LDA on the relabeled instances. Such algorithm can always convenge in around 10 iterations. After we learned matrix $\mathcal{P}$ and $\\{\lambda_{i}\\}_{i=1}^{C}$ by maximizing Eq. (3), we can use them to select representative and discriminative frames from video datasets by nearest neighbor searching. Algorithm 1 Input: training data ${\mathcal{X}}$ and its labels $\mathcal{L}$ at video level, $\beta$, $K$, $N$, $T$ and $\epsilon$. Output: $\mathcal{P}$, $\\{\lambda_{i}\\}_{i=1}^{C}$ 1: Initialize $\mathcal{P}$ and $w_{i}$; 2: for $Iter=1$ to $T$ do 3: for $i=1;i<=C;i++$ do 4: Project all the training data ${\mathcal{X}}$ into subspace ${\mathcal{Y}}$ using ${\mathcal{Y}}$=$\mathcal{P}{\mathcal{X}}$; 5: For each class $i\in[1,C]$, using Gaussian mixture model to partition its elements in the subspace, and compute $\lambda_{i}$ = $\\{\pi_{i},\mu_{i},\Sigma_{i}\\}$; 6: Maximize Eq. (3) to find $S_{i}^{j}$ with center $\mu_{i}^{j}$; 7: Relabel all elements positive in the cluster $S_{i}^{j}$ for class $i$ according to Eq. (LABEL:eq:Eq3sub); 8: end for 9: Update $z$ and construct the new subset ${\mathcal{X^{+}}}$ and its labels $\mathcal{L}^{+}$ for all $C$ classes; 10: Do Fisher linear discriminant analysis and update $\mathcal{P}$ 11: if $\mathcal{P}$ converge (change less than $\epsilon$), then break 12: Compute $N$ nearest neighbors for each training data, and calculate discriminative weight $w_{i}$ for each class $i$ according to Eq. (3.2). 13: end for 14: Return ${\mathcal{P}}$ and cluster centers $\\{\lambda_{i}\\}_{i=1}^{C}$ learned respectively for all $C$ classes; ### 3.3 Convergence analysis Our method updates latent variable $z$ and then $\mathcal{P}$ in an alternative manner. Such strategy can be attributed to the hard assignment of EM algorithm. Recall the EM approach: $\displaystyle\centering\mathcal{P^{}}=\operatornamewithlimits{arg\,max}_{\mathcal{P}}p(\mathcal{X},\mathcal{L}\|\mathcal{P})=\operatornamewithlimits{arg\,max}_{\mathcal{P}}\sum_{i=1}^{C}p(\mathcal{X},\mathcal{L}\|z_{i})p(z_{i}\|\mathcal{P})\@add@centering$ (6) then the likelihood can be optimized using iterative use of the EM algorithm. ###### Theorem 3.1 Assume the latent variable $z$ is inferred for each instance in $\mathcal{X}$, then to maximize the above function is equal to maximize the following auxiliary function $\displaystyle\centering\mathcal{P}=\operatornamewithlimits{arg\,max}_{\mathcal{P}}\sum_{i=1}^{C}p(z_{i}\|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln\bigg{(}p(\mathcal{X},\mathcal{L}\|z_{i})p(z_{i}\|\mathcal{P})\bigg{)}\@add@centering$ (7) This proof can be shown using Jensen’s inequality. ###### Lemma 3.2 The hard assignment of latent variable $z$ by maximizing Eq. (3) is a special case of EM algorithm. Proof $\displaystyle\sum_{i=1}^{C}p(z_{i}\|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln\bigg{(}p(\mathcal{X},\mathcal{L}\|z_{i})p(z_{i}\|\mathcal{P})\bigg{)}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{C}p(z_{i}\|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln(p(z_{i}\|\mathcal{X},\mathcal{L},P))+\sum_{i=1}^{C}p(z_{i}\|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln(p(\mathcal{X},\mathcal{L}\|\mathcal{P}))$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{C}p(z_{i}\|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})ln(p(z_{i}\|\mathcal{X},\mathcal{L},\mathcal{P}))+ln(p(\mathcal{X},\mathcal{L}\|\mathcal{P}))$ (8) Given $\mathcal{P}^{\prime}$, we can infer the latent variable $z$. Because the hard assignment of $z$, the first term in the right hand side of Eq. (Proof) assigns $z_{i}$ into one class. Note that $p(z\|\mathcal{X},\mathcal{L},\mathcal{P})ln(p(z\|\mathcal{X},\mathcal{L},\mathcal{P}))$ is a monotonically increasing function, which means that by maximizing the posterior likelihood $p(z\|\mathcal{X},\mathcal{L},\mathcal{P})$ for each instance, we can maximize Eq. (Proof) for the hard assignment case in Eq. (3). Thus, the updating strategy in our algorithm is a special case of EM algorithm, and it can converge into a local maximum as EM algorithm. Note that in our implement, we infer the latent variable in cluster level. In other words, to maximize $p(z_{i}\|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})$, we can include another latent variable $\pi_{j},j\in[1,K]$. In other words, we need to maximize $\sum_{j=1}^{K}p(z_{i},\pi_{j}\|\mathcal{X},\mathcal{L},\mathcal{P}^{\prime})$, which we can recursively determine the latent variable $\pi_{i}$ using an embedded EM algorithm. Hence, our algorithm use two steps of EM algorithm, and it can converge to a local maximum. Refer [23] for more details about the convergence of EM algorithm. ### 3.4 Probabilistic understanding for the model The latent SVM model [11, 1] propose to label instance $x_{i}$ in positive bag, by maximize $p(z(x_{i})=1\|x_{i})$, which is the optimal Bayes decision rule. Figure 1: Example of graphical representation only for one class (event). $h_{i}^{j}$ is the hidden variable, $x_{i}^{j}$ is the observable input, $y_{i}^{j}$ is the projection of $x_{i}^{j}$ in the subspace, $j\in[1,n_{i}]$, and $n_{i}$ is the number of total training data for class $i$. The $K$ cluster centers $\mu_{i}=\\{\mu_{i}^{1},\mu_{i}^{2},...,\mu_{i}^{K}\\}$ is determined by both $\pi_{i}$ and $w_{i}$. The graphical model of our method is similar to GMM model in vertical. By adding $z_{i}$ into LDA, the graphical model can handle latent variables. Similarly, Eq. (3a) takes the same strategy as latent SVM to maximize a posterior probability. Moreover, instead of only maximizing the $p(z=1\|x)$, we also maximize the joint probability $p(z=1,x)$, using the Bayes rule, $p(z=1,x)=p(x)p(z=1\|x)$. In this paper, we use Gaussian mixture model to approximate the prior $p(x)$ in this generative model. We argue that to maximize a joint probability is reasonable, because it considers both discriminative (posterior probability) and representative (prior) property in the video dataset. We give the graphical representation of our model in Fig. (1). ## 4 Experiments and results In this section, we perform experiments on various data sets to evaluate the proposed techniques and compare it to other baseline methods. For all the experiments, we set $T$ = 20 and $\beta=40$; and initialize uniformly weighted $w_{i}$ and projection matrix $\mathcal{P}$ with LDA. ### 4.1 Classification on toy data sets The MUSK data sets111www.cs.columbia.edu/ andrews/mil/datasets.html are the benchmark data sets used in virtually all previous approaches and have been described in detail in the landmark paper [8]. Both data sets, MUSK1 and MUSK2, consist of descriptions of molecules using multiple low-energy conformations. Each conformation is represented by a 166-dimensional feature vector derived from surface properties. MUSK1 contains on average approximately 6 conformation per molecule, while MUSK2 has on average more than 60 conformations in each bag. The Corel data set consists three different categories ( elephant , fox , tiger ) , and each instance is represented with 230 dimension features, characterized by color, texture and shape descriptors. The data sets have 100 positive and 100 negative example images. The latter have been randomly drawn from a pool of photos of other animals. We first use PCA reduce its dimension into 40 for our method. For parameter setting, we set $K$=3, $T$ = 20 and $N=4$ (namely the 4-Nearest-Neighbor (4NN) algorithm is applied for classification). The averaged results of ten 10-fold cross- validation runs are summarized in Table (1). We set LDA222we use the bag label as the instance label to test its performance and MI-SVM as our baseline. We can observe that both LFDA and JLFDA outperform MI-SVM on MUSK1 and Fox data sets, while has comparative performance as MI-SVM on the others. Data set \| inst/Dim \| MI-SVM \| LDA \| LFDA \| JLFDA ---\|---\|---\|---\|---\|--- MUSK1 \| 476/166 \| 77.9 \| 70.4 \| 81.4 \| 87.1 MUSK2 \| 6598/166 \| 84.3 \| 51.8 \| 76.4 \| 81.3 Elephant \| 1391/230 \| 81.4 \| 70.5 \| 74.5 \| 79.0 Fox \| 1320/230 \| 57.8 \| 53.5 \| 61.5 \| 59.5 Tiger \| 1220/230 \| 84.0 \| 71.5 \| 74.0 \| 80.5 Table 1: Accuracy results for various methods on MUSK and Corel data sets. Our approach outperform LDA on both datasets, and we get better result than MI-SVM on MUSK1 and Fox data set. ### 4.2 Semantic keyframe extraction We conduct experiments on the challenging TRECVID MED11 dataset333http://www.nist.gov/itl/iad/mig/med11.cfm. It contains five events: attempting a board trick feeding an animal, landing a fish, wedding ceremony and working on a woodworking project. All of five events consist of a number of human actions, processes, and activities interacting with other people and/or objects under different place and time. At this moment, we take 105 videos from 5 events for testing and the remaining 710 videos for training. For parameters, we set $K=10$ and $N=10$. We learned the representative clusters for each class, and then use them to find semantic frames in videos with the same labels. Then we evaluation the semantic frames for each video through human-factors analysis—the semantic keyframe extraction problem demands a human-in-the-loop for evaluation. We explain our human factors experiment in full detail in experiment setup. Our ultimate findings demonstrate that our proposed latent FDA with prior model is most capable of extraction semantically meaningful keyframes among latent FDA and competitive baselines. Video representation. For all videos, we extract HOG3D descriptors [16] every 25 frames (about sampling a frame per second). To represent videos using local features we apply a bag-of-words model, using all detected points and a codebook with 1000 elements. Benchmark methods. We make use of SVM as the benchmark method in the experiment. We take the one-vs-all strategy to train a linear MI-SVM classifier using $SVM^{light}$ [15], which is very fast in linear time, for each kind of event. Then we choose 10 frames for each video which are far from the margin and close to the margin on positive side. For the frames chosen farthest away from the margin, we refer it SVM(1), while for frames closest to the margin we refer it SVM(2). We also randomly select 10 frames from each video, and we refer it RAND in our experiments. Experiment setup Ten highly motivated graduate students (range from 22 to 30 years’ old) served as subjects in all of the following human-in-the-loop experiments. Each novel subject to the annotation-task paradigm underwent a training process. Two of the authors gave a detailed description about the dataset and problem, including its background, definition and its purpose. In order to indicate what representative and discriminative means for each event, the two authors showed videos for each kind of event to the subjects, and make sure all subjects understand what semantic keyframes are. The training procedure was terminated after the subject’s performance had stabilized. We take a pairwise ranking strategy for our evaluation. We extract 10 frames per video for 5 different methods (SVM(1), SVM(2), LFDA, JLFDA and RAND) respectively. For each video, we had about 1000 image pairs for comparison. We had developed an interface using Matlab to display two image pair and three options (Yes, No and Equal) to compare an image pair each time. The students are taught how to use the software; a trial requires them to give a ranking: If the left is better than the right, then choose ’Yes’; if the right is better than the left, choose ’No’. If the two image pair are same, then choose ’Equal’. The subjects are again informed that better means a better semantic keyframe. The ten subjects each installed the software to their computers, and conducted the image pair comparison independently. In order to speed up the annotation process, the interface can randomly sample 200 pairs from the total 1000 image pairs for each video, and we also ask subjects to random choose 10 videos from the test dataset. Experimental Results We have scores for each image pair. By sampling 10 videos from each event, we at last had annotations of 104 videos. It means our sampling videos got from 10 subjects almost cover all test data (105 videos). Table (2) shows the win-loss matrix between five methods by counting the pairwise comparison results on all 5 events. It shows that JLFDA and LFDA always beat the three baseline methods. Furthermore, JLFDA is better than LFDA because it considers a prior distribution from training data, which will help JLFDA to find more representative frames. See Fig. (3) for keyframes extracted with JLFDA. Figure 2: Comparison of 5 methods for five events. Higher value, better performance. Method \| Win-Loss matrix ---\|--- JLFDA \| FLDA \| RAND \| SVM(1) \| SVM(2) JLFDA \| - \| 3413 \| 2274 \| 2257 \| 3758 LFDA \| 2957 \| - \| 2309 \| 2230 \| 3554 RAND \| 2111 \| 2175 \| - \| 1861 \| 2274 SVM(1) \| 2088 \| 2270 \| 2010 \| - \| 2314 SVM(2) \| 3232 \| 3316 \| 2113 \| 2125 \| - Table 2: Win-Loss matrix for five methods. It represents how many times methods in each row win methods in column. We compared the five methods on the basis of Condorcet voting method. We treat ’Yes’, ’No’ and ’Equal’ are voters for each method in the image pairwise comparison process. If ’Yes’, we cast one ballot to the left method; else if ’No’, we add a ballot to the right method; else do nothing to the two methods. Fig. (2) shows ballots for each method on each event. It demonstrates our method JLFDA always beat other methods, except for E004 dataset. We also compared the five methods based on Elo rating system. For each video, we ranked the five methods according to Elo ranking system. Then, we counted the number of No.1 methods in each event. The results in Table (3) show that our method is better than others, except E004. Such results based on Elo ranking is consistent with Condorcet ranking method in Fig. (2). E004 is the wedding ceremony event and our method is consistently outperformed by the SVM baseline method. We believe this is due to the distinct nature of the E004 videos in which the video scene context itself distinguishes it from the other four events (the wedding ceremonies typically have very many people and are inside). Hence the discriminative component of the methods are taking over, and the SVM is able to outperform the Fisher discriminant. This effect seems more likely due to the nature of the five events in the data set than the proposed method intrinsically. Method \| the number of No.1 method in each event ---\|--- E001 \| E002 \| E003 \| E004 \| E005 JLFDA \| 6 \| 7 \| 7 \| 3 \| 7 LFDA \| 6 \| 4 \| 4 \| 5 \| 1 SVM(1) \| 4 \| 4 \| 4 \| 2 \| 4 SVM(2) \| 6 \| 3 \| 1 \| 7 \| 6 RAND \| 2 \| 2 \| 4 \| 3 \| 2 Table 3: For each video, we ranked the five methods according to Elo ranking system. Then, we counted the number of No.1 method one video level in each event. For example, E002 has total 20 videos, and JLFDA has rank first on 7 videos, while RAND has rank first on only two videos. Higher value, better results. It demonstrates that our method is more capable at extracting semantically meaningful keyframes. --- Figure 3: Sample keyframes from the first five events (each row from top to down): (a) snowboard trick, (b) feeding animal, (c) fishing, (d) marriage ceremony, and (e) wood making. Each row indicates sample results from the same videos for each event. It shows that our method can extract the representative and discriminative images for each kind of events. In other words, we can decide what’s happen when we scan the images. ## 5 Conclusion In this paper, we have presented a latent Fisher discriminant analysis model, which combines the latent variable inference and dimension reduction in an unified framework. Ongoing work will extend the kernel trick into the model. We test our method on classification and semantic keyframe extraction problem, and yield quite competitive results. To the best of our knowledge, this is the first paper to study the extraction of semantically representative and discriminative keyframes—most keyframe extraction and video summarization focus on representation summaries rather than jointly representative and discriminative ones. We have conducted a thorough ranking-based human factors experiment for the semantic keyframe extraction on the challenging TRECVID MED11 data set and found that our proposed methods are able to consistently outperform competitive baselines. ## References * [1] S. Andrews, I. Tsochantaridis, and T. Hofmann. Support vector machines for multiple-instance learning. In NIPS, pages 561–568, 2002. * [2] G. Baudat and F. Anouar. Generalized discriminant analysis using a kernel approach. Neural Computation, 12:2385–2404, 2000. * [3] P. N. Belhumeur, J. P. Hepanha, and D. J. Kriegman. Eigenfaces vs. fisherfaces: recognition using class specific linear projection. IEEE TPAMI, 19:711–720, 1997. * [4] A. Blum and T. Mitchell. Combining labeled and unlabeled data with co-training. In Proceedings of the Workshop on Computational Learning Theory, pages 92–100, 1998. * [5] D. Cai, X. He, and J. Han. Semi-supervised discriminant analysis. In ICCV. IEEE, 2007. * [6] L. Chen, H. Liao, M. Ko, J. 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Smeaton, W. Kraaij, and G. Quenot. Trecvid 2010 – an overview of the goals, tasks, data, evaluation mechanisms and metrics. In Proceedings of TRECVID 2010. NIST, USA, 2011. * [20] M. Sugiyama, T. Idé, S. Nakajima, and J. Sese. Semi-supervised local fisher discriminant analysis for dimensionality reduction. In Machine Learning, volume 78, pages 35–61, 2010. * [21] V. N. Vapnik. Statistical learning theory. John Wiley & Sons, 1998. * [22] X. Wang and X. Tang. Dual-space linear discriminant analysis for face recognition. In CVPR, pages 564–569, 2004. * [23] C. F. J. Wu. On the convergence properties of the EM algorithm. The Annals of Statistics, 11(1):95–103, 1983. * [24] J. Ye. Least squares linear discriminant analysis. In ICML, 2007. * [25] J. Ye and Q. Li. A two-stage linear discriminant analysis via qr-decomposition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27:929–941, 2005. * [26] Y. Zhang and D.-Y. Yeung. Semi-supervised discriminant analysis via cccp. In ECML. IEEE, 2008.	arxiv-papers	2013-09-21T03:42:04	2024-09-04T02:49:51.249837	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Gang Chen", "submitter": "Gang Chen", "url": "https://arxiv.org/abs/1309.5427" }
1309.5459	# Advances in computer architecture Irfan Uddin Irfan Uddin University of Amsterdam, The Netherlands [email protected] ###### Abstract In the past, efforts were taken to improve the performance of a processor via frequency scaling. However, industry has reached the limits of increasing the frequency and therefore concurrent execution of instructions on multiple cores seems the only possible option. It is not enough to provide concurrent execution by the hardware, software also have to introduce concurrency in order to exploit the parallelism. ###### Contents 1. 1 Introduction 2. 2 Performance via frequency scaling 3. 3 Performance via automatic exploitation of ILP 4. 4 Performance via concurrency in hardware 5. 5 Performance via explicit concurrency in software 6. 6 Modern multi-core and many-core systems 7. 7 Conclusion ## 1 Introduction In 1965 Intel’s co-founder Gorden Moore presented the now famous _Moore’s law_ stating that: _The number of transistors on a chip will roughly double every 18 months_. Later in 2000 the 18 months were changed to two years but the law is still followed by electrical engineers. To quote Prof. Yale N. Patt: _Moore’s law is not the law of physics, it is merely the psychology of electrical engineers what they can do with silicon. The law is alive only because it is used by industries to compete with each other_ [1]. Computer architects have consensus that in order to increase the performance of the processor will likely require more transistors on the chip, to support ever increasing software needs. While Moore’s law seems to enable the increase in the number of transistors, it is not yet clear how these transistors can be used to achieve the desired improvements. In this paper we summarize this argument and promotes multi-core design as a promising approach. We present some technological competition to the Microgrid. The rest of the paper is organized as follows. In Section 2 we describe the performance improvement in architecture by frequency scaling. We explain the automatic exploitation of ILP to increase performance in Section 3 and the use of concurrency in hardware to improve performance in Section 4. We show the explicit concurrency in software to achieve performance in Section 5. We give the details of modern many-core systems in Section 6 and conclude the paper in Section 7. ## 2 Performance via frequency scaling Since the advent of microprocessors, computer architects have always been in the quest to achieve the best performance by increasing the frequency of a single chip. To main focus is to achieve the lowest possible latency in the execution of individual instructions and highest possible throughput in the execution of the program. In order to achieve this improvement in performance via frequency scaling, there are two possibilities; reduce the line width of the technology and hence increase the switching speed of the transistors or reduce the amount of work done in a cycle by pipelining instruction execution. ### 2.1 Power consumption In order to achieve a higher frequency in a given technology a higher voltage is required which means more power is consumed. However, power consumption is at odds with ”Green computing” [23]. Also as the power consumption increases, all kinds of problems come up: atom migration in gates, change of electronic properties, wear and tear due to thermal expansion of the material itself, etc. Furthermore, devices are getting portable, and high voltage means constant access to power which is simply not practical in a portable environment. ### 2.2 Dissipation of heat The higher power consumed by a transistor is dissipated from the chip in the form of heat which must be removed from the chip to avoid damaging it, as silicon simply can not take heat. Increasing the frequency of the transistor will increase the quantity of heat, and decreasing the size of the transistor (decreases the energy required to switch a transistor) will generally increase the density of heat. It is expensive to provide cooling to a heated processor and requires extra space to install fans. Portable devices with this much heat can burn the body of the user. There is also the fact that batteries store limited energy and hence we want to minimise the power consumption in order to maximise lifetime. ### 2.3 Delay in wires The frequency of transistors can easily be improved, but wires are slow and wire delay becomes slower for a given amount of power dissipation. A wire is passive and can be made fast but then it will require more energy. It is required to look at the relative speed when driving a wire by a minimum sized transistor. Therefore the transaction from one part of the chip to another part is limited by wires i.e. the frequency of transistors is not limited by the number of transistors that can be manufactured on a chip, but actually limited by the number of gates reachable in a cycle [25, 2]. Increasing the frequency of the transistor means that there is a higher communication delay than the computation delay. In the past years the frequency of the processor has increased at the rate of 75% per year (not any more) and the frequency of the DRAM has increased at the rate of just 7% per year [26]. Because of this difference, the DRAM latency has become increasingly large compared to the processor’s cycle time which is very high from processor’s point of view. This divergence in latency is also known as ”Memory wall” [48]. To avoid the delay for DRAM access, concurrency is the next logical step i.e. at the time the processor sends request to DRAM and then has to wait for the request to complete, the processor should be able to perform some other activities concurrent to the DRAM access. ## 3 Performance via automatic exploitation of ILP An instruction stream has a combination of memory and computational operations. When the memory operations are issued, there can be some independent computational instructions that can be executed. But additional hardware is required to find these independent instructions and schedule them so that no data hazard occurs. This technique is called automatic exploitation of Instruction Level Parallelism (ILP) [47]; and a number of approaches are tried in this direction such as pipelining, out-of-order execution, branch prediction and cache prefetching. ### 3.1 Pipelining The function of each instruction (i.e. fetch, decode, execute etc.) can be broken down in multiple sub-functions. Like a conveyor belt in a factory line, these sub-functions can be pipelined in parallel where multiple instructions coming one after another. Pipelining have enabled the simultaneous execution of multiple instructions. The actual execution time of the individual instruction remain the same, but the throughput of several instructions is improved typically by $x$ times where $x$ is the length of the pipeline. e.g. in Pentium machines the throughput of instructions was increased up to 20 times. In addition, pipelining also enabled the processor to have a higher clock frequency because the operations executed at each pipeline stage is simpler and therefore takes a shorter cycle time. ### 3.2 Out-of-order execution The instruction stream has a limited number of independent instructions that can be executed in parallel. Often, an instruction is dependent on a previous operation (not only loads but also long latency operations like mul, floating point operations etc.) and must therefore stall the pipeline until the load operation completes. Out-of-order execution is used to allow instructions in the pipeline to overtake those issued earlier in order to avoid stalling the pipeline completely. However, hardware logic and energy is required for the dependency analysis to determine if instructions can overtake each other. Also, since instructions are executing out-of-order, a reorder buffer is required for the in-order completion to provide the expected results [46]. Out-of-order execution introduces additional dependencies e.g. Write-After- Read and Write-After-Write which are resolved by register renaming i.e. allowing the hardware to use more registers than will fit in the instruction format. However it increases the size of the register file, lengthens the pipeline and therefore increases the branch penalties [18]. ### 3.3 Branch prediction The number of instructions between branches is typically very small (average is less than 6 instructions [31]). Branch prediction is required to keep the pipeline full; while the branch and instructions before it are executing, we can fetch and decode instructions after the branch. If the branch was predicted accurately we can already have the next instructions in the pipeline, however if the branch was not predicted correctly it will result in a pipeline bubble or many pipeline bubbles in more complex processors. In addition, the effect of instructions must also be cancelled, which might involve the roll back of side effects. It means that a huge number of cycles and energy are lost on computing something that did not really participate in the required computation of the program. The issue with branch prediction is, if there are multiple tasks/processes the branch predictor has to predict branches not in only one but multiple programs interleaved over time. To keep the high accuracy across heterogeneous codes, the size of the branch predictor must grow. Because of multitasking branch prediction is an expensive approach. ### 3.4 Cache prefetching To avoid the delay of a cache miss, a processor can look at the memory access pattern of the program and guess what cache line will be accessed next and prefetch the line. If the guess is correct, the large delay required to access the memory is avoided, however if it is not correct a large amount of energy and memory bandwidth is wasted to fetch the data from the off-chip memory that will not be used. Intel Itanium, Pentium 4 and Transmeta Crusoe are some examples where cache prefetching is used. ### 3.5 Discussion Traditional and super-scalar machines [20] implement most of the techniques described above in order to achieve the best performance by the automatic exploitation of ILP. However, because of the additional hardware the microprocessors are getting complex, energy inefficient and not scalable [8]. Next to super-scalar machines, VLIW [11] machines have been introduced which in general are more scalable than super-scalar machines but have a complicated design and a complex code generation process. In addition the binary code for a VLIW machine can not be used on the same VLIW machine with a different issue width i.e. binary compatibility is not achieved [18]. Computer architects have tried hard to improve performance of the programs through implicit ILP without changing anything in the software. Because of implicit ILP, the programmers were not required to have a detailed knowledge of the hardware. The programmer used to wait to buy until a new more powerful machine is available in the market and then the software magically got faster. However, the industry is reaching the limits of increasing the frequency of processors to execute more instructions in one cycle [3]. It seems like the free lunch is over [33], programmers have to take responsibility of parallelization in order to get more performance and therefore they need to get familiar with the architecture and concurrent execution model. Some automatic parallelization techniques are taking the approach of abstracting away the architecture details [13], but can not exploit the architectural resources fully as programmers can do manually with the knowledge of the architecture. The introduction of concurrency in software engineering, increases the complexity of the software, decreases the productivity of the engineer, but increases the performance of the software. ## 4 Performance via concurrency in hardware The increased number of transistors on a chip has enabled computer architects to provide concurrent execution of independent instructions on multiple execution units. However, the implicit concurrency that can be extracted automatically in hardware is limited [3] i.e. parallelism is not possible unless concurrency is introduced explicitly in programs. ## 5 Performance via explicit concurrency in software Instead of using a single instruction stream and trying to improve the performance of the stream by implicit parallelism, we can have multiple instruction streams which can execute concurrently. These streams are called ”threads” and provide an independent flow of control within a process. A thread is also called a ”light-weight” process, as it has its own stack, local variables and program counter. Context switching in threads is much cheaper than in processes. Industry has realized that writing parallel programs to exploit concurrency of the hardware can not be avoided in future. Introducing concurrency in programs is difficult, also debugging these programs require a lot of effort from programmers. The order of execution is known but the order of execution of threads is not known which makes the behavior of multiple threads in a parallel program difficult to predict. The synchronization between threads to ensure that shared state is accessed atomically can make the concurrency difficult to define and can significantly affect the performance of the program in multi-core systems. It is true that threads in some form were quite commonly used in networking from 1980s e.g. multiple connections to HTTP servers. In networking, threads represent independent transaction and in applications, threads interact with each other in producing results. Because of the inherent difficulty, parallel programming never became a mainstream programming paradigm [34] in software engineering community in the past. Despite being difficult, parallel programing is the most desirable technology in the current state-of-the-art multi-cores processors. The performance of an application can be improved only from parallelization on contemporary hardware. A number of programming libraries have been developed to provide concurrency constructs to expose concurrency in programs. POSIX threads [12], OpenMP [27], SystemC [14] etc. are some of the libraries that provide constructs for creation, termination, synchronization and communication between threads. While parallelization is desirable, the management of threads in software by the contemporary hardware is expensive. Typically 10-100 thousand cycles are consumed in creation or synchronization of threads. There is also a cost of context switching. Therefore fine-grained parallelism can not be achieved by explicit concurrency in software. The next logical step is to introduce concurrency at multiple levels; applications, operating systems and hardware. We need threads in software, we need threads in hardware and we need the management of threads in hardware. The concurrency at all levels will exploit the maximum possible parallelism. ## 6 Modern multi-core and many-core systems Since parallelization is the only practical solution in current technology and the number of transistors on a single chip is growing [32], therefore we claim that in the future there will be large number of cores on a single chip where programmers have to put effort to write parallel programs with the knowledge of the underlying architecture. In this section we describe some state-of-the- art multi- and many- core systems. Some of these cores are available commercially while others are mainly in the research domain and are comparable to the Microgrid. There may exist some other many-core systems but these are not discussed in this section. ### 6.1 Nvidia’s GPGPU Nvidia’s GPGPU (General Purpose Graphical Processing Unit) [6] makes use of a GPU as a co-processor to accelerate the execution in CPUs for general-purpose computing. The acceleration happens by offloading some of the computationally intensive and time consuming portions of the program from CPU to GPU. The rest of the application still runs on the CPU. Therefore it is also known as ”heterogeneous” or ”hybrid” computing. A CPU generally consists of two, four or eight cores, while the GPU consists of hundreds/thousands of smaller cores. All these cores work together to crunch through the data in the application. From the user’s point of view, the application runs faster because it is using the massively parallel processing power of the GPU to boost performance. CUDA (Computer Unified Device Architecture) is a parallel programming model for GPU. It enables dramatic increase in computing performance by harnessing the power of GPUs. #### Discussion GPU is actually a specialized accelerator that is connected to traditional single- or multi-cores processor. Therefore a large amount of work is required for the transaction of data between processor and GPUs. Programmers have to divide the problem into coarse sub-problems that can be solved independently. In addition the programmers have to explicitly manage the memory and concurrency i.e. a complex model of the architecture is forced in the mind of the programmer. GPU architecture is based on the SIMD model and therefore can efficiently execute SIMD based applications. The SIMD architectures are very inefficient in branching, as each branch path must be executed sequentially. GPUs can achieve a very high performance in embarrassingly parallel applications. However applications with dense communication between threads (e.g. FFTs, compression algorithms, sorting etc.) can not achieve a very high performance compared to other multi-cores processors [21]. GPUs become really slow in functional calls, e.g. FFT is an embarrassingly parallel application, but it does not scale very well on GPUs because of function calls from the outer loop. ### 6.2 Sun/Oracle’s UltraSPARC Tx Sun Microsystems have introduced a RISC architecture named SPARC (Scalable Processor ARChitecture) in 1987. Oracle then bought Sun Microsystems and they together introduced UltraSPARC T1 microprocessor (code name Niagara) in 2005. This continued as a series of UltraSPARC T2, UltraSPARC T3 and in 2011 UltraSPARC T4. The UltraSPARC T4 [7] has a 16-stage integer pipeline, 11-stage floating point pipeline and 2 issue width. It has a thread priority mechanism where one thread can get preferential access to a core’s hardware to give increased performance. It has 4 processors on a single die and each processor consists of 8 cores therefore a total of 32 cores are available on a single chip. Each core has 8 hardware threads i.e. 64 hardware threads on a processor and 256 hardware threads on the chip. Each core can switch between eight threads using a modified LRU (Least Recently Used) algorithm for thread selection. Each core is associated with 16KB of L1 I-cache and D-cache and 128KB of L2-cache. Eight cores share 4MB L3-cache and the DDR is 1TB. Total transistors count is approximately 855 millions. The frequency of every core can be changed in the range of 2.85 and 3.0 GHz. The technology used is 40nm CMOS and the total die size is $403mm^{2}$. The UltraSPARC T4 processor has increased single-thread performance, while maintaining the high multi-thread throughput performance, therefore single- threaded applications can have an efficient execution. It automatically switches to single-thread mode when only a single thread is active, dedicating all resources to that thread’s execution. While software can activate up to eight threads on each core at a time, hardware dynamically and seamlessly allocates core resources such as instruction, data, L2-caches and TLBs, as well as out-of-order execution resources such as the 128-entry re-order buffer in the core. The cores provide sophisticated branch prediction and have the features for prefetching instructions and data. #### Discussion The UltraSPARC is addressing internet servers or any large scale systems. It generally addresses coarse-grained or embarrassingly parallel applications and therefore disregard desktop computing and fine-grained parallelism. A library is used to map software threads to hardware threads and it has an overhead of creation and synchronization [43]. It is based on SMP model and has an increased single thread performance. But it suffers from the scalability of the bandwidth and power consumption of the interconnection between processor and memory. The UltraSPARC T4 cores are complex (16 stage pipeline) and have out-of-order execution, branch prediction and cache prefetching, which are energy in- efficient features. It has some inefficiency coming from the use of a huge shared L2-cache which is necessary for server application (large number of synchronizations around data) but has a cost in silicon i.e. heat, energy and wiring complexity. ### 6.3 Tilera’s TILE64 Tilera has introduced TILE64 [29], based on MIMD model. It has 64 cores on a chip using 90nm CMOS. These cores are fully functional, programmable and each is capable of running its own operating system. A group of cores can also be used to run as a symmetrical multi-processing operating system. Every core has a frequency range of 600 to 900 MHz. The cores are in-order, three-way VLIW issue width and implement a MIPS-derived VLIW instruction set. The pipeline has many (more than 6) stages. Each core has 32 general-purpose registers and three functional units: two integer arithmetic logic units and a load-store unit. Every core has L1-cache and L2-cache as well as a distributed L3-cache. Tilera’s architecture eliminates the on-chip bus interconnection by placing a communications switch on each core and arranging them in a grid fashion on the chip to create an efficient two-dimensional traffic system for packets. This technology is named as intelligent mesh (iMesh). iMesh is similar to mesh network used in Intel’s SCC or NoC in embedded systems, with the innovation that the flow of the messages in the mesh network can dynamically be adapted based on the load of the network. Tilera’s Multicore Development Environment (MDE) provides programming framework that allows developers to scale their applications to large-scale multicore systems. It has enable the standard tools such as gcc, gdb and oprofile, to be multi-core aware so that the developer can easily debug, profile and optimize code running across dozens of cores. #### Discussion Tilera’s architecture does not include hardware support for floating point operation and therefore is not suitable for scientific computing. It mainly targets embedded applications e.g. video encoding/decoding and network packet processing. It is programmed in such a way that requires using registers for communication by the programmer. Which means there is more responsibility on the part of the programmer or compiler in creating threads. In a way programmers need to understand the architecture in detail in order to program it. Also the mapping of threads in the program to the hardware requires a software library and therefore the cost of creation, synchronization and mapping of software threads to hardware can not be avoided. ### 6.4 Intel’s SCC Intel’s SCC (Single-chip Cloud Computer) [15] is an experimental many-cores research platform designed to address hardware and software challenges by industry and academic institutions in the tera-scale project [28]. It consists of 48 Pentium 1 cores connected in a mesh network and on-chip message passing network for inter-thread communication. The cores are relatively simple but fully functional general-purpose cores. There is no hardware cache coherency protocol, which allowed Intel to place 48 cores on a chip using CMOS 45nm technology. It does not come as a stand-alone computer and a management PC (MCPC) is used to run the applications on the chip. Intel’s SCC has fine-grained power management where software applications are given control to turn cores on and off or to change their performance levels, continuously adapting to use the minimum energy needed at a given moment. It can run all 48 cores at one time over a range of 25W to 125W and selectively vary the voltage and frequency of the mesh network as well as a group of cores. Each tile (2 cores) can have its own frequency, and groupings of four tiles (8 cores) can each run at their own voltage. Every core uses the mainstream x86 (CISC) instruction set. The Linux operating system is available for the chip, as well as gcc and Fortran compilers. A small library RCC is used for the communication between cores. #### Discussion Intel’s SCC is a prototype which is designed for studying the parallel programming paradigm in general-purpose computers. Therefore it is not really a commercial product to be used for mainstream computing. It mainly addresses coarse-grained parallelism, and may not achieve a high performance improvement in fine-grained applications. The absence of hardware cache coherency protocol places more responsibility on the programmer and hence requires more effort from the programmers to manage the coherency of the caches. The Pentium 1 core is actually single-threaded machine and therefore can not achieve latency tolerance in long latency operations. ### 6.5 Microgrid The Microgrid [19, 4, 16] is a general-purpose, many-core architecture developed at the University of Amsterdam which implements hardware multi- threading using data flow scheduling and a concurrency management protocol in hardware to create and synchronize threads within and across the cores on chip. The suggested concurrent programming model for this chip is based on fork-join constructs, where each created thread can define further concurrency hierarchically. This model is called the microthreading model and is also applicable to current multi-core architectures using a library of the concurrency constructs called _svp-ptl_ [45] built on top of pThreads. In our work, we focus on a specific implementation of the microthreaded architecture where each core contains a single issue, in-order RISC pipeline with an ISA similar to DEC/Alpha, and all cores are connected to an on-chip distributed memory network [17, 5]. Each core implements the concurrency constructs in its instruction set and is able to support hundreds of threads and their contexts, called microthreads and tens of families (i.e. ordered collections of identical microthreads) simultaneously. A number of tools and simulators are added to the designer’s toolbox and used for the evaluation of the Microgrid from different perspective. The compiler for the Microgrid [22] can generate binary for different implementations of the Microgrid. We have software libraries that provide the run-time systems for the microthreading model on the shared memory SMP machines and referred as _svp-ptl_ [45] and distributed memory for clusters/grids and are referred as Hydra [24] and _dsvp-ptl_ [44] The SL compiler can generate binary for UTLEON3 [9, 10], MGSim [5, 30] and HLSim [38, 39, 40, 37, 41, 42, 35, 36]. ## 7 Conclusion The psychology of electrical engineers is that they can double the number of transistors on a single chip every second year, which has enabled computer architects to design more complex microprocessors. A number of approaches were tried to achieve improvements in the throughput of the program implicitly. But industry has reached the limits of implicit improvement in performance, and multi-core designs seem to be promising approaches to achieve performance explicitly. 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1309.5461	# Linear kernels for $k$-tuple and liar’s domination in bounded genus graphs Arijit Bishnu ACM Unit Indian Statistical Institute Kolkata, INDIA [email protected] Arijit Ghosh 111 Supported by the Indo-German Max Planck Center for Computer Science (IMPECS). D1: Algorithms & Complexity Max-Planck-Institut für Informatik Saarbrücken, Germany [email protected] Part of this work was done when the author was a visiting scientist in ACM Unit, Indian Statistical Institute, Kolkata. Subhabrata Paul ACM Unit Indian Statistical Institute Kolkata, INDIA [email protected] ###### Abstract A set $D\subseteq V$ is called a _$k$ -tuple dominating set_ of a graph $G=(V,E)$ if $\left\|N_{G}[v]\cap D\right\|\geq k$ for all $v\in V$, where $N_{G}[v]$ denotes the closed neighborhood of $v$. A set $D\subseteq V$ is called a _liar’s dominating set_ of a graph $G=(V,E)$ if * (i) $\left\|N_{G}[v]\cap D\right\|\geq 2$ for all $v\in V$, and * (ii) for every pair of distinct vertices $u,v\in V$, $\left\|(N_{G}[u]\cup N_{G}[v])\cap D\right\|\geq 3$. Given a graph $G$, the decision versions of $k$-Tuple Domination Problem and the Liar’s Domination Problem are to check whether there exists a $k$-tuple dominating set and a liar’s dominating set of $G$ of a given cardinality, respectively. These two problems are known to be NP-complete [LC03, Sla09]. In this paper, we study the parameterized complexity of these problems. We show that the $k$-Tuple Domination Problem and the Liar’s Domination Problem are W[2]-hard for general graphs but they admit linear kernels for graphs with bounded genus. ##### Keywords. $k$-tuple domination, liar’s domination, planar graphs, bounded genus graphs, kernelization, and W[2]-hard ## 1 Introduction Let $G=(V,E)$ be a graph. For a vertex $v\in V$, let $N_{G}(v)=\\{u\in V\|uv\in E\\}$ and $N_{G}[v]=N_{G}(v)\cup\\{v\\}$ denote the open and closed neighborhoods of $v$, respectively. A set $D\subseteq V$ is called a _dominating set_ of a graph $G=(V,E)$ if $\|N_{G}[v]\cap D\|\geq 1$ for all $v\in V$. The _domination number_ of a graph $G$, denoted by $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. The concept of domination has been well studied. Depending upon various applications, different variations of domination have appeared in the literature [HHS98a, HHS98b]. Among different variations of domination, $k$-tuple domination and liar’s domination are two important and well studied type of domination [HH00, LC02, LC03, RS09, Sla09]. A set $D\subseteq V$ is called a _$k$ -tuple dominating set_ of a graph $G=(V,E)$ if each vertex $v\in V$ is dominated by at least $k$ number of vertices in $D$, that is, $\|N_{G}[v]\cap D\|\geq k$ for all $v\in V$. The concept of $k$-tuple domination in graphs was introduced in [HH00]. For $k=2$ and $3$, it is called _double domination_ and _triple domination_ respectively. The _$k$ -tuple domination number_ of a graph $G$, denoted by $\gamma_{k}(G)$, is the minimum cardinality of a $k$-tuple dominating set of $G$. It is a simple observation that for the existence of a $k$-tuple dominating set, we need $\delta(G)\geq k-1$, where $\delta(G)$ is the minimum degree of $G$. On the other hand, liar’s domination is a new variation of domination and was introduced in 2009 by Slater [Sla09]. A set $D\subseteq V$ is called a _liar’s dominating set_ of a graph $G=(V,E)$ if the following two conditions are met: condition (i) $\|N_{G}[v]\cap D\|\geq 2$ for all $v\in V$ condition (ii) for every pair of distinct vertices $u,v\in V$, $\|(N_{G}[u]\cup N_{G}[v])\cap D\|\geq 3$ In a network guarding scenario, if sentinels are placed in the vertices of the dominating set, then the graph (network) is guarded. Consider the situation where a single sentinel is unreliable or lies and we do not know the exact sentinel that lies. We then need a liar’s dominating set to guard the network. The _liar’s domination number_ of a graph $G$, denoted by $\gamma_{LR}(G)$, is the minimum cardinality of a liar’s dominating set of $G$. Formally, the decision versions of $k$-Tuple Domination Problem and Liar’s Domination Problem are defined as follows. $k$-Tuple Domination Problem _Instance:_ A graph $G=(V,E)$ and a nonnegative integer $p$. _Question:_ Does there exist a $k$-tuple dominating set of cardinality at most $p$? Liar’s Domination Problem _Instance:_ A graph $G=(V,E)$ and a nonnegative integer $p$. _Question:_ Does there exist a liar’s dominating set of cardinality at most $p$? Note that, every liar’s dominating set is a double dominating set and every triple dominating set is a liar’s dominating set. Hence, liar’s domination number lies between double and triple domination number, that is, $\gamma_{2}(G)\leq\gamma_{LR}(G)\leq\gamma_{3}(G)$. The rest of the paper is organized as follows. Section $2$ introduces some pertinent definitions and preliminary results that are used in the rest of the paper and a brief review on the progress in the study of parametrization for domination problems. Section $3$ deals with the hardness results of both $k$-tuple domination problem and liar’s domination problem. In Section $4$, we show that both $k$-Tuple Domination Problem and Liar’s Domination Problem admit linear kernel in planar graphs. In Section $5$, we extend the results for bounded genus graphs. Finally, Section $6$ concludes the paper. ## 2 Preliminaries Let $G=(V,E)$ be a graph. Let $G[S]$, $S\subseteq V$ denote the induced subgraph of $G$ on the vertex set $S$. The _distance_ between two vertices $u$ and $v$ in a graph $G$ is the number of edges in a shortest path connecting them and is denoted as $d_{G}(u,v)$. The degree of a vertex $v\in V(G)$, denoted by $deg_{G}(v)$, is the number of neighbors of $v$. ### 2.1 Graphs on surfaces In this subsection, we recall some basic facts about graphs on surfaces following the discussion in [FT04]. The readers are referred to [MT01] for more details. A _surface_ $\Sigma$ is a compact $2$-manifold without boundary. Let $\Sigma_{0}$ denote the sphere $\\{(x,y,z)\|~{}x^{2}+y^{2}+z^{2}=1\\}$. A _line_ and _O-arc_ are subsets of $\Sigma$ that are homeomorphic to $[0,1]$ and a circle respectively. A subset of $\Sigma$ meeting the drawing only in vertices of $G$ is called _$G$ -normal_. If an O-arc is G-normal, then it is called a _noose_. The length of a noose is the number of its vertices. The _representativity_ of $G$ embedded in $\Sigma\neq\Sigma_{0}$ is the smallest length of a non-contractible noose in $\Sigma$ and it is denoted by _rep $(G)$_. The classification theorem for surfaces states that, any surface $\Sigma$ is homeomorphic to either a surface $\Sigma^{h}$ which is obtained from a sphere by adding $h$ handles (orientable surface), or a surface $\Sigma^{k}$ which is obtained from a sphere by adding $k$ crosscaps (non-orientable surface) [MT01]. The _Euler genus_ of a non-orientable surface $\Sigma$, denoted by _eg $(\Sigma)$_, is the number of crosscaps $k$ such that $\Sigma\cong\Sigma^{k}$ and for an orientable surface, _eg $(\Sigma)$_ is twice the number of handles $h$ such that $\Sigma\cong\Sigma^{h}$. Given a graph $G$, Euler genus of $G$, denoted by eg$(G)$, is the minimum eg$(\Sigma)$, where $\Sigma$ is a surface in which $G$ can be embedded. The _Euler characteristic_ of a surface $\Sigma$ is defined as $\chi(\Sigma)=2-\textup{eg}(\Sigma)$. For a graph $G$, $\chi(G)$ denotes the largest number $t$ for which $G$ can be embedded on a surface $\Sigma$ with $\chi(\Sigma)=t$. Let $G=(V,E)$ be a $2$-cell embedded graph in $\Sigma$, that is, all the faces of $G$ is homeomorphic to an open disk. If $F$ is the set of all faces, then _Euler’s formula_ tells that $V-E+F=\chi(\Sigma)=2-\textup{eg}(\Sigma)$. Next we define a process called _cutting along a noose $N$_. Although the formal defi is given in [MT01], we follow a more intuitive defi given in [FT04]. Let $N$ be a noose in a $\Sigma$-embedded graph $G=(V,E)$. Suppose for any $v\in N\cap V$, there exists an open disk $\Delta$ such that $\Delta$ contains $v$ and for every edge $e$ adjacent to $v$, $e\cap\Delta$ is connected. We also assume that $\Delta-N$ has two connected components $\Delta_{1}$ and $\Delta_{2}$. Thus we can define partition of $N_{G}(v)=N_{G}^{1}(v)\cup N_{G}^{2}(v)$, where $N_{G}^{1}(v)=\\{u\in N_{G}(v)\|uv\cap\Delta_{1}\neq\emptyset\\}$ and $N_{G}^{2}(v)=\\{u\in N_{G}(v)\|uv\cap\Delta_{2}\neq\emptyset\\}$. Now for each $v\in N\cap V$ we do the following: 1. 1. remove $v$ and its incident edges 2. 2. introduce two new vertices $v^{1},v^{2}$ and 3. 3. connect $v^{i}$ with the vertices in $N_{G}^{i}$, $i=1,2$. The resulting graph $\mathcal{G}$ is obtained from $\Sigma$-embedded graph $G$ by cutting along $N$. The following lemma is very useful in the proofs by induction on the genus. ###### Lemma 1 [FT04] Let $G$ be a $\Sigma$-embedded graph and let $\mathcal{G}$ be a graph obtained from $G$ by cutting along a non-contractible noose $N$. Then one of the following holds * • $\mathcal{G}$ is the disjoint union of graphs $G_{1}$ and $G_{2}$ that can be embedded in surfaces $\Sigma_{1}$ and $\Sigma_{2}$ such that $eg(\Sigma)=eg(\Sigma_{1})+eg(\Sigma_{2})$ and $eg(\Sigma_{i})>0,i=1,2$. * • $\mathcal{G}$ can be embedded in a surface with Euler genus strictly smaller than eg$(\Sigma)$. A _planar graph_ $G=(V,E)$ is a graph that can be embedded in the plane. We term such an embedding as a _plane graph_. ### 2.2 Parameterization and domination A _parameterized problem_ is a language $L\subseteq\Sigma^{}\times\mathds{N}$, where $\Sigma^{}$ denotes the set of all finite strings over a finite alphabet $\Sigma$. A parameterized problem $L$ is _fixed-parameter tractable_ if the question “$(x,p)\in L$” can be decided in time $f(p)\cdot\|x\|^{O(1)}$, where $f$ is a computable function on nonnegative integers, $x$ is the instance of the problem and $p$ is the parameter. The corresponding complexity class is called FPT. Next we define a reducibility concept between two parameterized problems. ###### Definition 2 [DF99, Nie06] Let $L,L^{\prime}\subseteq\Sigma^{}\times\mathds{N}$ be two parameterized problems. We say that $L$ reduces to $L^{\prime}$ by a _standard parameterized m-reduction_ if there are functions $p\mapsto p^{\prime}$ and $p\mapsto p^{\prime\prime}$ from $\mathds{N}$ to $\mathds{N}$ and a function $(x,p)\mapsto x^{\prime}$ from $\Sigma^{}\times\mathds{N}$ to $\Sigma^{}$ such that 1. 1. $(x,p)\mapsto x^{\prime}$ is computable in time $p^{\prime\prime}\|x\|^{c}$ for some constant $c$ and 2. 2. $(x,p)\in L$ if and only if $(x^{\prime},p^{\prime})\in L^{\prime}$. A parameterized problem is in the class W[i], if every instance $(x,p)$ can be transformed (in fpt-time) to a combinatorial circuit that has height at most $i$, such that $(x,p)\in L$ if and only if there is a satisfying assignment to the inputs, which assigns $1$ to at most $p$ inputs. A problem $L$ is said to be W[i]_-hard_ if there exists a standard parameterized m-reduction from all the problems in W[i] to $L$ and in addition, if the problem is in W[i], then it is called W[i]_-complete_. Next we define the reduction to problem kernel, also simply referred to as _kernelization_. ###### Definition 3 [Nie06] Let $L$ be a parameterized problem. By reduction to problem kernel, we mean to replace instance $I$ and the parameter $p$ of $L$ by a “reduced” instance $I^{\prime}$ and by another parameter $p^{\prime}$ in polynomial time such that • $p^{\prime}\leq c\cdot p$, where $c$ is a constant, * • $I^{\prime}\leq g(p)$, where $g$ is a function that depends only on $p$, and * • $(I,p)\in L$ if and only if $(I^{\prime},p^{\prime})\in L$. The reduced instance $I^{\prime}$ is called the _problem kernel_ and the size of the problem kernel is said to be bounded by $g(p)$. In parameterized complexity, domination and its variations are well studied problems. The decision version of domination problem is W[2]-complete for general graphs [DF99]. But this problem is FPT when restricted to planar graphs [AFN04a] though it is still NP-complete for this graph class [GJ79]. Furthermore, for bounded genus graphs, which is a super class of planar graphs, domination problem remains FPT [FT04]. It was proved that dominating set problem possesses a linear kernel in planar graphs [AFN04a] and in bounded genus graphs [FT04]. Also domination problem admits polynomial kernel on graphs excluding a fixed graph $H$ as a minor [Gut09] and on $d$-degenerated graphs [PRS12]. A search tree based algorithm for domination problem on planar graphs, which runs in $O(8^{p}n)$ time, is proposed in [AFF+05]. For bounded genus graphs, similar search tree based algorithm is proposed in [EFF04] and has a time complexity of $O((4g+40)^{p}n^{2})$, where $g$ is the genus of the graph. Algorithms with running time of $O(c^{\sqrt{p}}n)$ for domination problem on planar graphs have been devised in [ABF+02, AFN04b, FT03, FT04]. Like domination problem, $k$-Tuple Domination Problem and Liar’s Domination Problem are both NP-complete [LC03, Sla09] for general graphs. However, these problems have been polynomially solved for different graph classes [LC02, LC03, PP13a, PP13b]. But for planar graphs and hence for graphs with bounded genus, $k$-Tuple Domination Problem remains NP-complete [LC08]. In [Sla09], though the NP-completeness proof is given for general graphs, it can be verified that using the same construction one can find the NP-completeness of Liar’s Domination Problem in planar graphs, see Lemma 26 in Appendix 7. Some generalization of classical domination problem have been studied in the literature from parameterized point of view. Among those problems, $k$-dominating threshold set problem, $[\sigma,\rho]$-dominating set problem (also known as generalized domination) are generalized version of $k$-tuple dominating set problem. In [GV08], it is proved that $k$-dominating threshold set problem is FPT in $d$-degenerated graphs. $[\sigma,\rho]$-domination is studied in [CP14, TP97, vRBR09]. A set $D$ of vertices of a graph $G$ is $[\sigma,\rho]$-dominating set if for any $v\in D,\|N(v)\cap D\|\in\sigma$ and for any $v\notin D,\|N(v)\cap D\|\in\rho$ for any two sets $\sigma$ and $\rho$. It is known that $[\sigma,\rho]$-domination is FPT when parameterized by treewidth [vRBR09]. By Theorem 32 of [ABF+02], it follows that $k$-tuple domination is FPT on planar graphs. But there is no explicit kernel for both $k$-tuple domination and liar’s domination problem in the literature. There have been successful efforts in developing meta-theorems like the celebrated Courcelle’s theorem [Cou92] which states that all graph properties definable in monadic second order logic can be decided in linear time on graphs of bounded tree-width. This also implies FPT algorithms for bounded tree-width graph for these problems. In case of kernelization in bounded genus graphs, Bodlaender et al. give two meta-theorems [BFL+09]. The first theorem says that all problems expressible in counting monadic second order (CMSO) logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus and the second theorem says that all problems that have a finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. It is easy to see that both $k$-tuple and liar’s domination problems can be expressed in CMSO logic. Let $G=(V,E)$ be an instance of a graph problem $\Pi$ such that $G$ is embeddable in a surface of Euler genus at most $r$. The basic idea of quasi-coverable property for $\Pi$ is that there exists a set $S\subseteq V$ satisfying the conditions of $\Pi$ such that the tree-width of $G\setminus R^{r}_{G}(S)$ is at most $r$ where $R^{r}_{G}(S)$ is a special type of reachability set from $S$. In domination type of problems, this reachability set is actually the whole graph and hence these problems satisfy the quasi-coverable property. The basic idea of strong monotonicity for a graph problem $\Pi$ is roughly as follows: Let $\mathcal{F}_{i}$ be a class of graphs $G$ having a specific set of vertices $S$ termed as the boundary of $G$ such that $\|S\|=i$. The glued graph $G=G_{1}\oplus G_{2}$ of $G_{1}$ and $G_{2}$ is the graph which is obtained by taking the disjoint union of $G_{1}$ and $G_{2}$ and joining $i$ edges between the vertices of the boundary sets. A problem $\Pi$ is said to satisfy the strong monotonicity if for every boundaried graph $G=(V,E)\in\mathcal{F}_{i}$, there exists a set $W\subseteq V$ of a specific cardinality which satisfy the property of $\Pi$ such that for every boundaried graph $G^{\prime}=(V^{\prime},E^{\prime})\in\mathcal{F}_{i}$ with a set $W^{\prime}\subseteq V^{\prime}$, satisfying the property of $\Pi$, the vertex set $W\cup W^{\prime}$ satisfies the property of $\Pi$ for the glued graph $G=G\oplus G^{\prime}$. It can be verified easily that both $k$-tuple domination and liar’s domination problems satisfy the strongly monotone property. The strongly monotone property implies the finite integer index for these problems. Hence, by the second meta-theorem in [BFL+09], both $k$-tuple and liar’s domination problems admit linear kernels for graphs on bounded genus. Though these meta-theorems provide simple criteria to decide whether a problem admits a linear or polynomial kernel, finding a linear kernel with reasonably small constants for a specific problem is a worthy topic of further research [BFL+09]. In this paper, we have obtained linear kernels with small constants for both the problems on bounded genus graphs. We have also proved the W[2]-hardness for $k$-tuple and liar’s domination for general graphs. ## 3 Hardness results in general graphs In this section, we show that $k$-tuple Domination Problem and Liar’s Domination Problem are W[2]-hard. In [CP14], it is proved that $[\sigma,\rho]$-domination problem for any recursive sets $\sigma$ and $\rho$ is W[2]-hard. This implies the hardness for $k$-tuple domination in general graphs. But in this paper, we have come up with a simple W[2]-hardness proof for $k$-tuple domination in general graphs. To prove this, we show standard parameterized m-reductions from Domination Problem, which is known to be W[2]-complete [DF99], to $k$-Tuple Domination Problem and Liar’s Domination Problem, respectively. ###### Theorem 4 $k$-Tuple Domination Problem is $\mathsf{W[2]}$-hard. ###### Proof. We show a standard parameterized m-reduction from Domination Problem to $k$-Tuple Domination Problem. Let $<G=(V,E),p>$ be an instance of Domination Problem. We construct an instance $<G^{\prime}=(V^{\prime},E^{\prime}),p^{\prime}>$ of the $k$-Tuple Domination Problem as follows: $V^{\prime}=V\cup V_{k}$ where $V_{k}=\\{u_{1},u_{2},\ldots,u_{k}\\}$ and $E^{\prime}=E\cup\\{v_{i}u_{j}\|v_{i}\in V\mbox{ and }u_{j}\in V_{k}\setminus u_{k}\\}\cup\\{u_{i}u_{j}\|u_{i},u_{j}\in V_{k},i\not=j\\}$. Also set $p^{\prime}=p+k$. The construction of $G^{\prime}$ from $G$ in case of triple domination is illustrated in Figure 1. Figure 1: Construction of $G^{\prime}$ from $G$ for triple domination ###### Claim 5 $G$ has a dominating set of size at most $p$ if and only if $G^{\prime}$ has a $k$-tuple dominating set of size at most $p^{\prime}$. ###### Proof. Let $D$ be a dominating set of $G$ of cardinality at most $p$ and $D^{\prime}=D\cup V_{k}$. Each $v_{i}\in V$ is dominated by at least one vertex from $D$ and by $k-1$ vertices from $V_{k}$. Each $u_{i}\in V_{k}$ is dominated by $k$ vertices of $V_{k}$. Thus, $D^{\prime}$ is a $k$-tuple dominating set of $G^{\prime}$ of cardinality at most $p^{\prime}$. Conversely, let $D^{\prime}$ be a $k$-tuple dominating set of $G^{\prime}$ of cardinality at most $p^{\prime}$. Note that each $k$-tuple dominating set contains the set $V_{k}$ because to dominate $u_{k}$ by $k$ vertices we must select all the vertices of $V_{k}$. Let $D=D^{\prime}\setminus V_{k}$. Clearly $D\subseteq V$ and $\|D\|\leq p$. Now for each $v\in V$, $\|N_{G}[v]\cap D\|\geq 1$ because otherwise, there exists a vertex $v\in V$ such that $\|N_{G^{\prime}}[v]\cap D^{\prime}\|=k-1$. This is a contradiction because $D^{\prime}$ is a $k$-tuple dominating set of $G^{\prime}$. Thus $D$ is a dominating set of $G$ of cardinality at most $p$. Hence, $G$ has a dominating set of size at most $p$ if and only if $G^{\prime}$ has a $k$-tuple dominating set of size at most $p^{\prime}$. ∎ Thus, $k$-Tuple Domination Problem is W[2]-hard. ∎ Next we show the W[2]-hardness of Liar’s Domination Problem. ###### Theorem 6 Liar’s Domination Problem is $\mathsf{W[2]}$-hard. ###### Proof. We show a standard parameterized m-reduction from Domination Problem to Liar’s Domination Problem. Let $<G=(V,E),p>$ be an instance of Domination Problem. We construct an instance $<G^{\prime}=(V^{\prime},E^{\prime}),p^{\prime}>$ of the Liar’s Domination Problem as follows: $V^{\prime}=V\cup\\{u,u^{\prime},v,v^{\prime},w\\}$ and $E^{\prime}=E\cup\\{v_{i}u\|v_{i}\in V\\}\cup\\{v_{i}v\|v_{i}\in V\\}\cup\\{uu^{\prime},vv^{\prime},wu,wv\\}$. Also $p^{\prime}=p+4$. The construction of $G^{\prime}$ from $G$ is illustrated in Figure 2. Figure 2: Construction of $G^{\prime}$ from $G$ ###### Claim 7 $G$ has a dominating set of size at most $p$ if and only if $G^{\prime}$ has a liar’s dominating set of size at most $p^{\prime}$. ###### Proof. Let $D$ be a dominating set of $G$ of cardinality at most $p$ and $D^{\prime}=D\cup\\{u,u^{\prime},v,v^{\prime}\\}$. It is easy to verify that for each vertex $x\in V^{\prime}$, $\|N_{G^{\prime}}[x]\cap D^{\prime}\|\geq 2$ and for every pair of vertices $x,y\in V^{\prime}$, $\|(N_{G^{\prime}}[x]\cup N_{G^{\prime}}[y])\cap D^{\prime}\|\geq 3$. Hence $D^{\prime}$ is a liar’s dominating set of $G^{\prime}$ of cardinality at most $p+4=p^{\prime}$. Conversely, let $D^{\prime}$ be a liar’s dominating set of $G^{\prime}$ of cardinality at most $p^{\prime}$. Each liar’s dominating set must contain the set $\\{u,u^{\prime},v,v^{\prime}\\}$ because to doubly dominate $u^{\prime}$ and $v^{\prime}$ we must select the vertices $\\{u,u^{\prime},v,v^{\prime}\\}$. Let $X\subseteq V$ denote the set of vertices that are dominated by exactly two vertices ($u$ and $v$) from $D^{\prime}$. We claim $\|X\|\leq 1$. If there exists two such vertices $x,y\in V$, then $\|(N_{G^{\prime}}[x]\cup N_{G^{\prime}}[y])\cap D^{\prime}\|=2$ which violates condition (ii) of liar’s domination. We now deal with two cases: $\|X\|=1$: Let $X=\\{x\\}$. Here $\|N_{G^{\prime}}[x]\cap D^{\prime}\|=2$. This implies $w\in D^{\prime}$, otherwise the pair $x$ and $w$ violates condition (ii) of liar’s domination. We set $D^{\prime\prime}=(D^{\prime}\setminus\\{w\\})\cup\\{x\\}$. $D^{\prime\prime}$ is also a liar’s dominating set of $G^{\prime}$ of cardinality at most $p^{\prime}$. Note that all vertices in $V$ is triply dominated by $D^{\prime\prime}$ and it does not contain $w$. $\|X\|=0$: In this case each vertex of $V$ is triply dominated by $D^{\prime}$. Now if $w\notin D^{\prime}$, we are done. Otherwise the set $D^{\prime}\setminus\\{w\\}$ forms a liar’s dominating set of $G^{\prime}$ of cardinality at most $p^{\prime}$ such that each vertex of $V$ is triply dominated by $D^{\prime}\setminus\\{w\\}$. Hence without loss of generality, we assume that there is a liar’s dominating set $D^{\prime}$ of $G^{\prime}$ of cardinality at most $p^{\prime}$ such that every vertex in $V$ is triply dominated by $D^{\prime}$ and $w\notin D^{\prime}$. Let $D=D^{\prime}\setminus\\{u,u^{\prime},v,v^{\prime}\\}$. Clearly $D\subseteq V$ and $\|D\|\leq p$. Now for each $x\in V$, $\|N_{G}[x]\cap D\|\geq 1$ because otherwise, there exists a vertex $x\in V$ such that for the pair $x$ and $w$, condition (ii) of liar’s domination is violated. This is a contradiction because $D^{\prime}$ is a liar’s dominating set of $G^{\prime}$. Thus $D$ is a dominating set of $G$ of cardinality at most $p$. Hence, $G$ has a dominating set of size at most $p$ if and only if $G^{\prime}$ has a liar’s dominating set of size at most $p^{\prime}$. ∎ Thus, Liar’s Domination Problem is W[2]-hard. ∎ ## 4 Linear kernels for planar graphs Having seen that $k$-tuple and liar’s domination are W[2]-hard in general graphs, we focus on planar graphs in this section and show that they are FPT. ### 4.1 Double domination In this subsection we show that Double Domination Problem in planar graphs possesses a linear kernel. Our proof technique uses the region decomposition idea of Alber et al. [AFN04a]. First we describe the reduction rules for kernelization. #### 4.1.1 Reduction rule Let $G=(V,E)$ be the instance for Double Domination Problem. Consider a pair of vertices $u,v\in V$. Let $N_{G}(u,v)=N_{G}(u)\cap N_{G}(v)$. We partition the vertices of $N_{G}(u,v)$ in to three parts as follows: $\displaystyle N^{1}_{G}(u,v)$ $\displaystyle=$ $\displaystyle\\{x\in N_{G}(u,v)\|N_{G}(x)\setminus\\{N_{G}(u,v)\cup\\{u,v\\}\\}\neq\emptyset\\};$ $\displaystyle N^{2}_{G}(u,v)$ $\displaystyle=$ $\displaystyle\\{x\in N_{G}(u,v)\setminus N^{1}_{G}(u,v)\|N_{G}(x)\cap N^{1}_{G}(u,v)\neq\emptyset\\};$ $\displaystyle N^{3}_{G}(u,v)$ $\displaystyle=$ $\displaystyle N_{G}(u,v)\setminus(N^{1}_{G}(u,v)\cup N^{2}_{G}(u,v)).$ Reduction Rule: For every pair of distinct vertices $u,v\in V$, if $N^{3}_{G}(u,v)\neq\emptyset$, then * • delete all the vertices of $N^{2}_{G}(u,v)$ and * • delete all vertices of $N^{3}_{G}(u,v)$ except one vertex. ###### Lemma 8 Let $G=(V,E)$ be a graph and $G^{\prime}=(V^{\prime},E^{\prime})$ be the resulting graph after having applied the reduction rule to $G$. Then $\gamma_{2}(G)=\gamma_{2}(G^{\prime})$. ###### Proof. Let $u,v\in V$ such that $N^{3}_{G}(u,v)\neq\emptyset$. Now if $N^{2}_{G}(u,v)=\emptyset$ and $\|N^{3}_{G}(u,v)\|=1$, then $G^{\prime}$ is same as $G$. So, without loss of generality, assume that $\left\|N^{3}_{G}(u,v)\right\|>1$ and $N^{2}_{G}(u,v)\neq\emptyset$. Note that a vertex $x$ of $N^{3}_{G}(u,v)$ can be doubly dominated by any two vertices from $N_{G}[x]\subseteq\\{N^{2}_{G}(u,v)\cup N^{3}_{G}(u,v)\cup\\{u,v\\}\\}$. Again for any two vertices $x,y\in N^{2}_{G}(u,v)\cup N^{3}_{G}(u,v)\cup\\{u,v\\}$, $N_{G}[x]\cup N_{G}[y]\subseteq N_{G}[u]\cup N_{G}[v]$. This shows that we can double dominate each vertex of $N^{3}_{G}(u,v)$ in an optimal way by selecting $u$ and $v$ only. This selection of $u$ and $v$ was forced by the only vertex $w\in N^{3}_{G}(u,v)$ that remained in $G^{\prime}$. We claim that $G$ contains a minimum double dominating set $D$ which does not contain any vertex from $N^{2}_{G}(u,v)\cup N^{3}_{G}(u,v)$. First observe that there can not be three or more vertices from $N^{2}_{G}(u,v)\cup N^{3}_{G}(u,v)$ in $D$. If it were, then we could replace those three or more vertices by $u$ and $v$, thus contradicting the minimality of $D$. Now for those two (or one) vertices from $N^{2}_{G}(u,v)\cup N^{3}_{G}(u,v)$ in $D$, we can replace them by $u$ and (or) $v$. Therefore, $G$ contains a minimum double dominating set $D$ which does not contain any vertex from $N^{2}_{G}(u,v)\cup N^{3}_{G}(u,v)$. Clearly, this set $D$ also forms a minimum double dominating set of $G^{\prime}$. Hence, $\gamma_{2}(G)=\gamma_{2}(G^{\prime})$. ∎ In this reduction, for a pair of distinct vertices $u,v\in V$, we have actually deleted at most $\min\\{deg_{G}(u),deg_{G}(v)\\}$ vertices. So, the time taken is $\sum_{u,v\in V}\min\\{deg_{G}(u),deg_{G}(v)\\}$ for the whole reduction process. Since for a planar graph $\sum_{v\in V}deg_{G}(v)=O(n)$, where $n$ is the number of vertices, we have the following lemma. ###### Lemma 9 For a planar graph having $n$ vertices, the reduction rule can be carried out in $O(n^{3})$ time. #### 4.1.2 A linear kernel In this subsection, we show that the reduction rule given in the previous section yields a linear kernel for Double Domination Problem in planar graphs. For this proof, first we find a “maximal region decomposition” of the vertices $V^{\prime}$ of the reduced graph $G^{\prime}=(V^{\prime},E^{\prime})$ and then we show that $\|V^{\prime}\|=O(\gamma_{2}(G^{\prime}))$. We start with some definitions regarding maximal region decomposition following Alber et al. [AFN04a]. ###### Definition 10 Let $G=(V,E)$ be a plane graph. A closed subset of the plane is called a region $R(u,v)$ between two vertices $u,v$ if the following properties are met: 1. 1. the boundary of $R(u,v)$ is formed by two simple paths $P$ and $Q$ between $u$ and $v$ of length at most two edges, and 2. 2. all the vertices which are strictly inside the region $R(u,v)$ are from $N_{G}(u)\cap N_{G}(v)$. The definition of a region is slightly different from the definition given in [AFN04a], where all the vertices which are strictly inside the region $R(u,v)$ are from $N_{G}(u)\cup N_{G}(v)$. Note that by the above definition, paths of length one or two between $u$ and $v$ can form a region $R(u,v)$. For a region $R=R(u,v)$, let $\partial(R)$ denote the boundary of $R$ and $V(R)$ denote the vertices inside or on the boundary of $R$, i.e., $V(R)=\\{u\in V\|~{}u~{}\mbox{ is inside }R\mbox{ or on }\partial(R)\\}$. ###### Definition 11 Let $G=(V,E)$ be a plane graph and $D\subseteq V$. A $D$-region decomposition of $G$ is a set $\mathcal{R}$ of regions between pairs of vertices in $D$ such that 1. 1. for $R(u,v)\in\mathcal{R}$ no vertices of $D$ (except $u$ and $v$) lies in $V(R(u,v))$, and 2. 2. for two regions $R_{1},R_{2}\in\mathcal{R}$, $(R_{1}\cap R_{2})\subseteq(\partial(R_{1})\cup\partial(R_{2}))$, i.e., they can intersect only at the vertices on the boundary. For a $D$-region decomposition $\mathcal{R}$, we define $V(\mathcal{R})=\cup_{R\in\mathcal{R}}V(R)$. A $D$-region decomposition $\mathcal{R}$ is called maximal if there is no region $R$ such that $\mathcal{R^{\prime}}=\mathcal{R}\cup R$ is a $D$-region decomposition, where $V(\mathcal{R})$ is a strict subset of $V(\mathcal{R^{\prime}})$. First we observe an important property of a maximal $D$-region decomposition. ###### Lemma 12 Let $G=(V,E)$ be a plane graph with a double dominating set $D$ and let $\mathcal{R}$ be a maximal $D$-region decomposition. Then $V=V(\mathcal{R})$. ###### Proof. Let $v\in V$ be a vertex such that $v\notin V(\mathcal{R})$. There can be two cases – $v\in D$ and $v\notin D$. First, let us assume that $v\in D$. Since $D$ is a double dominating set of $G$, there exists another vertex $x\in D$ such that $vx\in E$. Now, the path $P=(x,v)$ forms a region $R$. Clearly $\mathcal{R}\cup R$ forms a $D$-region decomposition of $G$ which contradicts the maximality of $\mathcal{R}$. Let us now consider the other case $v\notin D$. Since $D$ is a double dominating set of $G$, there exists $x,y\in D$ such that $vx,vy\in E$. In this case, the path $P=(x,v,y)$ forms a region $R$. Here also, $\mathcal{R}\cup R$ forms a $D$-region decomposition of $G$ which contradicts the maximality of $\mathcal{R}$. Thus each vertex of $V$ is in $V(\mathcal{R})$, that is, $V\subseteq V(\mathcal{R})$. Thus $V=V(\mathcal{R})$. ∎ It is obvious that, for a plane graph $G=(V,E)$ with a double dominating set $D$, there exists a maximal $D$-region decomposition $\mathcal{R}$. Based on Lemma 12, we propose a greedy algorithm to compute a maximal $D$-region decomposition, which is given in Algorithm 1. The algorithm basically ensures the properties of the region decomposition mentioned in Definitions 10 and 11. Input: A plane graph $G=(V,E)$ and a double dominating set $D\subseteq V$. Output: A maximal $D$-region decomposition $\mathcal{R}$ of $G$. begin $V_{used}\leftarrow\emptyset$, $\mathcal{R}\leftarrow\emptyset$; while _$V_{used}\neq V$_ do Select a vertex $x$ from $V\setminus V_{used}$; Consider the set $\mathcal{R}_{x}$ of all regions $S$ with the following properties: 1. 1. $S$ is a region between $u$ and $v$, where $u,v\in D$. 2. 2. $S$ contains $x$. 3. 3. no vertex from $D\setminus\\{u,v\\}$ is in $V(S)$. 4. 4. $(S\cup R)\subseteq(\partial(S)\cup\partial(R))$ for all $R\in\mathcal{R}$. Choose a region $S_{x}\in\mathcal{R}_{x}$ which is maximal in terms of vertices; $\mathcal{R}\leftarrow\mathcal{R}\cup\\{S_{x}\\}$; $V_{used}\leftarrow V_{used}\cup V(S_{x})$; $\textnormal{{return}}(\mathcal{R})$; Algorithm 1 REGION$\\_$DECOMPOSITION$(G,D)$ Clearly Algorithm 1 output a maximal $D$-region decomposition in polynomial time. Next, we show that for a given plane graph $G$ with a double dominating set $D$, every maximal $D$-region decomposition contains at most $O(\|D\|)$ many regions. For that purpose, we observe that a $D$-region decomposition induces a graph in a very natural way. ###### Definition 13 The induced graph $G_{\mathcal{R}}=(V_{\mathcal{R}},E_{\mathcal{R}})$ of a $D$-region decomposition $\mathcal{R}$ of $G$ is the graph with possible multiple edges which is defined as follows: $V_{\mathcal{R}}=D$ and $E_{\mathcal{R}}=\\{(u,v)\|$there is a region $R(u,v)\in\mathcal{R}$ between $u,v\in D\\}$. Note that, since by Definition 11 the regions of a $D$-region decomposition do not intersect, the induced graph $G_{\mathcal{R}}$ of a $D$-region decomposition $\mathcal{R}$ is a planar graph with multiple edges. Next we bound the number of regions in a maximal $D$-region decomposition using the concept of _thin planar graph_ following Alber et al. [AFN04a]. ###### Definition 14 A planar graph $G=(V,E)$ with multiple edges is thin if there exists a planar embedding such that if there are two edges $e_{1},e_{2}$ between a pair of distinct vertices $v,w\in V$, then there must be two further vertices $u_{1},u_{2}\in V$ which sit inside the two disjoint areas of the plane that are enclosed by $e_{1}$ and $e_{2}$. ###### Lemma 15 Let $D$ be a double dominating set of a planar graph $G=(V,E)$. Then the induced graph $G_{\mathcal{R}}=(V_{\mathcal{R}},E_{\mathcal{R}})$ of a maximal $D$-region decomposition $\mathcal{R}$ of $G$ is a thin planar graph. ###### Proof. Let $R_{1}$ and $R_{2}$ be two regions between two vertices $v,w\in D$ and $e_{1}$ and $e_{2}$ be the corresponding multiple edges between two vertices $v,w\in V_{\mathcal{R}}$. Let $A$ be an area enclosed by $e_{1}$ and $e_{2}$. If $A$ contains a vertex $u\in D$, we are done. Suppose there is no vertex of $D$ in $A$. Now consider the following cases: There is no vertex from $V\setminus D$ in $A$: In this case, by combining the regions $R_{1}$ and $R_{2}$, we can form a bigger region which is a contradiction to the maximality of $\mathcal{R}$. There is a vertex $x\in(V\setminus D)$ in $A$: In this case, if $x$ is double dominated by $v$ and $w$, then again we can combine the two regions $R_{1}$ and $R_{2}$ to get a bigger region. So, assume that $x$ is dominated by some vertex $u$ other than $v$ and $w$. Since $G$ is planar, $u$ must be in $A$ which contradicts the fact that $A$ does not contain any vertex from $D$. Hence, combining both the cases we see that $G_{\mathcal{R}}$ is a thin planar graph. ∎ In [AFN04a], it is proved that for a thin planar graph $G=(V,E)$, we have $\|E\|\leq 3\|V\|-6$. Hence we have the following lemma. ###### Lemma 16 For a plane graph $G$ with a double dominating set $D$, every maximal $D$-region decomposition $\mathcal{R}$ contains at most $3\|D\|$ many regions. Now, if we can bound the number of vertices that belongs to any region $R(u,v)$ of a maximal $D$-region decomposition $\mathcal{R}$ by some constant factor, we are done. However, achieving this constant factor bound is not possible for any plane graph $G$. But in a reduced plane graph, we can obtain this bound, as shown in the following lemma. ###### Lemma 17 A region $R$ of a plane reduced graph contains at most $6$ vertices, that is, $\|V(R)\|\leq 6$. ###### Proof. Let $R$ be the region between $u$ and $v$ and $\partial(R)=\\{u,x,v,y\\}$. First note that $R$ contains at most two vertices from $N_{G^{\prime}}^{1}(u,v)$ and the only possibility of such vertices are $x$ and $y$. If there exists a vertex $w\in N_{G^{\prime}}^{1}(u,v)$, apart from $x$ and $y$, then $w$ has to have a neighbor $z\notin N_{G^{\prime}}(u,v)$. $z$ should be inside the region $R$ and hence, cannot be double dominated. Now, because of the reduction rule, we can say that $\left\|N_{G^{\prime}}^{3}(u,v)\right\|\leq 1$. We consider the two cases: Case I ($\left\|N_{G^{\prime}}^{3}(u,v)\right\|=1$): In this case, $\left\|N_{G^{\prime}}^{2}(u,v)\right\|=\emptyset$ by the reduction rule. Hence, $\|V(R)\|\leq 5$. Case II ($\left\|N_{G^{\prime}}^{3}(u,v)\right\|=0$): In this case, we claim that there can be at most two vertices from $N_{G^{\prime}}^{2}(u,v)$. If possible, let $p,q,r\in N_{G^{\prime}}^{2}(u,v)$. Now all these three vertices must be adjacent to either $x$ or $y$, which is not possible because of planarity. Hence, in this case $\|V(R)\|\leq 6$. ∎ First observe that, for a reduced graph $G^{\prime}$ with a minimum double dominating set $D$, by Lemma 16, there exists a maximal $D$-region decomposition $\mathcal{R}$ with at most $3\cdot\gamma_{2}(G^{\prime})$ regions. Also by Lemma 12, we have $V^{\prime}=V(\mathcal{R})$ and by Lemma 17, we have for each region $\|V(R)\|\leq 6$. Thus we have $\|V^{\prime}\|=\|V(\mathcal{R})\|=\|\cup_{R\in\mathcal{R}}V(R)\|\leq\sum_{R\in\mathcal{R}}\left\|V(R)\right\|\leq 6\cdot\|\mathcal{R}\|\leq 18\cdot\gamma_{2}(G^{\prime})$. Hence, we have the following theorem. ###### Theorem 18 For a reduced planar graph $G^{\prime}=(V^{\prime},E^{\prime})$, we have $\|V^{\prime}\|\leq 18\cdot\gamma_{2}(G^{\prime})$, that is, Double Domination Problem on planar graph admits a linear kernel. ### 4.2 Liar’s and $k$-tuple domination We first show that the number of vertices in a plane graph, $\|V\|=O(\gamma_{LR}(G))$. In this respect, first we note that both the results in Lemma 12 and Lemma 16 are valid for any plane graph $G$ and any double dominating set $D$. Since every liar’s dominating set is also a double dominating set, similar type of results hold for any plane graph $G$ and any liar’s dominating set $L$. We claim that the number of vertices in a region $R$ of a $L$-region decomposition is bounded above by a constant. Let $R$ be a region between $u$ and $v$ and $\partial(R)=\\{u,x,v,y\\}$. Note that in $V(R)$ there are two vertices ($u$ and $v$) from $L$. Now, if there exists two vertices $p,q\in V(R)\setminus\partial(R)$, then for the pair $p$ and $q$ condition (ii) of liar’s domination is violated. Hence, there is at most one vertex in $V(R)\setminus\partial(R)$. Therefore, $\|V(R)\|\leq 5$. Thus we have $\|V\|=\|V(\mathcal{R})\|=\|\cup_{R\in\mathcal{R}}V(R)\|\leq\sum_{R\in\mathcal{R}}\|V(R)\|\leq 5\cdot\|\mathcal{R}\|\leq 15\cdot\|L\|\leq 15\cdot\gamma_{LR}(G).$ Hence, we have the following theorem ###### Theorem 19 For a planar graph $G=(V,E)$, $\|V\|\leq 15\cdot\gamma_{LR}(G)$. Since every $k$-tuple dominating set for $k\geq 3$ is a liar’s dominating set, we can use Theorem 19. But, we can improve the constant a little bit. ###### Theorem 20 For a planar graph $G=(V,E)$, $\|V\|\leq 12\cdot\gamma_{k}(G)$, where $k\geq 3$. ###### Proof. Let $D$ be a minimum $k$-tuple dominating set of $G=(V,E)$. Since every $k$-tuple dominating set is a double dominating set, by Lemma 16 we can form a maximal $D$-region decomposition $\mathcal{R}$ of $G$ containing at most $3\cdot\|D\|$ many regions. Again by Lemma 12, we have $V=V(\mathcal{R})$. Since each region contains only two vertices of $D$, we have $\|V(R)\|\leq 4$. Otherwise there exists one vertex in $V(R)$ which is not dominated by $k$ vertices of $D$. Hence $\|V\|\leq 4\cdot\|\mathcal{R}\|\leq 12\cdot\|D\|\leq 12\cdot\gamma_{k}(G)$. ∎ ## 5 Linear kernels for bounded genus graphs In this section, we extend our results to bounded genus graphs to show that $k$-Tuple Domination Problem and Liar’s Domination Problem admit a linear kernel. The notations in this section follow Section 2.1. For double domination problem, we apply the same reduction rule on a graph $G$ with bounded genus $g$ to obtain the reduced graph $G^{\prime}$. Note that the reduced graph $G^{\prime}$ is also of bounded genus $g$. Let $G=(V,E)$ be an $n$-vertex $\Sigma$-embedded graph. It is easy to observe that, since $\sum_{v\in V}deg_{G}(v)=O(n+\textup{eg}(\Sigma))$, the reduced graph $G^{\prime}=(V^{\prime},E^{\prime})$ can be computed in $O(n^{3}+n^{2}\cdot\textup{eg}(\Sigma))$ time, where $\|V\|=n$. Next we show that $\|V^{\prime}\|=O(\gamma_{2}(G^{\prime})+g)$ which implies Double Domination Problem admits a linear kernel in bounded genus graphs. To prove the above, we consider two cases. In the first case, we assume that the reduced $\Sigma$-embedded graph has representativity strictly greater than $4$. In the case when $rep(G)\leq 4$, we go by induction on the Euler genus of surface $\Sigma$. In the first case, the graphs are locally planar, i.e., all the contractable noose are of length less or equal to $4$. Since the boundary of the regions in planar case is less than or equal to $4$, the boundary $\partial(R)$ of any region $R$ of a $D$-region decomposition $\mathcal{R}$ is contractible. Hence the proof in the planar case can be extended in this case. Hence we have the following lemma. ###### Lemma 21 Let $G^{\prime}=(V^{\prime},E^{\prime})$ be a reduced $\Sigma$-embedded graph where $\textup{rep}(G^{\prime})>4$. Then $\|V^{\prime}\|\leq 18(\gamma_{2}(G^{\prime})+\textup{eg}(\Sigma))$. ###### Proof. Let $D$ be a double dominating set of $G^{\prime}$ and $\mathcal{R}$ is a maximal $D$-region decomposition of $G^{\prime}$. Forming a induced graph, $G_{\mathcal{R}}$ as in case of double domination problem in planar graphs (Section 4.1.2), we have $\|\mathcal{R}\|\leq 3\cdot(\|D\|+eg(\Sigma))$. Also, in this case, every vertex of $V^{\prime}$ belongs to at least one region of $\mathcal{R}$ and for a region $R$, $\|V(R)\|\leq 6$. Hence, we have $\|V^{\prime}\|\leq 18(\gamma_{2}(G^{\prime})+\textup{eg}(\Sigma))$. ∎ Next consider the case where $3\leq\textup{rep}(G^{\prime})\leq 4$. For a noose $N$ in $\Sigma$, we define the graph $G_{N}=(V_{N},E_{N})$ as follows. First we consider the graph $\mathcal{G}$ obtained from $G^{\prime}=(V^{\prime},E^{\prime})$ by cutting along $N$. Then for every $v\in N\cap V^{\prime}$ if $v^{i}$, $i=1,2$, is not adjacent to a pendant vertex, then we add a pendant vertex $u^{i}$ adjacent to $v^{i}$ to form $G_{N}$. Clearly $G_{N}$ has genus less than that of $G^{\prime}$. If we add all the vertices of $V_{N}\setminus V^{\prime}$ to a double dominating set $D$ of $G^{\prime}$, then we clearly obtain a double dominating set of $G_{N}$ and as, $\textup{rep}(G^{\prime})\leq 4$, $\|V_{N}\setminus V^{\prime}\|\leq 16$. Hence, $\gamma_{2}(G_{N})\leq\gamma_{2}(G^{\prime})+\|N\cap V^{\prime}\|\leq\gamma_{2}(G^{\prime})+16$. Also note that if $G^{\prime}$ is a reduced graph, then so is $G_{N}$. Using these facts, we prove that Double Domination Problem possesses a linear kernel when restricted to graphs with bounded genus. ###### Lemma 22 For any reduced $\Sigma$-embedded graph $G^{\prime}=(V^{\prime},E^{\prime})$ with eg$(\Sigma)\geq 1$, $\|V^{\prime}\|\leq 18(\gamma_{2}(G^{\prime})+32\cdot\textup{eg}(\Sigma)-16)$. ###### Proof. We prove this result by induction on $\textup{eg}(\Sigma)$. Suppose $\textup{eg}(\Sigma)=1$. If $\textup{rep}(G^{\prime})>4$, then the result follows from Lemma 21. Otherwise Lemma 1 implies that the graph $G_{N}$, described above, is planar. Hence by Theorem 18, we have $\|V_{N}\|\leq 18\cdot\gamma_{2}(G_{N})$. Thus $\|V^{\prime}\|\leq\|V_{N}\|\leq 18(\gamma_{2}(G^{\prime})+16)$. Assume that $\|V^{\prime}\|\leq 18(\gamma_{2}(G^{\prime})+32\cdot\textup{eg}(\Sigma)-16)$ for any $\Sigma$-embedded reduced graph $G^{\prime}$ with eg$(\Sigma)\leq g-1$. Consider a reduced $\Sigma$-embedded graph $G^{\prime}$ with eg$(\Sigma)=g$. Now if rep$(G^{\prime})>4$, then again by Lemma 21, we are done. Hence assume that rep$(G^{\prime})\leq 4$. By Lemma 1, either $G_{N}$ is the disjoint union of graphs $G_{1}$ and $G_{2}$ that can be embedded in surfaces $\Sigma_{1}$ and $\Sigma_{2}$ such that eg$(\Sigma)=$ eg$(\Sigma_{1})+$ eg$(\Sigma_{2})$ and eg$(\Sigma_{i})>0$, $i=1,2$ (this is the case when $N$ is surface separating curve), or $G_{N}$ can be embedded in a surface with Euler genus strictly smaller than eg$(\Sigma)$ (this holds when N is not surface separating). Let us consider the case where $G_{N}$ is the disjoint union of graphs $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$ that can be embedded in surfaces $\Sigma_{1}$ and $\Sigma_{2}$. Since eg$(\Sigma_{i})\leq g-1$ for $i=1,2$, we can apply the induction hypothesis on $G_{i}$. Thus we have, $\displaystyle\|V^{\prime}\|\leq\|V_{N}\|$ $\displaystyle=\|V_{1}\|+\|V_{2}\|$ $\displaystyle\leq\sum_{i=1}^{2}18(\gamma_{2}(G_{i})+32\cdot\textup{eg}(\Sigma_{i})-16)$ $\displaystyle\leq 18(\gamma_{2}(G_{N})+32\cdot\textup{eg}(\Sigma)-32)$ as $G_{1}$ and $G_{2}$ are disjoint $\displaystyle\leq 18(\gamma_{2}(G^{\prime})+32\cdot\textup{eg}(\Sigma)-16).$ Next we consider the case where $G_{N}$ can be embedded in a surface $\Sigma^{\prime}$ with Euler genus strictly smaller than $g$. In this case too, we can apply induction hypothesis on $G_{N}$. Thus we have, $\displaystyle\|V^{\prime}\|\leq\|V_{N}\|$ $\displaystyle\leq 18(\gamma_{2}(G_{N})+32\cdot\textup{eg}(\Sigma^{\prime})-16)$ $\displaystyle\leq 18(\gamma_{2}(G_{N})+32\cdot(\textup{eg}(\Sigma)-1)-16)$ $\displaystyle\leq 18(\gamma_{2}(G^{\prime})+32\cdot\textup{eg}(\Sigma)-32)$ as $\gamma_{2}(G_{N})\leq\gamma_{2}(G^{\prime})+16$ $\displaystyle\leq 18(\gamma_{2}(G^{\prime})+32\cdot\textup{eg}(\Sigma)-16).$ Thus we have proved that, $\|V^{\prime}\|\leq 18(\gamma_{2}(G^{\prime})+32\cdot\textup{eg}(\Sigma)-16)$ for every $\Sigma$-embedded graph $G^{\prime}=(V^{\prime},E^{\prime})$. ∎ Hence by Lemma 21 and Lemma 22, we have the main result of this subsection. ###### Theorem 23 Double Domination Problem admits a linear kernel for bounded genus graphs. For liar’s domination problem, by Theorem 19, we have $\|V\|\leq 15\cdot\gamma_{LR}(G)$ in case of a planar graph $G=(V,E)$. Proceeding exactly in the same way as in the case of double domination, we can have the following theorem for a $\Sigma$-embedded graph $G$. ###### Theorem 24 Let $G=(V,E)$ be a $\Sigma$-embedded graph. Then $\|V\|\leq 15(\gamma_{LR}(G)+32\cdot\textup{eg}(\Sigma))$. Since for any graph that admits a $k$-tuple dominating set ($k\geq 3$), $\gamma_{LR}(G)\leq\gamma_{k}(G)$, we have the following corollary of Theorem 24. ###### Corollary 25 For a $\Sigma$-embedded graph $G=(V,E)$, $\|V\|\leq 15(\gamma_{k}(G)+32\cdot\textup{eg}(\Sigma))$. ## 6 Conclusion In this paper, we first have proved that $k$-Tuple Domination Problem and Liar’s Domination Problem are W[2]-hard for general graphs. Then we have shown that these two problems admit linear kernel for planar graphs and also for bounded genus graphs. It would be interesting to look for other graph classes where these problems admit efficient parameterized algorithms. ## Acknowledgements The authors want to thank Venkatesh Raman and Saket Saurabh for some of their nice introductory expositions to parametrization. ## 7 Appendix ###### Lemma 26 Liar’s Domination Problem is NP-complete for planar graphs. ###### Proof. The reduction is from Domination Problem in planar graphs, which is known to be NP-complete[GJ79]. Let $G=(V,E)$ be a planar graph with $V=\\{v_{1},v_{2},\ldots,v_{n}\\}$ and $k$ be an integer. We construct an instance $G^{\prime}=(V^{\prime},E^{\prime})$ and $k^{\prime}$ of Liar’s Domination Problem as follows: We add a set of $3n$ new vertices $S=\\{x_{i},y_{i},z_{i}\|1\leq i\leq n\\}$ to the vertex set of $V$, i.e., $V^{\prime}=V\cup S$ and the edge set of $G^{\prime}$ is given by $E^{\prime}=E\cup\\{v_{i}x_{i},x_{i}y_{i},y_{i}z_{i}\|1\leq i\leq n\\}$. Note that, since $G$ is planar, so is $G^{\prime}$. Also assume that $k^{\prime}=k+3n$. In [Sla09], it is proved that $G$ has a dominating set of cardinality at most $k$ if and only if $G^{\prime}$ has a liar’s dominating set of cardinality at most $k^{\prime}=k+3n$. Thus, Liar’s Domination Problem is NP-complete for planar graphs. ∎ ## References * [ABF+02] J. Alber, H. L. Bodlaender, H. Fernau, T. Kloks, and R. Niedermeier. Fixed parameter algorithms for dominating set and related problems on planar graphs. Algorithmica, 33(4):461–493, 2002. * [AFF+05] J. Alber, H. Fan, M. R. Fellows, H. Fernau, R. Niedermeier, F. A. Rosamond, and U. Stege. A refined search tree technique for dominating set on planar graphs. J. Comput. Syst. Sci., 71(4):385–405, 2005. * [AFN04a] J. Alber, M. R. Fellows, and R. Niedermeier. Polynomial-time data reduction for dominating set. J. ACM, 51(3):363–384, 2004. * [AFN04b] J. Alber, H. Fernau, and R. Niedermeier. Parameterized complexity: exponential speed-up for planar graph problems. J. 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Algorithms for vertex partitioning problems on partial $k$-trees. SIAM J. Disc. Math, 10(4):529–550, 1997. * [vRBR09] J. M. M. van Rooij, H. L. Bodlaender, and P. Rossmanith. Dynamic programming on tree decompositions using generalised fast subset convolution. In A. Fiat and P. Sanders, editors, Algorithms - ESA 2009, volume 5757 of Lecture Notes in Computer Science, pages 566–577. Springer Berlin Heidelberg, 2009.	arxiv-papers	2013-09-21T11:07:24	2024-09-04T02:49:51.268082	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Arijit Bishnu, Arijit Ghosh and Subhabrata Paul", "submitter": "Arijit Ghosh", "url": "https://arxiv.org/abs/1309.5461" }
1309.5468	11institutetext: Variable Energy Cyclotron Centre,1/AF Bidhannagar, Kolkata 700064, India S.N. Bose National Centre for Basic Sciences, Block JD, Salt Lake, Kolkata 700098, India Time series analysis in nonlinear dynamics Scaling phenomena in complex systems Levy flights # Universal Scaling Property of System Approaching Equilibrium P. Barat 11 A. Giri 11 M. Bhattacharya 11 Nilangshu K. Das 11 A. Dutta 221122 ###### Abstract In this Letter we show that the diffusion kinetics of kinetic energy among the atoms in non-equilibrium crystalline systems follows universal scaling relation and obey Levy-walk properties. This scaling relation is found to be valid for systems no matter how far they are driven out of equilibrium. ###### pacs: 05.45.Tp ###### pacs: 89.75.Da ###### pacs: 05.40.Fb Non-equilibrium systems are ubiquitous in nature. Equilibrium systems are ideal and can only be achieved in the laboratory. Attempts have been made to understand the dynamics of non-equilibrium systems by linear response theory [1] albeit its domains of validity are restricted to the linear response regime. There exists no general formalism to deal with systems that are far from equilibrium. For a system out of equilibrium, the probability of a given microstate evolves continuously with time. In the long time limit the system reaches a stationary state in which the probability measure over the configuration space converges to a constant distribution. Non-equilibrium systems display fluctuations which are less sensitive to the conditions of the surroundings and carry information about the dynamics of its present state. In the last decade certain general relations have been discovered which are valid for non-equilibrium systems and are independent of how far the system is driven out of equilibrium. These results include the Jarzynski equality [2, 3] and the fluctuation theorems [4, 5, 6, 7, 8, 9, 10, 11, 12]. They have been verified for a variety of systems theoretically as well as experimentally [13, 14, 15, 16, 17]. After the work by Crooks [7] and Seifert [8], it is now understood that many of these relations are closely related to the path probability of the system’s trajectory. In the absence of any general theory to understand the dynamics of a system far from equilibrium, one approach will be to take a simple but nontrivial model system and try to understand its dynamics when it approaches equilibrium from its non-equilibrium state. The general dynamical behavior of a non-equilibrium system will consist of superposition of various dynamics on well speared time scales which compel several thermodynamic parameters of the system like heat, work, internal energy, kinetic energy of the particles of the system to fluctuate. To understand the general features of the dynamics one has to apply statistical analysis to these fluctuations. In this Letter an attempt has been made to study a universal scaling relation for systems driven far from equilibrium by analyzing its fluctuation properties. The non-equilibrium systems of study are generated by molecular dynamics simulation technique. Dynamical systems arising from diversified disciplines of science can be quantified in a unified way from their scale invariance properties. Scafetta $et$ $al.$ [18] introduced two complementary scaling analysis methods: Diffusion Entropy Analysis (DEA) and Finite Variance Scaling Method (FVSM) to evaluate correct scaling that prevails in complex dynamical systems. DEA studies the scaling exponent $\delta$ of the probability distribution instead of its moment, based on the evaluation of Shannon entropy $S(t)$ of a time series $\\{\xi_{i}\\}$. The DEA has been successfully applied to time series of different kinds of dynamical systems [19]. In DEA analysis, the numbers in the time series, $\\{\xi_{i}\\}$, are thought to be the fluctuations of a diffusion trajectory [19] arising from the probability density function (PDF), $p(x,t)$, of the corresponding diffusion process. Here $x$ denotes the variable collecting the fluctuations and is referred as the diffusion variable. The scaling property of $p(x,t)$ is evaluated by means of the sub trajectories $x_{n}(t)=\sum_{i=0}^{t}\xi_{i+n}$ with n = 0,1,….etc. If the scaling condition $p(x,t)=t^{-\delta}F(xt^{-\delta})$ holds true, it is easy to prove that the entropy $S(t)$ is given by $S(t)=-\int^{+\infty}_{-\infty}p(x,t)ln[p(x,t)]dx=A+\delta ln(t)$, where $A$ is a constant. This indicates that in the case of a diffusion process with a scaling PDF, its entropy $S(t)$ increases linearly with $ln(t)$. One can also examine the scaling properties of the second moment of the same process by FVSM. One version of FVSM is the Standard Deviation Analysis (SDA) [19], which is based on the evaluation of the standard deviation $D(t)$ of the variable $x$, and yields $D(t)=\left[\langle x^{2};t\rangle-\langle x;t\rangle^{2}\right]^{\frac{1}{2}}\propto t^{H}$ [19]. The exponent $H$ is interpreted as the scaling exponent. For random noise with finite variance, the diffusion distribution $p(x,t)$ will converge, according to the central limit theorem, to a Gaussian distribution with $H=\delta=0.5$. If $H\neq\delta$, the scaling represents anomalous behavior. Levy-walk is a kind of anomalus diffusion which is obtained by generalizing the central limit theorem [20]. In this particular kind of diffusion process the scaling exponents $H$ and $\delta$ are found to obey the relation $\delta=(3-2H)^{-1}$ instead of being equal [19]. Table 1: The parameters used for the MD simulations. Material \| Si \| Ge \| Cu \| Solid Ar \| Mo \| Fe \| Al \| ---\|---\|---\|---\|---\|---\|---\|---\|--- Simulation cell size (in unit cell) \| $25^{3}$ \| $15^{3}$ \| $20^{3}$ \| $20^{3}$ \| $25^{3}$ \| $25^{3}$ \| $20^{3}$ \| Number of atoms in simulation cell \| 125000 \| 27000 \| 32000 \| 32000 \| 31250 \| 31250 \| 32000 \| Simulation time step (fs) \| 0.25 \| 0.5 \| 0.5 \| 2.0 \| 0.5 \| 0.5 \| 0.5 \| Nature of inter-atomic potential used \| SW111SW (Stillinger-Weber) \| SW \| EAM222 EAM (Embedded Atom Model) \| LJ333LJ (Lennard-Jones) \| FS444FS (Finnis-Sinclair) \| FS \| GLUE \| Total number of data taken \| 6000 \| 3000 \| 3000 \| 10000 \| 3000 \| 3000 \| 3000 \| Radius ‘r’ (Å) \| 8.8250 \| 9.0809 \| 5.4766 \| 7.9952 \| 6.2944 \| 5.7330 \| 6.1566 \| No of atoms in the sphere of radius ‘r’ \| 150 \| 147 \| 55 \| 55 \| 63 \| 63 \| 55 \| To generate the non-equilibrium model system we have utilized the molecular dynamics (MD) simulation technique in an innovative way. Depending on the interaction potential model, a typical MD simulation computes the trajectories of atoms in a system by solving Newton’s equations of motion numerically. In our study, the simulations have been performed for elements having different crystal structures like Silicon (Si) and Germanium (Ge) with diamond cubic structure, Iron (Fe) and Molybdenum (Mo) with body centered cubic structure and Copper (Cu), Aluminium (Al) and solid Argon (Ar) having face centered cubic structure. Thus, a variety of interaction potential models have been employed (refer Table 1) in the simulations. For all these elements, periodic boundary conditions are imposed along three directions (x,y,z) in the simulation cell. Initially, all the systems are equilibrated using constant energy and volume ensemble (NVE) at 100 K for 2.5 ps except for Ar, where the equilibration run is performed at 30K for 10 ps. The kinetic energy in the equilibrium state will follow Boltzmann distribution. The period of time for equilibration is chosen to be sufficient to bring the total energy of the system to divide approximately equally between kinetic energy (KE) and potential energy (PE). Details of the parameters used in the simulations like simulation cell size, number of atoms taken in the simulation cell, the time steps after which simulation data are recorded and the nature of the inter- atomic potentials used for the elements under study are given in Table I. To generate the non-equilibrium state, the three components of velocities of individual atom in the simulation cell were changed to random values such that the instantaneous KE of the atom increases however the average temperature of the system corresponds to T = 500 K (T = 70 K for Ar). The atoms in the simulation cell are then allowed to equilibrate. After sufficiently long period of time, the system is observed to attain equilibrium temperature of 300 K (50 K for Ar) and the average kinetic energy becomes equal to the average potential energy of the respective systems. The variations of temperature (equivalent to KE) with time of the systems of Ar and Cu leading to equilibration are shown in Fig. 1. Atomistic simulation data are recorded for each time step for the entire equilibration time to study the dynamic nature of the equilibration process. Figure 1: Variations of temperature (equivalent to KE ) with time of the systems of (a) Ar and (b) Cu leading to the equilibration The molecular dynamics simulations are carried out using MD++ simulation package [21]. Few atoms in the simulation cell are identified in a sphere of radius ‘r’ whose centre is chosen to be almost at the middle of the cell, far away from the surface. The values of ‘r’ and the number of atoms in the sphere for each simulation are given in Table 1. The KE in the consecutive time steps for the atoms in the sphere are used for further analysis. In these simulations the atoms in the simulation cell are kept isolated from the environment and hence there was no dissipation of energy, only there is a redistribution of KE and PE among the atoms. DEA and SDA analyses [22] were performed for the time series data of the KE for an individual atom. Typical plots of DEA and SDA analysis from which $\delta$ and $H$ are calculated for a Copper atom are shown in Fig. 2. The slopes of the curves evaluate $\delta$ and $H$. DEA and SDA analyses were performed for ten atoms randomly chosen from the atoms confined in the sphere of radius ‘r’ and the average of values of $\delta$ and $H$ are given in Table 2. Simulation cell sizes, number of atoms in the cell, sampling time, the crystal structure, and the nature of the interaction potential were varied to find the exact nature of the scaling. Table 2: Mean values of the scaling exponents $\delta$ and H obtained from ten atoms chosen randomly in the shell of radius ‘r’. Exponents $\delta$ and $H$ are evaluated from the time series of the evolution of KE for a single atom. Material \| Non-equilibrium \| $\delta$ \| $H$ \| $\left[\left(\delta-\frac{1}{3-2H}\right)/\delta\right]\times 100$ \| ---\|---\|---\|---\|---\|--- \| Temperature(K) \| \| \| \| Si \| 500 \| 0.953±0.016 \| 0.968±0.008 \| 1.380 \| Si \| 800 \| 0.898±0.025 \| 0.901±0.025 \| 7.046 \| Si ( single atom ) \| 500 \| 0.943 \| 0.970 \| 0.042 \| Ge \| 500 \| 0.930±0.021 \| 0.953±0.026 \| 1.712 \| Ge \| 800 \| 0.930±0.020 \| 0.960±0.015 \| 0.438 \| Ar \| 70 \| 0.968±0.010 \| 0.968±0.005 \| 2.908 \| Ar ( single atom ) \| 70 \| 0.963 \| 0.972 \| 1.665 \| Cu \| 500 \| 0.920±0.020 \| 0.955±0.022 \| 0.279 \| Cu ( single atom ) \| 500 \| 0.935 \| 0.978 \| 2.444 \| Fe \| 500 \| 0.925±0.018 \| 0.964±0.022 \| 0.847 \| Fe ( single atom ) \| 500 \| 0.959 \| 0.974 \| 0.879 \| Mo \| 500 \| 0.935±0.016 \| 0.964±0.021 \| 0.231 \| Al \| 500 \| 0.935±0.008 \| 0.966±0.013 \| 0.142 \| Figure 2: (a) DEA and (b) SDA of the variation of kinetic energy against time data obtained from a Copper atom in an ensemble when brought to a non- equilibrium state at 500 K from 100 K and allowed to equilibrate at 300 K To understand the process of equilibration when a single excited atom interacts with an ensemble of atoms that are in equilibrium, another kind of exercise was performed by MD simulation. In this case from the equilibrated ensembles of Si, Cu, Fe at 300 K and solid Ar at 50 K one atom in the middle of the cell was excited to a KE corresponding to 500 K and 70 K respectively. The excited atom is then allowed to equilibrate and the KE possesses by the atom at different time steps was monitored. In these simulations the energy of the simulation cells was also a constant of motion. All the equilibrated atoms in the cell form a heat bath and the excited atom equilibrate by interacting with this heat bath by sharing its excess KE. The time series thus obtained of the KE of the excited atom between consecutive time steps was used to understand the diffusion process of KE and the nature of the dynamical process responsible to bring the atom to the equilibrium state. The values of $\delta$ and $H$ obtained by DEA and SDA analysis from the time series are given in Table 2. Atoms in the simulation cell, when brought to a non-equilibrium state by enhancing their KE, try to redistribute their excess KE by the process of diffusion. This process of redistribution of KE among the atoms changes their mean positions of vibration and consequently their PE. Here all the atoms in the cell participate together in the diffusion process. Thus by analyzing the variation of the KE with time for a single atom in the cell one can account for the modality of this diffusion process and the nature of the equilibration. The change in the KE for an atom in the $i^{th}$ time step is given by $\Delta E=\left(\frac{2E_{i}}{m}\right)^{\frac{1}{2}}F_{i}\Delta t$ where $F_{i}$ is force on the atom at the $i^{th}$ time step. The force field experience by an atom in the cell will consist of three parts. The deterministic force, arising from the nearest neighbor interaction potential, and it plays the role of the external force $F(t)$ acting on the atom. There will be energy exchange between the atom and the surrounding atoms in a result of which the atom loses a part of its KE for exciting various degrees of freedom of the atoms in the ensemble as well there will be increase in the PE in the cost of KE of the atom. This can be described with help of a frictional force $F_{f}(t)$ acting on the atom. Besides loosing KE due to frictional effect there is a possibility of gaining KE in the form of random movement of the atom due to interactions from all other atoms in the cell and can be modeled by a random force $\Gamma(t)$ acting on the atom. $\Gamma(t)$ will have the property as $\langle\Gamma(t)\rangle=0$ and $\langle\Gamma(t_{1})\Gamma(t_{2})\rangle=Ag(\Delta t)$ where $\Delta t=(t_{2}-t_{1})$, the function $g(t)$ dies down rapidly with $t$ and $A$ is the strength of the random force and is a function of temperature T. Thus the dynamics of the atoms will follow Langevin type equation [23]. To see the effect of this random force field on the scaling behavior, the ensembles of Si and Ge atoms were exited to a higher temperature i.e. at 800 K and allowed to equilibrate at 450 K. The $\delta$ and $H$ values for the time series of the KE for these cases are given in Table 2. As the systems were allowed to equilibrate of its own without any defined protocol, forces acting on the atoms are time dependent through the dynamical process of equilibration. The force field experienced by an atom in the Figure 3: Probability distributions of the fluctuating forced fields experienced by (a) an Argon and (b) a Silicon atom when brought to a non- equilibrium state at 500 K from 100 K and allowed to equilibrate at 300 K simulation cell in the $i^{th}$ time step is calculated from its velocities, at $i-1$, $i$, $i+1^{th}$ time steps. Typical mean values of these forces are $1.011654\times 10^{-10}$ N and $24.7088\times 10^{-10}$ N for solid Argon and Si respectively. The fluctuating part of the force field at different time steps was obtained by subtracting the mean values. Typical probability distributions for the fluctuating part of the force field in case of solid Ar and Si atoms are shown in Fig. 3. The distributions are Gaussian with center $-6.69\times 10^{-12}$ N and width $9.59\times 10^{-11}$ N for Ar and the corresponding values for Si are $-4.98\times 10^{-11}$ N and $1.95\times 10^{-9}$ N respectively. To understand the correlation of the fluctuating force field, vectors $\\{\lvert X_{i}\rangle\\}$ are generated from the time series data of the fluctuating force field by taking consecutive n time steps for constructing each vector. The eigen value spectrum of the covariance matrix $\Sigma_{x}$ of these vectors are calculated to establish the nature of the correlation. Fig. 4 shows typical eigen value spectra for Ar and Si cases. The flatness of the spectra confirms that the fluctuating force fields are uncorrelated. Figure 4: Eigen value spectra of the covariance matrices constructed from the time series of the fluctuating forced fields experienced by (a) an Argon and (b) a Silicon atom when brought to a non-equilibrium state at 500 K from 100 K and allowed to equilibrate at 300 K The motion of the atoms in the simulation cell is thus governed by these forces and dictates the variation of KE of the atoms. The memory effect of the initial velocity will die down with time. In the long time limit the system equilibrates leading to equipartition of energy. The characteristic of the variation of KE in terms of time series indicates the evolution of the non- equilibrium state. To understand the universality of this evolution for different dynamical processes as generated by MD simulations, the Shannon entropy of the diffusion process of KE of an atom in the cell was obtained by calculating the PDF from the sub trajectories of this time series. The high values of $\delta$ and $H$ as given in Table 2 signifies a strong persistence in the fluctuations of the KE of the atoms. The values of $H$ are always larger than that of $\delta$ for all cases studied and are seen to fulfill the Levy-walk diffusion relation within the error bar as shown in Table 2. The exact inherent dynamics of the process of transmitting excess KE of an atom to its surroundings could not be revealed from this analysis. However, this analysis reflected the underlying generic features and physical principles that are independent of the detailed dynamics or characteristics of particular model. The diffusion of KE of an atom to its surroundings is a continuous stationary stochastic process as the probability of the diffusion trajectories follows a scaling relation. As the exponent $H$ is greater than 0.5 for all cases, the diffusion is anomalous super diffusion. Any diffusion is a kind of random walk and the Levy-walk is a mathematical model to describe anomalous super diffusion where the scaling exponent of variance against time is greater than one. Levy-walks have coupled space-time probability distributions and are characterized by a cluster of smaller variations of the random variable with a few large variations between them. This pattern repeats for all scales. As the process of equilibration of KE of an atom in the ensemble from its non-equilibrium state has to be very rapid, it cannot be Brownian type and it should be Levy-walk type as Levy-walk will outperform Brownian walk during the process of equilibration. The Levy-walk type of diffusion of KE among the atoms may be due to the following reason. Excess KE of an atom should disburse locally however there is a finite probability to transport its excess KE to a distant atom and consequently this atom again redistributes its KE in the same process. This process is much faster than normal diffusion when the mean squared value of fluctuation depends on t only. Mostly three dimensional systems show normal diffusion except in glassy systems [24]. The anomalous or superdiffusion and Levy-walks have been observed in various real-life phenomena like fluid flow in rotating annulus [25], low dimensional heat transport [26], light scattering in porous media [27] etc. However our findings show that the diffusion of KE for a single atom in its non-equilibrium state, when embedded in an environment of atoms that are either in equilibrium or in non-equilibrium state, show superdiffusion and Levy-walk properties throughout the process of equilibration. In the case when a single atom interacts with the ensemble of atoms that are in equilibrium we can assume that the atom is interacting with a thermal bath by absorbing or releasing KE without appreciable change in the bath state. In this case also we see the diffusion of KE from the atom is a Levy-walk process. Atoms of crystals when brought to a non-equilibrium state, try to redistribute its energy amongst them to converge to an equilibrium distribution. This fundamental process is governed by the increase in entropy of the trajectories of the individual atom and should be identical in nature for all types of elemental atoms and will be independent of the nature of interactions between the atoms and the perturbation which brought them to non-equilibrium state. The calculation of Shannon entropy of the diffusion process was made for one atom. However, the concept of entropy in statistical mechanics is for an ensemble. But the entropy production for a single trajectory has been addressed in the literature [8] concerning fluctuation theorems. Atoms in a lattice possessing KE different from its equilibrium values at any instant of time try to transport or accept from the nearest neighbors and arrive at a new value. The time series of this KE were translated in to a diffusion process in the form of diffusion trajectories and PDF of these trajectories were estimated. The estimated Shannon entropy production of this dynamical process with time is found to obey a universal scaling relation. This scaling relation is exact and valid for systems no matter how far they are driven out of equilibrium and is independent of the strength of perturbation that brought the system out of equilibrium. ## References * [1] L. Onsager, Phys. Rev. 37 (1931) 405 ; 38 (1931) 2265 . * [2] C. Jarzynski, Annu. Rev. Condens. Matter Phys. 2 (2011) 329 . * [3] C. Jarzynski, Phys. Rev. Lett. 78 (1997) 2690 . * [4] D. J. Evans, E G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71 (1993) 2401 . * [5] D. J. Evans and D. J. Searles, Phys. Rev. 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E 66 (2002) 036130, N. Scafetta and B.J.West, Phys. Rev. Lett. 92 (2004) 138501\. * [20] B.V. Gnedenko, A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, MA) 1954. * [21] http//micro.stanford.edu * [22] The programme for calculating DEA and SDA for the time series data was initially tested for synthetic time series like Gaussian White Noise generated by Box-Muller algorithm, Fractional Gaussian Noise generated based on the reference, Methodology and Computing in Applied Probability 1 (1999) 445 and Levy Noise generated based on the reference Phys. Rev. E 49 (1994) 4677. * [23] G.W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6 (1965) 504. * [24] L. Cipelletti and L. Ramos, J. Phys. : Condens. Matter 17 (2005) R253 and the references therein. * [25] T. H. Solomon, E. R. Weeks and H. L. Swinney, Phys. Rev. Lett. 71 (1993) 3975. * [26] A. Dhar, K.Saito and B. Derrida, Phys. Rev. E 87 (2013) 010103 (R) and the references therein. * [27] P. Barthelemy, J. Bertolotti and D. S. Wiersma, Nature 453 (2008) 495.	arxiv-papers	2013-09-21T12:27:48	2024-09-04T02:49:51.278288	{ "license": "Public Domain", "authors": "P. Barat, A. Giri, M. Bhattacharya, Nilangshu K. Das and A. Dutta", "submitter": "Amal Giri", "url": "https://arxiv.org/abs/1309.5468" }
1309.5507	# Microgrid - The microthreaded many-core architecture Irfan Uddin University of Amsterdam, The Netherlands [email protected] ###### Abstract Traditional processors use the _von Neumann_ execution model, some other processors in the past have used the _dataflow_ execution model. A combination of _von Neuman_ model and _dataflow_ model is also tried in the past and the resultant model is referred as _hybrid dataflow_ execution model. We describe a hybrid dataflow model known as the microthreading. It provides constructs for creation, synchronization and communication between threads in an intermediate language. The microthreading model is an abstract programming and machine model for many-core architecture. A particular instance of this model is named as the microthreaded architecture or the Microgrid. This architecture implements all the concurrency constructs of the microthreading model in the hardware with the management of these constructs in the hardware. ###### Contents 1. 1 Introduction 2. 2 Latency tolerance 3. 3 The microthreading model 4. 4 The Microgrid 5. 5 Programming the Microgrid 6. 6 The microthreading model in the context of the Microgrid 7. 7 I/O in the Microgrid 8. 8 Conclusion ## 1 Introduction Traditional processors are based on the _von Neumann_ execution model. In this model a sequence of instructions is executed one-by-one and the state of the program is identified by a single program counter. However, performance can not be improved by executing instructions sequentially. In order to improve the performance, _dataflow_ scheduling is used where instructions are executed based on the availability of data. Theoretically, dataflow models are parallel execution models, because instructions can be scheduled only with the availability of the data, and the scheduled instructions can be executed independently. Most of the out-of-order execution techniques are derived from dataflow scheduling. However, the execution of the program is still determined largely by the instruction sequence as dataflow scheduling is only applied to a few instructions in a small window over the sequential code. In contrast, Moonsoon [29] and Wavescalar [40] are based on dataflow models. The _von Neumann_ model is simple but sequential, the _dataflow_ model is parallel and improves the efficiency of the execution of the program but with the cost of adding complexity in the hardware design, therefore a _hybrid dataflow_ model [6, 17] was tried to combine the advantages of both. The issue with generalized dataflow model is the requirement of a matching store to detect which operations become ready to execute when its dependencies are satisfied. The efficient organization of the matching store is not so clear to the designers of the execution models. With a hybrid dataflow model, the ordering of instructions can be sequentialized to reduce the need for a big matching store. In hybrid dataflow models instructions are executed based on the _von Neumann_ model within a thread but there is support for dynamic execution of threads based on the _dataflow_ model. This model allowed multicore architectures to exploit parallelism in programs. P-RISC [33], Multiscalar [38] and DDM-CMP [39] are based on the _hybrid dataflow_ model. Modern many-core systems one way or other provide concurrency constructs to exploit parallelism. To quote Prof. Chris Jesshope _By 2020 we could see up to $10^{4}$ cores and $10^{6}$ hardware threads on a single chip_. The concurrency constructs introduced by the microthreading model are implemented in the instruction set of the microthreaded architecture which is also referred as the Microgrid. This architecture assumes a lot of concurrency in the applications and provides the concurrent architecture to exploit parallelism. The rest of the paper is organized as follows. In Section 2 we define the latency tolerance feature in computer architecture. We explain the details of the microthreading model in Section 3 and the details of the architecture based on this model in Section 4. We describe the programming of the Microgrid in Section 5. We explain the microthreading model in the context of the Microgrid in Section 6. We present the I/O in the Microgrid in Section 7 and conclude the paper in Section 8. ## 2 Latency tolerance In any program, a computation is preceded and followed by memory operation which takes a variable amount of time as it depends on the locality of the data i.e. the data is located in L1-, L2-,L3- caches or off-chip memory. In single-threaded programs the processor has to wait for memory operations to complete and then continue with the computation. In multi-threaded programs, when a memory operation is issued, the thread may be suspended and execution is switched to another available thread. Because of dataflow scheduling, the memory operation completes asynchronously and wakes up the suspended thread. The execution of program is tolerant to the long latency operations and is shown in fig. 1 assuming multiple instructions issued and executed in a single core. The use of multiple threads per core in order to hide latency has been understood for long time [53]. For instance, software time sharing is adequate to tolerate external I/O latencies but can not tolerate other long latency operations (e.g. memory access) because of the absence of fine-grained latency tolerance. Hardware multi-threading with dynamic scheduling is used to tolerate long latency operations e.g. MTA cores can switch on every miss operation but it requires a long pipeline flush. Niagara cores switch on the issue of a memory operation [37] and therefore tolerate long latency operations, but can not tolerate the latency in FPUs or other asynchronous operations (e.g. management of threads). Figure 1: Latency tolerance in multi-threaded programs assuming multiple instructions issued and executed in the core. ## 3 The microthreading model The microthreading model [18] is based on a type of _hybrid dataflow_ model and has evolved from DRISC [4] (Dynamically-scheduled RISC) which was proposed in 1996 with the goal of separating computation from communication. DRISC provides dataflow scheduling in RISC core, which executes instructions asynchronously and with multiple threads it can tolerate the latency of long latency operations. The innovation in this model is that instead of multiple threads, an ordered set of threads referred as families are used, which provide composability in programs. The operations to create families resemble the fork/join operations found in most parallel programming models. The microthreading model is actually a hybrid of von Neumann model, dataflow scheduling of threads. The microthreading model has been refined over the decade from a single processor model to an abstract machine model for many-cores and can capture as much concurrency as possible using families of threads in a dynamically evolving concurrency tree. A family is comparable to loop or function call in traditional programming. Any thread can create further families (any heterogeneous combination is supported by the model) showing a hierarchy of threads. An example concurrency tree of the microthreading model is shown in fig. 2, where every family is composed of some number of threads and every thread can create another family. Figure 2: An example concurrency tree created by the microthreading model. A family can create any combination of threads and the hierarchy of families can go until any level. As the overhead of creating and synchronizing threads in software is 10 to 100 thousands cycles on contemporary hardware [49]. The microthreading model shifts the perspective from software threads to hardware threads with concurrency management in hardware in order to reduce the cost from few thousands cycles to just a few cycles. The number of cycles taken by an instruction depends on the dynamic state of the architecture, but because of asynchronous completion and fine-grained latency tolerance the model has the potential to achieve the goal of RISC i.e. _one cycle per instruction_. In the microthreading model the throughput of the program can potentially demonstrate that every instruction takes one cycle to complete, assuming single instruction issued and executed in the core. ### 3.1 Communication and synchronization The microthreading model supports hybrid dataflow scheduling by using I-structures [3]. An I-structure is a data structure with the semantics of dataflow i.e. every element has a state of either _full_ or _empty_. Any operation accessing an element with _empty_ state is suspended. The state of the element is changed to _full_ asynchronously, and the suspended operation is released to access the element. This process is termed as _split-phase_ operation [54] which has two phases; requesting and consuming. A request to the desired data is issued, but if the data is not available, the request is suspended. The execution model can continue executing other computations while the request is in the progress. When the data arrives, the suspended operation is released and the instruction can consume the data. The microthreading model uses I-structure as a set of channels; globals and shared to support communication between threads. These channels are actually registers address using the register names of the underlying ISA. But for the sake of generality these registers are referred as channels. These channels have blocking read and non-blocking writes. By blocking we mean that the operation is suspended because of the unavailability of the data, and by non- blocking we mean that the operation is performed and the writing of the data complete asynchronously. Global channels are mapped to all threads in the family and have read only access. Shared channels provide the mechanism for the communication only between adjacent threads uni-directionally in the ordered sequence of threads in the family. The sharing of synchronizers between threads enables fast thread-to-thread synchronization, for example to implement dependencies in a loop. The creating thread communicate with the created threads using messages, which can access the synchronizing memory of a family remotely. The one-way communication is restricted but ensures a deadlock free communication [52]. The communication and synchronization between threads are shown in fig. 3 and more details can be found in [21, Sec. 4.3.3.3]. Figure 3: The communication between threads and the synchronization of the family. ### 3.2 Memory consistency model The microthreading model addresses consistency using two models for threads and families. A single thread is sequentially consistent, as all the instructions within a thread appear to execute in sequential order. A family is weakly consistent, as it follows the three properties of Weak Consistency: * • Any created thread can perform a read or write operation only after all the writes by the creating thread prior to the creation of the family. * • The family can be synchronized only when all previous write operations by the created threads are performed. * • The creating thread can perform a read or write operation only after the created family is synchronized. Memory consistency models [11, 28] have been defined for different programming and machine models. We present only two consistency models which are relevant to the microthreading model. * • Sequential Consistency (SC) : A system is sequentially consistent if the result of the execution of instructions is the same as if instruction of all the cores were executed in some sequential order. In addition, the instructions of each individual core appear in the same sequence as specified by the program. SC has two cases: 1. 1. A read operation by any processor/thread is allowed to perform only if all previous read and write accesses are performed globally. 2. 2. A write operation by any processor/thread is allowed to perform only if all previous read and write accesses are performed globally. * • Weak Consistency (WC) : A system is weakly consistent if the synchronizing operation is performed only if there is no pending ordinary (non-synchronizing) read or write operations, and any ordinary read or write operations can not be performed if there is a pending synchronizing operation. The weak consistency is defined in [11] but we are slightly modifying the terminology to match with the microthreading model. WC has three cases: 1. 1. All previous synchronizing operations must be performed before an ordinary operation is performed. 2. 2. All previous ordinary operations must be performed before the synchronizing variable. 3. 3. Synchronizing operations are sequentially consistent with respect to one another. ### 3.3 The notion of resources in the microthreading model The microthreading model is an abstract machine and programming model and therefore does not have physical resources, but it is designed with the notion of resources as the model addresses many-core architectures. A family of threads is allocated to a group of resources and are referred as _place_. These resources can be one or more cores, single or many threads etc. A family created on a _place_ will execute there until synchronized and a thread allocated to a core will execute there until terminated. In order to control the number of threads on a core per family, _windowsize_ is introduced as a run-time parameter. A programmer can carefully use this parameter to use only the requested number of resources, leaving resources for other families (or may be more important families). More importantly this parameter is used to avoid deadlock from dependencies down the concurrency tree. It can wisely be used by the programmer or compiler to ensure that at least one leaf of the concurrency tree can continue execution by using the available resources and hence avoid deadlocks. The _windowsize_ limits the number of threads executing in a family, similar to the concept as k-bounded loops [8] in dataflow scheduling. The parameter ’k’ is comparable to the _windowsize_ in the microthreading model. However, it should be noted that _windowsize_ is per core and ’k’ is per loop which may be distributed on many cores. The microthreading model introduces _break_ used by a thread to terminate the execution of the family. The _break_ will let the already created threads of the family to complete and will stop the creation of any new threads in the family. It is comparable to the _exit_ statement in loops or functions in traditional programming, and is useful when the synchronization of the family depends on some dynamic conditions. ### 3.4 Concurrency constructs The microthreading model defines concurrency constructs to use the concurrency provided by the hardware. But these constructs need to be introduced in some intermediate language in order to write programs. In this section we explain some of the concurrency constructs of the microthreading model and show that these constructs are defined in an intermediate language referred as SL [21, 22]. In Section 5 we will show an example program that will demonstrate the use of concurrency constructs of the microthreading model in an application. ### Allocation and deallocation of place A group of cores i.e. _place_ is required to be allocated before the creation of a family and released when the family is synchronized. * • $sep\\_alloc(<parameters>)$ and $sep\\_free(<parameters>)$ are used to allocate and de-allocate cores through a software API. The $<parameters>$ are a list of parameters passed for allocation and de- allocation e.g. specifying the strategy of allocation or number of cores etc. and can be found in [19]. ### Creation and synchronization of family A family of threads can be created by using $sl\\_create$ constructs similar to a loop in traditional programming. The created family can be synchronized using $sl\\_sync$ construct. * • $sl\\_create(fid,pid,start,limit,step,windowsize,options,thread[,<arguments>])$ defines the creation of a parameterized family. * • $fid$ is the identifier of the family of type $sl\\_family\\_t$. * • $pid$ is the identifier of the _place_ of type $sl\\_place\\_t$. * • $start$, $limit$ and $step$ indicate the starting, ending and iteration step counter of threads (similar to the iterators in a loop). * • $windowsize$ determines the upper limit on the number of threads that can be created on a core. * • $option$ decides the way the family should be created an executed. e.g. $sl\\_forceseq$ will force the family to execute sequentially. * • $thread$ is the name of the thread which defines the code to be executed by created threads in the family. * • $<argument>$ is a comma separated list of global and shared arguments passed to the thread. * • $sl\\_sync()$ is used for the synchronization of the family. ### Global and shared channels The global and shared channels introduced by the microthreading model are implemented as global and shared variable in the intermediate language. They are used as parameters/arguments to the creation of a family of thread. * • $sl\\_glarg(type,variable,value)$ and $sl\\_sharg(type,variable,value)$ indicates the global and shared $variable$ of $type$ with some $value$ passed from the creating thread to the created thread. * • $sl\\_glparm(type,variable)$ and $sl\\_shparm(type,variable)$ defines the global and shared $variable$ of $type$ received by the created thread from the creating thread. * • $sl\\_getp(variable)$ is used to read from global or shared $variable$. * • $sl\\_setp(variable,value)$ is used to write the shared $variable$ with a modified $value$ to be read by the next thread. * • $sl\\_seta(variable,value)$ is used by the creating thread to write a $value$ to the shared variable to the first thread of the family. * • $sl\\_geta(variable)$ is used by the creating thread to read the shared $variable$ from the last thread in the family. ### Starting and terminating threads Threads are implicitly created with the creation of a parameterized family and terminated when all of their instructions are executed. But programmers need to define the starting and terminating of a thread similar to the way functions are defined in traditional programming. * • $sl\\_def(type,<parameter>)$ is used to define a thread with return of $type$ and $<parameter>$ of a list of the global or shared variables used by the thread. * • $sl\\_enddef$ defines the terminating of the thread. * • $sl\\_index(index)$ can be used to retrieve the $index$ of the ordered set of threads in the family. ### Breaking family Any thread in a family can use the break statement to stop creation of new threads in the family. The syntax is given below:. * • $sl\\_break$ is used to stop creating new threads in the family. ### 3.5 More about families The microthreading model has evolved over the years and have introduced different types of families as per the requirements of the programs. All the families are created using $sl\\_create$ construct (except detached family, see below). The family can be of any of the type (described in this section) based on the way the created threads are executing. However, some types of families are need to be explicitly defined by passing a parameter to the create construct. In this section we give a brief overview of the different types of families supported by the microthreading model with the aim that programmers can write a parallel program using different combination of threads and families. Families that are required to be explicitly defined at the time of creation are explicitly stated in their respective subsection. #### Independent family A family is called independent family when its created threads do not require any communication between threads. Embarrassingly parallel applications generally do not require communication between created threads and the microthreading model supports these types of applications by creating them as independent families. #### Dependent family Threads in dependent family communicate with each other. Fine-grained parallel applications require a lot of communication and coarse-grained parallel applications communicate a little between threads. The microthreading model supports these types of application by creating dependent families. These families communicate with each other through shared channels introduced earlier (c.f. Section 3.1). Dependent families are inherently sequential and therefore do not get any speedup by distributing the threads on many cores. But in the microthreading model these threads get benefit from latency tolerance and asynchronous completion as a dependent family can be executed along with other families. #### Homogeneous family All families are statically homogeneous but can be made dynamically heterogeneous if the programmer calls/creates different functions in a thread dependent on the index of the thread. #### Heterogeneous family A family can be composed of threads that are not identical to each other. The programmer need to explicitly write the code for different threads based on the index of the thread to perform different operations. #### Detached family A family is required to be synchronized before continuing further. But there may be situations when a family may not want to synchronize and is called _detached_ family. For instance, creating a family that prints some characters on terminal may not require synchronization as all the created threads perform only read operations and do not modify any memory. The programmer has to explicitly specify the _detached_ family by creating a family with $sl\\_detach(<parameters>)$ instead of $sl\\_create(<parameters>)$, where $parameters$ are defined in Section 3. #### Exclusive family To support mutual exclusion in the microthreading model, exclusive families are introduced. Exclusive families are treated differently to regular families because of the requirement of being mutually exclusive. More details about mutual exclusion in the microthreading model can be found in [21, Sec. 14.1]. To create an exclusive family the programmer have to explicitly use $sl\\_exclusive$ in the place of $options$ in the create construct. Generally exclusive families consist of a single thread. #### Sequentialized family A family can be forced to execute sequentially, to avoid using resources by creating many threads. The programmer has to explicitly use $sl\\_forceseq$ at the place of $options$ in the create construct. The sequentialized family will execute as regular code of the parent thread. ## 4 The Microgrid Microgrid is actually a blueprint with parameters defined for the architecture. There are many Microgrids; with 1 core, 128 cores, 1000 cores, random bank memory, COMA etc. We use the term Microgrid to generalize all instances of the blueprint. At some time we talk about a particular instance, but then we explicitly give the details of the parameters. In the context of the Microgrid, a core is explicitly termed as microthreaded core to differentiate from traditional cores. In addition, a thread is termed as microthread to distinguish it from traditional threads. The basic component of the Microgrid is the microthreaded core, and it is important that we understand the execution of instructions in the core in order to simulate the core at the high-level. The concurrency constructs of the microthreading model are implemented in the instruction set (ISA) of the core. The cores are designed to be simple; single issue, in-order, 6-stage pipeline [20] and based on RISC. Some components of the operating system are implemented in the silicon of the core [7] e.g. scheduler, resource allocation, mapping etc. Some energy inefficient features are removed from the design of the core e.g. speculation, out-of-order execution and cache prefetching. Figure 4: The microthreaded core. The 6 stage pipeline of the core is shown in fig. 4, where we are not showing the scale of components, but mainly the layout of components. It is a classic RISC core with some modifications to existing components and adding some more components to support the concurrency constructs of the microthreading model in the core. The instruction fetch (IF), instruction decode (ID), register read (RR), execute (EX), memory (MEM), write back (WB), I-cache, D-cache, Integer register file, Floating register file and shared FPU are existing components but slightly modified in order to support the ISA of the Microgrid. The scheduler, allocator, network, thread table and family table are the newly added components to support the concurrency constructs of the microthreading model. The thread table and family table are used to store the thread contexts and family contexts. Every core also has a single exclusive context for creating an exclusive family and thread. The shared channels in the microthreading model are implemented in the registers of the cores. These registers are synchronizing and provide two ports for synchronous and asynchronous completion e.g. D-cache and FPU operations are completed asynchronously and therefore connected to the asynchronous ports of the registers. The bus interface connects the L1-cache to the snoopy bus and the network interface connects the core to the network of other cores on the chip. The threads in hardware are supported by the thread management unit through the instruction set of the core for creation and synchronization of threads. It has lowered the overhead of creation and synchronization of software threads from more than ten thousands of cycles to just few cycles. It also has extremely low overhead of context switching (zero cycle) and provides fine- grained interleaving i.e. interleaving at every cycle. However, this interleaving is not as strict as in traditional threads. Threads can execute multiple instructions from a single thread until a context switch is required. Interleaving can be bypassed to maximize pipeline usage if there is no other available thread. In single-threaded programs interleaving does not make sense anyway. To ensure fairness in the execution of threads, interleaving is enforced so that no thread can monopolize the execution time of the core. The binary code of a program generated for the microthreaded core can be executed on any other core that supports the same instruction set as Microgrid. The other core may not support concurrent execution and the code will be executed as single threaded. The code can also be executed on different number of thread slots per core. This is called binary code compatibility i.e. the binary code may not get the same performance but at least it will not fail, its performance will be gracefully degraded. As Microgrid provides many cores on a single chip, we show a group of four cores in fig. 5 to demonstrate the interconnection of large number of cores. The delegation network is a Network-on-Chip (NoC) where all cores are addressable from all other cores. Currently it is implemented as a fully connected network in MGSim, but could be implemented as a mesh network in the hardware implementation. A core is connected to the previous and the next core by a distribution network. Two cores share an FPU, and every core has an L1-cache which are connected to L2-cache by a snoopy bus. L2-caches are connected with each other in the distributed cache network. In order to handle deadlock in memory network, every L2-cache has an incoming and outgoing message buffer. An example layout of the 128-core Microgrid in a single chip is shown in fig. 6. Figure 5: A group of four cores, each core has an L1-cache, two cores share an FPU and four cores share an L2-cache. Figure 6: The layout of 128 cores on the Microgrid chip. ### 4.1 The communication network The Microgrid chip has two communication networks on the chip, in addition to the distributed memory network. #### Delegation network Every core is connected to every other core by a bi-directional fully connected network. This network is highly efficient because a message travels in 10 or even less cycles from source to destination. The current implementation of the Microgrid assumes a single cycle, node-to-node routing in a lightly loaded network, but in a loaded network it may take more cycles due to contention and buffering delays. Since delegation network is very efficient, it is used only for the concurrency management between cores that are not adjacent. #### Distribution network All the cores are connected in a single bi-directional daisy-chained network in a Moore curve. The curve can be chosen in a way to preserve locality in L1-caches and L2-caches. It takes two cycles to travel from one core to an adjacent core. The cycles taken by a return trip of a message from a core to any other core on the distribution network is given in Eq: 1, where $c$ is the number of cores. $delay=2\times 2\times c$ (1) The distribution network is used for the logical partitioning of the chip i.e. different parts of the program can be executed on different parts of the chip. It provides grouping of cores to be used for delegating a family, where the distribution network knows the starting core in the group and the size of the group. ### 4.2 Resource management in the hardware of the Microgrid An integer value in the source program is used to identify a group of adjacent cores where a family can be delegated. We refer to the group of cores as _place_ and the integer value that identify the place as _placeid_. The identifier can identify the starting core of the place and the size of the place. More details about places can be found in [24, 22] and [21, Chap. 11 & App. E]. A family can execute on the same core as the parent thread or the same place as the parent thread. The former is called _local_ place and the later is called _default_ place and are identified by place id as $0$ and $1$ respectively. The execution of a family on a place different than used by the parent thread is called _delegation_. We can perform some arithmetic operations on _placeid_ to derive the starting core and the size in the place. The starting core can be calculated as given in Eq: 2 and the size of the place can be calculated as given in Eq: 3. The size of the place is always in power of 2, and the starting core must have the _coreid_ which is multiple of the size of the place. This is a restriction, but easy in implementation as this information can be derived from integers. $starting\\_core\;\;=\;\;(placeid\;\;\&\;\;(placeid\;-\;1))\;>>\;1$ (2) $size\;\;=\;\;placeid\;\;\&\;\;-placeid$ (3) ### 4.3 Proposed software service to access resources in the Microgrid A software layer for the allocation of resources is defined in a protocol referred as SEP [19, 27] which provides an easy to implement and efficient management of resources. In the initial research work, a single core on the Microgrid is reserved for the operations of SEP. It works in mutually exclusive manner, and can soon become a bottleneck when a lot of requests are coming to this single core. Some research is ongoing in avoiding bottleneck and making the management hierarchical. SEP is implemented using the well-known binary buddy allocation [30] of the memory management. An example of buddy allocation used for cores in the Microgrid is shown in fig. 7. It shows a system having a total of eight cores with their identifiers given on the top. The time the system is initialized, all the cores are in one group. Then a request is issued to allocate one core and involves these steps: * • Divide 8 cores in half. * • Still bigger than required, divide 4 cores into half. * • Still bigger than required, divide 2 cores into half. * • We found a single core as requested, allocate the single core. A request for a given number of cores, will allocate the group of cores if available. In case the available group is larger than requested, divide the bigger group into smaller group until the group of requested cores is obtained. When the allocated cores are released, they are grouped into a larger group. In the given example when the last 4 cores are released it is grouped into larger group of 8 cores, bringing the system to the initial state. Figure 7: An example of the binary buddy allocation of cores by SEP. A programmer can choose different numbers of cores in a place. Every time a place is requested, a group of cores which are not allocated to any other family is allocated. The allocated cores can be used by the family (and sub families if any) to which the place is allocated, and can not be allocated to any other family until explicitly de-allocated. SEP provides two API to programmers; $sep\\_alloc()$ and $sep\\_free()$ for allocation and de- allocation respectively similar to $malloc$ and $free$ in C. The allocation and de-allocation of SEP can be performed asynchronously, but in the current implementation these processes are not asynchronous. The moment the allocation or de-allocation is issued by a thread, the thread will wait until the operation is completed. The allocation of place depends on the policy given below: * • Minimum: Allocate at least the number of cores specified (may be more if available). * • Maximum: Allocate at most the number of cores specified (may be less if unavailable). * • Exact: Allocate the exact number of cores. The allocation will fail if the given number of cores can not be allocated. * • Any size: Any available size in the power of 2 will be allocated, starting from the lowest available size of the group. ## 5 Programming the Microgrid In this section we show an example program i.e. Matrix Multiplication of equal sized matrices, to demonstrate the way programs are written for the Microgrid. The objective is to show that a sequential C program can be transformed to a microthreaded program easily by high-level programming languages, a compiler or with little effort by the programmer. We also show the concurrency constructs in the generated assembly to demonstrate the way the concurrency constructs are supported in the ISA of the Microgrid. #### Sequential C program The sequential C code for Matrix Multiplication of size $1000\times 1000$ is shown in listing LABEL:code:seq_matmul. We allocate three arrays; two for source matrices and one for the result matrix. Once the memory is allocated, we can fill these arrays with some numbers, but for saving space in the page, we assume the existing values present in those memory locations. To perform matrix multiplication we write three loops; outer loop, middle loop and inner loop. The inner loop is the one which multiplies the two elements of the source matrices and stores in the result matrix. After the multiplication is completed, we free the allocated memory. Listing 1: Sequential execution of Matrix Multiplication. ⬇ #include <stdio.h> int N = 1000; int main() { // Allocate memory for matrices int A = (int) malloc(N * N * sizeof(int)); int B = (int) malloc(N * N * sizeof(int)); int C = (int) malloc(N * N * sizeof(int)); int i, j, k; // Perform the multiplication // matmul_outer for(i=0; i<N; i++) { // matmul_middle for(j=0; j<N; j++) { //matmul_inner for(k=0; k<N; k++) { C[iN+j] += A[iN+k] * B[kN+j]; } } } free(A), free(B), free(C); return 0; } #### Microthreaded program The microthreaded program for the Matrix Multiplication of size $1000\times 1000$ written in SL is shown in listing LABEL:code:par_matmul. In the $t\\_main$ function, we first allocate a group of 8 cores in the Microgrid. Then we allocate three arrays for source and result matrices. Then we replace the outer loop by creating a family of $N$ threads. The threads in the outer family create further middle families and then each of those creates the inner family. The inner family perform the multiplication of elements in source arrays and store the multiplication in the resultant array. The transformation from sequential program to microthreaded program involves the creation of families and shared and global parameters. Once all the families are synchronized, the allocated memory to arrays and allocated cores are released. Listing 2: The microthreaded version of Matrix Multiplication. ⬇ #include <svp/sep.h> #include <stdio.h> int N = 1000; //matmul_inner sl_def(matmul_inner, void, sl_shparm(long, sum), sl_glparm(void, A), sl_glparm(void, B), sl_glparm(size_t, i), sl_glparm(size_t, j)) { sl_index(k); int (A)[N][N] = (int ()[N][N])(void) sl_getp(A); int (B)[N][N] = (int ()[N][N])(void) sl_getp(B); int v = (A)[sl_getp(i)][k] * (B)[k][sl_getp(j)]; sl_setp(sum, v + sl_getp(sum)); }sl_enddef //matmul_middle sl_def(matmul_middle, void, sl_glparm(void, A), sl_glparm(void, B), sl_glparm(void, C), sl_glparm(size_t, i)) { sl_index(j); sl_create(,,0,N,,,,matmul_inner, sl_sharg(long, sum, 0), sl_glarg(void, , sl_getp(A)), sl_glarg(void, , sl_getp(B)), sl_glarg(size_t, , sl_getp(i)), sl_glarg(size_t, , j)); sl_sync(); int (C)[N][N] = (int ()[N][N])(void) sl_getp(C); (C)[sl_getp(i)][j] = sl_geta(sum); }sl_enddef //matmul_outer sl_def(matmul_outer, void, sl_glparm(void, A), sl_glparm(void, B), sl_glparm(void, C)) { sl_index(i); sl_create(,,0,N,,,,matmul_middle, sl_glarg(void, , sl_getp(A)), sl_glarg(void, , sl_getp(B)), sl_glarg(void, , sl_getp(C)), sl_glarg(size_t, , i)); sl_sync(); }sl_enddef sl_def(t_main, void) { // Allocate a place using SEP int core = 8; sl_place_t pid; if (sep_alloc(root_sep, &pid, SAL_EXACT, core) == -1) { printf(”cannot␣allocate␣a␣place.”); exit(1); } // Allocate memory for matrices int A = (int) malloc(N * N * sizeof(int)); int B = (int) malloc(N * N * sizeof(int)); int C = (int) malloc(N * N * sizeof(int)); // Perform the multiplication sl_create(,pid,0,N,,,,matmul_outer, sl_glarg(void, , A), sl_glarg(void, , B), sl_glarg(void, , C) ); sl_sync(); free(A); free(B); free(C); sep_free(root_sep, &pid); } sl_enddef #### Microthreaded assembly code The microthreaded assembly code is shown in listing LABEL:code:assembly_matmul. We can see the assembly instructions of concurrency constructs i.e. $allocate$ for the allocation of family, $crei$ for creation of family, $sync$ for synchronization of family, $puts$ writing a shared variable, $gets$ reading from a shared variable etc. This is the innovation in the Microgrid, which avoids the mapping of software threads to hardware threads, which reduces the overhead from more than 10-100 thousands cycles of software threads to just few cycles in hardware threads. The ISA of program is scheduled by the microthreaded core with the hardware support for concurrency management. Listing 3: A snipper of the assembly program generated from the microthreaded program of Matrix Multiplication for the Microgrid. ⬇ … # Assembly code of thread matmul_inner .ent matmul_inner .registers 4 1 4 0 0 0 matmul_inner: ldpc $l3 ldah $l3, 0($l3) lda $l3, 0($l3) $matmul_inner..ng: ldah $l1,N($l3) ldl $l1,N($l1) mulq $l0,$l1,$l2 mulq $g2,$l1,$l1 addq $l2,$g3,$l2 addq $l1,$l0,$l0 s4addq $l2,$g1,$l2 s4addq $l0,$g0,$l0 ldl $l2,0($l2) ldl $l0,0($l0) mull $l2,$l0,$l0 addq $l0,$d0,$l0 mov $l0, $s0 end .end matmul_inner … # to show the concurrency constructs in the ISA .ent t_main … allocate/s $l0, 0, $l0 setstart $l0, 0 ldq $l1,matmul_outer($l17) setlimit $l0, $l9 setstep $l0, 1 setblock $l0, 0 wmb crei $l0, 0($l1) putg $l12, $l0, 0 putg $l11, $l0, 1 putg $l10, $l0, 2 sync $l0, $l1 mov $l1, $31 release $l0 … .end t_main #### SL tool-chain Figure 8: The SL tool-chain. The SL tool-chain is shown in fig. 8 and is built around SL [22] which is a C based language but is extended to express the concurrency constructs of the microthreading model. SL is an intermediate language intended for higher level programming languages (such as Single Assignment C or SAC [36, 13, 12, 14, 15], FastFlow [2, 1] etc.) and parallelizing C compiler [34, 35]. SAC is Matlab-like programming language, and it provides concurrency from a very high view. Programmer writes program using arrays and let the compiler decide to exploit concurrency in the programs. A number of tools and simulators are added to the designer’s toolbox and used for the evaluation of the Microgrid from different perspective. The SL compiler can generate binary for different implementations of the Microgrid. We have software libraries that provide the run-time systems for the microthreading model on the shared memory SMP machines and referred as _svp- ptl_ [51] and distributed memory for clusters/grids and are referred as Hydra [26] and _dsvp-ptl_ [50] The SL compiler can generate binary for UTLEON3 [9, 10], MGSim [5, 32] and HLSim [44, 45, 46, 43, 47, 48, 41, 42]. Unless specified otherwise, the SL compiler generates two implementations for every family in the given SL program; concurrent and sequential. Either one is used as per the dynamic state of the chip. In case resources can not be allocated to a family, the sequential version can be used. The sequential version of the program is used to avoid deadlock and if the deadlock can be avoided by static analysis, then suspension on resource allocation is safe. The programmer can force the program to use the concurrent version all the time i.e. threads are suspended until resources become available. ## 6 The microthreading model in the context of the Microgrid We gave details of the Microgrid, and would like to revisit the microthreading model in the context of the Microgrid. We need to understand all the details of the model and the architecture in order to simulate the architecture at a high level, as the high-level simulator have to exhibit the same behavior. ### 6.1 A concrete example Figure 9: A concrete example of the microthreaded program of the microthreading model executing in the single core of the Microgrid. We explicitly use a single core to show the coordination of different events in different threads over time. A concrete example of the microthreaded program in the microthreading model executing in a single core of the Microgrid is shown in fig. 9. All concurrency constructs are shown in the figure in the form of rectangles of different sizes/color. We also show the effect of concurrency constructs on the execution of threads. Every thread has an implicit _start_ and _end_ event. Threads are created at the rate of one thread per cycle. The creation process takes 4 cycles (with additional latency to load the cache-line from memory if not in the I-cache, also potential queueing delays if another family is being created). The instructions of a thread can be executed as soon as the thread is created which means the first instruction of the first thread can execute along with the creation of the other threads [23]. The process of creation of threads is implemented sequential on a single core, and is decided based on the trade-off between simple design and efficient creation of family. In the figure we can see that two _allocate_ events execute concurrently, but the _create_ events are sequentialized on a single core. We show a heterogeneous family of threads i.e. Threads $0$ and $1$ in Family $0$ in the first level is different from each other. We show a dependent family where threads are waiting for the previous threads to write the shared variable/register. We also show that the sync event is waiting for all the threads to complete. We show a homogeneous independent family where the parent threads is executing along with the child threads and therefore the sync does not wait as the time the sync is issued where all the created threads are completed already. ### 6.2 Family’s Life cycle A family passes through different stages in the Microgrid during the execution. In this section we consider an example when a parent thread is executing on a core and it delegates a family to a place of four cores shown in fig. 10. A detailed explanation of these stages is given in below section. Figure 10: The life cycle of a family during the execution in the Microgrid. #### Allocation It is implemented as try-to-allocate mechanism (see below for various options on allocate). In this process; family context, thread contexts and registers on all cores of the place are tried to be allocated. The parent thread sends a message on the delegation network to the first core. The first core checks the availability of at least one thread context, one family context and 31 registers. In case of success the message is passed to the next core of the place using distribution network. When all the cores succeed, the message comes back from the last to the first core on distribution network, and the contexts are asynchronously allocated. The first core then notifies the parent core using delegation network. The allocate protocol has three different modes: • Normal (or default): The allocation fails immediately after the allocation fails on any core in the place. * • Suspend: The allocation waits for the availability of resources. * • Exclusive: The allocation allocates exclusive contexts and in case of unavailability, keeps waiting until resources are acquired. The allocate protocol has four different strategies: * • Normal (or default): It will try to allocate as many cores as requested, possibly down to one core (in power of 2). * • Exact: It will allocate the family on all the requested cores. * • Single: It will allocate the entire family to the first core in the place. * • Balanced: It will allocate the entire family to the least loaded core of the place, in terms of family contexts. #### Configuration Once the contexts are allocated on the cores, the next message is sent on the delegation network to the first core to store parameters in the allocated family table entry for the bulk creation. It consists of $setstart$, $setlimit$, $setstop$, $setblock$ etc. which are derived from the create construct and defines the number of threads to be created per core. The configuration process is completed asynchronously and does not need any acknowledgement. Therefore as soon as the configuration message is issued, the next message can also be sent. In the current implementation, the delegation network preserves the order of messages i.e. configure messages will reach to the destination before create message. #### Creation The create message is sent from the parent core to the first core in the place via delegation network. As soon as the message is received by a core, thread creation is started asynchronously and the message is forwarded to the next core in the place. The first core will acknowledge the parent core when all the threads on the first core are created. The parent core may then write the global and/or shared registers to the first core through inter-context communication and can continue with its instructions (if any) or wait for the synchronization of the family. When all the threads are created, the core can de-allocate the contexts that are not really required e.g. at the time of allocation 31 registers were allocated, but suppose a thread is created which only uses 15 registers therefore the core will de-allocate the extra 16 registers. All the threads continue their execution on the allocated core, and as the threads get terminated the context get de-allocated asynchronously. Every core shares the information when it completes executing the allocated threads of a given family with the next core. The last core has the information that all the threads in the family are completed. #### Synchronization When all threads in the family complete, the last core will send a message on the delegation network to the parent core. The parent thread is activated so that the family can be synchronized. All the modified state from the family becomes defined. In the case of a dependent family, the parent core will also read the value from the register updated by the last thread through inter- context communication. #### Release A message is sent from the parent core to the first core on delegation network to release the family. This process will complete asynchronously and the parent core can continue its execution. ### 6.3 Thread’s life cycle Figure 11: The lifecycle of a thread during the execution on the core of the Microgrid. A thread passes through different stages during the execution and is shown in fig. 11. A newly created thread is allocated thread context, registers and PC (Program counter) is configured to place the thread in ready queue. In case the thread gets an I-cache miss the thread waits until the instructions are loaded from cache. The thread then passes to the active queue. The context of the thread is moved from the active queue to running where the instructions actually execute in the execute stage of the pipeline. A thread can be suspended when it is active or running at the write back stage of the pipeline e.g. in case of dependency. The asynchronous completion of a thread moves the thread from the suspended stage to ready queue. At the termination of the thread the allocated entries are cleaned up which can be used by any newly created threads. ### 6.4 Distribution of threads Figure 12: Distribution of independent threads on 4 cores with 5 window size. The distribution of threads to the cores in a place depends on the type of family i.e. dependent or independent. The threads in a dependent family are restricted to single core only, as no significant speedup can be obtained by distributing them on multiple cores. However instructions in the thread will get benefit from latency tolerance on asynchronous completion in loads, store and floating point operations. Threads of an independent family are distributed by an equal distribution of threads per core. In fig. 12 we show an example of a family with 40 threads on 4 cores with window size of 5. Threads are distributed as 40/4 and every core can create 10 threads. But the window size is 5 and therefore only 5 threads can be activated at a time on a given core. As soon as a thread terminates, a new thread can use the context of the terminated thread. For instance on core x, threads 0,1 and 2 are terminated, threads 3-7 are executing and threads 8 and 9 are still waiting to be created. ### 6.5 Communication through registers Figure 13: Allocation of registers to threads and their communication through registers. Every thread created in the Microgrid uses a set of registers in the register file. The channels of the microthreading model used for the communication and synchronization of the family introduced in Section 3.1 are implemented in registers of the Microgrid. The registers allocated to a thread are categorized as; globals, locals, shareds and dependents. The mapping of registers to threads and their communication through registers is shown in fig. 13. Some explanation of these registers are given as: * • Global registers implements global channels and are visible to all threads in the family. The parent thread writes to the global registers and all created threads in the family can read from them. * • Local registers are only visible to the individual thread only. * • Dependent registers implements shared channels and have read-only access to the shared registers from the previous threads. * • Shared registers implements shared channels and have the write access by the current threads, and carry the modified value to be read by the next thread. ## 7 I/O in the Microgrid The Microgrid requires a decentralized approach where not every core is connected to I/O but some specialized microthreaded cores are used to support I/O and referred as I/O cores. I/O cores have limited/extended instruction set [25] compared to the regular microthreaded core and has no floating point operations. The instruction set of the I/O core supports the I/O infrastructure and are connected to the delegation network of the Microgrid. We do not simulate I/O cores in the current implementation of HLSim, therefore we refer readers to [16, 21, 31] for the I/O management in the Microgrid. ## 8 Conclusion The microthreading model is a hybrid dataflow model and provides the simplicity of _von Neumann_ model and asynchronous completion of the _dataflow_ execution model. It is an abstract machine model for many-cores architecture and shifts the perspective from software threads to hardware threads. Because of asynchronous completion and fine-grained latency tolerance the microthreading model can potentially achieve the goal of RISC i.e. _one instruction per cycle_ in the throughput of the program assuming single issue width. 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1309.5551	# Design space exploration in the microthreaded many-core architecture Irfan Uddin University of Amsterdam, The Netherlands [email protected] ###### Abstract Design space exploration is commonly performed in embedded system, where the architecture is a complicated piece of engineering. With the current trend of many-core systems, design space exploration in general-purpose computers can no longer be avoided. Microgrid is a complicated architecture, and therefor we need to perform design space exploration. Generally, simulators are used for the design space exploration of an architecture. Different simulators with different levels of complexity, simulation time and accuracy are used. Simulators with little complexity, low simulation time and reasonable accuracy are desirable for the design space exploration of an architecture. These simulators are referred as high-level simulators and are commonly used in the design of embedded systems. However, the use of high-level simulation for design space exploration in general-purpose computers is a relatively new area of research. ###### Contents 1. 1 Introduction 2. 2 Design space exploration in embedded systems 3. 3 Application -dependent and -independent DSE 4. 4 Design space exploration in general-purpose systems 5. 5 Design space exploration in the Microgrid 6. 6 Conclusion ## 1 Introduction Simulators with high simulation speed and less complexity are desirable for early Design Space Exploration (DSE) of the architecture. Any decision to improve the architecture becomes more expensive and requires more effort at the later stage and requires more effort, time and budget. DSE is performed in all kinds of computer systems. However, in the embedded systems domain the use of high-level simulation for DSE purposes has been accepted as an efficient approach for more than a decade. In that sense, the DSE in embedded systems pioneered the high-level simulation techniques. Therefore, in this paper we will give details about DSE in embedded systems using high-level simulators. We will explain that general-purpose computers are getting more complex and therefore high-level simulators are also required for the DSE. Since Microgrid is a complex architecture therefore we need to know its design space before we present the high-level simulation techniques for the Microgrid. The rest of the paper is organized as follows. In Section 2 we give an explanation of DSE in embedded systems. In Section 3 we differentiate the application dependent DSE and independent DSE. We give the DSE for the Microgrid in 5 and conclude the paper in 6. ## 2 Design space exploration in embedded systems Embedded systems perform predefined tasks and therefore have particular design requirements. They are constrained in terms of performance, power, size of the chip, memory etc. They generally address mass products and often run on batteries, and therefore need to be cheap to be realized in silicon and power efficient. Modern embedded systems, typically have a heterogeneous MultiProcessor-System-on-Chip (MP-SoC) architecture, where a component can be fully programmable processor for general-purpose application or a fully dedicated hardware for time critical applications. This heterogeneity makes embedded systems more complex and therefore designers use high-level simulator to perform DSE at an early stage, because high-level simulators take less effort to develop and less time in executing applications. In this section we describe the high-level simulation technique used for the DSE in embedded systems. Different high-level simulation techniques are introduced for the DSE in embedded systems, and are often based on the separation of concerns [26] between application, architecture and mapping functions. DSE in embedded systems is generally application-dependent or scenario-based. Traditional embedded systems are targeting one particular architecture and application, the aim is to explore the design for improvement based on certain objective. Scenario-based DSE [43] is the process of mapping every individual process of an application to every architecture component with different configurations. This mapping results in an exponential number of mapping choices i.e. design space. We show an example in fig. 1 (taken from [21]) to demonstrate that only three processes are mapped to three architecture components, but the resulted design space is large. Figure 1: The mapping of three application processes to three architecture components resulting into a large number of design space to be explored. Ideally DSE would like to considers all possible mappings, but an exhaustive search is infeasible. Therefore computer architects use design pruning to optimize the search through the design space to speed up the DSE. A smart DSE intelligently evaluates a small fraction of the design space to come up with a sub-optimal solution. These choices have a crucial impact on the success of the final product. DSE addresses multiple objectives [15] e.g. maximum performance, minimum power consumption and less complex components. It is very difficult to have a single solution that meets all the objectives simultaneously. The main problem is that the objectives are conflicting e.g. low power generally means bad performance or good performance means high power usage. Therefore a set of solutions are selected based on a Pareto optimal front [1], where solutions are not dominated by any other solution looking for the same objectives. ### 2.1 Related work High-level simulators have been used for the DSE in embedded systems domain for more than a decade, and are used in the research of academia and industries. Below are some of the research groups using high-level simulation for DSE in embedded systems. There might exist other areas of research in using high-level simulation for the DSE of embedded systems. * • Sesame, University of Amsterdam [16]. * • (Metro)Polis, University of California, Berkeley [50]. * • Mescal, University of California, Berkeley [12]. * • Milan, University of Southern California, Los Angeles [2]. * • The octopus toolset, University of Eindhoven [4]. * • SystemC-based environment, STMicroelectronics [47]. ## 3 Application -dependent and -independent DSE We want to clearly distinguish between application-dependent DSE in traditional embedded systems and application-independent DSE in modern embedded system or general-purpose computers. In traditional embedded systems, applications are statically mapped to different configurations of an architecture using some mapping functions. Based on the simulation results, innovative ideas can be generated which can improve application, mapping and architecture separately. In modern embedded systems we do not have one particular application or scenario, but a range of applications targeted to a different configurations of the architecture. For instance in smart phones it is not only one type of application that can statically be mapped, but a range of different types of applications are required to be explored on the different configuration of the architecture. In a way modern embedded systems are converging to the general- purpose systems. The range of applications increases in general-purpose computers, where a variety of applications can be executed on the given architecture. In these situations, the mapping of the application to architectural component can not be analyzed statically but instead the code patterns in algorithms are analyzed, and then different processes of an application are dynamically mapped to different parts of the chip based on certain objectives. Because of the dynamic mapping, application-independent DSE is not as trivial as scenario-based DSE. Design pruning is more structured in traditional embedded systems. For instance, genetic algorithms, simulated annealing etc. are some of the structured techniques that are commonly used in design pruning. However, for design pruning in modern embedded systems or general-purpose systems, there exists no structured solution that can dynamically determine a reduction in the design space to optimize the search. ## 4 Design space exploration in general-purpose systems The growing number of cores and size of the on-chip memory are creating significant challenges for evaluating the design space of future general- purpose computers. We need scalable and fast simulators for the exploration of large number of cores on a chip within limited development time and budget. Commercially available processors available in the market have few cores on a chip e.g. Intel E708800 Series, IBM’s POWER7 and AMD’s Opteron 600 Series. In the near future we believe there will be hundreds of cores per chip and DSE at the early stage can no longer be avoided [7] in general-purpose computers, as the number of mapping an application explodes as the number of cores increases. The use of high-level simulators for the DSE in general-purpose computers is relatively new compared to embedded systems domain. A number of simulation techniques are in research to develop high-speed simulators for the DSE of general-purpose computers with less complexity and shorter development time then conventional cycle accurate simulator. These simulation techniques are diverse and do not follow one particular pattern. In this section we give details of some high-level simulation techniques. There might exist other high-level simulations targeting general-purpose computers. ### 4.1 Interval simulation Interval simulation [7, 18] is a high-level simulation technique for the DSE of super-scalar single- and multi- core processors. It raises the level of abstraction from detailed simulation by using analytical models to derive the timing simulation of individual cores without the detailed execution of instructions in all the stages of the pipeline. The model is based on deriving the execution of an instruction stream in intervals. An interval is decided based on the miss events e.g. branch misprediction, cache misses, TLB misses etc. With interval analysis, execution time is partitioned into discrete intervals using miss events. The analytical models of every core cooperate with miss events in the system, and can be extended to model the tight interleaving of threads in multi-core processors. Interval simulation framework has two parts; functional simulation and timing simulation, and are connected with each other through a queue. The functional simulator feeds instructions into the tail of the queue and the timing simulator reads those instructions from the head of the queue. The functional simulator generates a dynamic instruction stream, including user-level and system-level code and is subsequently fed into the timing simulator. The timing simulator analyzes the code and advances the simulation time as per the time required to execute an instruction stream. In case of I-cache miss, branch misprediction and long latency load operations the simulation time is advanced by the miss latency, branch resolution time plus the front-end pipeline depth and long latency operations respectively. #### Discussion Interval simulator only simulates a small number of cores in super-scalar machines which disregards hardware microthreading and therefore the complexity of simulating latency tolerance is not encountered. In the Microgrid we can have more than 100 cores on the chip, and the architecture is completely different than super-scalar machine, as it provides fine-grained latency tolerance based on data-flow scheduling. The way programs can be written for the Microgrid is also different. Therefore interval simulation can not directly be used for the DSE of the Microgrid. However, we have learned some techniques from interval simulation and have used these in HLSim. For instance, in interval simulation in case of a cache miss the simulation time is advanced with the addition of cache miss latency. In HLSim we advance the simulation time with the cache miss latency but adjusted with a latency tolerance factor based on the number of active threads. Because in case of latency tolerance the cache miss latency can be shorter than the latency without any latency tolerance. ### 4.2 Statistical simulation Statistical simulation has gained interest over the past few years, as it speeds up simulation by providing short running synthetic traces. The execution of the original benchmarks is profiled and the key execution characteristics are captured in a synthetic trace, which closely exhibits similar execution characteristic as original benchmarks. The key benefit of statistical simulation is that the synthetic trace clones the dynamic instruction count with several orders of magnitude smaller than in the original benchmarks, and therefore reduces the simulation time dramatically. Nussbaum and Smith [29] and Hughes and Li [19] use statistical simulation paradigm to evaluate multithreaded programs running on shared-memory multiprocessor (SMP) systems. They have extended the statistical simulation to model synchronization and accesses to shared memory. Genbrugge and Eeckhout [13, 17] use statistical simulation to measure some execution characteristics in the statistical profile to be able to accurately simulate shared resources in multi-core processors. #### Discussion Statistical simulation is a trace driven simulation technique. A synthetic trace is generated which can be reduced to a shorter trace and is representative of the large trace of the benchmarks. The problem with this technique is that the original trace files can be very large which consume space and this technique can not consider the dynamic adaptation of multiple applications on the chip. The high-level simulation of the Microgrid, is execution driven i.e we dynamically generate events which are representative of the instruction count in the basic block in a thread. These events are mapped to the architecture and represent the execution of the application with fine-grained interleaving. The events have information of a short piece of code and therefore statistical simulation techniques wee not a suitable choice to be used in HLSim. ### 4.3 Sampled simulation The basic idea of sampled simulation is to simulate a number of sampling units rather than the entire dynamic instruction stream. The sampling units are selected either randomly [9], periodically [49] or based on phase analysis [33]. Different research in the multithreaded and multi-core processors simulation is using sampled simulation. Van Biesbrouck et al. [42] propose the co-phase matrix for speeding up sampled simultaneous multithreading (SMT) processor simulation running multi-program workloads. Stenstrom et al. [14] are researching the premise that fewer sampling units are enough to estimate overall performance for larger multi-processor systems than for smaller multi- processor system in case one is interested in aggregate performance only. Wenisch et al. [48] have obtained similar conclusions of throughput in server workloads. Barr et al. [3] proposes the Memory Timestamp Record (MTR) to store micro-architecture state (cache and directory state) at the beginning of the sampling unit as a checkpoint. #### Discussion Sampled simulation is also a trace-based simulation technique which suffers from the large trace files to be processed and changing an application results in producing and analysing a different trace file. Every time there is some optimization in the application, a new trace needs to be generated and analyzed. ### 4.4 Related works There are other simulation techniques used in the design space exploration of general-purpose computers given below. * • FPGA prototypes: They have low little simulation time, high accuracy and are useful in DSE. However these simulations require more development time and are more complex. They also suffer from combinatoric explosion of considering many low level parameters during design space exploration. Some examples are: [30, 31, 8, 46]. * • Trace simulation: These simulation techniques generate large execution traces from benchmarks, and are used for the evaluation of the architecture. They avoid the extremely large analysis of the application, by executing the program only one time, generating the trace and mapping it to the trace to different configuration of the architecture. However a large storage is required in order to store the large traces and a change in the application requires a different trace to be generated. Statistical simulations and sampled simulations are some of the techniques that addresses the reduction of the large trace files. Some example are: [9, 20, 27]. ## 5 Design space exploration in the Microgrid ### 5.1 Microgrid The Microgrid [24, 5, 22] is a general-purpose, many-core architecture developed at the University of Amsterdam which implements hardware multi- threading using data flow scheduling and a concurrency management protocol in hardware to create and synchronize threads within and across the cores on chip. The suggested concurrent programming model for this chip is based on fork-join constructs, where each created thread can define further concurrency hierarchically. This model is called the microthreading model and is also applicable to current multi-core architectures using a library of the concurrency constructs called _svp-ptl_ [45] built on top of pthreads. In our work, we focus on a specific implementation of the microthreaded architecture where each core contains a single issue, in-order RISC pipeline with an ISA similar to DEC/Alpha, and all cores are connected to an on-chip distributed memory network [23, 6]. Each core implements the concurrency constructs in its instruction set and is able to support hundreds of threads and their contexts, called microthreads and tens of families (i.e. ordered collections of identical microthreads) simultaneously. A number of tools and simulators are added to the designer’s toolbox and used for the evaluation of the Microgrid from different perspective. The compiler for the Microgrid [25] can generate binary for different implementations of the Microgrid. We have software libraries that provide the run-time systems for the microthreading model on the shared memory SMP machines and referred as _svp-ptl_ [45] and distributed memory for clusters/grids and are referred as Hydra [28] and _dsvp-ptl_ [44] The SL compiler can generate binary for UTLEON3 [10, 11], MGSim [6, 32] and HLSim [37, 38, 39, 36, 40, 41, 34, 35]. HLSim is a high-level simulation technique aimed for the DSE of the Microgrid and is based on discrete event simulation technique. It is execution driven simulator and therefore does not suffer from the large size of trace files. The events are dynamically mapped to the architecture at run time. We have built the simulator from scratch without using any off-the-shelf code, but some simulation techniques from Sesame and Interval simulation were used during the development for inspiration. ### 5.2 Design space in the Microgrid The Microgrid is a complex many-cores architecture and therefore has a huge design space for complex application. In order to have an efficient and validated system in the silicon we need to perform DSE in the Microgrid to explore the performance of different applications on the different configurations of the architecture. After DSE we can perform design pruning to change these parameters that affect the performance. We categorize the design space of the Microgrid as: * • Static architectural parameters: 1. 1. Thread table size 2. 2. Family table size 3. 3. Frequency of cores and memory 4. 4. Number of cores sharing an FPU 5. 5. Frequency of delegation and distribution network 6. 6. Size of L1-cache and L2-cache 7. 7. Associativity of L1-cache and L2-cache 8. 8. Number of L1-caches sharing L2-cache 9. 9. Number of L2-caches in low-level ring 10. 10. Number of low-level rings associated in the top-level ring 11. 11. Distribution of address space of RAM into banks 12. 12. Size of directory and root directory 13. 13. The memory architecture 14. 14. Synchronization-aware protocol * • Dynamic application parameters: 1. 1. Place size 2. 2. Window size 3. 3. Cold caches There are some other parameters that are very low-level e.g. size of the chip, FPU frequency, pipeline stages, the way cores are distributed on the chip etc. We have shown only the parameters that we will simulate in the current implementation of HLSim for the design space exploration in the Microgrid. ## 6 Conclusion DSE is required in all kind of computer systems. The use of high-level simulators for DSE is pioneered in embedded systems and getting popular in general-purpose systems. 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1309.5589	# A generalization of Ćirić fixed point theorems Nguyen Van Dung Faculty of Mathematics and Information Technology Teacher Education, Dong Thap University, Cao Lanh, Dong Thap 871200, Viet Nam [email protected], [email protected] , Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Khru, Bangkok 10140, Thailand [email protected] and Kanokwan Sitthithakerngkiet Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of North Bangkok (KMUTNB), Wongsawang, Bangsue, Bangkok 10800, Thailand [email protected] ###### Abstract. In this paper, we state and prove a generalization of Ćirić fixed point theorems in metric space by using a new generalized quasi-contractive map. These theorems extend other well known fundamental metrical fixed point theorems in the literature (Banach [1], Kannan [11], Nadler [13], Reich [15], etc.) Moreover, a multi-valued version for generalized quasi-contraction is also established. ###### Key words and phrases: Ćirić fixed point, metric space ###### 2000 Mathematics Subject Classification: Primary 47H10, 54H25; Secondary 54D99, 54E99 ## 1\. Introduction The Banach’s contraction principle [1] which was first appeared in 1922 is one of the most useful and important theorems in classical functional analysis. Its utility is not only to prove that, in a complete metric space $X$, the contraction map $T$ (i.e., $d(Tx,Ty)\leq\alpha d(x,y)$ for some $0\leq\alpha<1$ and for all $x,y\in X$) has a unique fixed point but also to show that the Picard iteration converges to the fixed point. For the reason that the contraction must be continuous, there are many researchers establish the fixed point theorems on various classes of operators that are weaker than contractive conditions but are not continuous, see for example [11, 15]. One of the most well-known results in generalizations of Banach’s contraction principle which the Picard iteration still converges to the fixed point of map is the Ćirić fixed point theorem [4]. Before providing the Ćirić fixed point theorem, we recall that a self-map $T$ on a metric space $(X,d)$, is said to be a _quasi-contraction_ iff there exists a nonnegative number $q<1$ such that for all $x,y\in X$, (1.1) $\displaystyle d(Tx,Ty)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\big{\\}}.$ The Ćirić fixed point theorem is given by the following theorem. ###### Theorem 1.1 ([4], Theorem 1). Let the metric space $X$ be $T$-orbitally complete and let $T$ be a quasi- contraction. Then we have 1. (1) $T$ has a unique fixed point $x^{}$ in $X$. 2. (2) $\lim\limits_{n\rightarrow\infty}T^{n}x=x^{}$ for all $x\in X$. 3. (3) $d(T^{n}x,x^{})\leq\cfrac{q^{n}}{1-q}d(x,Tx)$ for all $x\in X$. This result was generalized to many results, such as a common fixed point theorem of nonlinear contraction [17, Theorem 4], a generalized $\varphi$-contraction [2, Section 2.6], a Ćirić almost contraction [3, Theorem 3.2] and see also [5, 10, 12]. But from the well-known result of Rhoades [14] in 1977 to recent surveys, in Berinde [2] and Collaco and Silva [6] for instance, there were no any other value added to quasi-contraction condition. On the other hand, the Banach’s contraction principle has been extended to multi-valued contractions by Nadler [13] and see also [7, 8, 9, 16]. In this paper, we define a new generalized quasi-contraction by adding four new values $d(T^{2}x,x)$, $d(T^{2}x,Tx)$, $d(T^{2}x,y)$, $d(T^{2}x,Ty)$ to a quasi-contraction condition. Also, an example is presented. After that we state and prove unique fixed point theorems which are the generalization of Ćirić fixed point theorem in [4]. Moreover, we also establish fixed point theorems for multi-valued generalized quasi-contraction. ## 2\. Preliminaries First, we recall some notions which will be used in what follows. Let $(X,d)$ be a metric space and $A,B$ be any two subsets of $X$. We denote $\displaystyle D(A,B)$ $\displaystyle=$ $\displaystyle\inf\big{\\{}d(a,b):a\in A,b\in B\big{\\}}$ $\displaystyle\rho(A,B)$ $\displaystyle=$ $\displaystyle\sup\big{\\{}d(a,b):a\in A,b\in B\big{\\}}$ $\displaystyle BN(X)$ $\displaystyle=$ $\displaystyle\big{\\{}A:\emptyset\neq A\subset X\text{ and }\delta(A)<+\infty\big{\\}},$ where $\delta(A):=\sup\big{\\{}d(a,b):a,b\in A\big{\\}}$. ###### Definition 2.1 ([4]). Let $T:X\longrightarrow X$ be a map on metric space. For each $x\in X$ and for any positive integer $n$, put $O_{T}(x,n)=\\{x,Tx,\ldots,T^{n}x\\}~{}and~{}O_{T}(x,+\infty)=\\{x,Tx,\ldots,T^{n}x,\ldots\\}.$ The set $O_{T}(x,+\infty)$ is called the _orbit_ of $T$ at $x$ and the metric space $X$ is called _$T$ -orbitally complete_ if every Cauchy sequence in $O_{T}(x,+\infty)$ is convergent in $X$. Note that every complete metric space is $T$-orbitally complete for all maps $T:X\longrightarrow X$. The following example shows that there exists a $T$-orbitally complete metric space but it is not complete. ###### Example 2.2. Let $(X,d)$ be a metric space which is not complete and $T:X\longrightarrow X$ be the map defined by $Tx=x_{0}$ for all $x\in X$ and some $x_{0}\in X$. Then $(X,d)$ is a $T$-orbitally complete metric space which is not complete. ###### Definition 2.3 ([4]). Let $F:X\longrightarrow BN(X)$ be a multi-valued mapping. Let $x_{0}\in X$, an _orbit_ of $F$ at $x_{0}$ is a sequence $\big{\\{}x_{n}:x_{n}\in Fx_{n-1},n\in\mathbb{N}\big{\\}}.$ A space $X$ is called to be _$F$ -orbitally complete_ if every Cauchy sequence which is a subsequence of an orbit of $F$ at $x$ for some $x\in X$, converges in $X$. Next, the definitions of generalized quasi-contraction for single-valued and multi-valued are given as follows; ###### Definition 2.4. Let $T:X\longrightarrow X$ be a mapping on metric space $X$. The mapping $T$ is said to be a _generalized quasi-contraction_ iff there exists $q\in[0,1)$ such that for all $x,y\in X$, $\displaystyle d(Tx,Ty)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx),$ $\displaystyle d(T^{2}x,x),d(T^{2}x,Tx),d(T^{2}x,y),d(T^{2}x,Ty)\big{\\}}.$ ###### Example 2.5. Let $X=\\{1,2,3,4,5\\}$ with $d$ defined as $d(x,y)=\left\\{\begin{array}[]{ll}0&\hbox{ if }x=y\\\ 2&\hbox{ if }(x,y)\in\big{\\{}(1,4),(1,5),(4,1),(5,1)\big{\\}}\\\ 1&\hbox{ otherwise.}\end{array}\right.$ Let $T:X\longrightarrow X$ be defined by $T1=T2=T3=1,T4=2,T5=3.$ Then, we have $d(Tx,Ty)=d(1,1)=0~{}\text{ if }~{}x,y\in\\{1,2,3\\};$ $d(T1,T4)=d(T2,T4)=d(T3,T4)=d(1,2)=1;$ $d(T1,4)=d(T2,4)=d(T3,4)=d(1,4)=2;$ $d(T1,T5)=d(T2,T5)=d(T3,T5)=d(1,3)=1;$ $d(T1,5)=d(T2,5)=d(T3,5)=d(1,5)=2;$ $d(T4,T5)=d(2,3)=1;$ $d(4,5)=d(4,T4)=d(5,T5)=d(4,T5)=d(5,T4)=1;$ $d(T^{2}4,4)=d(T2,4)=d(1,4)=2;$ $d(T^{2}5,5)=d(T3,5)=d(1,5)=2.$ The above calculations show that $T$ is not quasi-contraction for $x=4$ and $y=5$ because there is no a nonnegative number $q<1$ satisfying the equation (2.4). However, $T$ is generalized quasi-contraction since the (2.4) holds for some $q\in\big{[}0.5,1\big{)}$ and for all $x,y\in X$. ## 3\. The main results On the following results, we state and prove the new fixed point theorems which are general cases of the Ćirić fixed point theorem. ###### Theorem 3.1. Let $(X,d)$ be a metric space. Suppose that $T:X\longrightarrow X$ is a generalized quasi-contraction and $X$ is $T$-orbitally complete. Then we have 1. (1) $T$ has a unique fixed point $x^{}$ in $X$. 2. (2) $\lim\limits_{n\rightarrow\infty}T^{n}x=x^{}$ for all $x\in X$. 3. (3) $d(T^{n}x,x^{})\leq\cfrac{q^{n}}{1-q}d(x,Tx)$ for all $x\in X$ and $n\in\mathbb{N}$. ###### Proof. (1). Step 1. _$T$ has a fixed point._ For each $x\in X$ and $1\leq i\leq n-1$ and $1\leq j\leq n$, we have $\displaystyle d(T^{i}x,T^{j}x)$ $\displaystyle=$ $\displaystyle d(TT^{i-1}x,TT^{j-1}x)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(T^{i-1}x,T^{j-1}x),d(T^{i-1}x,TT^{i-1}x),d(T^{j-1}x,TT^{j-1}x),d(T^{i-1}x,TT^{j-1}x),$ $\displaystyle d(T^{j-1}x,TT^{i-1}x),d(T^{2}T^{i-1}x,T^{i-1}x),d(T^{2}T^{i-1}x,TT^{i-1}x),d(T^{2}T^{i-1}x,T^{j-1}x)$ $\displaystyle d(T^{2}T^{i-1}x,TT^{j-1}x)\big{\\}}$ $\displaystyle=$ $\displaystyle q\max\big{\\{}d(T^{i-1}x,T^{j-1}x),d(T^{i-1}x,T^{i}x),d(T^{j-1}x,T^{j}x),d(T^{i-1}x,T^{j}x),$ $\displaystyle d(T^{j-1}x,T^{i}x),d(T^{i+1}x,T^{i-1}x),d(T^{i+1}x,T^{i}x),d(T^{i+1}x,T^{j-1}x)$ $\displaystyle d(T^{i+1}x,T^{j}x)\big{\\}}$ $\displaystyle\leq$ $\displaystyle q\delta\big{[}O_{T}(x,n)\big{]}$ where $\delta\big{[}O_{T}(x,n)\big{]}=\max\big{\\{}d(T^{i}x,T^{j}x):0\leq i,j\leq n\big{\\}}$. From (3), since $0\leq q<1$, there exists $k_{n}(x)\leq n$ such that (3.2) $d(x,T^{k_{n}(x)}x)=\delta\big{[}O_{T}(x,n)\big{]}.$ Then we have $\displaystyle d(x,T^{k_{n}(x)}x)$ $\displaystyle\leq$ $\displaystyle d(x,Tx)+d(Tx,T^{k_{n}(x)}x)$ $\displaystyle\leq$ $\displaystyle d(x,Tx)+q\delta\big{[}O_{T}(x,n)\big{]}$ $\displaystyle=$ $\displaystyle d(x,Tx)+qd(x,T^{k_{n}(x)}x).$ It implies that (3.3) $\delta\big{[}O_{T}(x,n)\big{]}=d(x,T^{k_{n}(x)}x)\leq\cfrac{1}{1-q}d(x,Tx).$ For all $n,m\leq 1$ and $n<m$, it follows from the generalized quasi- contractive condition of $T$ and (3.3) that $\displaystyle d(T^{n}x,T^{m}x)$ $\displaystyle=$ $\displaystyle d(TT^{n-1}x,T^{m-n+1}T^{n-1}x)$ $\displaystyle\leq$ $\displaystyle q\delta\big{[}O_{T}(T^{n-1}x,m-n+1)\big{]}$ $\displaystyle=$ $\displaystyle qd(T^{n-1}x,T^{k_{m-n+1}(T^{n-1}x)}T^{n-1}x)$ $\displaystyle=$ $\displaystyle qd(TT^{n-2}x,T^{k_{m-n+1}(T^{n-1}x)+1}T^{n-2}x)$ $\displaystyle\leq$ $\displaystyle q^{2}\delta\big{[}O_{T}(T^{n-2}x,k_{m-n+1}(T^{n-1}x)+1)\big{]}$ $\displaystyle\leq$ $\displaystyle q^{2}\delta\big{[}O_{T}(T^{n-2}x,m-n+2)\big{]}$ $\displaystyle\leq$ $\displaystyle\ldots$ $\displaystyle\leq$ $\displaystyle q^{n}\delta\big{[}O_{T}(x,m)\big{]}$ $\displaystyle\leq$ $\displaystyle\cfrac{q^{n}}{1-q}d(x,Tx).$ Since $\lim\limits_{n\rightarrow\infty}q^{n}=0$, $\\{T^{n}x\\}$ is a Cauchy sequence in $X$. Since $X$ is $T$-orbitally complete, there exists $x^{}\in X$ such that (3.5) $\lim\limits_{n\rightarrow\infty}T^{n}x=x^{}.$ By using the generalized quasi-contractive condition of $T$ again, we have $\displaystyle d(x^{},Tx^{})$ $\displaystyle\leq$ $\displaystyle d(x^{},T^{n+1}x)+d(T^{n+1}x,Tx^{})$ $\displaystyle=$ $\displaystyle d(x^{},T^{n+1}x)+d(TT^{n}x,Tx^{})$ $\displaystyle\leq$ $\displaystyle d(x^{},T^{n+1}x)+q\max\big{\\{}d(T^{n}x,x^{}),d(T^{n}x,TT^{n}x),d(x^{},Tx^{}),d(T^{n}x,Tx^{}),$ $\displaystyle d(x^{},TT^{n}x),d(T^{2}T^{n}x,T^{n}x),d(T^{2}T^{n}x,TT^{n}x),d(T^{2}T^{n}x,x^{}),d(T^{2}T^{n}x,Tx^{})\big{\\}}$ $\displaystyle=$ $\displaystyle d(x^{},T^{n+1}x)+q\max\big{\\{}d(T^{n}x,x^{}),d(T^{n}x,T^{n+1}x),d(x^{},Tx^{}),d(T^{n}x,Tx^{}),$ $\displaystyle d(x^{},T^{n+1}x),d(T^{n+2}x,T^{n}x),d(T^{n+2}x,T^{n+1}x),d(T^{n+2}x,x^{}),d(T^{n+2}x,Tx^{})\big{\\}}.$ Taking the limit as $n\rightarrow\infty$ in (3), and using (3.5), we get $d(x^{},Tx^{})\leq qd(x^{},Tx^{}).$ Since $q\in[0,1)$, we obtain $d(x^{},Tx^{})=0$, that is, $x^{}=Tx^{}$. Then $T$ has a fixed point. Step 2. _The fixed point of $T$ is unique._ Let $x^{},y^{}$ be two fixed points of $T$. Since $T$ is generalized quasi-contraction, we have $\displaystyle d(x^{},y^{})$ $\displaystyle=$ $\displaystyle d(Tx^{},Ty^{})$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x^{},y^{}),d(x^{},Tx^{}),d(y^{},Ty^{}),d(x^{},Ty^{}),d(y^{},Tx^{}),$ $\displaystyle d(T^{2}x^{},x^{}),d(T^{2}x^{},Tx^{}),d(T^{2}x^{},y^{}),d(T^{2}x^{},Ty^{})\big{\\}}$ $\displaystyle=$ $\displaystyle qd(x^{},y^{}).$ Since $q\in[0,1)$, we obtain $d(x^{},y^{})=0$. That is, $x^{}=y^{}$. Then the fixed point of $T$ is unique. (2). It is proved by (3.5). (3). Taking the limit as $m\rightarrow\infty$ in (3), we get $d(T^{n}x,x^{})\leq\cfrac{q^{n}}{1-q}d(x,Tx).$ ∎ ###### Corollary 3.2. Let $(X,d)$ be a metric space and $T:X\longrightarrow X$ be a map satisfying the following: 1. (1) $X$ is $T$-orbitally complete. 2. (2) There exists $k\in\mathbb{N}$ and $q\in[0,1)$ such that for all $x,y\in X$, $\displaystyle d(T^{k}x,T^{k}y)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x,y),d(x,T^{k}x),d(y,T^{k}y),d(x,T^{k}y),d(y,T^{k}x),$ $\displaystyle d(T^{2k}x,x),d(T^{2k}x,T^{k}x),d(T^{2k}x,y),d(T^{2k}x,T^{k}y)\big{\\}}.$ Then we have 1. (1) $T$ has a unique fixed point $x^{}$ in $X$. 2. (2) $d(T^{n}x,x^{})\leq\cfrac{q^{m}}{1-q}\max\big{\\{}d(T^{i}x,T^{i+k}x):i=0,1,\ldots,k-1\big{\\}}$ for all $x\in X$ and $n\in\mathbb{N}$ where $m$ is the greatest integer not exceeding $\cfrac{n}{k}$. 3. (3) $\lim\limits_{n\rightarrow\infty}T^{n}x=x^{}$ for all $x\in X$. ###### Proof. (1). By the conclusion of Theorem 3.1, $T^{k}$ has a unique fixed point $x^{}$ and $T^{k}(Tx^{})=T(T^{k}x^{})=Tx^{}$. It implies that $Tx^{}=x^{}$, that is, $T$ has a fixed point $x^{}$. The uniqueness of the fixed point of $T$ is easy to see. (2). Let $n\in\mathbb{N}$. Then $n=mk+j$, $0\leq j<k$ and for each $x\in X$, $T^{n}x=(T^{k})^{m}T^{j}x$. It follows from Theorem 3.1.(3) that $\displaystyle d(T^{n}x,x^{})$ $\displaystyle\leq$ $\displaystyle\cfrac{q^{m}}{1-q}d(T^{j}x,T^{k}T^{j}x)$ $\displaystyle\leq$ $\displaystyle\cfrac{q^{m}}{1-q}\max\big{\\{}d(T^{i}x,T^{i+k}x):i=0,1,\ldots,k-1\big{\\}}.$ (3). It is a direct consequence of (2). ∎ ###### Corollary 3.3 ([4], Theorem 2). Let $(X,d)$ be a metric space and $T:X\longrightarrow X$ be a map satisfying the following: 1. (1) $X$ is $T$-orbitally complete. 2. (2) There exists $k\in\mathbb{N}$ and $q\in[0,1)$ such that for all $x,y\in X$, (3.8) $\displaystyle d(T^{k}x,T^{k}y)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x,y),d(x,T^{k}x),d(y,T^{k}y),d(x,T^{k}y),d(y,T^{k}x)\big{\\}}.$ Then we have 1. (1) $T$ has a unique fixed point $x^{}$ in $X$; 2. (2) $d(T^{n}x,x^{})\leq\cfrac{q^{m}}{1-q}\max\big{\\{}d(T^{i}x,T^{i+k}x):i=0,1,\ldots,k-1\big{\\}}$ for all $x\in X$ and $n\in\mathbb{N}$ where $m$ is the greatest integer not exceeding $\cfrac{n}{k}$; 3. (3) $\lim\limits_{n\rightarrow\infty}T^{n}x=x^{}$ for all $x\in X$. Now, we denote the multi-valued mapping $F:X\longrightarrow BN(X)$ of generalized quasi-contraction by $\displaystyle\rho(Fx,Fy)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x,y),\rho(x,Fx),\rho(y,Fy),D(x,Fy),D(y,Fx),$ $\displaystyle D(F^{2}x,x),D(F^{2}x,Fx),D(F^{2}x,y),D(F^{2}x,Fy)\big{\\}},$ for some $q\in[0,1)$ and for all $x,y\in X$. The following theorem presents the fixed point theorem for multi-valued version of generalized quasi- contractive mapping. ###### Theorem 3.4. Let $(X,d)$ be a metric space and $F:X\longrightarrow BN(X)$ be a multi-valued map. Suppose that $F$ is a generalized quasi-contraction and $X$ is $F$-orbitally complete. Then we have 1. (1) $F$ has a unique fixed point $x^{}$ in $X$ and $Fx^{}=\\{x^{}\\}$. 2. (2) For each $x_{0}\in X$, there exists an orbit $\\{x_{n}\\}_{n}$ of $F$ at $x_{0}$ such that $\lim\limits_{n\rightarrow\infty}x_{n}=x^{}$ for all $x\in X$, and 3. (3) $d(x_{n},x^{})\leq\cfrac{(q^{1-a})^{n}}{1-q^{1-a}}d(x_{0},x_{1})$ for all $n\in\mathbb{N}$, where $a<1$ is any fixed positive number. ###### Proof. (1). Given $a\in(0,1)$ and defined a single-valued mapping $T:X\longrightarrow X$ by the following statement: $for~{}each~{}x\in X,~{}Tx\in Fx~{}~{}satisfies~{}~{}d(x,Tx)\geq q^{a}\rho(x,Fx).$ By the Definition 2.3 and the condition of $F$, we have for every $x,y\in X$, $\displaystyle d(Tx,Ty)$ $\displaystyle\leq$ $\displaystyle\rho(Fx,Fy)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x,y),\rho(x,Fx),\rho(y,Fy),D(x,Fy),D(y,Fx),$ $\displaystyle D(F^{2}x,x),D(F^{2}x,Fx),D(F^{2}x,y),D(F^{2}x,Fy)\big{\\}}$ $\displaystyle=$ $\displaystyle qq^{-a}\max\big{\\{}q^{a}d(x,y),q^{a}\rho(x,Fx),q^{a}\rho(y,Fy),q^{a}D(x,Fy),q^{a}D(y,Fx),$ $\displaystyle q^{a}D(F^{2}x,x),q^{a}D(F^{2}x,Fx),q^{a}D(F^{2}x,y),q^{a}D(F^{2}x,Fy)\big{\\}}$ $\displaystyle\leq$ $\displaystyle q^{1-a}\max\big{\\{}d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx),$ $\displaystyle d(T^{2}x,x),d(T^{2}x,Tx),d(T^{2}x,y),d(T^{2}x,Ty)\big{\\}}.$ By Theorem 3.1, we conclude that $T$ has a unique fixed point $x^{}$. Then $\rho(x^{},Fx^{})\leq q^{a}d(x^{},Tx^{})=0$ implies that $\rho(x^{},Fx^{})=0$. Then $x^{}$ is a fixed point of $F$ and $Fx^{}=\\{x^{}\\}$. From the direct consequences of Theorem 3.1 where $x_{n}=T^{n}x$ for all $n\in\mathbb{N}$, we obtain that (2) and (3) hold. ∎ ###### Corollary 3.5 ([4], Theorem 3). Let $(X,d)$ be a metric space and $F:X\longrightarrow BN(X)$ be a multi-valued map satisfying the following: 1. (1) $X$ is $F$-orbitally complete. 2. (2) There exists $q\in[0,1)$ such that for all $x,y\in X$, (3.10) $\displaystyle\rho(Fx,Fy)$ $\displaystyle\leq$ $\displaystyle q\max\big{\\{}d(x,y),\rho(x,Fx),\rho(y,Fy),D(x,Fy),D(y,Fx)\big{\\}}.$ Then we have 1. (1) $F$ has a unique fixed point $x^{}$ in $X$ and $Fx^{}=\\{x^{}\\}$. 2. (2) For each $x_{0}\in X$, there exists an orbit $\\{x_{n}\\}_{n}$ of $F$ at $x_{0}$ such that $\lim\limits_{n\rightarrow\infty}x_{n}=x^{}$ for all $x\in X$, and 3. (3) $d(x_{n},x^{})\leq\cfrac{(q^{1-a})^{n}}{1-q^{1-a}}d(x_{0},x_{1})$ for all $n\in\mathbb{N}$, where $a<1$ is any fixed positive number. ###### Example 3.6. Let $(X,d)$ and $T:X\longrightarrow X$ be defined by Example 2.5. It is easy to see that $X$ is $T$-orbitally complete metric space. By the definition of the distance $d$ and mapping $T$ , we conclude that $X$ and $T$ satisfy all of the conditions in Theorem 3.1. Clearly, $x^{}=1$ is a unique fixed point of $T$. Note that, if $x\in\\{1,2,3\\}$ then $T^{n}x=1$ for $n=1,2,3,...$ and if $x\in\\{4,5\\}$ then $T^{n}x=1$ for $n=2,3,4,...$. That is $\lim\limits_{n\rightarrow\infty}T^{n}x=x^{}$ for all $x\in X$. Let $q\in\big{[}0.5,1\big{)}$ be fixed by the generalized quasi-contraction of $T$ which arises from Example 2.5. We see that the inequality $d(T^{n}x,x^{})\leq\cfrac{q^{n}}{1-q}d(x,Tx)$ holds for all $x\in X$ and $n\in\mathbb{N}$. Therefore, this example is presented to certify the results of Theorem 3.1. However, it is not applicable to Theorem 1.1. ###### Remark 3.7. Example 3.6 shows that our results are proper generalizations of Ćirić fixed point theorems in [4]. Then our results are exactly a new form of fixed point theorems in metric spaces. Moreover, we may generalize other fixed point theorems contained at most five mentioned values in the literature to that contain $d(T^{2}x,x)$, $d(T^{2},Tx)$, $d(T^{2}x,y)$, $d(T^{2}x,Ty)$ in addition. Acknowledgement. This research of is supported by King Mongkut’s University of Technology North Bangkok, Thailand ## References * [1] S. Banach, _Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales_ , Fund. Math. 3 (1922), 133 – 181. * [2] V. Berinde, _Iterative approximation of fixed points_ , Lecture Notes in Mathematics, Springer, 2006. * [3] V. Berinde, _General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces_ , Carpathian J. 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Samet, _The Kannan’s fixed point theorem in a cone rectangular metric space_ , J. Nonlinear Sci. Appl. 2 (2009), no. 3, 161 – 167. * [11] R. Kannan, _Some results on fixed points_ , Bull. Culcutta Math. Soc. 60 (1968), 71 – 76. * [12] V. Lakshmikantham and L. Ćirić, _Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces_ , Nonlinear Anal. 70 (2009), no. 12, 4341 – 4349. * [13] S. B. Nadler, _Multi-valued contraction mappings_ , Pacific J. Math. 30 (1969), no. 2, 475 – 488. * [14] B. E. Rhoades, _A comparison of various definition of contractive mappings_ , Trans. Amer. Math. Soc. 226 (1977), 257 – 290. * [15] S. Rolewicz, _Metric linear spaces_ , PWN-Polish Scientific Publishers, Warszawa, 2002. * [16] P. V. Semenov, _Fixed points of multivalued contractions_ , Funct. Anal. Appl. 36 (2002), no. 2, 159 – 161. * [17] S. Sessa, _On a weak commutativity condition of mappings in fixed point considerations_ , Publ. Inst. Math. (Beograd)(N.S.) 32 (1982), no. 46, 14 – 153.	arxiv-papers	2013-09-22T10:53:43	2024-09-04T02:49:51.311516	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Nguyen Van Dung and Poom Kumam and Kanokwan Sitthithakerngkiet", "submitter": "Dung Nguyen Van", "url": "https://arxiv.org/abs/1309.5589" }
1309.5649	# Project Description Gunnar Carlsson 111Research supported in part by NSF DMS-0406992 Department of Mathematics, Stanford University Stanford, California 94305 # A geometric reformulation of the representational assembly conjecture for the $K$-theory of fields with pro-$l$ absolute Galois group Gunnar Carlsson 111Research supported in part by NSF DMS-0406992 Department of Mathematics, Stanford University Stanford, California 94305 ## 1 Introduction In recent years there have been striking developments in the study of the algebraic $K$-theory of fields (see [41], [16], [9]). These developments have resulted in the identification of the $E_{2}$-term of a spectral sequence converging to the algebraic $K$-theory of fields (rather a mod-$p$ version of it) in terms of Galois cohomology, and have resolved long standing conjectures of Milnor and Bloch-Kato in the area. In particular the $E_{2}$-term depends only on the absolute Galois group of the field. These results raise a pair of questions. * • For a field $F$, is there a construction depending only on the absolute Galois group $G_{F}$ of $F$, which reconstructs the full $K$-theory spectrum of $F$, rather than an $E_{2}$-term of a spectral sequence? Can such a construction be made functorial for inclusions of subfields and the corresponding inclusions of absolute Galois groups? Is the construction readily describable in terms of well understood constructions on the absolute Galois group? * • The conjectures of Milnor and Bloch-Kato are cohomological statements about the absolute Galois group, specifically that the cohomology is generated in degree one and that the relations are quadratic. Are there properties more closely connected to the structure of the absolute Galois group as a group which are relevant? A conjectural answer to the first question in the case of geometric fields (i.e. fields containing an algebraically closed subfield $k$) was proposed in [8], and proved in the case of abelian absolute Galois groups. This conjecture is formulated as follows. We assume we are given a field $F$ containing an algebraically closed subfield $k\subseteq F$, let $\overline{F}$ denote its algebraic closure, and let $G_{F}$ denote its absolute Galois group $Gal(\overline{F}/F)$, in general a profinite group. One can construct a version of the equivariant $K$-theory spectrum, $K^{G}$, which applies to profinite groups, and observe that $K^{G}(\overline{F})\cong K(F)$. On may also consider the natural map $K^{G}(k)\rightarrow K^{G}(\overline{F})$ of spectra. This map is far from being an equivalence since, for example, $\pi_{0}K^{G}(k)$ is the free abelian group on the continuous $k$-linear representations of $G_{F}$, where $k$ is given the discrete topology, and on the other hand $\pi_{0}K^{G}(\overline{F})\cong\pi_{0}K(F)\cong\mathbb{Z}$ However, the spectra $K^{G}(k)$ and $K^{G}(\overline{F})$ both carry multiplicative structures, which makes $K^{G}(\overline{F})$ into a module over the commutative ring spectrum $K^{G}(k)$. See [15] and [23] for a discussion of these multiplicative theories. In [7], a derived version of the completion construction for commutative rings was developed, which associates to a homomorphism of commutative ring spectra $f:A\rightarrow B$ and an $A$-module $M$ a completion of $M$ along the homomorphism $f$, denoted by $M^{\wedge}_{f}$, which agrees with the standard completion construction for finitely generated modules over Noetherian commutative rings. We will denote by $\varepsilon$ the homomorphism $K^{G}(k)\rightarrow\mathbb{H}_{l}$ which is induced by the functor which assigns to every representation of $G_{F}$ its dimension mod $l$, where $\mathbb{H}_{l}$ is the mod-$l$ Eilenberg-MacLane spectrum, regarded as the result of an infinite loop space machine applied to the symmetric monoidal category with object set equal to $\mathbb{Z}/l\mathbb{Z}$, and with only identity morphisms. At this point, we may construct derived completions along the homomorphism $\varepsilon$, to obtain a map $\alpha_{F}:K^{G}(k)^{\wedge}_{\varepsilon}\rightarrow K^{G}(\overline{F})^{\wedge}_{\varepsilon}$ called the representational assembly, and the conjecture is that this map is an equivalence of spectra. The difficulty in approaching this conjecture is that we do not have simple direct methods for computing with the completions. We will instead approach the problem indirectly as follows. We will construct an affine scheme ${\cal E}{G_{F}}$, with associated ring equipped with the discrete topology, and equipped with a continuous action by the topological group $G_{F}$. Equivariant $K$-theory can be suitably defined for such actions. The scheme ${\cal E}{G_{F}}$ has the following properties. 1. 1. The maps $K^{G_{F}}(Spec(k))^{\wedge}_{\varepsilon}\rightarrow K^{G_{F}}({\cal E}{G_{F}})^{\wedge}_{\varepsilon}$ and $K^{G_{F}}(Spec(\overline{F}))^{\wedge}_{\varepsilon}\rightarrow K^{G_{F}}(Spec(\overline{F})\times{\cal E}{G_{F}})^{\wedge}_{\varepsilon}$ are equivalences. 2. 2. There is a well-defined notion orbit scheme for ${\cal E}F$, and there are equivalences $K({\cal E}{G_{F}}/G_{F})\rightarrow K^{G_{F}}({\cal E}{G_{F}})$ and $K(Spec(\overline{F})\mathrel{\mathop{\times}\limits_{G_{F}}}{\cal E}{G_{F}})\rightarrow K^{G_{F}}(Spec(\overline{F})\times{\cal E}{G_{F}})$ This follows from a freeness property of ${\cal E}{G_{F}}$. 3. 3. There is a natural equivalence $K^{G_{F}}({{\cal E}G_{F})}^{\wedge}_{\epsilon}\cong K^{G_{F}}({\cal E}G_{F})^{\wedge}_{l}$, where $(-)^{\wedge}_{l}$ denotes Bousfied-Kan $l$-adic completion. These properties give rise to the following diagram. $\begin{diagram}$ The goal is to prove that $\alpha_{F}$ is an equivalence. Properties (1) and (3) above show that all the horizontal maps are equivalences, and it therefore follows that it will suffice to prove that the right hand vertical map $\eta$ is an equivalence. Proving that the map $\eta$ is an equivalence can be thought of as a parametrized rigidity theorem. We will use this reduction in the paper [10] to prove the conjecture. Some comments on the construction are in order. * • ${\cal E}G$ is not functorial for homomorphisms of groups, and it depends on a number of choices. Moreover, ${\cal E}G_{F}$ is not even $\mathbb{A}^{1}$-contractible, as its algebraic $K$-theory is not isomorphic to the algebraic $K$-theory of the base field $k$. What can be proved is that it is potentially $l$-adically contractible, in that $K({\cal E}G_{F})$ has $l$-adic completion equivalent to the $l$-adic completion of $K(k)$. We believe that there should ultimately be canonical constructions for the universal space of a profinite group, and that our construction is a particular model which is $l$-adically equivalent to it. * • The equivariant $K$-theory of ${\cal E}G_{F}$ is interesting in that the natural action of the representation ring of $G_{F}$, $R[G_{F}]$, on $K_{}({\cal E}G_{F})$ has the property that the augmentation ring of $R[G_{F}]$ acts trivially on it. This fact is crucial in the proof of Property (3) above. • The construction of ${\cal E}{G_{F}}$ relies on a structural property shared by all absolute Galois groups of geometric fields, that of total torsion freeness, which asserts that the abelianization of any closed subgroup of $G_{F}$ is torsion free. This is a surprisingly restrictive property for groups, and permits the construction of free actions on varieties of the form $\mathbb{G}_{m}^{n}$, which turn out to be extremely useful. * • ${\cal E}G_{F}$ is of the form $Spec(A)$, where $A=\bigcup_{n}k[t_{s}^{\pm\frac{1}{l^{n}}};s\in S]$, where $S$ is some indexing set. * • There has been a great deal of work on reconstructing the arithmetic of a fields from its Galois group, see [30] and [4], for example. In the abstract, this asserts that one is able to reconstruct the $K$-theory spectrum of the field from its absolute Galois group. The point of the present work is to identify the $K$-theory spectrum in terms of well understood constructions on the absolute Galois group, with the expectation that such understanding will make applying $K$-theory to arithmetic problems will be made easier. * • There is another formulation of the main conjecture, which would replace the ring spectra which occur by their counterparts within equivariant spectra, and would therefore naturally replace all homotopy groups which occur by their corresponding Mackey functors, and further replace the representation rings by their Green functor counterparts (see [5]). The two conjectures are not a priori equivalent, since the relationship between the completion in the category of rings and the completion in the Green functor context is not at all clear. Indeed, our results are proved for fields whose absolute Galois group is a pro-$l$ group, although we expect to deal with the general case in future work. Moreover, the evaluation of the Mackey functor completion would be expected in general to be very difficult, since very simple questions about, for example, derived functors of tensor products are quite difficult in the Green functor situation [40]. C. Barwick has developed methods for proving the Mackey/Green functor version of the conjecture. Presumably one will find that our methods will permit the computation of the derived completions in the Mackey/Green functor version, in terms of derived completions of ordinary rings. The paper is organized as follows. Section 2 records preliminary material, including a discussion of actions of profinite groups on affine schemes which will be important for the construction of ${\cal E}{G_{F}}$. Section 3 defines totally torsion free profinite groups, and proves that absolute Galois groups all enjoy this property. Section 4 develops a method for constructing profinite group actions on certain (non-Noetherian) affine schemes. The schemes in question are the inverse limits of systems of the form $\cdots\stackrel{{\scriptstyle\times l}}{{\longrightarrow}}\mathbb{G}_{m}^{n}\stackrel{{\scriptstyle\times l}}{{\longrightarrow}}\mathbb{G}_{m}^{n}\stackrel{{\scriptstyle\times l}}{{\longrightarrow}}\mathbb{G}_{m}^{n}\stackrel{{\scriptstyle\times l}}{{\longrightarrow}}\mathbb{G}_{m}^{n}$ and ultimately ${\cal E}{G_{F}}$ will be constructed as an infinite product of such schemes. We are able to construct actions on such varieties from affine representations over $\mathbb{Z}_{l}$ of $G_{F}$. Section 5 actually constructs ${\cal E}{G_{F}}$, using the totally torsion free property to construct sufficiently many such affine representations. Section 6 is a technical section, which studies the representation rings of profinite $l$-groups which are an extension of a torsion free, topologically finitely generated, abelian profinite $l$-group by a finite $l$-subgroup of the symmetric group. Sections 7 and 8 now use this preparatory work to prove the main result. The author wishes to thank B. Conrad, P. Diaconis, D. Dugger, L. Hesselholt, T. Lawson, G. Lyo, I. Madsen, D. Ramras, R. Vakil, and K. Wickelgren for numerous helpful conversations on this work. ## 2 Preliminaries ### 2.1 General We will assume that the reader is familiar with the theory of ring spectra and modules over them, as well as the notion of a commutative ring spectrum. We will use the version of this theory which uses symmetric spectra, which is presented in [33]. In particular, we will assume the reader is familiar with the standard choices of model structures on this category, and the associated notions of cofibrant objects, as well as $Hom_{A}(M,N)$, $M\mathrel{\mathop{\wedge}\limits_{A}}N$ and universal coefficient and Künneth spectral sequences converging to them. We will also assume familiarity with homotopy limits and colimits in this category. Throughout the paper, a spectrum will mean a symmetric spectrum, a ring spectrum will mean a symmetric ring spectrum, and a commutative ring spectrum will mean a commutative symmetric ring spectrum. ### 2.2 $K$-theory There are numerous versions of the construction of the $K$-theory spectra associated with various kinds of categorical input data. We will require two of them. The first is the construction to A. Elmendorf and M. Mandell [14]. It takes as input small permutative categories, and produces from that construction a spectrum. It advantage is that the category theoretic information required to impose a commutative ring spectrum structure on the resulting spectrum, and to produce module spectra over such spectra, is codified in [14]. This kind of multiplicative structure is at the core of the results of this paper. The second construction (due to Waldhausen [42]) takes as input data a category with cofibrations and weak equivalences, and produces a $K$-theory spectrum from this data. It has the advantage that it has the excision properties proved to exist by Quillen ([31]), such as localization and devissage. Each Waldhausen category can be regarded as a small permutative category, to which the Elmendorff-Mandell construction applies, and there is a sequence of natural equivalences of spectra relating the two constructions which are described in [8], section 4. We will denote the Elmendorff-Mandell construction by $\mathfrak{K}$ and the Waldhausen construction by $K$. We also define a weakening of the notion of a ring spectrum as follows. ###### Definition 2.1 By a weak ring spectrum, we mean a spectrum $R$ equipped with a map $R\wedge R\rightarrow R$ which is homotopy associative in the evident sense. By a module over a weak ring spectrum $R$, we will mean a spectrum $M$ equipped with a map $R\wedge M\rightarrow M$, where the diagrams involving the structure maps commute up to homotopy. The Waldhausen construction readily produces weak ring structures from tensor products on modules, as well as modules from appropriate notions of tensor products. Remark: The reason we need both constructions is the following. In order to define our derived completion construction, we will need to have the precise commutative ring structure on our spectra afforded by the Elmendorf-Mandell model. On the other hand, for computation behavior of homotopy groups, it is particularly important to realize constructions such as homotopy fibers as the $K$-theory of categories themselves, and this can be carried out effectively using the Waldhausen construction. We note that the structure of a weak ring spectrum is sufficient to compute behavior on homotopy groups. The natural equivalences allow us to go back and forth between models to obtain the required information. ###### Proposition 2.1 Let $\underline{C}$ an abelian category, equipped with a strictly associative symmetric monoidal product, which turns $\underline{C}$ into a bipermutative category in the sense of [14]. Then the following hold. 1. 1. $\mathfrak{K}(\underline{C})$ is naturally a commutative ring spectrum in the sense of [14]. 2. 2. Let $\underline{D}\subseteq\underline{C}$ be a Serre subcategory, with the property that it is closed under tensor product with any object in $\underline{C}$. Then the quotient category $\underline{C}/\underline{D}$ is also bipermutative, and $\mathfrak{K}(\underline{C}/\underline{D})$ is also a commutative ring spectrum. 3. 3. The natural equivalence between $\mathfrak{K}$ and $K$ respects the weak ring spectrum structures, where the weak ring spectrum structure on $\mathfrak{K}$ is simply the forgetful one obtained from the full ring spectrum structure. 4. 4. Given $\underline{C}$ and $\underline{D}$ as above, the homotopy fiber of the evident map $\mathfrak{K}(C)\rightarrow\mathfrak{K}(\underline{C}/\underline{D})$ is a $\mathfrak{K}(\underline{C})$-module. As a weak $K(\underline{C})$-module, it is equivalent to $K(\underline{D})$. ### 2.3 Equivariant algebraic $K$-theory We will fix an algebraically closed ground field $k$ throughout this paper. In this section we will define the equivariant algebraic-$K$-theory of finite group actions on certain commutative rings. We will be addressing the fact that in order to build the schemes we require, we will have to leave the Noetherian setting. The suitable setting will be that of coherent commutative rings, for which the category of finitely presented modules is an abelian category. We will let $R[G]$ denote the representation ring of the group $G$ in an algebraically closed field $k$, which is a fixed ground field throughout the paper. We also let $I_{G}$ denote the augmentation ideal, i.e. the kernel of the natural augmentation $R[G]\rightarrow\mathbb{Z}$. We recall from [19] the following definition. ###### Definition 2.2 A commutative ring $A$ is said to be coherent if for every homomorphism $f:F_{0}\longrightarrow F_{1}$ of finitely generated free $A$-modules, the kernel of $f$ is a finitely generated $A$-module. A standard fact about coherent rings is the following. ###### Proposition 2.2 The category of finitely presented modules over a coherent ring $A$ is an abelian category. The following result is an extension of Proposition 4.5 of [8]. ###### Proposition 2.3 Let $D$ be a directed set, and let $F$ be a functor from the associated category to the category of rings. Suppose further that $F(d\leq d^{\prime})$ is faithfully flat for all $d\leq d^{\prime}$, and that all the rings $F(d)$ are coherent. Then the ring $\mathrel{\mathop{colim}\limits_{D}}F$ is coherent. Proof: The proof is identical to the proof of Proposition 4.5 of [8], and we omit it. $\square$ ###### Definition 2.3 We say a ring is Noetherian approximable if it can be described as a colimit over a directed set of Noetherian rings, with the transition maps in the colimit being faithfully flat. Given a coherent commutative ring $A$, it is not known whether or not a polynomial ring $A[x_{1},\ldots,x_{n}]$ is also coherent. However, this does hold for Noetherian approximable rings $A$. ###### Proposition 2.4 Let $A$ be a Noetherian approximable commutative ring. Then the ring $B=A[x_{s};s\in S]$, where $S$ is some indexing set, is Noetherian approximable and therefore coherent. Proof: $B$ can be described as the colimit of rings of the form $R[x_{1},\ldots,x_{t}]$, where $R$ is Noetherian, and where the transition maps in the colimit are faithfully flat. $R[x_{1},\ldots,x_{n}]$ is Noetherian, and the maps in the system are clearly faithfully flat, hence the result. $\square$ We let $A$ be a coherent commutative ring, equipped with an action by a finite group $G$. ###### Definition 2.4 By a $G$-twisted $A$-module, we will mean a finitely presented $A$-module $M$ equipped with a $k$-linear $G$-action so that $g\cdot(\alpha\cdot m)=\alpha^{g}\cdot(g\cdot m)$ for all $g\in G,\alpha\in A$, and $m\in M$. The $G$-twisted $A$-modules form a category whose morphisms are the equivariant $A$-linear maps, which we denote $\mbox{\em Mod}^{G}(A)$. ###### Proposition 2.5 Given $A$ and $G$ as above, the category $\mbox{\em Mod}^{G}(A)$ is an abelian category. Proof: Clear. $\square$ Although we will primarily be considering affine schemes, we will use ring and scheme terminology interchangeably. ###### Definition 2.5 We will define the $G$-equivariant $K$-theory on the category of coherent commutative rings to be the algebraic $K$-theory of the full subcategory of $\mbox{Mod}^{G}(A)$ on the finitely presented $G$-twisted modules, and denote it by $K^{G}(A)$. $K^{G}$ is functorial for flat homomorphisms of coherent commutative rings with $G$ action. In the case of Noetherian rings, this is what would usually be referred to as $G$-theory or $K^{\prime}$-theory. In the case of a point, i.e. $A=k$, we have that $K^{G}(k)$ is simply the category of finite dimensional $k$-linear representations of $G$. $\pi_{0}(K^{G}(k))$ is then the $k$-linear representation ring $R[G]$. In particular, if either (1) $Char(k)=0$ or (2) $Char(k)$ is relatively prime to $\\#(G)$, then $\pi_{0}K^{G}(k)$ is isomorphic to the usual complex representation ring or character ring of $G$. $K^{G}(k)$ is a commutative ring spectrum. It is also equipped with an augmentation $\epsilon:K^{G}(k)\rightarrow K(k)$, which forgets the $G$ action, and for which applying $\pi_{0}$ induces the usual augmentation on representation rings, which we also refer to as $\epsilon$. We need a brief discussion of orbit schemes. From [21], it is known that for finite group actions on affine schemes, there is always a categorical orbit scheme which is the spectrum of the fixed point subring of the action. ###### Proposition 2.6 Suppose that the action of $G$ on an affine scheme $X=Spec(A)$ is free in the sense that it is free on the set of all points, where a point denotes a morphism $Spec(\Omega)\rightarrow X$ for some algebraically closed field $\Omega$ containing $k$. Then there is a natural equivalence $K^{G}(A)\cong K(A^{G})$. It is induced by the composite $K(A^{G})\rightarrow K^{G}(A^{G})\rightarrow K^{G}(A)$ Equivalently, for free $G$-actions we have an equivalence $K(S/G)\rightarrow K^{G}(X)$. Proof: The equivalence of categories from which this equivalence results is proved in [22]. $\square$ We call group actions satisfying the above conditions etale. The following is well known. ###### Proposition 2.7 The category of affine schemes admits arbitrary inverse limits. The ring of all global sections of an inverse limit is the corresponding colimit of the rings of global sections of the component schemes. ###### Proposition 2.8 Let $G$ be a finite group, and let $\underline{X}$ denote a diagram of affine schemes with $G$-action. If all the $G$ actions composing the diagram are etale, then so is the $G$-action on $lim\underline{X}$. Proof: For any algebraically closed field $\Omega$, the set of maps $Spec(\Omega)\rightarrow lim\underline{X}$ is the inverse limit of the sets of maps from $Spec(\Omega)$ to the component schemes. An inverse limit of free $G$-sets over a directed set is always free. $\square$ Given an action of a finite group $G$ on a scheme $X$, and a $G$-invariant subscheme $Y\subseteq X$, we define the category $\mbox{Nil}^{G}(X,Y)$ to be the category of $G$-twisted $X$-modules whose support lies entirely in $Y$. ### 2.4 Profinite group actions on commutative rings We begin with the definition of profinite groups. The basic material on profinite groups is contained in [32]. ###### Definition 2.6 A topological group $G$ is a profinite group if it is Hausdorff, compact, and totally disconnected. It follows from this definition that $G$ is the inverse limit of the inverse system of finite groups $\\{G/N\\}_{N}$, where the indexing ranges over the closed normal subgroups of $G$ of finite index. ###### Proposition 2.9 Let $G$ be a profinite group. Then a subgroup of $G$ is open only if it is closed and of finite index. We next consider continuous actions of a profinite group $G$ on discrete sets $X$. ###### Proposition 2.10 Let $X$ be a discrete set, and suppose that we are given an action of a profinite group $G$ on $X$. Then the action is continuous if and only if the stabilizers $G_{x}$ are closed subgroups of finite index of $G$ for all $x\in X$. Proof: From the discreteness of $X$, it follows that $\alpha^{-1}(x)$ is an open set in $G\times X$ for all $x\in X$, where $\alpha:G\times X\rightarrow X$ is the action map. A subset of $G\times X$ is open if and only if it is of the form $\coprod_{x\in X}U_{x}\times\\{x\\}$ where each $U_{x}$ is an open set in $G$. On the other hand, $\alpha^{-1}(x)=\coprod_{\gamma\in\Gamma}G_{x}\gamma^{-1}\times\gamma\cdot x$ It is now clear that this set is open if and only if $G_{x}$ is an open subgroup of $G$, i.e. that it is closed and of finite index by Proposition 2.9. $\square$ We also have the following result on transitive $G$-spaces. ###### Proposition 2.11 Let $X$ be a topological space with a continuous transitive action of a profinite group $G$, and let $G_{x}$ denote the stabilizer of a point $x\in X$. Then if $X$ is Hausdorff, we have that $X$ is $G$-homeomorphic to the quotient space $G/G_{x}$, and $G_{x}$ is a closed subgroup of $G$. Proof: It is immediate that $X$ is compact, since it is the surjective image of the compact space $G$. The natural map $G/G_{x}\rightarrow X$ is a bijective continuous map from one compact Hausdorff space to another, and is therefore a homeomorphism. It is direct that $G/G_{x}$ is Hausdorff if and only if $G_{x}$ is closed. $\square$ ###### Definition 2.7 Let $G$ be a profinite group. By a continuous affine $G$-scheme $A$, we will mean a continuous action of $G$ on a commutative ring $A$ equipped with the discrete topology. Example: Let $G$ and $G_{sep}$ be the Galois groups of $\overline{k}$, the algebraic closure of a field $k$, and $k_{sep}$, its absolute closure. Then the actions of $G$ and $G_{sep}$ on $Spec(\overline{k})$ and $Spec(k_{sep})$ are continuous affine $G$ (respectively $G_{sep}$)-schemes. Example: Let $G$ denote the $l$-adic integers, and let $A=\bigcup_{n}k[t^{\pm\frac{1}{l^{n}}}]$. The group of automorphisms of $A$ leaving $k[t^{\pm 1}]$ elementwise fixed is isomorphic to $\mathbb{Z}_{l}$, and the action of $\mathbb{Z}_{l}$ on $A$ is a continuous affine $\mathbb{Z}_{l}$-scheme. Let $G$ be a profinite group, and let $A$ be a continuous affine $G$-scheme. Let $B$ denote the fixed point subring $A^{G}$. For every closed normal subgroup $N$ of finite index in $G$, we may consider the scheme $Spec(A^{N})$. These schemes fit together into an inverse system of affine schemes $\\{Spec(A^{N})\\}_{N}$, and we may regard the inverse system as an inverse system of sets, by simply considering the points of the schemes. We also have the group actions $G/N\times Spec(A^{N})\longrightarrow Spec(A^{N})$ which fit together into an action of a pro-group on a pro-set. For every $\mathfrak{p}\in Spec(B)$, we may consider the fiber over $\mathfrak{p}$ in $\mathrel{\mathop{lim}\limits_{\longleftarrow}}Spec(A^{N})$, denoted by $F_{\mathfrak{p}}$, and find that we obtain a continuous action of $G$ on $F_{\mathfrak{p}}$. ###### Proposition 2.12 Let $A$ be a continuous affine $G$-scheme, with $B=A^{G}$. Then the map $\pi:Spec(A)\rightarrow Spec(B)$ is surjective, and all the fibers $\pi^{-1}(\mathfrak{p})$ are transitive $G$-sets. Moreover, $\pi^{-1}(\mathfrak{p})$ is homeomorphic to a $G$-space of the form $G/G^{\prime}$, where $G^{\prime}$ is a closed subgroup of $G$. Proof: For every closed subgroup of finite index $K\subseteq G$, we have the subring $A^{K}$ of elements of $A$ fixed by $K$. The ring $A$ is equal to the union $\bigcup A^{K}$ as $K$ varies over smaller and smaller closed subgroups of finite index in $G$, because the stabilizer of every element of $A$ is a closed subgroup of finite index in $G$ by Proposition 2.10. We will now show that the natural map $\alpha:Spec(A)\longrightarrow\mbox{ }\mathrel{\mathop{lim}\limits_{\longleftarrow}}Spec(A^{K})$ is a bijection. To show that $\alpha$ is injective, consider two prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ in $A$. If $\mathfrak{p}\neq\mathfrak{q}$, then there is an element $a\in A$ with $a$ an element of one of $\mathfrak{p}$ and $\mathfrak{q}$ and not in the other, say $a\in\mathfrak{p}$ and $a\notin\mathfrak{q}$. But we have seen that $a\in A^{K}$ for some closed subgroup $K$ of finite index in $G$. It follows that the images of $\mathfrak{p}$ and $\mathfrak{q}$ in $Spec(A^{K})$ are distinct, which gives the injectivity of $\alpha$. For surjectivity, we first note that elements $\mathrel{\mathop{lim}\limits_{\longleftarrow}}Spec(A^{K})$ correspond exactly to families $\\{\mathfrak{p}_{K}\\}_{K}$, where the system varies over the closed subgroups of finite index, where each $\mathfrak{p}_{K}$ is a prime ideal in $A^{K}$, and where we have the consistency condition $\pi^{K}_{K^{\prime}}(\mathfrak{p}_{K})=\mathfrak{p}_{K^{\prime}}$ whenever $K\subseteq K^{\prime}$. Here $\pi^{K}_{K^{\prime}}$ denotes the restriction map $Spec(A^{K})\rightarrow Spec(A^{K^{\prime}})$. Given a family ${\cal F}=\\{\mathfrak{p}_{k}\\}_{K}$ as above, we define a subset $\mathfrak{p}_{{\cal F}}\subseteq A$ to be the set of all elements $a\in A$ such that there exists a closed subgroup $K\subseteq G$ of finite index so that $a\in\mathfrak{p}_{K}\subseteq A^{K}$. We will verify that $\mathfrak{p}_{{\cal F}}$ is a prime ideal in $A$. We first make the elementary observation that if $a\in\mathfrak{p}_{K}$ for some $K$, then $a\in\mathfrak{p}_{K^{\prime}}$ for all closed subgroups of finite index $K^{\prime}\subseteq K$. We verify that $\mathfrak{p}_{{\cal F}}$ is closed under addition. Suppose $a,a^{\prime}\in\mathfrak{p}_{{\cal F}}$, and fix closed subgroups of finite index $K$ and $K^{\prime}$ so that $a\in\mathfrak{p}_{K}$ and $a^{\prime}\in\mathfrak{p}_{K^{\prime}}$. Now, the subgroup $K\cap K^{\prime}$ is also a closed subgroup of finite index, and both $a$ and $a^{\prime}$ are elements of $\mathfrak{p}_{K\cal K^{\prime}}$, consequently $a+a^{\prime}\in\mathfrak{p}_{K\cal K^{\prime}}$, which means that $a+a^{\prime}\in\mathfrak{p}_{{\cal F}}$. To see that $\mathfrak{p}_{{\cal F}}$ is an ideal, we consider $a\in A$ and $i\in\mathfrak{p}_{{\cal F}}$. Select closed subgroups of finite index $K$ and $K^{\prime}$ so that $a\in A^{K}$ and $i\in\mathfrak{p}_{K^{\prime}}$. We then have $a\in A^{K\cap K^{\prime}}$ and $i\in\mathfrak{p}_{K\cap K^{\prime}}$, and therefore that $a\cdot i\in\mathfrak{p}_{K\cap K^{\prime}}$. Finally, to see that $\mathfrak{p}_{{\cal F}}$ is prime, consider $a,a^{\prime}\in A$ for which $a\cdot a^{\prime}\in\mathfrak{p}_{{\cal F}}$. Select a closed subgroup of finite index $K$ so that $a,a^{\prime}\in A^{K}$. Since $a\cdot a^{\prime}\in\mathfrak{p}_{{\cal F}}$, there is another closed subgroup of finite index $K^{\prime}$ so that $a\cdot a^{\prime}\in\mathfrak{p}_{K^{\prime}}$. We then have that $a,a^{\prime}$, and $a\cdot a^{\prime}$ all lie in $A^{K\cap K^{\prime}}$ and that $a\cdot a^{\prime}\in\mathfrak{p}_{K\cap K^{\prime}}$. Because of the primeness of $\mathfrak{p}_{K\cap K^{\prime}}$, it follows that either $a$ or $a^{\prime}$ is an element of $\mathfrak{p}_{K\cap K^{\prime}}$, and consequently that either $a$ or $a^{\prime}$ lies in $\mathfrak{p}_{{\cal F}}$. Under the identification $Spec(A)\longrightarrow\mathrel{\mathop{lim}\limits_{\longleftarrow}}Spec(A^{N})$ The fiber $\pi^{-1}(\mathfrak{p})$ is identified with $F_{\mathfrak{p}}$ above. We next note that the maps $\pi^{K}_{K^{\prime}}:Spec(A^{K})\rightarrow Spec(B)$ are surjective for any closed normal subgroup of finite index $K\subseteq G$. This result is immediate from Proposition 1.1 of [21], applying it to the action of the finite group $G/K$ on $Spec(A^{K})$. The surjectivity statement above now follows immediately. For the transitivity of the $G$ action on the fibers of the map $\pi$, we proceed as follows. We first note that $Spec(A)$ can also be described as $\mathrel{\mathop{lim}\limits_{\longleftarrow}}Spec(A^{N})$ where $N$ ranges over the closed normal subgroups of finite index in $G$, due to the finality of the closed normal subgroups of finite index in the set of all closed subgroups of finite index. For each such $N$, and pair of elements $x,y\in Spec(A)$, we consider the subset $\Phi_{N}(x,y)\subseteq G$ consisting of all $g\in G$ such that $gx_{N}=y_{N}$, where $x_{N}$ and $y_{N}$ denote the projections of $x$ and $y$ in $Spec(A^{N})$. We are letting $G$ act on $Spec(A^{N})$ in the obvious way through the projection $G\rightarrow G/N$. The set $\Phi_{N}(x,y)$ is non-empty due to the transitivity statement for finite group actions on rings made in Proposition 1.1 of [21]. Moreover, if we have $N^{\prime}\subseteq N$, then $\Phi_{N^{\prime}}(x,y)\subseteq\Phi_{N}(x,y)$. It is clear that the sets $\Phi_{N}(x,y)$ are closed subsets of $G$, and for a fixed choice of $x$ and $y$, have the finite intersection property. Therefore, due to the compactness of $G$, $\bigcap_{N}\Phi_{N}(x,y)\neq\emptyset$ Any element in $\bigcap_{N}\Phi_{N}(x,y)$ will now carry $x$ to $y$, giving the transitivity statement. We now know that $\pi^{-1}(\mathfrak{p})$ is a Hausdorff transitive continuous $G$-set, and conclude from Proposition 2.11 that $\pi^{-1}(\mathfrak{p})$ is $G$-homeomorphic to a $G$-space of the form $G/G^{\prime}$, where $G^{\prime}$ is a closed subgroup of $G$. $\square$ ###### Corollary 2.1 For continuous affine $G$-schemes, a categorical orbit space exists, and $Spec(A^{G})$ is a choice of such an orbit space. Proof: This is proved as in the finite case, performed in [21]. $\square$ Let $A$,$B$, and $G$ be as above. Let $\mathfrak{p}$ be a prime ideal of $A$, and let $G_{\mathfrak{p}}$ denote its stabilizer, which will be called its decomposition group. $G_{\mathfrak{p}}$ is a closed subgroup of $G$. Let $k({\mathfrak{p}})$ denote the field of fractions of $A/\mathfrak{p}$. Similarly, let $\mathfrak{q}=\mathfrak{p}\cap B$, and write $k(\mathfrak{q})$ for the field of fractions of $B/\mathfrak{q}$. There is an evident inclusion $k(\mathfrak{q})\subseteq k(\mathfrak{p})$. The group $G_{\mathfrak{p}}$ acts continuously on $k(\mathfrak{p})$, when $k(\mathfrak{p})$ is equipped with the discrete topology. There is therefore a homomorphism from $G_{\mathfrak{p}}$ to the Galois group $Gal(k(\mathfrak{p}),k(\mathfrak{q}))$, which is in general a profinite group. This homomorphism is surjective. This is stated explicitly in [21] in the case where $G$ is finite, and the result in this context follows immediately from this case by passing to inverse limits over Hausdorff finite quotients. The kernel of this homomorphism is called the inertia group of $\mathfrak{p}$, and is clearly a closed subgroup of $G_{\mathfrak{p}}$. We denote it by $I_{\mathfrak{p}}$. We will also study the notion of etale actions in the context of profinite actions. ###### Definition 2.8 Let $G$ be a profinite group, and let $A$ be a continuous affine $G$-scheme. We say the action is etale if each of the inclusions $A^{G}\hookrightarrow A^{K}$ is an etale map for every closed subgroup of finite index $K$. This is clearly equivalent to the condition that each inclusion $A^{G}\hookrightarrow A^{N}$ is a Galois cover with group $G/N$, where $N$ varies over all the closed normal subgroups of finite index. We have the following analogue of Corollary 2.4 of [21] for verifying that an action is etale. ###### Proposition 2.13 Let $A$ be a commutative ring equipped with the discrete topology, and let a profinite group $G$ act continuously on $A$. Then the action is etale if and only if the inertia group $I_{\mathfrak{p}}$ is trivial for all prime ideals $\mathfrak{p}$ of $A$. Proof: This is a straightforward consequence of the case of a finite group $G$, for which a proof is given in §2 of [21]. $\square$ There is a version of this criterion which will be useful to us. Let $A$ and $G$ be as above, and let $\mathfrak{q}$ denote any prime ideal of $A^{G}$. Let $\Omega$ denote any algebraically closed field which admits an embedding $i:k(\mathfrak{q})\rightarrow\Omega$. We let $\mathfrak{H}(A,\Omega,i)$ denote the set of all ring homomorphisms $A\rightarrow\Omega$ extending the composite $A^{G}\rightarrow A^{G}/\mathfrak{q}\hookrightarrow k(\mathfrak{q})\stackrel{{\scriptstyle i}}{{\longrightarrow}}\Omega$ For any closed normal subgroup $N$ of $G$ of finite index, we similarly have the set $\mathfrak{H}(A^{N},\Omega,i)$ and we have restriction maps $\mathfrak{H}(A^{N^{\prime}},\Omega,i)\longrightarrow\mathfrak{H}(A^{N},\Omega,i)$ whenever $N^{\prime}\subseteq N$. It is readily verified that $\mathfrak{H}(A,\Omega,i)\cong\mathrel{\mathop{lim}\limits_{\leftarrow}}\mathfrak{H}(A^{N},\Omega,i)$ and therefore further that $\mathfrak{H}(A,\Omega,i)$ is equipped with a continuous $G$ action. ###### Proposition 2.14 The $G$-space $\mathfrak{H}(A,\Omega,i)$ is transitive, and is $G$-homeomorphic to the quotient $G/I_{\mathfrak{p}}$, where $\mathfrak{p}$ is the kernel of some ring homomorphism $A\rightarrow\Omega$ extending $i$. Proof: For the finite case, this is proved in §2 of [21]. The extension to the profinite case is direct. $\square$ ###### Corollary 2.2 A continuous affine $G$-scheme is etale if and only if the $G$ actions on all sets $\mathfrak{H}(A,\Omega,i)$ are free, for all prime ideals $\mathfrak{p}$ of $A^{G}$ and homomorphisms $i:k(\mathfrak{p})\hookrightarrow\Omega$ for all algebraically closed fields $\Omega$. In order to define the equivariant algebraic $K$-theory of continuous affine $G$-schemes, we extend the quasi-Noetherian condition to such actions. ###### Definition 2.9 We say that a continuous affine $G$-scheme $(A,G)$ is coherent if $A$ is a coherent ring. We say it is totally coherent if $A^{N}$ is a coherent ring for all closed subgroups $N\subseteq G$. We now define the category of twisted $G$ modules for coherent continuous affine $G$-schemes. ###### Definition 2.10 Let $(A,G)$ be a ring $A$ equipped with a continuous action by a profinite group $G$, where we assume that $A$ is equipped with the discrete topology. By a $G$-twisted module over $A$, we mean an $A$ -module $M$, endowed with the discrete topology, together with a continuous action by $G$, which is compatible with the $G$ action on $A$ via the formula $g\cdot(\alpha m)=\alpha^{g}\cdot gm$ The $G$-twisted $A$-modules form an abelian category. A $G$-twisted $A$-module is said to be finitely presented if the underlying module is a finitely presented $A$-module. In the case of coherent continuous affine $G$-schemes, the finitely presented $G$-twisted $A$-modules also form an abelian category. We will write $\mbox{\em Mod}^{G}(A)$ for the category of finitely presented $G$-twisted $A$-modules. We now define the equivariant algebraic $K$-theory of coherent continuous affine $G$-schemes. ###### Definition 2.11 Given a coherent continuous affine $G$-scheme $(A,G)$, by the equivariant $K$-theory of $A$, denoted $K^{G}(A)$, we will mean the $K$-theory of the Waldhausen category associated to the abelian category $\mbox{\em Mod}^{G}(A)$. With this definition, we would now like to provide an analogue to Proposition 2.6 for actions of a profinite group. Begin with some results on coherent rings. ###### Proposition 2.15 Let $f:A\rightarrow B$ be a faithfully flat morphism of rings. Then an $A$-module $M$ is finitely generated (respectively finitely presented) if and only if $B\mathrel{\mathop{\otimes}\limits_{A}}M$ is finitely generated (finitely presented). Proof: This is Corollary 1.11 of [22] $\square$ ###### Corollary 2.3 Let $f:A\rightarrow B$ be a faithfully flat morphism of rings. If $B$ is coherent, then so is $A$. Proof: We will need to prove that the kernel of a homomorphism $f:F\rightarrow F^{\prime}$ between finitely generated $A$-modules is finitely generated. Because of the flatness, we have an exact sequence $0\rightarrow B\mathrel{\mathop{\otimes}\limits_{A}}Ker(f)\rightarrow B\mathrel{\mathop{\otimes}\limits_{A}}F\rightarrow B\mathrel{\mathop{\otimes}\limits_{A}}F^{\prime}$ The coherence of $B$ now gives that $B\mathrel{\mathop{\otimes}\limits_{A}}Ker(f)$ is finitely generated. Proposition 2.15 now gives the result. $\square$ ###### Corollary 2.4 Let $A$ be a coherent commutative ring equipped with the discrete topology, and let a profinite group $G$ act continuously on $A$. If the $G$-action is etale, then the fixed point subring $A^{G}$ is coherent. Proof: The action being etale implies that the inclusion $A^{G}\hookrightarrow A$ is faithfully flat. $\square$ ###### Corollary 2.5 For any continuous etale action of a profinite group $G$ on a coherent ring $A$, $A^{G}$ is coherent there is a natural map $K(A^{G})\rightarrow K^{G}(A^{G})\rightarrow K^{G}(A)$ In the terminology of schemes $X=Spec(A)$, there is a map $\eta_{X}:K(X/G)\rightarrow K^{G}(X)$ when $X$ is a coherent affine scheme, natural for flat maps of schemes. We wish to prove that $\eta_{X}$ is an equivalence. In the case of a finite group $G$, this result follows from Theorem II.5.1 of [25], by proving that the functors $M\rightarrow M^{G}$ from $\mbox{Mod}^{G}(A)$ to $\mbox{Mod}^{\\{e\\}}(A^{G})$ and $M\rightarrow B\mathrel{\mathop{\otimes}\limits_{A}}M$ from $\mbox{Mod}^{\\{e\\}}(A^{G})$ to $\mbox{Mod}^{G}(A)$ are inverse equivalences of categories. The fact that both functors preserve the subcategories of finitely presented modules follows from the faithful flatness of the inclusion $A^{G}\hookrightarrow A$ together with Proposition 2.15 above. ###### Proposition 2.16 The natural transformation $\eta_{X}$ is an equivalence for any continuous etale action of a profinite group $G$ on an affine scheme $X=Spec(A)$, where $A$ is coherent. Proof: We may construct the categories $\mbox{Mod}^{G/N}(A^{N})$ for any closed normal subgroup $N\subseteq G$ of finite index. We note that all the rings $A^{N}$ are coherent by Corollary 2.3, and that all $N\subseteq N^{\prime}$ the inclusions $A^{N}\hookrightarrow A^{N^{\prime}}$ are faithfully flat, because they are etale ring extensions. It follows that we have a directed system $N\rightarrow\mbox{Mod}^{G/N}(A^{N})$, where the functors $\mbox{Mod}^{G/N}(A^{N})\rightarrow\mbox{Mod}^{G/N^{\prime}}(A^{N^{\prime}})$ are defined to be $A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}-$. There is therefore a functor $j:\mathrel{\mathop{colim}\limits_{\mathrel{\mathop{\longrightarrow}\limits_{N}}}}\mbox{Mod}^{G/N}(A^{N})\rightarrow\mbox{Mod}^{G}(A)$ which we claim is an equivalence of categories. This will suffice to prove the result, because each of the functors $\mbox{Mod}^{\\{e\\}}(A^{G})\rightarrow\mbox{Mod}^{G/N}(A^{N})$ is an equivalence of categories by the validity of the result for finite group actions mentioned above. We proceed to prove that $j$ is an equivalence of categories. We first observe that $j$ is surjective on isomorphism classes of objects. In order to do this, we will first give a description of $G$-twisted modules over $A$ whose underlying $A$-module is free in terms of matrices. Suppose $F$ is a free module with continuous $G$-action twisted compatibly with the action of $G$ on $A$. We fix a basis $B=\\{b_{1},\ldots,b_{t}\\}$ for $F$. Because of the continuity of the action, it is clear that there is a closed normal subgroup $N$ of finite index in $G$ so that $N$ fixes the set $B$ elementwise. It follows that the action of an element $g\in G$ on an element of $B$ depends only on its coset in $G/N$. For each coset $\gamma$ in $G/N$, we then obtain a $t\times t$ matrix $M(\gamma)$ with entries in $A$, defined by $M(\gamma)(b_{j})=\sum_{i}M(\gamma)_{ij}b_{i}$. Let $N$ be such that $M(\gamma)_{ij}\in A^{N}$ for all $\gamma$ and pairs $(i,j)$. The coefficients of all the matrices $M(\gamma)$ are therefore acted on trivially by $N$, and therefore the action of $G$ on the free $A^{N}$-submodule generated by $B$ factors through $G/N$. The matrices $M(\gamma)$ clearly satisfy the relations $M(\gamma_{1}\gamma_{2})=M(\gamma_{1})\cdot M(\gamma_{2})^{\gamma_{1}}$ (2–1) where the superscript indicates entrywise action of $G$ on the matrix $M(\gamma_{2})$. Sets of matrices $\\{M_{\gamma}\\}_{\gamma}\in G$ satisfying these relations in turn determine a $G/N$-twisted module over $A^{N}$, and after applying $A\mathrel{\mathop{\otimes}\limits_{A^{N}}}-$, over $A$. It is furthermore clear that the result is isomorphic to the original $G$-twisted $A$-module, so we have proved that any continuous twisted $G$-module over $A$, with free underlying module, is obtained by extension of scalars from a free twisted $G/N$-module over $A$, for some closed normal subgroup of finite index $N$. Next, we will prove that given any homomorphism of $G$-twisted $A$-modules $f:A\mathrel{\mathop{\otimes}\limits_{A^{N}}}F_{0}\rightarrow A\mathrel{\mathop{\otimes}\limits_{A^{N}}}F_{1}$ where $F_{0}$ and $F_{1}$ are $G/N$-twisted $A^{N}$-modules whose underlying modules are finitely generated and free, there is an $N^{\prime}\supseteq N$ and homomorphism of $G/N^{\prime}$-twisted $A^{N^{\prime}}$-modules $\overline{f}:A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}F_{0}\rightarrow A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}F_{1}$ so that $f=A\mathrel{\mathop{\otimes}\limits_{A^{N^{\prime}}}}\overline{f}$. To see this, we observe that given bases for $F_{0}$ and $F_{1}$, $f$ determines a matrix $M_{f}$ with entries in $A$. The $G$-twisted modules $F_{0}$ and $F_{1}$ determine families of matrices $\\{M_{0}(\gamma)\\}_{\gamma\in G}$ and $\\{M_{1}(\gamma)\\}_{\gamma\in G}$ with entries in $A^{N}$, and the fact that $f$ is a homomorphism of $G$-twisted $A$-modules means that the equations $M_{1}({\gamma})M_{f}^{\gamma}=M_{f}M_{0}(\gamma)$ (2–2) hold for all $\gamma\in G$, when $M_{0}(\gamma)$ and $M_{1}(\gamma)$ are regarded as matrices with entries in $A$ via the homomorphism $A^{N}\rightarrow A$. As above, there is a $N^{\prime}\supseteq N$ so that all the entries of $M_{f}$ lie in the image of the homomorphism $A^{N^{\prime}}\rightarrow A$, and we let $\overline{M}_{f}$ denote the matrix $M_{f}$ regarded as a matrix with entries in $A^{N^{\prime}}$. We know that the relations (2–2) hold in $A$, but this means that the relations $M_{1}({\gamma})\overline{M}_{f}^{\gamma}=\overline{M}_{f}M_{0}(\gamma)$ hold in $A^{N^{\prime}}$ by the faithful flatness of the homomorphism $A^{N^{\prime}}\rightarrow A$. The required homomorphism $\overline{f}$ is now the homomorphism represented by the matrix $\overline{M}_{f}$. Next consider any finitely presented $G$-twisted $A$-module $M$, with a presentation $F_{1}\stackrel{{\scriptstyle f}}{{\rightarrow}}F_{0}\rightarrow M\rightarrow 0$ (2–3) where $F_{0}$ and $F_{1}$ are $G$-twisted $A_{F}$-modules with finitely generated free underlying modules. Our first argument above shows that there is closed normal subgroup of finite index $N$ so that $F_{0}$ and $F_{1}$ are extended from $A^{N}$, and the second shows that there is a $N^{\prime}\supseteq N$ so that $f$ is extended from $A^{N^{\prime}}$. Let $\overline{F}_{0},\overline{F}_{1}$ be $G/N^{\prime}$-twisted $A^{N^{\prime}}$-modules (with free underlying modules) and $\overline{f}$ be a homomorphism $\overline{f}:\overline{F}_{1}\rightarrow\overline{F}_{0}$ of $G/N^{\prime}$-twisted $A^{N^{\prime}}$-modules so that the presentation 2–3 is obtained by applying $A\mathrel{\mathop{\otimes}\limits_{A^{N^{\prime}}}}-$ to the exact sequence $\overline{F}_{1}\stackrel{{\scriptstyle\overline{f}}}{{\rightarrow}}\overline{F}_{0}\rightarrow C\rightarrow 0$ of $G/N^{\prime}$-twisted $A^{N^{\prime}}$-modules, where $C$ denotes the cokernel of $\overline{f}$. It follows by flatness that $M\cong A\mathrel{\mathop{\otimes}\limits_{A^{N^{\prime}}}}C$. It follows that the functor $j$ above is surjective on isomorphism classes of objects. To prove injectivity on isomorphism classes of objects, we suppose that we are given a closed normal subgroup of finite index $N\subseteq G$ and finitely presented $G/N$-twisted $A^{N}$-modules $M$ and $M^{\prime}$, so that $A\mathrel{\mathop{\otimes}\limits_{A^{N}}}M$ and $A\mathrel{\mathop{\otimes}\limits_{A^{N}}}M^{\prime}$ are isomorphic. We suppose we are given an isomorphism $f:A\mathrel{\mathop{\otimes}\limits_{A^{N}}}M\rightarrow A\mathrel{\mathop{\otimes}\limits_{A^{N}}}M^{\prime}$ and its inverse $f^{-1}$. Both $f$ and $f^{-1}$ are of the form $A\mathrel{\mathop{\otimes}\limits_{A^{N^{\prime}}}}\overline{f}$ and $A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}\overline{f^{-1}}$ for some $N^{\prime}$ and homomorphisms $\overline{f}:A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}M\rightarrow A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}M^{\prime}$ and $\overline{f^{-1}}:A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}M^{\prime}\rightarrow A^{N^{\prime}}\mathrel{\mathop{\otimes}\limits_{A^{N}}}M$ But $\overline{f}$ and $\overline{f^{-1}}$ are also inverses, because they become so over $A$, and because of the faithful flatness of $A^{N^{\prime}}\rightarrow A$. It follows that $\overline{f}$ is an isomorphism, which gives the injectivity result for isomorphism classes of objects. Finally we will need to check that for any pair of objects $x,y$ in $\underline{C}=\mathrel{\mathop{colim}\limits_{\mathrel{\mathop{\longrightarrow}\limits_{N}}}}\mbox{Mod}^{G/N}(A^{N})$, $j$ induces a bijection $Hom_{\underline{C}}(x,y)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}Hom_{\mbox{Mod}^{G}(A)}(j(x),j(y))$ The injectivity of this map is immediate from the faithful flatness of all the maps $A^{N}\rightarrow A$. To see the surjectivity, we assume that the objects $x$ and $y$ are represented by $G/N$-twisted $A^{N}$-modules $M$ and $M^{\prime}$, respectively, equipped with presentations $F_{1}\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}F_{0}\rightarrow M\rightarrow 0$ and $F_{1}^{\prime}\stackrel{{\scriptstyle\beta}}{{\rightarrow}}F_{0}^{\prime}\rightarrow M^{\prime}\rightarrow 0$ A homomorphism $f$ from $M$ to $M^{\prime}$ can be represented by a pair of homomorphisms $f_{0}:F_{0}\rightarrow F_{0}^{\prime}$ and $f_{0}:F_{1}\rightarrow F_{1}^{\prime}$ so that $f_{0}\alpha=\beta f_{1}$. These homomorphisms can be represented by matrices, and by arguments similar to the ones above we find that these matrices can be represented over $A^{N}$ for some closed normal subgroup of finite index $N\subseteq G$. This gives the result. $\square$ ### 2.5 Homotopy properties Let $G$ be any finite group, and let $G$ act by automorphisms on a vector space $V$ over a field $k$ via a representation $\rho$f. We suppose that $\\#(G)$ is relatively prime to $char(k)$. We may form the skew polynomial ring $k[V]_{\rho}[G]$ (see [20]). The following is now a formal observation. ###### Proposition 2.17 The category $\mbox{Mod}^{G}(V)$ ($V$ is regarded as an affine scheme) is equivalent to the category of finitely generated modules over the regular Noetherian ring $k[V]_{\rho}[G]$. Let $\mbox{Rep}[G]$ denote the category of finite dimensional $k$-linear representations of $G$, which can be identified with the category of finitely generated modules over the group ring $k[G]$. We have an inclusion $i:k[G]\hookrightarrow k[V]_{\rho}[G]$ and a projection $\pi:k[V]_{\rho}[G]\rightarrow k[G]$ of rings, sending the vector space $V$ identically to $0$. The following result is now an immediate consequence of Theorem 7 of [31]. ###### Theorem 2.1 The ring homomorphisms $i$ and $\pi$ give rise to exact functors $k[V]_{\rho}[G]\mathrel{\mathop{\otimes}\limits_{k[G]}}-:\mbox{\em Rep}[G]\rightarrow\mbox{ \em Mod}^{G}(V)$ and $k[G]\mathrel{\mathop{\otimes}\limits_{k[V]_{\rho}[G]}}-:\mbox{\em Mod}^{G}(V)\rightarrow\mbox{\em Rep}[G]$ Both functors induce equivalences on $K$-theory spectra. We will also need to describe how induction of representations works in this context. Let $H\subseteq G$ be an inclusion of finite groups, and let $\rho$ denote a $k$-linear representation of $H$, with $V_{\rho}$ as the representation space. The induced representation $i_{H}^{G}(\rho)$ can be defined via the operation $k[G]\mathrel{\mathop{\otimes}\limits_{k[H]}}-:\mbox{Mod}(k[H])\rightarrow\mbox{Mod}(k[G])$ where for a ring $A$ the category of finitely generated left $A$ -modules is denoted by $\mbox{Mod}(A)$. We denote the representation space of $i_{H}^{G}(\rho)$ by $V_{i_{H}^{G}(\rho)}$. There is a natural $H$-equivariant inclusion $V_{\rho}\hookrightarrow V_{i_{H}^{G}(\rho)}$, and consequently an inclusion of rings $k[V_{\rho}]_{\rho}[H]\hookrightarrow k[V_{i_{H}^{G}(\rho)}]_{{i_{H}^{G}(\rho)}}[G]$ ###### Proposition 2.18 Given $H,G$, and $\rho$ as above, the following diagram of functors commutes. $\begin{CD}\mbox{{\em Mod}}(k[H])@>{\phantom{\mbox{xxxxxxx}}{k[G]\mathrel{\mathop{{\otimes}}\limits_{k[H]}}-}\phantom{\mbox{xxxxxxxxxx}}}>{}>\mbox{\em Mod}(k[G])\\\ @V{k[V_{\rho}]_{\rho}[H]\mathrel{\mathop{\otimes}\limits_{k[H]}}-}V{}V@V{}V{k[V_{i^{G}_{H}(\rho)}]_{i_{H}^{G}(\rho)}[G]\mathrel{\mathop{\otimes}\limits_{k[G]}}-}V\\\ \mbox{\em Mod}(k[V_{\rho}]_{\rho}[H])@>{k[V_{i^{G}_{H}(\rho)}]_{i_{H}^{G}(\rho)}[G]\mathrel{\mathop{\otimes}\limits_{k[V_{\rho}]_{\rho}[H]}}-}>{}>\mbox{\em Mod}(k[V_{i^{G}_{H}(\rho)}]_{i_{H}^{G}(\rho)}[G])\end{CD}$ ### 2.6 Completions We recall from [7] that given a homomorphism of commutative ring spectra $f:A\rightarrow B$, and an $A$-module $M$, one may construct the derived completion $M^{\wedge}_{f}$. Here are some examples of this construction. Example: Any commutative ring may be thought of as a commutative ring spectrum via the Eilenberg-MacLane construction. If $f:A\rightarrow B$ is a surjective homomorphism of commutative Noetherian rings, and $M$ is a finitely generated $A$-module, then the derived completion $M^{\wedge}_{f}$ is the Eilenberg- MacLane construction on the usual completion $M^{\wedge}_{I}$, where $I$ is the kernel of $f$. Example Let $f:S^{0}\rightarrow\mathbb{H}_{l}$ be the mod $l$ Hurewicz map, and let $X$ be any spectrum (and therefore an $S^{0}$-module). Then $X^{\wedge}_{f}$ is the usual Bousfield-Kan notion of completion of the spectrum at the prime $l$ (see [6]). Example: Let $A=R[\mathbb{Z}_{l}]$ be the representation ring of the additive group of the $l$-adic integers, defined as the direct limit of the representation rings of the finite cyclic $l$-th power order groups. Let $f:A\rightarrow\mathbb{F}_{l}$ be the augmentation followed by reduction mod $l$. Then $A^{\wedge}_{f}$ is equivalent to the $l$-adic completion of the integral group ring of the simplicial group of points on the circle group. In particular, the homotopy groups are $\cong\mathbb{Z}_{l}$ for dimensions 0 and 1, and are $\cong 0$ otherwise. It has a number of useful properties which can be found in [7], and we record some of them here. Let $f:A\rightarrow B$ be a homomorphism of commutative ring spectra. ###### Proposition 2.19 Suppose that $M\rightarrow N\rightarrow P$ is a cofibration sequence of $A$-modules. Then the naturally defined sequence $M^{\wedge}_{f}\rightarrow N^{\wedge}_{f}\rightarrow P^{\wedge}_{f}$ is also a cofibration sequence. ###### Proposition 2.20 Suppose we are given a homomorphism $\lambda:M\rightarrow N$ of $A$-modules. Suppose further that the derived smash product map $id_{B}\wedge\lambda:B\mathrel{\mathop{\wedge}\limits_{A}}M\rightarrow B\mathrel{\mathop{\wedge}\limits_{A}}N$ is an equivalence of spectra. Then the natural map $\lambda^{\wedge}_{f}:M^{\wedge}_{f}\rightarrow N^{\wedge}_{f}$ is an equivalence of spectra. In particular, if $B\mathrel{\mathop{\wedge}\limits_{A}}M\simeq$, then $M^{\wedge}_{f}\simeq$. Remark: By the derived smash product, we mean the smash product construction applied to cofibrant replacements for $M$ and $N$. If they are already cofibrant, then one can apply the smash product directly. ###### Proposition 2.21 Suppose that $A$ is a (-1)-connected commutative ring spectrum, and that $M$ is a connective $A$-module. Suppose further that we have a homomorphism of commutative ring spectra $A\rightarrow\mathbb{H}_{l}$, and that for all $n$ and $i$, we have that the groups $Tor^{\pi_{0}A}_{i}(\pi_{n}M,\mathbb{F}_{l})$ vanish. Then we have $M\mathrel{\mathop{\wedge}\limits_{A}}\mathbb{H}_{l}\simeq$, and in particular $M^{\wedge}_{\mathbb{H}_{l}}\simeq$. Proof: We have the Künneth spectral sequence converging to $\pi_{}(M\mathrel{\mathop{\wedge}\limits_{A}}\mathbb{H}_{l})$, whose $E_{2}$-term is $Tor_{\pi_{}A}(\pi_{}M,\mathbb{F}_{l})$. We also have the inclusion $\pi_{0}A\hookrightarrow\pi_{}A$, as the inclusion of the degree zero component in the graded ring $\pi_{}A$. We let $B_{}$ denote the commutative graded ring $\mathbb{F}_{l}\mathrel{\mathop{\otimes}\limits_{\pi_{0}A}}\pi_{}A$. We can now describe the functor $\mathbb{F}_{l}\mathrel{\mathop{\otimes}\limits_{\pi_{}A}}-$ from graded $\pi_{}A$-modules to graded $\mathbb{F}_{l}$-vector spaces as the composite of the functor $\mathbb{F}_{l}\mathrel{\mathop{\otimes}\limits_{\pi_{0}A}}-$ from graded $\pi_{}A$-modules to graded $B_{}$-modules with the functor $\mathbb{F}_{l}\mathrel{\mathop{\otimes}\limits_{B_{}}}-$ from the category of graded $B_{}$-modules to graded $\mathbb{F}_{l}$-vector spaces. The Grothendieck spectral sequence now shows that the vanishing of the derived functors of the first functor forces the vanishing of the derived functors of the composite. $\square$ ###### Proposition 2.22 Let $f:A\rightarrow B$ be a homomorphism of $(-1)$-connected commutative ring spectra, with $\pi_{0}(f)$ surjective, and let $M$ be an $A$-module. Let $I$ denote the kernel of $\pi_{0}(f)$. Suppose that the $\pi_{0}A$-modules $\pi_{j}M$ are such that for each $j$, there is an integer $e_{j}$ so that $I^{e_{j}}\cdot\pi_{j}M=\\{0\\}$. Then the natural map $\eta:M\rightarrow M^{\wedge}_{f}$ is an equivalence. Proof: Theorem 7.1 of [7] reduces us to the case where $A$ is the ring $\pi_{0}A$, $B$ is the ring $\pi_{0}B$, and $M$ is a module for which $I^{e}M=\\{0\\}$ for some $e$. We then obtain the finite filtration $\\{0\\}=I^{e}M\subseteq I^{e-1}M\subseteq\cdots\subseteq I^{2}M\subseteq IM\subseteq M$ each of whose quotients is annihilated by $I$. It follows from (6) of Proposition 3.2 of [7] that $\eta:I^{s}M/I^{s+1}M\rightarrow(I^{s}M/I^{s+1}M)^{\wedge}_{f}$ is an equivalence. A straightforward application of Proposition 2.19 now gives the result. $\square$ The following technical criterion will be useful in later sections. ###### Proposition 2.23 Let $M_{0}\stackrel{{\scriptstyle f_{0}}}{{\rightarrow}}M_{1}\stackrel{{\scriptstyle f_{1}}}{{\rightarrow}}M_{2}\stackrel{{\scriptstyle f_{2}}}{{\rightarrow}}\cdots$ be a directed system of module spectra over the commutative ring spectrum $A$. Suppose further that we are given a homomorphism $f:A\rightarrow B$ of commutative ring spectra. Finally, suppose that the $Ind$-groups $\pi_{t}(B\mathrel{\mathop{\wedge}\limits_{A}}M_{0})\rightarrow\pi_{t}(B\mathrel{\mathop{\wedge}\limits_{A}}M_{1})\rightarrow\pi_{t}(B\mathrel{\mathop{\wedge}\limits_{A}}M_{2})\rightarrow\cdots$ are $Ind$-trivial. Then the $A$-module $M_{\infty}=\mbox{ }\mathrel{\mathop{hocolim}\limits_{r}}M_{r}$ satisfies $B\mathrel{\mathop{\wedge}\limits_{A}}M_{\infty}\simeq$, and therefore $(M_{\infty})^{\wedge}_{f}\simeq$. Proof: Because smash products are colimit constructions, it is possible to commute the constructions $B\mathrel{\mathop{\wedge}\limits_{A}}-$ and colim, which shows that $B\mathrel{\mathop{\wedge}\limits_{A}}M_{\infty}\simeq$. $\square$ One situation in which one can prove that the hypotheses of Proposition 2.23 are satisfied is treated in the following result. ###### Proposition 2.24 Suppose $f:A\rightarrow B$ is a homomorphism of $(-1)$-connected commutative ring spectra, which is surjective on $\pi_{0}$, and suppose $L$ is a connective commutative $A$-algebra. , equipped with a homomorphism of commutative $A$-algebras $\epsilon_{L}:L\rightarrow B$ so that the diagram $\begin{diagram}$ commutes. Let $I_{A}\subseteq\pi_{0}A$ denote the ideal $Ker(f)$. Consider now an Ind-$A$-module of the form $L\stackrel{{\scriptstyle\times\alpha_{0}}}{{\longrightarrow}}L\stackrel{{\scriptstyle\times\alpha_{1}}}{{\longrightarrow}}L\stackrel{{\scriptstyle\times\alpha_{2}}}{{\longrightarrow}}\cdots$ (2–4) where $\alpha_{i}$ is an element of the ideal $I_{A}\cdot\pi_{0}L$. Then Ind- groups $\pi_{t}(B\mathrel{\mathop{\wedge}\limits_{A}}L)\stackrel{{\scriptstyle\pi_{t}(\times\alpha_{0})}}{{\longrightarrow}}\pi_{t}(B\mathrel{\mathop{\wedge}\limits_{A}}L)\stackrel{{\scriptstyle\pi_{t}(\times\alpha_{1})}}{{\longrightarrow}}\pi_{t}(B\mathrel{\mathop{\wedge}\limits_{A}}L)\stackrel{{\scriptstyle\pi_{t}(\times\alpha_{2})}}{{\longrightarrow}}\cdots$ (2–5) are Ind-trivial, and therefore by Proposition 2.23, the homotopy colimit $L_{\infty}=\mbox{}\mathrel{\mathop{hocolim}\limits_{i}}L$ satisfies $B\mathrel{\mathop{\wedge}\limits_{A}}L_{\infty}\simeq$. Proof: From the theory of derived tensor products for modules over ring spectra, as developed in [33], there is a spectral sequence with $E_{2}$-term $Tor^{\pi_{}A}_{}(\pi_{}B,\pi_{}L)$ converging to $\pi_{}(B\mathrel{\mathop{\wedge}\limits_{A}}L)$. We now have an inductive system of spectral sequences corresponding to diagram 2–5 above, and claim that maps in the system are identically zero. To prove this, we must show that any $\lambda\in I_{A}\cdot\pi_{0}L$ induces the zero map on all $Tor$-groups. To see this, we write such a $\lambda$ as $\lambda=\sum_{s}i_{s}\lambda_{s}$ where $i_{s}\in I_{A}$, and $\lambda_{s}\in\pi_{0}L$. If we can prove that each $i_{s}$ induces the zero map on $Tor$-groups, we will have verified the claim. But this is clear, since multiplication by elements of $I_{A}$ induce the zero map on $\pi_{}B$. Further, the colimit of the spectral sequences converges to the colimit of the homotopy groups due to the strong convergence of the spectral sequences which is guaranteed by the connectivity hypothes on $A$ and $B$, which gives the result. $\square$ ###### Corollary 2.6 Suppose we are given $A,B,L,\mbox{ and }I_{A},$ are as in Proposition 2.24, and that $\epsilon_{L}:L\rightarrow B$ is a homomorphism of commutative ring spectra so that the diagram $\begin{diagram}$ commutes. Let $I_{L}$ be the kernel of $\pi_{0}(\epsilon_{L})$. Suppose further that there is an integer $N$ so that $I_{L}^{N}\subseteq I_{A}\cdot\pi_{0}(L)$. We suppose we are given an Ind-$A$-module as in (2–4) above, but with the assumption that $\alpha_{i}\in I_{L}$ rather than that $\alpha_{i}\in I_{A}$. Then the Ind-groups in (2–5) are trivial, and we again have that $B\mathrel{\mathop{\wedge}\limits_{A}}L_{\infty}\simeq$ Proof: We note that any $N$-fold composite in the system (2–5) has its image in $I_{A}\cdot\pi_{0}(L)$, and a standard cofinality argument gives the result. $\square$ ## 3 Totally torsion free groups We recall that a topological group $G$ is said to be profinite if it is Hausdorff, compact, and totally disconnected. These properties imply that as a topological group, it can be expressed as an inverse limit of finite groups. Let $l$ denote a prime. A profinite $l$-group is a profinite group whose only finite quotients are finite $l$-groups. For any profinite group $G$, we define the commutator subgroup of $G$, $[G,G]$ to be the closure of the subgroup generated by elements of the form $ghg^{-1}h^{-1}$ in $G$. The subgroup $[G,G]$ is normal, and the quotient $G/[G,G]$ will be denoted by $G^{ab}$. It is the universal profinite abelian quotient of $G$. One also has the following. ###### Proposition 3.1 Any closed subgroup of a profinite group is itself a profinite group. The quotient of a profinite group by a closed normal subgroup is also a profinite group. In particular, any subgroup containing a closed subgroup of finite index is itself a closed subgroup. This is in $\S$2.1 of [32]. We say a profinite $l$-group $G$ is totally torsion free if for every closed subgroup $K\subseteq G$ the quotient group $K^{ab}$ is torsion free. Free abelian pro-$l$ groups and free pro-$l$ groups are totally torsion free, and free products (in the category of profinite pro-$l$ groups) of totally torsion free pro-$l$ groups are totally torsion free. However, the condition is quite restrictive, as the following example shows. Example: The $2$-adic Heisenberg group, i.e. the group of $3\times 3$ upper triangular matrices with entries in $\mathbb{Z}_{2}$ and ones on the diagonal is not TTF, although it is easily seen to be torsion free. To see this, we note that the subgroup of matrices of the form $\left[\begin{array}[]{ccc}1&2k&l\\\ 0&1&m\\\ 0&0&1\end{array}\right]$ has abelianization isomorphic to $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}/2\mathbb{Z}$ The matrix $\left[\begin{array}[]{ccc}1&0&1\\\ 0&1&0\\\ 0&0&1\end{array}\right]$ is not a commutator, but its square is. ###### Proposition 3.2 Let $k$ be a field containing all roots of unity, with absolute Galois group $G_{k}$. Then $G_{k}$ is totally torsion free. Proof: It clearly suffices to prove that $G_{k}^{ab}$ is torsion free for all fields $k$ containing all the roots of unity. Kummer theory (see e.g. [26], §VI.8) tells us that $G_{k}^{ab}$ is the Galois group of the infinite extension $k^{ab}=\bigcup_{m}k((k^{})^{\frac{1}{m}})$, and therefore that any non-identity element $g\in G_{k}^{ab}$ acts non-trivially on an element of the form $\sqrt[m]{\kappa}$ for some $m$ and $\kappa\in k$. Suppose now that $g^{n}=e$ in $G_{k}^{ab}$. We have $g\cdot\sqrt[m]{\kappa}=\zeta\sqrt[m]{\kappa}$ for some $\zeta\in\mu_{m}(k)$. Now choose any $n$-th root $\xi$ of $\sqrt[m]{\kappa}$. We see that $g\cdot\xi=\eta\xi$ for some $mn$-th root of unity $\eta$ satisfying $\eta^{n}=\zeta$. Now, $g^{n}\cdot\xi=\eta^{n}\xi=\zeta\xi\neq\xi$, which contradicts the assumption that $g$ is $n$-torsion. $\square$ Remark: This structural property of Galois groups will allow us to identify completion constructions based on representation theory with $K$-spectra of certain schemes which appear to be the algebraic geometric version of the classifying space construction in topology. It would be interesting to consider how this property relates with the Bloch-Kato conjecture. ## 4 Construction of some continuous affine $G$-schemes Let $G$ be a profinite $l$-group. We will need to construct continuous affine $G$-actions so as to be able to assemble the scheme ${\cal E}G$ we require. In this section, we will construct such actions attached to affine representations over $\mathbb{Z}_{l}$ of $G$. By taking (infinite) products of schemes constructed in this way, we will obtain the scheme ${\cal E}G$ discussed in the introduction when $G$ is totally torsion free. ###### Definition 4.1 An affine $\mathbb{Z}_{l}$-module is a pair $(F,X)$, where $F$ is a free finitely generated $\mathbb{Z}_{l}$-module, and $X$ is a continuous simply transitive $F$-space. Here $F$ is regarded as a topological abelian group, and the simple transitivity means that for every $x\in X$, the natural map $F\rightarrow X$ given by $f\rightarrow f\cdot x$ is a homeomorphism. A based affine $\mathbb{Z}_{l}$-module is a triple $(F,X,B)$ so that the pair $(F,X)$ is an affine $\mathbb{Z}_{l}$-module and $B$ is a basis for the $\mathbb{Z}_{l}$-module $F$. An isomorphism of affine $\mathbb{Z}_{l}$-modules from $(F,X)$ to $(F^{\prime},X^{\prime})$ is a pair $(\varphi,\chi)$, where $\varphi:F\rightarrow F^{\prime}$ is a continuous isomorphism of $\mathbb{Z}_{l}$-modules and $\chi:X\rightarrow X^{\prime}$ is a homeomorphism, and where the relationship $\chi(f\cdot x)=\varphi(f)\cdot\chi(x)$ holds for all $f$ and $x$. An isomorphism $\varphi$ of based affine $\mathbb{Z}_{l}$-modules is an isomorphism of the underlying affine $\mathbb{Z}_{l}$-modules which preserves $\mathbb{Z}_{l}$-multiples of basis elements, i.e. for all $z\in\mathbb{Z}_{l}$ and $b\in B$ there exist $z^{\prime}\in\mathbb{Z_{l}}$ and $b^{\prime}\in B$ so that $\varphi({zb})=z^{\prime}b^{\prime}$. If for all $z$ and $b$, there is an element $z^{\prime}\in\mathbb{Z}_{l}$ so that $\varphi(zb)=z^{\prime}b$, then the isomorphism is said to be a translation. ###### Proposition 4.1 For $F$ a free, finitely generated $\mathbb{Z}_{l}$-module and a simply transitive $G$-space $X$, we have an isomorphism $\mbox{Aut}(F,X)\cong GL(F)\ltimes F$ For a based affine $\mathbb{Z}_{l}$-module $(F,X,B)$ we have $\mbox{Aut}(F,X,B)\cong\Sigma_{B}\ltimes F$ where $\Sigma_{B}$ denotes the symmetric group of permutations of the set $B$. Proof: Clear from the definitions of the morphisms. $\square$ Remark: The group $\mbox{Aut}(F,X,B)$ is a profinite group. For every $n>0$, we have the quotient of $X$ by the equivalence relation $\simeq_{n}$ given by $x\simeq_{n}y$ if and only if there is an element $f\in l^{n}F$ so that $f\cdot x=y$. These equivalence relations are respected by the action of $\mbox{Aut}(F,X,B)$, and the quotients are finite sets. Let $(F,X,B)$ be a based affine $\mathbb{Z}_{l}$-module, and let $\mathfrak{A}=\mbox{Aut}(F,X,B)$. Also, let $\mathfrak{T}\subseteq\mathfrak{A}$ denote the subgroups of translations. We now construct a family $\\{\rho_{s}\\}$ of $k$-linear representations of $\mathfrak{A}$. Let $T^{\infty}$ denote the inverse limit $\mathrel{\mathop{lim}\limits_{\leftarrow}}\mu_{l^{n}}$, and choose a continuous identification $\alpha:\mathbb{Z}_{l}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T^{\infty}$. Let $\pi_{s}:T^{\infty}\rightarrow\mu_{l^{s}}$ be the projection. Fixing an element $b\in B$, we define a subgroup $\mathfrak{A}^{\prime}\subseteq\mathfrak{A}$ to be $p^{-1}(\Sigma_{B}(b))$, where $p:\mathfrak{A}\rightarrow\Sigma_{B}$ is the natural projection, and where $\Sigma_{B}(b)\subseteq\Sigma_{B}$ denotes the stabilizer of the element $b$. We define a one dimensional character $\sigma_{s}$ of $\mathfrak{A}^{\prime}$ by setting $\sigma_{s}(b)=\pi_{s}\alpha(1)$, $\sigma_{s}(b^{\prime})=1$ for $b^{\prime}\neq b$, and $\sigma_{s}(\Sigma_{B}(b))=\\{1\\}$. We define $\rho_{s}$ to be the induced representation $\mbox{Ind}_{\mathfrak{A}^{\prime}}^{\mathfrak{A}}(\sigma_{s})$ Of course, each $\rho_{s}$ now gives an action of $\mathfrak{A}$ on the $n$-dimensional affine space over $k$ by algebraic automorphisms. We denote this variety with $\mathfrak{A}$-action by $\mathbb{A}^{n}(k,\rho_{s})$. Note that the affine space also is equipped with a fixed choice of coordinates, since it is an induced representation. The coordinates correspond to the cosets $\Sigma_{B}/\Sigma_{B}(b)$, which in turn correspond to the basis $B$ of $F$. We define a self map $\theta:\mathbb{A}^{n}(k)\rightarrow\mathbb{A}^{n}(k)$ by requiring that $\theta(\sum x_{b}b)=\sum x_{b}^{l}b$ ###### Proposition 4.2 The map $\theta$ is $\mathfrak{A}$-equivariant as a map from $\mathbb{A}^{n}(k,\rho_{s+1})$ to $\mathbb{A}^{n}(k,\rho_{s})$ Proof: Immediate. $\square$ We now consider the pro-scheme $\cdots\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathbb{A}^{n}(k,\rho_{s+1})\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathbb{A}^{n}(k,\rho_{s})\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathbb{A}^{n}(k,\rho_{s-1})\rightarrow\cdots\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathbb{A}^{n}(k,\rho_{1})\stackrel{{\scriptstyle\theta}}{{\rightarrow}}\mathbb{A}^{n}(k,\rho_{0})$ It is equipped with the action by $\mathfrak{A}$, and we will denote its inverse limit by ${\cal A}_{(F,X,B)}$. This inverse limit exists because the system consists of affine varieties, and is itself the affine scheme associated to the ring $\bigcup_{n}k[t^{\frac{1}{l^{n}}}]$. The $\mathfrak{A}$-action is a continuous affine $\mathfrak{A}$-action. We also construct the related continuous affine $\mathfrak{A}$-action ${\cal T}_{(F,X,B)}$, which is obtained by deleting the coordinate hyperplanes from each $\mathbb{A}^{n}(k,\rho_{s})$ for each $s$. We also define ${\cal A}^{s}_{(F,X,B)}$ and ${\cal T}^{s}_{(F,X,B)}$ to be the representation $\mathbb{A}^{n}(k,\rho_{s})$ and the result of removing the coordinate hyperplanes from $\mathbb{A}^{n}(k,\rho_{s})$, respectively. Remark: The construction of ${\cal A}_{(F,X,B)}$ and ${\cal T}_{(F,X,B)}$ depends on the choice of $b\in B$. It is easy to verify that varying this choice does not affect ${\cal A}_{(F,X,B)}$ or ${\cal T}_{(F,X,B)}$ up to isomorphism. ###### Proposition 4.3 The restriction of the $\mathfrak{A}$ action on ${\cal T}_{(F,X,B)}$ to $\mathfrak{T}$ is an etale $\mathfrak{T}$-action. Proof: Consider the direct system of rings $k[t_{1}^{\pm 1},\ldots,t_{n}^{\pm 1}]\subseteq k[t_{1}^{\pm\frac{1}{l}},\ldots,t_{n}^{\pm\frac{1}{l}}]\subseteq k[t_{1}^{\pm\frac{1}{l^{2}}},\ldots,t_{n}^{\pm\frac{1}{l^{2}}}]\subseteq\cdots$ where $n$ is the rank of $F$, and let ${\cal L}_{n}$ denote its colimit. This system is identified with the system of rings one obtains by applying the affine coordinate ring functor to the system defining ${\cal T}_{(F,X,B)}$. Further, ${\cal L}_{n}$ is an infinite Galois extension of $k[t_{1}^{\pm 1},\ldots,t_{n}^{\pm 1}]$, with Galois group $\Gamma=\mathbb{Z}_{l}^{n}$. The restriction of the $\mathfrak{A}$-action on ${\cal T}_{(F,X,B)}$ constructed above using the linear representations $\rho_{s}$, when restricted to $\mathfrak{T}$, clearly provide an identification of $\mathfrak{T}\cong F$ with $\Gamma$, and the result now follows. $\square$ ###### Definition 4.2 For any pro-$l$ group $G$, an affine $l$-adic representation (or $l$-BAR) is a continuous homomorphism from $G$ to the automorphism group of a based affine $\mathbb{Z}_{l}$-module. For any affine $l$-adic representation $\eta$ of $G$ on an affine $\mathbb{Z}_{l}$ module $(F,X,B)$, we obtain actions of $G$ on the schemes ${\cal A}_{(F,X,B)}$ and ${\cal T}_{(F,X,B)}$ by pulling back the action of the automorphism group of $(F,X,B)$ on these schemes. We will denote these $G$-schemes by ${\cal A}_{\eta}$ and ${\cal T}_{\eta}$. We also write ${\cal A}^{s}_{\eta}$ and ${\cal T}_{\eta}^{s}$ for the pullback along $\eta$ of the actions ${\cal A}_{(F,X,B)}^{s}$ and ${\cal T}_{(F,X,B)}^{s}$. There is a useful description of ${\cal T}_{\eta}^{s}$ which we will need. ###### Proposition 4.4 The scheme ${\cal T}_{\eta}^{s}$ can be identified with the quotient ${\cal T}_{\eta}/l^{s}\cdot\mathfrak{T}$, where $\mathfrak{T}$ denotes the full translation subgroup of $\mathfrak{A}$. $G$ acts on this quotient, since the image of $G$ under $\eta$ normalizes all the subgroups $l^{s}\cdot\mathfrak{T}$. Proof: This is just the ring theoretic result that $\bigcup_{n}k[t_{1}^{\pm\frac{1}{l^{n}}},\ldots,t_{k}^{\pm\frac{1}{l^{n}}}]^{l^{s}\cdot\mathfrak{T}}=k[t_{1}^{\pm\frac{1}{l^{s}}},\ldots,t_{k}^{\pm\frac{1}{l^{s}}}]$ $\square$ ###### Proposition 4.5 Let $\eta$ be any $l$-BAR. Then the continuous affine $G$-actions ${\cal A}_{\eta}\times Spec(L)$ and ${\cal T}_{\eta}\times Spec(L)$ are both coherent, where $L$ is any field equipped with a continuous action by $G$. As always, it is assumed that $L$ is equipped with the discrete topology. Proof: It suffices to prove that the rings $\bigcup_{n}L[x_{1}^{\frac{1}{l^{n}}},\ldots,x_{s}^{\frac{1}{l^{n}}}]$ and $\bigcup_{n}L[x_{1}^{\pm\frac{1}{l^{n}}},\ldots,x_{s}^{\pm\frac{1}{l^{n}}}]$ are coherent rings for any field $L$. This follows immediately from Proposition 2.3 since the inclusions $L[x_{1}^{\frac{1}{l^{n}}},\ldots,x_{s}^{\frac{1}{l^{n}}}]\hookrightarrow L[x_{1}^{\frac{1}{l^{n+1}}},\ldots,x_{s}^{\frac{1}{l^{n+1}}}]$ and $L[x_{1}^{\pm\frac{1}{l^{n}}},\ldots,x_{s}^{\pm\frac{1}{l^{n}}}]\hookrightarrow L[x_{1}^{\pm\frac{1}{l^{n+1}}},\ldots,x_{s}^{\pm\frac{1}{l^{n+1}}}]$ are all flat. $\square$ Given a possibly infinite set of $l$-BAR’s $\\{\eta_{i}\\}$, the schemes $\prod{\cal A}_{\eta_{i}}$ and $\prod{\cal T}_{\eta_{i}}$ both are also continuous affine $G$-schemes. ###### Corollary 4.1 Both $\prod{\cal A}_{\eta_{i}}$ and $\prod{\cal T}_{\eta_{i}}$ are coherent. Proof: It will suffice to prove that the rings $\bigcup_{n}L[x_{1}^{\frac{1}{l^{n}}},\ldots,x_{s}^{\frac{1}{l^{n}}},\ldots]$ and $\bigcup_{n}L[x_{1}^{\pm\frac{1}{l^{n}}},\ldots,x_{s}^{\pm\frac{1}{l^{n}}},\ldots]$ are coherent. This follows easily from Proposition 2.3 since the inclusions $\bigcup_{n}L[x_{1}^{\frac{1}{l^{n}}},\ldots,x_{s}^{\frac{1}{l^{n}}}]\hookrightarrow\bigcup_{n}L[x_{1}^{\frac{1}{l^{n}}},\ldots,x_{s+1}^{\frac{1}{l^{n}}}]$ and $\bigcup_{n}L[x_{1}^{\pm\frac{1}{l^{n}}},\ldots,x_{s}^{\pm\frac{1}{l^{n}}}]\hookrightarrow\bigcup_{n}L[x_{1}^{\pm\frac{1}{l^{n}}},\ldots,x_{s+1}^{\pm\frac{1}{l^{n}}}]$ are all flat. $\square$ We now have the following definition. ###### Definition 4.3 Let $G$ be a profinite group, and suppose $K$ is any closed subgroup of finite index. We say an affine $l$-adic representation $\eta$ of $G$ is $K$-principal if the stabilizer in $G$ of any $x\in X_{\eta}$ is equal to $K$. Note that it follows that $K$ is normal. Clearly, the condition is equivalent to the condition that $X$ is a free $G/K$-set. The following lemma is an $l$-adic analogue of the results of Auslander and Kuranishi concerning the structure of fundamental groups of flat manifolds [3]. ###### Lemma 4.1 Let $K\subseteq G$ be a closed normal subgroup of a profinite group $G$. Then a based affine $l$-adic representation $\eta$ of a $G$ group is $K$-principal if and only if the image of $\eta$ in the automorphism group of $X_{\eta}$ is torsion free. Proof: We let $F$ be a free finitely generated $\mathbb{Z}_{l}$-module with basis $B$, so that $(F_{\eta},X_{\eta},B_{\eta})\cong(F,F,B)$ where $F$ acts on itself by translation. The automorphism group of $(F,F,B)$ is a semidirect product $\Sigma_{B}\ltimes F$. The group $\Sigma_{B}$ acts on $F$ by permuting the basis $B$, and we describe the action of an element $(\sigma,f)$, where $\sigma\in\Sigma_{B}$ and $f\in F$ by $(\sigma,f)\cdot f^{\prime}=\sigma\cdot f^{\prime}+f$. Given an element $(\sigma,f)$ of $\Sigma_{B}\ltimes F$, the action of $(\sigma,f)$ on $F$ has a fixed point if and only if $f=v-\sigma v$ for some $v\in F$. This condition is in turn equivalent to the requirement that any $\mathbb{Z}_{l}$-linear function which satisfies $f\cdot\sigma=f$ vanishes on $f$. Functions satisfying this condition form a free $\mathbb{Z}_{l}$-module of rank equal to the number of cycles in the cycle decomposition of the permutation $\sigma$, with the characteristic functions on orbits forming a basis. On the other hand, $(\sigma,f)$ has finite order $n$ if and only if $\sigma^{n}=e$ and $f+\sigma\cdot f+\sigma^{2}\cdot f+\cdots+\sigma^{n-2}\cdot f+\sigma^{n-1}\cdot f=0$ Let $\\{\mathfrak{o}_{i}\\}_{i}$ denote the collections of orbits under action of $\sigma$, and let $\pi_{i}$ denote the projection on the set of basis elements in $\mathfrak{o}_{i}$. The summands corresponding to the orbits are invariant under $\sigma$, and we obtain a $\sigma$-invariant direct sum decomposition $F\cong\bigoplus_{i}F_{i}$ where $F_{i}$ is the span of the basis elements belonging to $\mathfrak{o}_{i}$. The above equation now demonstrates that $\sum_{s=0}^{n-1}\sigma^{s}(\pi_{i}(f))=0$ It follows that all the $\sigma$-invariant $\mathbb{Z}_{l}$-linear functions on $F$ vanish on $f$. So we have proved that for any element $(\sigma,f)\in Aut(F,F,B)$, the condition that $(\sigma,f)$ as a fixed point under its action on $F$ is equivalent to its having finite order. The result now follows. $\square$ We will write ${\cal T}_{\eta}/G$ for the orbit scheme of ${\cal T}_{\eta}$. We now have the following result. ###### Proposition 4.6 Let $G$ be a profinite group, with $N\subseteq G$ a closed normal subgroup. Let $\eta$ denote a based affine $l$-adic representation of $G$. If $\eta$ is $N$-principal, then ${\cal T}_{\eta}$ is an etale $G/N$-scheme. Proof: Let $(F_{\eta},X_{\eta},B_{\eta})$ be the based $l$-adic representation space, so $\eta$ is a continuous homomorphism $G\rightarrow\mathfrak{A}$, where $\mathfrak{A}$ denotes the automorphism group of $(F_{\eta},X_{\eta},B_{\eta})$. As before, let $\mathfrak{T}$ denote the translation subgroup of $\mathfrak{A}$. The subgroup $\eta^{-1}(\mathfrak{T})$ is now a closed subgroup of finite index in $G$, containing $N$, and therefore projects to a closed subgroup $T$ of finite index in $G/K$. Let $\mathfrak{X}$ denote the set of all morphisms from schemes of the form $Spec(F)$, where $F$ is an algebraically closed field, to the inverse limit ${\cal T}_{\eta}$. In order to prove the freeness of the $G/N$-scheme, we must prove the freeness of he $G/N$ action on the set $\mathfrak{X}$. We know from Proposition 4.3 that the restriction of the action on $\mathfrak{X}$ to the subgroup of finite index $T$ is free, and that the group $G/N$ is torsion free. In order to prove the freeness of the full $G/N$-action, we suppose that there is are elements $\gamma\in G/N$, with $\gamma\neq e$, and $x\in\mathfrak{X}$, so that $\gamma\cdot x=x$. It would follow that for any $n$, the element $\gamma^{n}$ would also fix $x$. But for sufficiently large $n$, $\gamma^{n}\in T$, and further that $\gamma^{n}\neq e$, because of the torsion freeness proved in Proposition 4.1. The freeness of the $N$ -action now contradicts the existence of $\gamma$. $\square$ We actually need a slightly stronger result, namely that for a fixed $N$-principal $l$-BAR, there is an $s\geq 0$ so that the scheme ${\cal T}^{s}_{\eta}$ is acted on freely by the quotient group $G/N\cdot\eta^{-1}(l^{s^{\prime}}\mathfrak{T})$ for all $s^{\prime}\geq s$. This will require a few algebraic facts. Consider any extension of groups $0\rightarrow A\rightarrow H\rightarrow Q\rightarrow 0$ where $Q$ is a finite cyclic group of order $l$, and $A$ is abelian. The extension gives an action of $Q$ on $A$ by automorphisms. Let $N(A)\subseteq A$ denote the subgroup of elements of the form $a+\sigma\cdot a+\sigma^{2}\cdot a+\cdots\sigma^{p-1}a$, where $\sigma\in Q$ is a generator. Of course the group is independent of the choice of generator. We define a function $c:Q\rightarrow A/N(A)$ as follows. For any element $\sigma\in Q$, select a lift $\overline{\sigma}\in H$, i.e. so that $\pi\overline{\sigma}=\sigma$, and define $c(\sigma)=\overline{\sigma}^{l}$. It is easy to check that $c(\sigma)$ is independent of the choice of $\overline{\sigma}$. ###### Lemma 4.2 An element $\sigma\in Q$ lifts to an element of order $l$ if and only if $c(\sigma)=0$. Proof: The “only if” part is clear, since if $\sigma$ lifts to an element of order $l$, $c(\sigma)$ is by definition equal to zero. Suppose on the other hand that $c(\sigma)=0$, which means that there is a lift $\overline{\sigma}$ and an element $\alpha\in A$ so that $\overline{\sigma}^{l}=\alpha+\sigma\alpha+\cdots\sigma^{l-1}\alpha$ If we now set $\overline{\sigma}^{\prime}=\overline{\sigma}\cdot(-\alpha)$, then $(\overline{\sigma}^{\prime})^{l}=e$. $\square$ ###### Lemma 4.3 Let $\Gamma$ denote a pro-$l$ group which fits in an exact sequence $\\{e\\}\rightarrow L\rightarrow\Gamma\stackrel{{\scriptstyle\pi}}{{\rightarrow}}Q\rightarrow\\{e\\}$ where $Q$ is a finite $l$-group, and $L\cong\mathbb{Z}_{l}^{n}$. Let $\Gamma^{\prime}\subseteq\Gamma$ be a subgroup so that the composite $\Gamma^{\prime}\rightarrow\Gamma\rightarrow Q$ is surjective. Suppose further that $\Gamma^{\prime}$ is torsion free, and that we are given a continuous $\Gamma$-set $X$ on which $\Gamma^{\prime}$ and $L$ both act freely. Then there is an $s\geq 0$ so that $\Gamma^{\prime}/\Gamma^{\prime}\cap l^{s^{\prime}}\cdot L$ acts freely on the set $X/l^{s^{\prime}}\cdot L$ for all $s^{\prime}\geq s$. The number $s$ depends only on the groups involved, not the $\Gamma$-set $X$. Proof: We first claim that it suffices to prove the result for $Q$ a cyclic finite $l$-group. To see this, we suppose we are able to prove the result for each subgroup $\Gamma^{\prime}_{C}=(\pi\|\Gamma^{\prime})^{-1}(C)\subseteq\Gamma^{\prime}$. This means that for every cyclic subgroup $C\subseteq Q$, there is an $s(C)\geq 0$ so that $\Gamma_{C}^{\prime}/\Gamma_{C}^{\prime}\cap l^{s^{\prime}}L$ acts freely on $X/l^{s^{\prime}}L$ for all $s^{\prime}\geq s(C)$ . Since there are only finitely many cyclic subgroups of $Q$, we may set $S=\mathrel{\mathop{max}\limits_{C}}s(C)$. It now follows that $\Gamma^{\prime}/\Gamma^{\prime}\cap l^{s^{\prime}}L$ acts freely on $X/l^{s^{\prime}}L$ whenever $s^{\prime}\geq S$, since all the groups $\Gamma_{C}^{\prime}/\Gamma_{C}^{\prime}\cap l^{s^{\prime}}L$ do, and since these the union of these subgroups is all of $\Gamma^{\prime}/\Gamma^{\prime}\cap l^{s^{\prime}}L$. From this point on, we assume that $Q$ is cyclic of prime power order. The next reduction is that it will suffice to prove the result for $Q$ a cyclic group of order $l$. To see this, we first assume that the result holds for all groups $\Gamma^{\prime}$ with $Q$ of order $l$, and let $\Gamma^{\prime\prime}\subseteq\Gamma^{\prime}$ denote the subgroup $\pi^{-1}(Q_{l})$, where $Q_{l}$ is the unique subgroup of order $l$. Suppose $\gamma\in\Gamma^{\prime}-\Gamma^{\prime\prime}$, and let $l^{k}$ denote the order $\pi(\gamma)\in Q$. We then have that $\gamma^{l^{k-1}}\in Q_{l}$, and $\gamma^{l^{k-1}}\neq e$. By our assumption, there is an $s\geq 0$ so that the group $\Gamma^{\prime\prime}/\Gamma^{\prime\prime}\cap l^{s}L$ acts freely on $X/l^{s^{\prime}}\cdot LX$ for all $s^{\prime}\geq s$. This means that $\gamma$ also acts freely, since a power of it acts freely. This give the reduction to the case where $Q$ has order $l$. To handle this case, we note that we have the function $c:Q\rightarrow L\cap\Gamma^{\prime}$ defined above. By Lemma 4.2 above, the torsion freeness of $\Gamma^{\prime}$ means that $c(\sigma)\neq 0$ in the group $L\cap\Gamma^{\prime}/N(L)\cap\Gamma^{\prime}$. But it is clear that $L\cap\Gamma^{\prime}/N(L)\cap\Gamma^{\prime}$ is identified with the inverse limit $\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\longleftarrow}\limits_{s}}}}L\cap\Gamma^{\prime}/(N(L)\cap\Gamma^{\prime}+l^{s}L\cap\Gamma^{\prime})$ so for sufficiently large $s$, we have $\overline{\sigma}^{l}\neq 0$ in $L\cap\Gamma^{\prime}/l^{s}L\cap\Gamma^{\prime}$, for every possible lift $\overline{\sigma}$ of $\sigma$. Since the $L$ action on $X$ is assumed to be free, it follows that $L\cap\Gamma^{\prime}/l^{s}L\cap\Gamma^{\prime}$ acts freely on $X/l^{s}\cdot L$ for sufficiently large $s$. We now have that every lift of $\sigma$ acts freely for sufficiently large $s$. It is clear also that the same calculation applies to all non-identity powers of $\sigma$, which together with the fact that $L/l^{s}\cdot L$ acts freely on $X/l^{s}\cdot L$, as always for sufficiently large $s$, gives the result. $\square$ ###### Proposition 4.7 Let $\eta$ be a $N$-principal $l$-BAR for some closed normal subgroup of $G$. Then there exists an $s\geq 0$ so that for every $s^{\prime}\geq s$, the action of the finite group $G/(N\cdot\eta^{-1}(l^{s^{\prime}}\mathfrak{T}))$ on ${\cal T}^{s^{\prime}}_{\eta}$ is etale. Proof: The $G/N$-action on ${\cal T}_{\eta}$ is etale. Let $\Omega$ denote an algebraically closed field and let $i:Spec\Omega\rightarrow{\cal T}_{\eta}/G$ be a morphism. Then we have the set of all maps $\Omega\rightarrow{\cal T}_{\eta}$ for which the diagram $\begin{diagram}$ commutes, which we have denoted by $\mathfrak{H}(A[{\cal T}_{\eta}],\Omega,i)$. The etaleness of the $G/N$-action on ${\cal T}_{\eta}$ means that the $G/N$-action on $\mathfrak{H}(A[{\cal T}_{\eta}],\Omega,i)$ is free. The group $G/N$ is identified by the representation with a subgroup of $\Sigma_{n}(l)\ltimes\mathbb{Z}_{l}^{n}$, where $\Sigma_{n}(l)$ denotes an $l$-Sylow subgroups,and is torsion free, so satisfies the hypotheses of Lemma 4.3 above. Note that the hypothesis that $L$ acts freely in Lemma 4.3 is satisfied due to Proposition 4.3. The group $\Gamma$ will be $I\ltimes\mathbb{Z}_{l}^{n}$, where $I$ is the image of the composite $G/K\rightarrow\Sigma_{n}(l)\times\mathbb{Z}_{l}^{n}\rightarrow\Sigma_{n}(l)$ and the affine representation $\eta$ identifies $G/N$ with a subgroup of $\Gamma$. It now follows that there is an $s$ so that for all $s^{\prime}\geq s$, the group $G/(N\cdot\eta^{-1}(l^{s^{\prime}}\mathfrak{T}))$ acts freely on the orbit set $\mathfrak{H}(A[{\cal T}_{\eta}],\Omega,i)/(l^{s^{\prime}}\mathfrak{T})$ But, because of the description of ${\cal T}_{\eta}^{s}$ given in Proposition 4.4, this orbit set is just the set $\mathfrak{H}(A[{\cal T}_{\eta}^{s}],\Omega,i)$, which gives the result. Note that we are using the self evident fact that if a group $G$ acts freely on a set $X$, and $N$ is a normal subgroup of $G$, then the naturally defined action of $G/N$ on $X/N$ is also free. $\square$ We record a useful property of $K$-principal actions. ###### Proposition 4.8 Let $\eta_{1}$ and $\eta_{2}$ be affine $l$-adic representations of a profinite group $G$. Then we have the evident product affine $l$-adic representation $\eta_{1}\times\eta_{2}$. Let $\\{e\\}\subseteq K^{\prime}\subseteq K\subseteq G$ be closed subgroups. Suppose that $\eta_{1}$ is $K$-principal, and that $\eta_{2}\|K$ is $K^{\prime}$-principal. Then $\eta_{1}\times\eta_{2}$ is $K^{\prime}$-principal. In particular, if $\eta_{i}$ is $K_{i}$-principal, for $K_{i}\subseteq G$, then $\eta_{1}\times\eta_{2}$ is $K_{1}\cap K_{2}$-principal. Proof: Clear. $\square$ We also consider the notion of induction of affine based $l$-adic representations. Let $\eta=(F_{\eta},X_{\eta},B_{\eta})$ be a based affine $l$-adic representation of $K$, where $K$ is a closed subgroup of finite index in a profinite group $G$. Then we define an induced based affine $l$-adic representation $i_{K}^{G}(\eta)=(F_{i_{K}^{G}(\eta)},X_{i_{K}^{G}(\eta)},B_{i_{K}^{G}(\eta)})$ as follows. The sets $F_{i_{K}^{G}(\eta)}$ and $X_{i_{K}^{G}(\eta)}$ are defined as the sets of equivariant functions $F^{K}(G,F_{\eta})$ and $F^{K}(G,X_{\eta})$, respectively, where $K$ acts by left multiplication on $G$ and via $\eta$ on $F_{\eta}$ and $X_{\eta}$. The group $G$ acts by right multiplication on $G$, yielding a left action on the two sets. The free action of $F^{K}(G,F_{\eta})$ on $F^{K}(G,X_{\eta})$ is given by pointwise multiplication. To define $B_{i_{K}^{G}(\eta)}$, we first define elements $f_{b}\in F_{i_{K}^{G}(\eta)}$ for each $b\in B_{\eta}$ by setting $f_{b}(e)=b$. The remainder of $B_{i_{K}^{G}(\eta)}$ is obtained by translating the elements using the $G$ action. It is clear that this set will give a basis for $F_{i_{K}^{G}(\eta)}$. We say an affine based $l$-adic representation $\eta$ of a profinite group $G$ is a translation representation if for every $g\in G$, the automorphism $\eta(g)$ is a translation. ###### Proposition 4.9 Let $G$ be a profinite group, and let $K\lhd G$ be a normal subgroup of finite index. Suppose we are given a one-dimensional based affine $l$-adic representation $\eta$ of $K$. Then we have the following. 1. 1. The restriction of $i_{K}^{G}(\eta)$ to $K$ is a translation representation. 2. 2. The restriction of $i_{K}^{G}(\eta)$ to $K$ is $K^{\prime}$-principal, where $K^{\prime}$ is defined as follows. For each element $\gamma\in G/K$, we denote by $\eta^{\gamma}$ the result of composing $\eta$ with the conjugation action of a coset representative of $\gamma$. Then $K^{\prime}$ is given by $K^{\prime}=\bigcap_{\gamma\in G/K}Ker(\eta^{\gamma})$ ## 5 Construction of ${\cal E}G$ In this section, we will use products of the actions constructed in the last section to build pro-objects in the category of continuous $G$-actions of a profinite $l$-group which will satisfy the requirements formulated in the introduction. Throughout this section, we will allow ourselves to refer to a based affine $l$-adic representation as an $l$-BAR, in the interest of brevity. We begin with a technical lemma. ###### Lemma 5.1 Let $G$ be a totally torsion free pro-$l$-group. Let $G_{0}\subseteq G$ be any closed subgroup of finite index. Then there exists a closed normal subgroup $K\subseteq G_{0}$ and a $K$-principal $l$-BAR $\eta$ of $G$. Proof: We first observe that it suffices to prove the result for $G_{0}$ normal in $G$, since there is a always a normal subgroup of finite index contained in $G_{0}$. To construct an $l$-BAR which is $K$-principal for some $K\subseteq G_{0}$, we first note that the set of subgroups of $G$ containing $G_{0}$ consists entirely of closed subgroups, by Proposition 3.1. We now assume by induction that we are able to construct such $l$-BARs for all closed subgroups of index less than the index of $G_{0}$ in $G$. We construct such $l$-BAR’s for every normal subgroup containing $G_{0}$ properly. There are obviously only finitely many such subgroups. We consider the finite list $\\{N_{i}\\}_{i=1}^{s}$ of all normal subgroups of $G$ which contain $G_{0}$ properly (they are necessarily closed), and for each one we construct a $K_{i}$-principal $l$-BAR $\eta_{i}$ of $G$, where $K_{i}\subseteq N_{i}$. The $l$-BAR $\eta_{1}\times\eta_{2}\times\cdots\times\eta_{s}$ is now $K_{1}\cap\cdots\cap K_{s}$-principal by repeated application of Proposition 4.8. We now consider the subgroup $G_{0}\cdot(K_{1}\cap\cdots\cap K_{s})$. This subgroup contains $G_{0}$, and therefore corresponds to a subgroup of the quotient $G/G_{0}$, contained in $(N_{1}/G_{0})\cap(N_{2}/G_{0})\cap\cdots\cap(N_{s}/G_{0})$. This group can be characterized as the intersection $\sigma(G/N)$ of all non- trivial normal subgroups in $G/G_{0}$. Since $G_{0}$ is a closed normal subgroup of the pro-$l$ group $G$, we know that $G/G_{0}$ is an $l$-group, and hence its center $Z(G/G_{0})$ is non-trivial. Therefore, $\sigma(G/G_{0})\subseteq Z(G/G_{0})$ is a central subgroup, which implies that every subgroup of $\sigma(G/G_{0})$ is normal in $G/G_{0}$. By the definition of $\sigma(G/G_{0})$, this means that $\sigma(G/G_{0})$ has no non- trivial proper subgroups, and the only such $l$-groups are $\\{e\\}$ and $\mathbb{Z}/l\mathbb{Z}$. Consequently, either $G_{0}\cdot(K_{1}\cap\cdots\cap K_{s})/G_{0}$ is trivial, in which case $K_{1}\cap\cdots\cap K_{s}\subseteq G_{0}$, and we are done, or it is cyclic of order $l$. In this case, we are given a homomorphism from $\overline{G}_{0}=G_{0}\cdot(K_{1}\cap\cdots\cap K_{s})$ to $\mathbb{Z}/l\mathbb{Z}$, and therefore a character $\chi_{1}$ of $\overline{G}_{0}$. Since the group $G$ is totally torsion free, we find that the group $\overline{G}_{0}^{ab}$ is torsion free, and therefore that the character group is divisible. Construct a sequence of characters $\chi_{i}$ so that $\chi_{i}^{l}=\chi_{i-1}$, with $\chi_{1}$ the previously defined character. This sequence constructs a one-dimensional $l$-BAR of $\overline{G}_{0}$, which we now induce up to an $l$-BAR ${\eta}$ of dimension equal to the index of $G_{0}$ in $G$. We claim that the $l$-BAR $\eta\times\eta_{1}\times\cdots\times\eta_{s}$ is $Ker(\eta)\cap(K_{1}\cap K_{2}\cap\cdots\cap K_{s})$-principal. To see this we note that the stabilizer of any point in $X_{\eta_{1}}\times\cdots\times X_{\eta_{s}}$ is equal to $K_{1}\cap\cdots\cap K_{s}\subseteq\overline{G}_{0}$, and therefore by Proposition 4.8 it suffices to prove that the restriction of the $G$-action on $X_{\eta}$ to $K_{1}\cap\cdots\cap K_{s}$ is $Ker(\eta)\cap K_{1}\cap\cdots\cap K_{s}$-principal. Proposition 4.9 asserts that the restriction to $G_{0}$ of $\eta$ is $Ker(\eta)$-principal. The result now follows from Proposition 4.8. $\square$ We need a second fact concerning representations of a profinite $l$-group. ###### Proposition 5.1 Let $\sigma$ denote any irreducible continuous representation of a totally torsion free profinite $l$-group $G$, over an algebraically closed field of characteristic prime to $l$. Then $\sigma$ is isomorphic to ${\cal T}^{s}_{\eta}$ for some $l$-$BAR$ $\eta$ and some $s$. Proof: The continuity of the $\sigma$ gives that $\sigma$ factors through a finite quotient $G/N$ of $G$, where $N$ is a closed normal subgroup of finite index in $G$. By Blichfeldt’s theorem, $\sigma$ (regarded as a representation of $G/N$) is monomial, and is therefore of the form $i_{K}^{G}(\chi)$ for some closed subgroup of finite index $K\subseteq G/N$, and some one-dimensional character $\chi$ of $K$. Let $\overline{K}$ denote the inverse image of $K$ in $G$. The group $K$ is also totally torsion free, and therefore any homomorphism from $K$ to a finite cyclic $l$-group can be lifted to a continuous homomorphism $\hat{\chi}$ from $K$ to $\mathbb{Z}_{l}$. This homomorphism is a one-dimensional $l$-BAR, and it is clear from the construction that $\chi$ is equal to $\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\hat{\chi}$. It now readily follows that the induced (from $\overline{K}$ to $G$) $l$-BAR is the required $\eta$. $\square$ We now construct the continuous affine $G$-scheme ${\cal E}G$. ###### Definition 5.1 Let $G$ be any profinite group, which is acting continuously on an affine scheme $X=Spec(A)$. We say $X$ is cofinite if there is a closed normal subgroup $N\subseteq G$ of finite index so that the $G$ action on $X$ factors through the quotient $G/N$. For any cofinite affine $G$-scheme $X$, we define the subgroup ${\cal N}(X)\subseteq G$ to be the maximal closed normal subgroup so that the $G$-action on $X$ factors through $G/{\cal N}(X)$. It is clear that this defines ${\cal N}(X)$ uniquely, and moreover that if we have a map $X\rightarrow Y$ of cofinite affine $G$-schemes, then ${\cal N}(X)\subseteq{\cal N}(Y)$. We say a cofinite affine $G$-scheme $X$ is etale if the group action of $G/{\cal N}(X)$ on $X$ is a free action in the sense of Proposition 2.6. We define $\mathfrak{C}^{cof}(G)$ to be the category whose objects are coherent cofinite affine $G$-schemes, and whose morphisms are the faithfully flat equivariant maps of such $G$-schemes. The full subcategory of etale actions will be denoted by $\mathfrak{C}^{cof}_{et}(G)$. ###### Proposition 5.2 Suppose that we have a $G$-equivariant map $f:X\rightarrow Y$ of cofinite etale affine $G$-schemes, which is faithfully flat. Then for any closed subgroup $K$ of finite index in $G$, the natural map $X/K\rightarrow Y/K$ is also faithfully flat. Proof: Since the etale property of an action clearly descends to closed subgroups, it suffices to prove this for the case $K=G$. We now have the diagram $\begin{diagram}$ The morphism $\pi_{Y}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}f$ is faithfully flat, since $f$ is and $\pi_{Y}$ is since it is etale and surjective. $\pi_{X}$ is faithfully is also faithfully flat by the same reasoning. It follows that $f/G$ is faithfully flat. $\square$ For each closed normal subgroup of finite index $N\lhd G$ we apply Lemma 5.1 to construct an $l$-BAR $\eta_{N}$ which is $N^{\prime}$-principal for some closed normal subgroup $N^{\prime}\subseteq N$. Let $\mathfrak{N}^{G}$ denote the partially ordered set of closed normal subgroups of $G$. For each $(s,N)\in\mathbb{N}\times\mathfrak{N}^{G}$, we let $\theta^{\prime}(s,N)$ denote the (finite) product $\theta^{\prime}(s,N)=\prod_{\overline{N}\supseteq N,\overline{N}\in\mathfrak{N}}{\cal T}_{\eta_{\overline{N}}}^{s}$ The construction $\theta^{\prime}$ clearly gives a functor from the category $\mathbb{N}\times\mathfrak{N}^{G}$ to $\mathfrak{C}^{cof}(G)$, where the effect on morphisms consists on bonding maps ${\cal T}_{\eta}^{s+1}\rightarrow{\cal T}_{\eta}^{s}$ together with product projections. We will need to build a more complicated pro-object. Let $\mathfrak{R}^{G}$ denote the set of all isomorphism classes of finite dimensional continuous $k$-linear representations of $G$. For each $N\in\mathfrak{N}^{G}$, we let $\mathfrak{R}^{G}(N)\subseteq\mathfrak{R}^{G}$ denote the subset of all representations which factor through the quotient $G/N$. For each $\alpha\in\mathfrak{R}^{G}$, we select an $l$-BAR $\eta_{\alpha}$ so that for some $s$, $\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\eta_{\alpha}$ belongs to the isomorphism class $\alpha$. This is possible because of Proposition 5.1. We define $\theta^{\prime\prime}(s,N)=\prod_{\alpha\in\mathfrak{R}^{G}(N)}{\cal T}_{\eta_{\alpha}}^{s}$ $\theta^{\prime\prime}$ defines a functor from $\mathbb{N}\times\mathfrak{N}^{G}$ to $\mathfrak{C}^{cof}(G)$, again with structure maps arising from those in the pro objects ${\cal T}_{\eta}$ and product projections. We let ${\cal E}G$ denote the inverse limit of $\theta=\theta^{\prime}\times\theta^{\prime\prime}$ in the category of affine $G$-schemes. ${\cal E}G$ is thus an affine scheme whose affine coordinate ring is the colimit of the rings $A(\theta(s,N))$ over the category $\mathbb{N}\times\mathfrak{N}^{G}$, where $A(X)$ denotes the affine coordinate ring of $X$. ###### Proposition 5.3 For any pair $N\subseteq N^{\prime}\subseteq G$ of closed normal subgroups of finite index, and any $s\leq s^{\prime}$, the map of schemes $\theta(s^{\prime},N^{\prime})\rightarrow\theta(s,N)$ is faithfully flat. It follows that ${\cal E}G$ is coherent. Proof: On affine coordinate rings, each map is of the form $A=k[x_{1}^{\pm\frac{1}{l^{s}}},\ldots,x_{t}^{\pm\frac{1}{l^{s}}}]\hookrightarrow k[x_{1}^{\pm\frac{1}{l^{s^{\prime}}}},\ldots,x_{t^{\prime}}^{\pm\frac{1}{l^{s^{\prime}}}}]=B$ where $t^{\prime}\geq t$. In this case, it is an easy computation that $B$ is a free module over $A$. Both rings are Noetherian, therefore coherent, and the result now follows from Proposition 2.3. $\square$ ###### Proposition 5.4 The $G$-action on the affine coordinate ring $A[{\cal E}G]$ is continuous, when $A[{\cal E}G]$ is equipped with the discrete topology. Proof: Each ring $A[\theta(s,N)]$ is acted on by $G$ via an action which factors through a finite quotient of $G$. Since $A[{\cal E}G]$ is the colimit of the rings $A[\theta(s,N)]$, the result follows. $\square$ We will now prove that the $G$-action on ${\cal E}G$ is etale. Let $\overline{\theta}$ denote the functor $(s,N)\rightarrow\theta(s,N)/N$. $\overline{\theta}(s,N)$ is equipped with a $G/N$-action. ###### Lemma 5.2 The natural map $\mathrel{\mathop{lim}\limits_{\leftarrow}}\theta\rightarrow\mathrel{\mathop{lim}\limits_{\leftarrow}}\overline{\theta}$ is an isomorphism of affine schemes. Proof: We prove that the map induces an isomorphism on affine coordinate rings. The map $A[\mathrel{\mathop{lim}\limits_{\leftarrow}}\overline{\theta}]\rightarrow A[\mathrel{\mathop{lim}\limits_{\leftarrow}}\theta]$ is clearly injective, since for every $(s,N)$, the map $A[\overline{\theta}(s,N)]\rightarrow A[\theta(s,N)]$ can be identified with the inclusion $A[\theta(s,N)]^{N}\rightarrow A[\theta(s,N)]$. Given an element $x\in A[\mathrel{\mathop{lim}\limits_{\leftarrow}}\theta]$, it is in the image of $A[\theta(s,N)]$ for some $s$ and $N$. This means that it is fixed by a normal subgroup of finite index $N^{\prime}\subseteq N$, and consequently that it is in the image of $A[\theta(s,N^{\prime})]^{N^{\prime}}=A[\overline{\theta}(s,N^{\prime})]$. This proves surjectivity. $\square$ ###### Lemma 5.3 Let $N\subseteq G$ be a closed subgroup of finite index. Then $\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\leftarrow}\limits_{s}}}}\overline{\theta}(s,N)$ an etale $G/N$-action for all $N$. Proof: Proposition 4.6 shows that $\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\leftarrow}\limits_{s}}}}{\theta}(s,N)$ is an etale action of $G/N^{\prime}$ for some closed normal subgroup $N^{\prime}\subseteq N$. It follows that the action of $G/N$ on $\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\leftarrow}\limits_{s}}}}{\theta}(s,N)/N=\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\leftarrow}\limits_{s}}}}\overline{\theta}(s,N)$ is etale. $\square$ ###### Corollary 5.1 For every closed normal subgroup $N\subseteq G$ of finite index, the $G/N$-action on $\overline{\theta}(s^{\prime},N)$ is etale for sufficiently large $s$. Proof: We let $N^{\prime}\subseteq N\subseteq G$ be the closed normal subgroup of $G$ so that $\eta_{N}$ is $N^{\prime}$-principal, and therefore that $\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\leftarrow}\limits_{s}}}}{\theta}(s,N)$ is an etale $G/N^{\prime}$-action. It follows from Proposition 4.7 that there is an $s\geq 0$ so that the action of $G/\eta^{-1}(l^{s^{\prime}}\mathfrak{T})$ on $\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\leftarrow}\limits_{s}}}}{\theta}(s,N)/(\eta^{-1}(l^{s^{\prime}}\mathfrak{T}))$ is etale for all $s^{\prime}\geq s$. Because $N$ is of finite index, there is also a $t\geq 0$ so that $\eta^{-1}(l^{t^{\prime}}\mathfrak{T})\subseteq N$ for all $t^{\prime}\geq t$. Letting $S=max(s,t)$, we have that for every $s^{\prime}\geq S$, the $G/N$-action on $\overline{\theta}(s^{\prime},N)$ is defined as the natural action of $G/N$ on the $N/\eta^{-1}(l^{s^{\prime}}\mathfrak{T})$-orbits of the action of the $G/\eta^{-1}(l^{s^{\prime}}\mathfrak{T})$-action on $\mathrel{\mathop{lim}\limits_{\mathrel{\mathop{\leftarrow}\limits_{s}}}}{\theta}(s,N)/(\eta^{-1}(l^{s^{\prime}}\mathfrak{T}))$. Since the $G/\eta^{-1}(l^{s^{\prime}}\mathfrak{T})$-action is etale, so is the $G/N$-action. $\square$ ###### Proposition 5.5 ${\cal E}G$ is etale. Proof: We must show that for any closed subgroup of finite index $N$, the $G/N$ action on $({\cal E}G)/N$ is etale. We have that $({\cal E}G)/N$ is the inverse limit of the functor $(s,N)\rightarrow\theta(s,N)/N$. This is a diagram, parametrized by a directed set, with values in the category of affine schemes with $G/N$-action. For every closed normal subgroup of finite index $N^{\prime}\subseteq G$, let $s(N^{\prime})\geq 0$ be an integer so that the $G/N$-action on $\theta(s,N^{\prime})/N$ is etale for all $s\geq s(N^{\prime})$. The subset $\hat{\mathfrak{N}}=\\{(s,N^{\prime})\mbox{ such that }s\geq s(N^{\prime})\\}$ contained in $\mathbb{N}\times\mathfrak{N}^{G}$ is clearly final, and so ${\cal E}G/N$ can be described as the inverse limit of the restriction of the functor $N^{\prime}\rightarrow\theta(s,N^{\prime})/N$ to the subset $\hat{\mathfrak{N}}$. Proposition 2.8 now gives the desired result. $\square$ ###### Proposition 5.6 Let $W$ denote any continuous affine $G$-scheme, and assume that $A[W]$ is Noetherian approximable. Then ${\cal E}G\times W$ is coherent, and the $G$-action is etale. The natural map $K({\cal E}G\mathrel{\mathop{\times}\limits_{G}}W)\rightarrow K^{G}({\cal E}G\times W)$ therefore exists and is an equivalence. Proof: ${\cal E}G\times W$ clearly is an etale action, using the base change property for etale actions. The affine coordinate ring of ${\cal E}G\times W$ can be written as the colimit of rings of the form $A[W][x_{1}^{\pm\frac{1}{l^{s}}},\ldots,x_{t}^{\pm\frac{1}{l^{s}}}]$, which are coherent by Proposition 2.4 since $A[W]$ is Noetherian approximable. Further, each of the inclusions in the colimit system is of the form $A[W][x_{1}^{\pm\frac{1}{l^{s}}},\ldots,x_{t}^{\pm\frac{1}{l^{s}}}]\hookrightarrow A[W][x_{1}^{\pm\frac{1}{l^{s^{\prime}}}},\ldots,x_{t^{\prime}}^{\pm\frac{1}{l^{s^{\prime}}}}]$ and are therefore clearly faithfully flat. This demonstrates the coherence of $A[{\cal E}G\times W]$. The result now follows by Proposition 2.16. $\square$ Remark: The conditions on $\theta^{\prime}$ provide a freeness property of ${\cal E}G$. Extending by creating the product with $\theta^{\prime\prime}$ enlarges the pro-scheme so as to trivialize the action of $R[G]$ on the equivariant $K$-groups of ${\cal E}G$. We prove the above mentioned technical result on the action of $R[G]$ on $K^{G}_{}({\cal E}G)$. which will be useful in analyzing completions of $K^{G}({\cal E}G)$. ###### Proposition 5.7 Let $G$ be a totally torsion free $l$-profinite group. Then the action of $\pi_{0}K^{G}(k)\cong R[G]$ satisfies $I_{G}\cdot\pi_{}K^{G}({\cal E}G)=\\{0\\}$. Proof: It clearly suffices show that $j(I_{G})=\\{0\\}$. This means that we will need to show that for any representation $\rho$ of $G$, we have that $[\rho]-dim(\rho)$ vanishes in $K^{G}_{0}({\cal E}G)$. Blichfeldt’s theorem (see [35]) tells us that any irreducible representation of $G$ is monomial, i.e it is isomorphic to an induced representation $i_{G_{0}}^{G}(\chi)$ where $G_{0}$ is a subgroup of index $dim(\rho)$ and $\chi$ is a one dimensional character of $G_{0}$. Because of the totally torsion free property of $G$, $\chi$ is of the form $\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\eta_{\chi}$ where $\eta_{\chi}$ is a one-dimensional $l$-BAR of $G_{0}$, and the $\rho_{s}$’s are defined in Section 4. Inducing $\eta_{\chi}$ up to $G$ we obtain an $l$-BAR $\eta$ of $G$, and it is clear from the definition that $\rho$ can be identified with $\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\eta$. The conclusion is that every irreducible representation of $G$ is of the form $\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\eta$ for some $l$-BAR of $G$, and some $s$. Next, we observe that any $l$-BAR is a factor of a principal one. This is an immediate consequence of Lemma 5.1. For any $l$-BAR $\eta$ we have the associated permutation representation $\overline{\eta}$. We will first prove that $[\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\eta]-[\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\overline{\eta}]=0$ in $K_{0}^{G}({\cal E}G)$. Fix $s\geq 0$. Given an $l$-BAR $\eta$, we let $K_{\eta}$ denote the kernel of $\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\eta$. Similarly, let $\eta^{\prime}$ be a $K_{\eta^{\prime}}$-principal $l$-BAR containing $\eta$ as a factor, where $K_{\eta^{\prime}}\subseteq K_{\eta}$. We now have a factorization $R[G/K_{\eta}]\cong K_{0}^{G/K_{\eta}}(k)\rightarrow K_{0}^{G/K_{\eta}}({\cal T}^{s}_{\eta})\stackrel{{\scriptstyle j}}{{\longrightarrow}}K_{0}^{G/K_{\eta^{\prime}}}({\cal T}^{s}_{\eta^{\prime}})\rightarrow K_{0}^{G}({\cal E}G)$ since ${\cal T}_{\eta}^{s}$ is one of the factors occurring in the inverse system defining ${\cal T}_{\eta}$, and therefore one of the factors occurring in ${\cal E}G$. The map $j$ is the composite $K_{0}^{G/K_{\eta}}{\cal T}_{\eta}^{s}\rightarrow K_{0}^{G/K_{\eta^{\prime}}}{\cal T}_{\eta}^{s}\rightarrow K_{0}^{G/K_{\eta^{\prime}}}{\cal T}_{\eta^{\prime}}^{s}$ where the left hand map is the inflation map associated to the projection $G/K_{\eta^{\prime}}\rightarrow G/K_{\eta}$, and the right hand map is the map induced by the equivariant projection ${\cal T}_{\eta^{\prime}}^{s}\rightarrow{\cal T}_{\eta}^{s}$. Letting $\rho=\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\eta$ and $\overline{\rho}=\rho_{s}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}\overline{\eta}$ it will suffice to prove that the image of $[\rho]-[\overline{\rho}]=0$ in $K_{0}^{G}\mathbb{T}(k,\rho)$ vanishes. Let $N$ be the kernel of $\rho$, a normal subgroup of finite index in $G$. Note that $\rho$ and $\overline{\rho}$ may be regarded as elements in $K^{G/N}_{0}\mathbb{A}(k,\rho)$. We observe that there is a factorization $R[G/N]\cong K^{G/N}(k)\rightarrow K^{G/N}_{0}\mathbb{A}(k,\rho)\rightarrow K_{0}^{G}\mathbb{T}(k,\rho)\mbox{ }(=K_{0}^{G}{\cal T}_{\eta}^{s})$ We now select a basis $\\{x_{1},\ldots,x_{n}\\}$ for the representation space of $\rho$, where $n=dim(\rho)$, so that $\rho$ is monomial with respect to this basis, and we let $\overline{\rho}$ denote the associated permutation representation. The group $K^{G/N}_{0}\mathbb{A}(k,\rho)$ is the group completion of the commutative monoid of isomorphism classes of finitely generated left modules over the skew polynomial ring $A_{\rho}=k[x_{1},\ldots,x_{n}]_{\rho}[G/N]$, by Proposition 2.17. We construct a short exact sequence of left $A_{\rho}$-modules as follows. First, let $G_{0}\subseteq G$ be the stabilizer of the line spanned by the element $x_{1}$, so $\rho$ is induced from a one-dimensional representation of $G_{0}/N$, so that $\rho$ is induced from a one-dimensional representation $\chi$ of $G_{0}$. We let $F$ be the module $k[x_{1}]_{\chi}[G_{0}/N]\mathrel{\mathop{\otimes}\limits_{k[G_{0}/N]}}\epsilon$, where $\epsilon$ denotes the trivial representation of $G_{0}/N$. The module $F$ is free of rank one over $k[x_{1}]$, and we may write $F=k[x_{1}]b$, where $b=1\otimes 1$ is the basis element which is acted on trivially by $G_{0}/N$. We let $k$ denote the left $k[x_{1}]_{\chi}[G_{0}/N]$-module which as a $k$-module is cyclic of rank one, and on which all elements of $G_{0}/N$ act by the identity and such that $x_{1}\cdot k=0$. There is a natural homomorphism $\pi:F\rightarrow k$, which is the identification $F/xF\cong k$. Let $F^{\prime}$ denote the kernel of $\pi$. It is the free cyclic $k[x]$-module generated by $xb$, and from this description it is clear that $F^{\prime}$ is isomorphic to $k[x_{1}]_{\chi}[G_{0}/N]\mathrel{\mathop{\otimes}\limits_{k[G_{0}/N]}}\chi$. The conclusion is that the equation $[k]=[\epsilon]-[\chi]$ holds in $K_{0}^{G_{0}/N}\mathbb{A}(k,\chi)\cong R[G_{0}/N]$, where we are interpreting this group as the projective class group of finitely generated modules over $k[x_{1}]_{\chi}[G_{0}/N]$. Performing induction from $G_{0}/N$, and using Proposition 2.18, we find that the equation $[M]=[i_{G_{0}/N}^{G/N}(\epsilon)]-[i_{G_{0}/N}^{G/N}(\chi)]$ holds in $K_{0}^{G/N}\mathbb{A}(k,\rho)\cong R[G/N]$, where $M=k[x_{1},\ldots,x_{n}]_{\rho}[G/N]\mathrel{\mathop{\otimes}\limits_{K[x_{1}]_{\chi}[G_{0}/N]}}k$ We observe that $i_{G_{0}/N}^{G/N}(\epsilon)$ is the permutation representation $\overline{\rho}$ of $G$. It is also readily checked that the element $x_{1}x_{2},\cdots x_{n}$ annihilates the module $M$, and therefore that it becomes trivial on tensoring up to $k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$, which means that the equation $0=[i_{G_{0}/N}^{G/N}(\epsilon)]-[i_{G_{0}/N}^{G/N}(\rho]$ holds in $K_{0}^{G/N}\mathbb{T}(k,\rho)\cong R[G/N]$. This gives the fact that $[\rho]-[\overline{\rho}]=0$ holds in $K_{0}({\cal E}G)$ for every irreducible representation $\rho$ of $G$. Since every permutation representation of a finite group is decomposable (it contains a copy of the trivial representation of $G$), this means that for every irreducible representation $\rho$ of $G$, the class $[\rho]\in K_{0}({\cal E}G)$ is equal to a sum of images of representations of smaller dimension. Note that the special case $n=1$ of this result shows that one-dimensional representations $\chi$ satisfy the condition that $[\chi]-dim(\chi)$ vanishes in $K^{G}({\cal E}G)$. An induction on $n$ now shows that $[\rho]$ is equal to $dim(\rho)$, which was to be shown. $\square$ ###### Corollary 5.2 Let $W$ be any affine scheme with continuous $G$-action. Then the action of $R[G]$ satisfies $I_{G}\cdot K^{G}({\cal E}G\times W)=\\{0\\}$. Proof: This follows since $K^{G}({\cal E}G\times W)=\\{0\\}$ is a ring spectrum, and we have that $I_{G}$ maps trivially into $\pi_{0}K^{G}({\cal E}G\times W)$ since there is a factorization $R[G]\rightarrow K^{G}({\cal E}G)\rightarrow K^{G}({\cal E}G\times W)$ and we already know from Proposition 5.7 that the map $R[G]\rightarrow K^{G}({\cal E}G)$ is trivial. $\square$ Now we consider the diagram of commutative ring spectra $\begin{diagram}$ where $r$ denotes the ring map induced by the fundamental class in the sphere spectrum, and $\epsilon$ is the augmentation. The functorality of the derived completion construction means that the diagram induces a map $\beta:K^{G}({\cal E}G\times W)^{\wedge}_{l}=K^{G}({\cal E}G\times W)^{\wedge}_{r}\rightarrow K^{G}({\cal E}G\times W)^{\wedge}_{\epsilon}$ ###### Corollary 5.3 The map $\beta$ is an equivalence of spectra. Proof: The algebraic-to-geometric spectral sequence (Theorem 7.1 of [7]) and the preceding Proposition show that it suffices to prove that the map on derived completions $\hat{\beta}:M^{\wedge}_{l}\rightarrow M^{\wedge}_{\epsilon}$ induced by the diagram $\begin{diagram}$ is an equivalence for modules $M$ with trivial action by $I_{G}\subseteq R[G]$, or equivalently which are obtained by restricting scalars along the homomorphism $\epsilon:R[G]\rightarrow\mathbb{Z}$. In order to prove this, we regard the ring $R[G]$ as a commutative ring spectrum, and let $\tilde{\mathbb{Z}}$ denote a cofibrant replacement for the $R[G]$-algebra $\mathbb{Z}$, with algebra structure given by $\epsilon$, in the model category of commutative ring spectra discussed in [36]. We have the natural structure map $R[G]\rightarrow\tilde{\mathbb{Z}}$, as well as a homomorphism $\tilde{\mathbb{Z}}\rightarrow R[G]$ of commutative ring spectra obtained by extending the homomorphism of rings $\mathbb{Z}\rightarrow R[G]$ to the cofibrant replacement $\tilde{\mathbb{Z}}$. We also have a homomorphism $\tilde{\mathbb{Z}}\rightarrow\mathbb{H}_{l}$ which is compatible with the mod-$l$ reduction of the augmentation on $R[G]$. We now have two monads $S$ and $T$ on the category of $\tilde{\mathbb{Z}}$-module spectra, where $S(M)=\mathbb{H}_{l}\mathrel{\mathop{\wedge}\limits_{\tilde{\mathbb{Z}}}}M$ and $T(M)=\mathbb{H}_{l}\mathrel{\mathop{\wedge}\limits_{R[G]}}M$. Further, there are natural transformations $T\rightarrow S$ and $S\rightarrow T$ induced by the ring homomorphisms $\tilde{\mathbb{Z}}\rightarrow R[G]\rightarrow\tilde{\mathbb{Z}}$. It follows that the natural transformations of triples induce equivalence on the total spectra of the cosimplicial objects of the triples $S$ and $T$, by Theorem 2.15 of [7]. These total spectra attached to $S$ and $T$ are the derived completions over $\tilde{\mathbb{Z}}$ and $R[G]$ respectively. The fact that weak equivalences of commutative ring spectra induce equivalences of derived completions now gives the result. $\square$ We also need a similar result for $K^{G}(\overline{F})$, where $G$ is the absolute Galois group of a field $F$, containing $k$ as a subfield. ###### Proposition 5.8 With $G,F,\overline{F}$ as above, the action of $R[G]$ on $K^{G}_{}(\overline{F})$ is trivial in the sense that $I_{G}\cdot K^{G}_{}(\overline{F})=\\{0\\}$. Proof: It is proved in [22] that the category of finitely generated $G$-twisted $\overline{F}$-modules is equivalent to the category of finite dimensional vector spaces over $F$, and that the equivalence can be taken by applying the functor $V\rightarrow\overline{F}\mathrel{\mathop{\otimes}\limits_{F}}V$. There is a natural twisted $G$-action on $\overline{F}\mathrel{\mathop{\otimes}\limits_{F}}V$. Since $\pi_{0}K(F)$ is a free group of rank one, given only by the dimension, it is therefore clear that for any finite dimensional $k$-linear representation $(W,\rho)$ of $G$, $\overline{F}\mathrel{\mathop{\otimes}\limits_{F}}W$ is isomorphic to $\overline{F}\mathrel{\mathop{\otimes}\limits_{F}}W^{0}$, where $W^{0}$ denotes $W$ equipped with trivial action. This gives the result. $\square$ ###### Corollary 5.4 There is a natural equivalence $K^{G}(\overline{F})^{\wedge}_{\epsilon}\cong K(F)^{\wedge}_{l}$. Proof: The proof is identical to that given in Proposition 5.3 above. That proof uses only the triviality of the $I_{G}$-action on $\pi_{0}$, and so applies here as well. $\square$ We conclude this section by proving that the equivariant $K$-theory of ${\cal E}G\times W$ can be computed as a particular kind of colimit of equivariant $K$-theory spectra of smooth schemes of finite type over $k$. This will be important to us because the rigidity results we prove in [10] will apply only to such systems. ###### Definition 5.2 Let $G$ be a profinite group, and as usual let $\mathfrak{N}^{G}$ denote the directed set of closed normal subgroups of finite index in $G$. Let $F:\mathfrak{N}^{G}\rightarrow\mathfrak{C}^{cof}_{et}(G)$ denote any functor ($\mathfrak{C}^{cof}_{et}(G)$ was defined in Definition 5.1 above). We say $F$ is adapted if ${\cal N}(F(N))=N$ for all closed normal $N\subseteq G$. Let $\rho$ be any continuous homomorphism from $G$ to the absolute Galois group of a field $K$. Let $L$ denote the absolute closure of $K$. We define spectrum valued functors $K(F,L)$ and $K^{G}(F,L)$ associated to this data by $K(F,L)=\mathrel{\mathop{hocolim}\limits_{N\in\mathfrak{N}^{G}}}K(F(N)\mathrel{\mathop{\times}\limits_{G/N}}L^{N})$ and $K^{G}(F,L)=\mathrel{\mathop{hocolim}\limits_{N\in\mathfrak{N}^{G}}}K^{G/N}(F(N)\times L^{N})$ These colimits are naturally equivalent by Proposition 2.6. $F$ is said to be smooth (respectively of finite type) if each $F(N)$ is smooth (respectively of finite type). We are now able to express the $K$-theory of ${\cal E}G$ in these terms. ###### Proposition 5.9 Let $G$ be a totally torsion free profinite $l$-group, and let $\rho$ be a homomorphism into the absolute Galois group of a field $K$. Let $L$ denote the algebraic closure of $K$. Then there is a smooth adapted functor $F(G,\rho):\mathfrak{N}^{G}\rightarrow\mathfrak{C}^{cof}_{et}(G)$ of finite type, and an equivalence $K^{G}(F(G,\rho),L)\rightarrow K^{G}({\cal E}G\times Spec(L))$. This equivalence is natural for inclusions of fields with continuous $G$-action. Proof: For every closed normal subgroup of finite index $N\subseteq G$, Corollary 5.1 asserts that there is an integer $s(N)$ so that the action of $G/N$ on $\overline{\theta}(s^{\prime},N)$ for every $s\geq s(N)$, is etale. We now define a function $j:\mathfrak{N}^{G}\rightarrow\mathbb{N}\times\mathfrak{N}^{G}$ by $j(N)=\mathrel{\mathop{max}\limits_{N^{\prime}\subseteq N}}s(N)$ It is clear from the definition that $J:\mathfrak{N}^{G}\rightarrow\mathbb{N}\times\mathfrak{N}^{G}$, defined by $J(N)=(j(N),N)$ is a map of directed sets, that $\overline{\theta}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}J$ is an adapted functor, and further that the image of $J$ is a final subset of $\mathfrak{N}^{G}$. It follows that we have the functors $K^{G}(\overline{\theta}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}J,L)$ for any $\rho$ and $L$ as in the statement of the Proposition. Moreover, we have an equivalence of colimits from $K^{G}(\overline{\theta}\mbox{\hskip 1.42271pt}\raisebox{1.13791pt}{\tiny{$\circ$ }}J,L)$ to $K^{G}({\cal E}G\times Spec(L))$. Finally, each of the schemes $F(N)$ are of finite type and smooth, since they are quotients by a finite etale group action on a torus. $\square$ Remark: Note that the cases we are interested in are $K=L=k$ and $K=F$, $L=\overline{F}$. We write $F^{G}_{L}$ for $l$-adic completion of the the functor $N\rightarrow\theta(\nu(N))$ given in the proof of the Proposition above. We will prove that the maps $K^{G}(k)\rightarrow K^{G}({\cal E}G)$ and $K^{G}(Spec(\overline{F}))\rightarrow K^{G}({\cal E}G\times Spec(\overline{F}))$ induce equivalences on derived completions along the homomorphism $\epsilon:K^{G}(k)\rightarrow\mathbb{H}_{l}$, and further that the completions of $K^{G}({\cal E}G)$ and $K^{G}({\cal E}G\times Spec(\overline{F}))$ are just $l$-adic completions. Consider the diagram $\begin{diagram}$ We will be proving that the left hand horizontal arrows are equivalences. The right hand horizontal arrows are equivalences by Proposition 5.9 above. It follows that in order to prove that the left hand vertical arrow is an equivalence, it will suffice to prove that the right hand vertical arrow is one. This is what will be proved in [10]. ## 6 Algebraic properties of certain representation rings In this section, we analyze some particular properties of the representation rings of certain semidirect products of symmetric groups with profinite groups of the form $\mathbb{Z}_{l}^{n}$. The results of this section are the key technical tools which allow us to prove our main results. ###### Theorem 6.1 Let $k$ be a field of characteristic $l$, and let $\epsilon:A\rightarrow k$ be an augmented $k$-algebra with augmentation ideal $I_{A}$. Let $G\subseteq G^{\prime}\subseteq\Sigma_{n}$ denote subgroups of $l$-power order. Let $\displaystyle B=\mathop{\otimes}^{n}_{k}A$, equipped with the action of $\Sigma_{n}$, and therefore of $G$ and $G^{\prime}$. The algebra $B$ is given the tensor product augmentation $\epsilon_{B}$, and the algebras $B^{G}$ and $B^{G^{\prime}}$ both obtain augmentations by restriction of $\epsilon_{B}$, and we denote the restrictions by $\epsilon_{B^{G}}$ and $\epsilon_{B^{G^{\prime}}}$ respectively. Further, we let $I_{B^{G}}$ and $I_{B^{G^{\prime}}}$ denote the corresponding augmentation ideals. Finally, suppose that every element in $A$ has an $l$-th root. Then the equality of ideals $I_{B^{G}}=I_{B^{G^{\prime}}}\cdot B^{G}$ holds. Proof: We first prove that the $k$-algebra $B^{G}$ also has the property that every element admits an $l$-th root. To see this, we let $\mathfrak{A}$ denote a $k$-basis for $A$. Then we form the product $\mathfrak{B}=\mathfrak{A}^{n}$, note that it forms a $k$-basis for $B$, and let $G$ act on $\mathfrak{B}$ via the inclusion $G\hookrightarrow\Sigma_{n}$. For every orbit $\mathfrak{o}=\\{\beta_{1},\ldots,\beta_{s}\\}$, we let $t(\mathfrak{o})$ denote the element $\beta_{1}+\cdots+\beta_{s}\in B^{G}$. It is clear that the elements $t(\mathfrak{o})$, as $\mathfrak{o}$ ranges over the orbits of the $G$ action on $\mathfrak{B}$, form a $k$-basis for $B^{G}$, and therefore that it suffices to prove that each element $t(\mathfrak{o})$ admits an $l$-th root in $B^{G}$. We suppose that we are given an orbit $\mathfrak{o}$, with orbit representative $\beta\in\mathfrak{o}$. Let $G_{\beta}$ denote the stabilizer of $\beta$. The group $G_{\beta}$ now acts on the set $\\{1,\ldots,n\\}$ via its inclusion into $\Sigma_{n}$, and we let $\\{1,\ldots,n\\}=S_{1}\cup S_{2}\cup\cdots\cup S_{l}$ denote the orbit decomposition of this action. Since $G_{\beta}$ stabilizes $\beta$, it is clear that there are elements $\\{\alpha_{j}\\}_{j=1}^{l}$, with $\alpha\in\mathfrak{A}$, so that $\beta=\alpha_{j(1)}\otimes\alpha_{j(2)}\otimes\cdots\otimes\alpha_{j(n)}$ where $j(i)$ denotes the integer so that $i\in S_{j(i)}$. For each $j=1,\ldots,l$, we select an element $\overline{\alpha}_{j}$ so that $(\overline{\alpha}_{j})^{l}=\alpha_{j}$. Also, select left coset representatives $\\{g_{1},\ldots,g_{k}\\}$ for the collection of left cosets $G/G_{\beta}$. The element $\sum_{m=1}^{k}g_{m}(\overline{\alpha}_{j(1)}\otimes\overline{\alpha}_{j(2)}\otimes\cdots\otimes\overline{\alpha}_{j(l)})$ is now clearly an element in $B^{G}$, and it is equally clearly an $l$-th root of $t(\mathfrak{o})$, which gives us the claim. We now proceed to the proof of the theorem. We consider first the case where $G\lhd G^{\prime}$ and where the index of $G$ in $G^{\prime}$ is $l$. Consider any element $b\in I_{B^{G}}$. We let $\overline{b}$ denote any $l$-th root of $b$ in $B^{G}$. It is immediate that $\overline{b}\in I_{B^{G}}$ as well. Now construct the polynomial $\varphi(X)=\prod_{\gamma\in G^{\prime}/G}(X-\overline{b}^{\gamma})$ Note that the notation $\overline{b}^{\gamma}$ is well defined for $\gamma\in G^{\prime}/G$ because $\overline{b}\in B^{G}$. From the definition of $\varphi$, it is clear that $\varphi(\overline{b})=0$, and that $\varphi$ is of the form $\varphi(X)=X^{l}+\sum_{i=1}^{l}(-1)^{i}c_{i}X^{l-i}$ where $c_{i}$ is the $i$-th elementary symmetric function in the elements $\overline{b}^{\gamma}$, as $\gamma$ varies over all the elements of $G^{\prime}/G$. Note that $c_{i}\in I_{B^{G^{\prime}}}$. Since $\overline{b}^{l}=b$, it follows that $b=\sum_{i=1}^{l}(-1)^{i+1}c_{i}\overline{b}^{l-i}$ which gives the result in this case. For the general case, we may form a sequence of subgroups $G=G_{t}\subseteq G_{t-1}\subseteq G_{t-2}\subseteq\cdots\subseteq G_{1}\subseteq G_{0}=G^{\prime}$ where each inclusion is the inclusion of a normal subgroup of index $l$. This is an immediate consequence of the fact that $G^{\prime}$ is an $l$-group. The result for the general case can now be obtained from the case of a normal subgroup of index $l$ via an induction on $t$. We wish to prove that $I_{B^{G_{0}}}B^{G_{t}}=I_{B^{G_{t}}}$, and by induction we suppose that we have proved that $I_{B^{G_{0}}}B^{G_{t-1}}=I_{B^{G_{t-1}}}$ and $I_{B^{G_{t-1}}}B^{G_{t}}=I_{G_{t}}$. We obtain $I_{B^{G_{t}}}=I_{B^{G_{t-1}}}B^{G_{t}}=I_{B^{G_{0}}}B^{G_{t-1}}B^{G_{t}}=I_{B^{G_{0}}}B^{G_{t}}$ $\square$ We now obtain the result which allows us to apply Corollary 2.6. ###### Theorem 6.2 Let $k$ be a field of characteristic $l$, and $A$ an augmented $k$-algebra in which every element has an $l$-th root. Let $G\subseteq G^{\prime}\subseteq\Sigma_{n}$ be an inclusion of $l$-groups, and let $\displaystyle B=\mathop{\otimes}^{n}_{k}A$ be equipped with the natural action of $\Sigma_{n}$ and therefore of $G$ and $G^{\prime}$. Also equip $B$ with the tensor product augmentation. Suppose we have a diagram of augmented (to $k$) rings $\begin{diagram}$ in which $\pi$ and $\pi^{\prime}$ are surjective, and so that the kernel $\mathfrak{I}_{R}$ of $\pi$ satisfies $(\mathfrak{I}_{R})^{N}=\\{0\\}$ for some positive integer $N$. Then there is an integer $M$ so that $(I_{R})^{M}\subseteq I_{R^{\prime}}\cdot R$, where $I_{R}$ and $I_{R^{\prime}}$ denote the augmentation ideals of $R$ and $R^{\prime}$ respectively. Proof: Let $r\in I_{R}$. Then $\pi(r)\in I_{B^{G}}$, and consequently, by Proposition 6.1, $\pi(r)\in I_{B^{G^{\prime}}}B^{G}$. We can therefore write $\pi(r)=\sum_{s}i(y_{s})z_{s}$ where $y_{s}\in I_{B^{G^{\prime}}}$ and $z_{s}\in B^{G}$. Since $\pi$ and $\pi^{\prime}$ are surjective, we can write $y_{s}=\pi^{\prime}(\overline{y}_{s})$ and $z_{s}=\pi(\overline{z}_{s})$ for each $s$. We let $\overline{r}=\sum_{s}f(\overline{y}_{s})\overline{z}_{s}$ It is clear from the construction that $\pi(r-\overline{r})=0$, and of course that $\overline{r}\in I_{R^{\prime}}R$. We now let $M$ be any power of $l$ which is greater than $N$. Then we have $0=(r-\overline{r})^{M}=r^{M}-\overline{r}^{M}$ so $r^{M}=\overline{r}^{M}$, which gives that $r^{M}\in I_{R^{\prime}}R$. $\square$ We will use this result to study certain representation rings. A first step is to study the behavior of the mod-$l$ representation rings of certain wreath products. Recall that for any groups $G$ and $H$, with $G$ finite, the wreath product $G\wr H$ is the semidirect product $G\ltimes H^{\\#(G)}$, where the $G$ action on $H^{\\#(G)}$ is by permutation of coordinates. For any group $G$, we will write $\mathfrak{R}_{l}[G]$ for $R[G]\mathrel{\mathop{\otimes}\limits_{\mathbb{Z}}}\mathbb{F}_{l}$. ###### Proposition 6.1 Let $C_{l}$ denote the cyclic group of order $l$, where $l$ is a prime. Then for any profinite group $K$, we construct $C_{l}\wr K$, with the inclusion $i:K^{l}\hookrightarrow C_{l}\wr K$. The then following results hold. 1. 1. The restriction map $i^{}:\mathfrak{R}_{l}[C_{l}\wr K]\rightarrow\mathfrak{R}_{l}[K^{l}]$ is surjective onto the invariant subring $\mathfrak{R}_{l}[K^{l}]^{C_{l}}$. 2. 2. The kernel $\mathfrak{I}$ of $i^{}$ satisfies $\mathfrak{I}^{l}=0$. Proof: Consider first the case where $K$ is finite. Then the results of section 4.3 of [24] show that an additive basis of $\mathfrak{R}_{l}[C_{l}\wr K]$ can be constructed as follows. Let $\mbox{Irr}(K)$ denote the set of irreducible representations of $K$. Then we have the permutation action of $C_{l}$ on $\mbox{Irr}(K)^{l}$, which can then be decomposed into orbits, which either consist of one element or of $l$ elements. Letting the orbit set be denoted by $\mathfrak{O}$, we write $\mathfrak{O}=\mathfrak{O}_{(1)}\coprod\mathfrak{O}_{(l)}$ for this decomposition. A basis for $\mathfrak{R}_{l}[C_{l}\wr K]$ is now in bijective correspondence with the set $\mathfrak{O}_{(l)}\coprod\mathfrak{O}_{(1)}\times\mbox{Irr}(C_{l})$. The correspondence is given by assigning to elements $\\{\rho_{1},\ldots,\rho_{l}\\}$ of $\mathfrak{O}_{(l)}$ the induced representation $i_{K^{l}}^{C_{l}\wr K}(\rho_{1})$, and on $\mathfrak{O}_{(l)}\times\mbox{Irr}(C_{l})$ by assigning to elements $(\rho,e)$ the representation whose underlying $K^{l}$ representation is $\otimes^{l}\rho$, and whose action by $C$ is given by permuting the tensor product, and to $(\\{\rho\\},\chi)$ the representation $(\rho,e)\otimes\chi$. The restriction map $i^{}$ is now clearly surjective onto the invariant subring $\mathfrak{R}_{l}[K^{l}]$, which is the first result. The kernel ideal is clearly spanned by differences of the form $\langle\rho,\chi\rangle-\langle\rho,\chi_{0}\rangle$, where $\chi_{0}$ denotes the trivial character of $C_{l}$. Such elements lie in $I_{\mathfrak{R}_{l}[C_{l}]}\cdot\mathfrak{R}_{l}[C_{l}\wr K]$, where $\mathfrak{R}_{l}[C_{l}\wr K]$ is an $\mathfrak{R}_{l}[C_{l}]$-algebra via the homomorphism induced by the group homomorphism $C_{l}\wr K\rightarrow C_{l}$. Since the augmentation ideal in $\mathfrak{R}_{l}[C_{l}]$ has trivial $l$-th power, the second result follows. The extension to profinite groups is a straightforward direct limit argument. $\square$ Consider an $l$-Sylow subgroup $\mathbb{S}(l,n)$ of $\Sigma_{l^{n}}$. It is well-known that this group can be described as an iterated wreath product $\mathbb{S}(l,n)=\wr^{i}C_{l}$ where $C_{l}$ denotes a cyclic group of order $l$. We also consider the semidirect product $\mathfrak{S}(l,n)=\mathbb{S}(l,n)\ltimes\mathbb{Z}_{l}^{l^{n}}$. ###### Proposition 6.2 The image of the homomorphism $i^{}:\mathfrak{R}_{l}[\mathfrak{S}(l,n)]\rightarrow\mathfrak{R}_{l}[\mathbb{Z}_{l}^{l^{n}}]$ is the invariant subring $\mathfrak{R}_{l}[\mathbb{Z}_{l}^{l^{n}}]^{\mathbb{S}(l,n)}$. If we let $\mathfrak{I}$ denote the kernel of the homomorphism, then $\mathfrak{I}^{l^{n}}=\\{0\\}$. Proof: We first note that $\mathfrak{S}(l,n)\cong C_{l}\wr\mathfrak{S}(l,n-1)$. We now use this observation to perform an induction over $n$, as follows. The $n=1$ case is Proposition 6.1. For the case $n\geq 1$, the observation above show us the restriction map $\mathfrak{R}_{l}[\mathfrak{S}(l,n)]\rightarrow\mathfrak{R}_{l}[\mathfrak{S}(l,n-1)^{l}]\cong\otimes^{l}\mathfrak{R}_{l}[\mathfrak{S}(l,n-1)]$ surjects on to the invariant subring $(\otimes^{l}\mathfrak{R}_{l}[\mathfrak{S}(l,n-1)])^{C_{l}}$. We observe the general fact that if $f:V\rightarrow W$ is a homomorphism of $\mathbb{F}_{l}$-vector spaces, then the map $\otimes^{l}f:(\otimes^{l}V)^{C_{l}}\rightarrow(\otimes^{l}W)^{C_{l}}$ is also surjective, since the map $f$ admits a right inverse. By the inductive hypothesis, the map $\mathfrak{R}_{l}[\mathfrak{S}(l,n-1)]\rightarrow\mathfrak{R}_{l}[\mathbb{Z}^{l^{n-1}}_{l}]$ surjects onto the ring of invariants under the $\mathfrak{S}(l,n-1)$ action on $\mathfrak{R}_{l}[\mathbb{Z}^{l^{n-1}}_{l}]$. What the induction now shows is that the image of the restriction map $\mathfrak{R}_{l}[\mathfrak{S}(l,n)]\rightarrow\mathfrak{R}_{l}[\mathbb{Z}^{l^{n}}_{l}]$ surjects onto the $C_{l}$ invariant part of $(\otimes^{l}(\mathfrak{R}_{l}[\mathbb{Z}^{l^{n-1}}_{l}])^{C_{l}}$. Now it is easy to check that this invariant part is exactly identified with the $\mathfrak{S}(l,n)$-invariant part of $\mathfrak{R}_{l}[\mathbb{Z}_{l}^{l^{n}}]$ under the identification $\mathfrak{R}_{l}[\mathbb{Z}_{l}^{l^{n}}]\cong\otimes^{l}\mathfrak{R}_{l}[\mathbb{Z}_{l}^{l^{n-1}}]$ which concludes the proof, which proves the first assertion. The second is an easy induction using Proposition 6.1. $\square$ We will need to work with some composite order finite groups as well. We will need some terminology. ###### Definition 6.1 For any finite group $G$, we will write $\hat{\mathfrak{R}}_{l}[G]$ for the completion of the ring $\mathfrak{R}_{l}[G]$ at its augmentation ideal. For any profinite group $G$ and finite quotient $Q$ of $G$, we define $\hat{\mathfrak{R}}_{l}[G;Q]=\hat{\mathfrak{R}}_{l}[Q]\mathrel{\mathop{\otimes}\limits_{\mathfrak{R}_{l}[Q]}}\mathfrak{R}_{l}[G]$ We collect some elementary properties. ###### Proposition 6.3 Let $G$ be a finite group. 1. 1. If $G$ is an $l$-group, then the natural homomorphism $\mathfrak{R}_{l}[G]\rightarrow\hat{\mathfrak{R}}_{l}[G]$ is an isomorphism. More generally, let $\Gamma$ denote a profinite group, with a quotient $G$ which is a finite $l$-group, then the natural homomorphism $\mathfrak{R}_{l}[G]\rightarrow\hat{\mathfrak{R}}_{l}[\Gamma;G]$ is an isomorphism. 2. 2. For any subgroup $H\subseteq G$, the induction homomorphism $i_{H}^{G}:R[H]\rightarrow R[G]$ naturally extends to a homomorphism $\hat{i}_{H}^{G}:\hat{\mathfrak{R}}_{l}[H]\rightarrow\hat{\mathfrak{R}}_{l}[G]$. 3. 3. We let $G_{l}$ denote the $l$-Sylow subgroup of $G$. Then the restriction map $\hat{\mathfrak{R}}_{l}[G]\rightarrow\hat{\mathfrak{R}}_{l}[G_{l}]$ is an injection. More generally, let $\Gamma$ denote a profinite group, with a finite quotient $G$. Let $G_{l}\subseteq G$ be a $l$-Sylow subgroup of $G$. Then the natural map $\hat{\mathfrak{R}}_{l}[\Gamma;G]\rightarrow\hat{\mathfrak{R}}_{l}[\Gamma_{l},G_{l}]$ is an injection, where $\Gamma_{l}$ denotes the inverse image of $G_{l}$ in $\Gamma$ under the projection $\Gamma\rightarrow G$. 4. 4. For any finite group $G$, the kernel of the (surjective) homomorphism $R[G]\rightarrow\hat{\mathfrak{R}}_{l}[G]$ is the ideal $(l)+((l)+I_{G})^{\infty}$ where for any ideal $J^{\infty}$ denotes the intersection $\bigcap_{n}J^{n}$. More generally, given a profinite group $\Gamma$ and a finite quotient $G$, the kernel of the homomorphism $R[\Gamma]\rightarrow\hat{\mathfrak{R}}_{l}[\Gamma;G]$ is the ideal $((l)+((l)+I_{G})^{\infty})R[\Gamma]$. 5. 5. For any finite $l$-group $G$, the ideal $((l)+I_{G})^{\infty}\subseteq R[G]$ is the zero ideal. Proof: Part (1) follows from Proposition 3.5 of [34]. For part (2), we first quote Corollary 3.9 of [34], which asserts that the $I_{H}$ topology and $I_{G}$ topology on $R[H]$ are the same. It is easy to check that this extend directly to $\mathfrak{R}_{l}$. To prove that we have a natural extension of $i_{H}^{G}$, we need to check that for every $n$, there is a $d(n)$ so that $i_{H}^{G}(I_{H}^{n(d)})\subseteq I_{G}^{n}$. The observation above now shows that it suffices to prove that for every $n$, there is some number $N(n)$ so that $i_{H}^{G}(I_{G}^{N(n)}\mathfrak{R}_{l}[H])\subseteq I_{G}^{n}\mathfrak{R}_{l}[G]$. But Frobenius induction has the property that $i_{H}^{G}(I_{G}^{n}\mathfrak{R}_{l}[H])\subseteq I_{G}^{n}\mathfrak{R}_{l}[G]$, so we are done. For part (3), we find that the composite $\hat{\mathfrak{R}}_{l}[G]\rightarrow\hat{\mathfrak{R}}_{l}[H]\stackrel{{\scriptstyle\hat{i}_{H}^{G}}}{{\rightarrow}}\hat{\mathfrak{R}}_{l}[G]$ has the property that $1$ is carried to an element $\lambda$ whose augmentation is equal to $\\#(G/G_{l})$. This number is prime to $l$, and so is a unit in $\mathbb{F}_{l}$. Since the augmentation ideal in $\hat{\mathfrak{R}}_{l}[G]$ is nilpotent, it follows that $\lambda$ is a unit, so $\hat{\mathfrak{R}}_{l}[G]$ is in fact a summand (as $\hat{\mathfrak{R}}_{l}[G]$-modules) of $\hat{\mathfrak{R}}_{l}[H]$. Part (4) follows immediately from the definitions. Part (5) Is a direct consequence of Part (1). $\square$ ###### Corollary 6.1 The ring homomorphism $\hat{\mathfrak{R}}_{l}[\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n};\Sigma_{n}]\rightarrow\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]$ has image the invariant subring $\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]^{\Sigma_{n}}$. Further, its kernel $\mathfrak{I}$ satisfies $\mathfrak{I}^{M}=\\{0\\}$, where $M$ is the smallest power of $l$ which is greater than $n$. Proof: Let $\Sigma_{n}(l)\subseteq\Sigma_{n}$ denote an $l$-Sylow subgroup. Letting $\\{\sigma_{1},\ldots,\sigma_{t}\\}$ denote a complete set of coset representatives for $\Sigma_{n}(l)$ in $\Sigma_{n}$, we have a homomorphism $\tau:\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]^{\Sigma_{n}(l)}\rightarrow\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]^{\Sigma_{n}}$ defined by $\tau(x)=\sum_{i}\sigma_{i}(x)$ The composite $\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]^{\Sigma_{n}}\stackrel{{\scriptstyle i}}{{\rightarrow}}\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]^{\Sigma_{n}(l)}\stackrel{{\scriptstyle\tau}}{{\rightarrow}}\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]^{\Sigma_{n}}$ is equal to multiplication by the index of $\Sigma_{n}(l)$ in $\Sigma_{n}$, from which it follows that $\tau$ is a surjection onto a summand. It is now easy to check that there is a commutative diagram $\begin{diagram}$ where the left hand vertical arrow is induction from $\Sigma_{n}(l)\ltimes\mathbb{Z}_{l}^{n}$ to $\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n}$. The upper horizontal arrow is surjective by Proposition 6.2, and we have observed that $\tau$ is surjective. It follows that the lower horizontal arrow is also surjective, which is the first assertion in the Corollary. The second statement follows directly from (3) in Proposition 6.3. $\square$ ###### Corollary 6.2 Let $n$ be a positive integer, and let $s$ and $t$ be positive integers so that $s+t=n$. The the homomorphism $\hat{\mathfrak{R}}_{l}[(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n}]\rightarrow\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]$ has image equal to the invariant subring $\mathfrak{R}_{l}[\mathbb{Z}_{l}^{n}]^{\Sigma_{s}\times\Sigma_{t}}$, and there is an integer $M$ so that the $M$-th power of the kernel ideal is $=\\{0\\}$. Proof: Immediate from Corollary 6.1, since this example is simply a tensor product of two of the examples to which it applies. $\square$ ###### Corollary 6.3 Let $n,s$, and $t$ be as in the preceding corollary. Then $\hat{\mathfrak{R}}_{l}[(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n};\Sigma_{s}\times\Sigma_{t}]$ becomes an $\hat{\mathfrak{R}}_{l}[\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n};\Sigma_{n}]$-algebra under the ring homomorphism induced by the inclusion $(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n}\rightarrow\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n}$. We let $\hat{I}[-]\hookrightarrow\hat{\mathfrak{R}}_{l}[-]$ denote the augmentation ideal. Then there is an integer $M$ so that $\hat{I}[(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n};\Sigma_{s}\times\Sigma_{t}]^{M}\subseteq\hat{I}[\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n};\Sigma_{n}]\cdot\hat{\mathfrak{R}}_{l}[(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n};\Sigma_{s}\times\Sigma_{t}]$ Proof: This is a direct application of Theorem 6.2, with $R=\hat{\mathfrak{R}}_{l}[(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n};\Sigma_{s}\times\Sigma_{t}]$, $R^{\prime}=\hat{\mathfrak{R}}_{l}[\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n};\Sigma_{n}]$, $A=\mathfrak{R}_{l}[\mathbb{Z}_{l}]$, $f$ is the homomorphism induced by the inclusion $(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n}\hookrightarrow\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n}$, and where $\pi$ and $\pi^{\prime}$ are the homomorphisms induced by the inclusions $\mathbb{Z}_{l}^{n}\hookrightarrow(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n}$ and $\mathbb{Z}_{l}^{n}\hookrightarrow\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n}$ respectively. That the hypothesis of Theorem 6.2 is satisfied is guaranteed by Corollary 6.2. $\square$ ###### Corollary 6.4 Fix $n,s$, and $t$ as in the preceding corollary. Then there is an integer $M$ so that $I_{(\Sigma_{s}\times\Sigma_{t})\ltimes\mathbb{Z}_{l}^{n}}^{M}\subseteq((l)+I_{\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n}}+((l)+I_{\Sigma_{s}\times\Sigma_{t}})^{\infty})R[\Sigma_{s}\times\Sigma_{t}\ltimes\mathbb{Z}_{l}^{n}]$ Proof: Immediate consequence of Part (4) of Proposition 6.3, together with Corollary 6.3. $\square$ ## 7 Equivariant $K$-theory of ${\cal A}_{\eta}$ and ${\cal T}_{\eta}$ We wish to study the map $K^{G}{\cal A}_{\eta}\rightarrow K^{G}{\cal T}_{\eta}$, with an eye to proving that it induces an equivalence on completions along the homomorphism of commutative ring spectra $\mathfrak{K}^{G}(k)\rightarrow\mathbb{H}_{l}$. More generally, we will want to study the map $K^{G}({\cal A}_{\eta}\times W)\rightarrow K^{G}({\cal T}_{\eta}\times W)$, and prove that it induces an equivalence on completions, where $W$ is a Noetherian approximable Affine scheme with continuous $G$-action. We will carry out the analysis in the case $W=Spec(k)$, and then indicate any necessary modifications in the general case. In order to do this, we will create a filtration by abelian subcategories on the category of all finitely generated modules over affine space whose support consists of coordinate hyperplanes (including the full affine space). Of course, this filtration will induce a filtration of the $K$-theory spectra associated to the categories, via application of the localization sequence. This section is entirely devoted to the analysis of the subquotients in this filtration as modules over the weak ring spectra $K(k)$ and $K^{G}(k)$. We emphasize that all calculations in this section involve only modules over weak ring spectra. These calculations will then be used in the next section to draw conclusions about the derived completions of modules over $\mathfrak{K}^{G}(k)$, via comparisons such as the ones performed in [8], section 4. Let $\mathbb{A}=\mathbb{A}(k)=Spec(k[x_{1},x_{2},\ldots,x_{n}])$ be the $n$-dimensional affine space over $k$, with explicit choice of coordinates $x_{i}$. Of course, Theorem 7 of [31] tells us that $K(\mathbb{A})\cong K(k)$. For any $1\leq s\leq n$, we let $V(s)$ denote the hyperplane defined by $x_{s}=0$. More generally, let $S\subseteq\\{x_{1},\ldots,x_{n}\\}$, and define $V(S)=\bigcap_{s\in S}V(s)$ For any $i$, we will let $V_{i}$ denote the union $V_{i}=\bigcup_{\\#(S)\geq i}V(S)$ Note that $V_{i+1}\subseteq V_{i}$. We will also denote by $\mathbb{T}$ the scheme $\mathbb{A}-\cup_{s}V_{s}$, which is $Spec(k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$. We will refer to subvarieties of the form $V(S)$ as coordinate subspaces, and the corresponding ideals as coordinate ideals. Note that the case $S=\emptyset$ is included, with the full $\mathbb{A}$ as corresponding variety. ###### Definition 7.1 Let $\mbox{Mod}_{i}(\mathbb{A})$ to be the subcategory of finitely generated modules whose support is contained in $V_{i}$. For a set $S\subseteq\\{1,\ldots,n\\}$ of cardinality $i$, we also define $\mbox{Mod}_{i}(\mathbb{A};S)\subseteq\mbox{Mod}_{i}(\mathbb{A})$ to be the subcategory of modules supported on the subvariety $V(S)\cup V_{i+1}$. We note that all the categories $\mbox{Mod}_{i}(\mathbb{A})$ and $\mbox{Mod}_{i}(\mathbb{A};S)$ are abelian categories, and all inclusions are inclusions of Serre subcategories. Let $\mathbb{A}_{S}$ denote the subscheme $\mathbb{A}-\bigcup_{s\notin S}V(s)$, and $\overline{\mathbb{A}}_{S}\subseteq\mathbb{A}_{S}$ denote the subscheme $\\{(x_{1},\ldots,x_{n})\|x_{i}=0\mbox{ for }i\in S\\}$. We also let $\mbox{\em Nil}(\mathbb{A};S)$ denote the category of coherent $\mathbb{A}_{S}$-modules on which the generators $\\{x_{i}\\}_{i\in S}$ act nilpotently. We clearly have an increasing sequence of subcategories $\\{0\\}=\mbox{Mod}_{n+1}(\mathbb{A})\subseteq\mbox{Mod}_{n}(\mathbb{A})\subseteq\mbox{Mod}_{n-1}(\mathbb{A})\subseteq\cdots\subseteq\mbox{Mod}_{1}(\mathbb{A})\subseteq\mbox{Mod}_{0}(\mathbb{A})=\mbox{Mod}(\mathbb{A})$ ###### Proposition 7.1 There is a natural functor from the quotient abelian category $\mbox{\em Mod}(\mathbb{A})/\mbox{\em Mod}_{1}(\mathbb{A})$ to the category $\mbox{\em Mod}(\mathbb{T})$, which induces an equivalence of $K$-theory spectra. Proof: This is simply the localization sequence associated to the removal of $V_{1}$ from $V_{0}\ \mathbb{A}$, which is $\mathbb{T}$. $\square$ We let $\Phi(\mathbb{A},\mathbb{T})$ denote the homotopy fiber of the map of $K$-theory spectra $K(\mathbb{A})\rightarrow K(\mathbb{T})$. It follows readily from Proposition 7.1 above that $\Phi(\mathbb{A},\mathbb{T})$ is equivalent to the $K$-theory spectrum $K(\mbox{Mod}_{1})$. ###### Proposition 7.2 There is a filtration of $\Phi=\Phi(\mathbb{A},\mathbb{T})$ by spectra $\Phi_{i}=\Phi_{i}(\mathbb{A},\mathbb{T})$, with $\simeq\Phi_{n+1}\subseteq\Phi_{n}\subseteq\Phi_{n-1}\subseteq\cdots\subseteq\Phi_{2}\subseteq\Phi_{1}=\Phi$ and so that $\Phi_{i}/\Phi_{i+1}\cong K(\mbox{\em Mod}_{i}(\mathbb{A})/\mbox{\em Mod}_{i+1}(\mathbb{A}))$. Proof: The spectra $\Phi_{i}$ can be taken to be the homotopy fibers of the maps of spectra $K(\mathbb{A})\rightarrow K(\mbox{Mod}(\mathbb{A})/\mbox{Mod}_{i}(\mathbb{A}))$ $\square$ Each inclusion $\mbox{Mod}_{i+1}(\mathbb{A})\hookrightarrow\mbox{Mod}_{i}(\mathbb{A})$ is an inclusion of an abelian subcategory, and it is clear from the discussion of quotient abelian categories in [38] that the quotient $\mbox{Mod}_{i}(\mathbb{A})/\mbox{Mod}_{i+1}(\mathbb{A})$ can be analyzed as follows. For each $S$ with $\\#(S)=i$, we may consider the inclusion $i_{S}:\mbox{Mod}_{i}(\mathbb{A};S)\hookrightarrow\mbox{Mod}_{i}(\mathbb{A})$, and therefore the inclusion of subquotients $j_{S}:\mbox{Mod}_{i}(\mathbb{A};S)/\mbox{Mod}_{i+1}(\mathbb{A})\hookrightarrow\mbox{Mod}_{i}(\mathbb{A})/\mbox{Mod}_{i+1}(\mathbb{A})$. We may then sum over all $S$ of cardinality $i$ and apply $K$-theory to obtain $\bigvee_{\\#(S)=i}j_{S}:\bigvee_{\\#(S)=i}K(\mbox{Mod}_{i}(\mathbb{A};S)/\mbox{Mod}_{i+1}(\mathbb{A}))\longrightarrow K(\mbox{Mod}_{i}(\mathbb{A})/\mbox{Mod}_{i+1}(\mathbb{A}))$ (7–6) This functor is an equivalence since as in the proof of Theorem 5.4 of [31] it can be checked that the underlying functor is an equivalence of categories. Further, there is a natural equivalence of abelian categories $\mbox{Mod}_{i}(\mathbb{A};S)/\mbox{Mod}_{i+1}(\mathbb{A})\simeq\mbox{Nil}(\mathbb{A};S)$ and hence equivalences of $K$-theory spectra of these categories $K(\mbox{Mod}_{i}(\mathbb{A};S)/\mbox{Mod}_{i+1}(\mathbb{A}))\simeq K(\mbox{Nil}(\mathbb{A};S))\simeq K(\mbox{Mod}(\overline{\mathbb{A}}_{{S}}))$ The second of the equivalences is proved as in [31] by the devissage principle. We will also consider the situation of a product of a fixed Noetherian affine scheme $W$ with $\mathbb{A}$ and $\mathbb{T}$. In this case, we define the subcategory $\mbox{Mod}_{i}(\mathbb{A},W)\subseteq\mbox{Mod}(\mathbb{A}\times W)$ to consist of all $\mathbb{A}\times W$-modules whose support is contained in $\pi^{-1}(V_{i})$, where $\pi:\mathbb{A}\times W\rightarrow\mathbb{A}$ is the projection. There are obvious analogues $\mbox{Mod}_{i}(\mathbb{A},W;S)$ and $\mbox{Nil}(\mathbb{A},W;S)$ of $\mbox{Mod}_{i}(A;S)$ and $\mbox{Nil}(\mathbb{A};S)$. It is further clear from Section 2.2 that all the spectra $K(\mbox{Mod}_{i}(\mathbb{A},W)),K(\mbox{Mod}_{i}(\mathbb{A},W;S)),\mbox{ and }K(\mbox{Nil}(\mathbb{A},W;S))$ are module spectra over the weak ring spectrum $K(\mathbb{A}\times W)$, that all the inclusion and restriction functors induce maps of $K(\mathbb{A}\times W)$-module spectra, and therefore that the quotient spectra $K(\mbox{Mod}_{i}(\mathbb{A},W))/K(\mbox{Mod}_{i+1}(\mathbb{A},W))$ are also $K(\mathbb{A}\times W)$-module spectra. It is also clear that $K(\mbox{Nil}(\mathbb{A},W;S))$ is naturally a $K(\mathbb{A}_{S}\times W)$-module spectrum, in a way which extends the $K(\mathbb{A}\times W)$-module structure via the map of weak ring spectra $K(\mathbb{A}\times W)\rightarrow K(\mathbb{A}_{S}\times W)$. Moreover, it is clear that the decomposition of spectra $K(\mbox{Mod}_{i}(\mathbb{A},W))/K(\mbox{Mod}_{i+1}(\mathbb{A},W))\cong\bigvee_{\\#(S)=i}K(\mbox{Nil}(\mathbb{A},W;S))\cong\bigvee_{\\#(S)=i}K(\mbox{Mod}(\overline{\mathbb{A}}_{{S}}\times W))$ given by the homotopy inverse to the map $\bigvee_{\\#(S)=i}K(j_{s})$ from 7–6 above is also an equivalence of $K(\mathbb{A}\times W)$-module spectra, where the $K(\mathbb{A}\times W)$ action on the summand $K(\mbox{Nil}(\mathbb{A},W;S))$ and $K(\mbox{Mod}(\overline{\mathbb{A}}_{{S}}\times W))$ are obtained by restriction of scalars along the map of weak ring spectra $K(\mathbb{A}\times W)\rightarrow K(\mathbb{A}_{S}\times W)$. We next begin the analysis of how this filtration works equivariantly. Let $\rho$ be any monomial $k$-linear representation of a finite group $G$, and let $\\{x_{1},\ldots,x_{n}\\}$ denote any basis for the representation space $V_{\rho}$ in which the representation is monomial. Let $\overline{\rho}:G\rightarrow\Sigma_{n}$ denote the corresponding permutation representation. As usual, we write $\mathbb{A}=\mathbb{A}(k,\rho)$ for the associated affine variety, and therefore on the associated module category. It is clear that the filtration of $\mbox{Mod}(\mathbb{A})$ by the subcategories $\mbox{Mod}_{i}(\mathbb{A})$ is invariant under the action defined by $\rho$, and we therefore easily obtain a filtration of $\mbox{Mod}^{G}(\mathbb{A})$ by Serre subcategories $\mbox{Mod}_{i}^{G}(\mathbb{A})$. The equivariant version of Proposition 7.2 holds. Let $\Phi^{G}=\Phi^{G}(\mathbb{A},\mathbb{T})$ denote the homotopy fiber of the map of spectra $K^{G}(\mathbb{A})\rightarrow K^{G}(\mathbb{T})$ ###### Proposition 7.3 There is a filtration of $\Phi^{G}$ by spectra $\Phi_{i}^{G}=\Phi_{i}^{G}(\mathbb{A},\mathbb{T})$, with $\simeq\Phi_{n+1}^{G}\subseteq\Phi_{n}^{G}\subseteq\Phi_{n-1}^{G}\subseteq\cdots\subseteq\Phi_{2}^{G}\subseteq\Phi_{1}^{G}=\Phi^{G}$ and so that $\Phi^{G}_{i}/\Phi^{G}_{i+1}\cong K^{G}(\mbox{\em Mod}_{i}(\mathbb{A})/\mbox{\em Mod}_{i+1}(\mathbb{A}))$. We will have several categories of $\mathbb{A}$-modules which are closed under the action of $G$ or of subgroups of $G$. For each such category, we can then denote its equivariant version with the superscript $G$, and define its equivariant $K$-theory as the $K$-theory of the equivariant version of the category. For instance, we will write $K^{G}(\mbox{Mod}_{i}(\mathbb{A}))$ for $K(\mbox{Mod}^{G}_{i}(\mathbb{A}))$. We now identify the subquotients $\mbox{Mod}_{i}^{G}(\mathbb{A})/\mbox{Mod}_{i+1}^{G}(\mathbb{A})$. Let ${\cal P}_{i}$ denote the set of subsets of $\\{1,\ldots,n\\}$ of cardinality $i$, and let ${\cal P}_{i}=\mathfrak{o}_{1}\cup\cdots\cup\mathfrak{o}_{t}$ be an orbit decomposition, with orbit representatives $S_{j}\in\mathfrak{o}_{j}$. Note that for each $j$, the subcategory $\mbox{Mod}_{i}(\mathbb{A};\mathfrak{o}_{j})$ is invariant under the action of $G$. Then an argument entirely analogous to the non-equivariant one above shows that $\mbox{Mod}_{i}^{G}(\mathbb{A})/\mbox{Mod}_{i+1}^{G}(\mathbb{A})\cong\bigvee_{j=1}^{t}\mbox{Mod}^{G}_{i}(\mathbb{A};\mathfrak{o}_{j})/\mbox{Mod}^{G}_{i+1}(\mathbb{A})$ Moreover, it is now easy to see that each summand is described by $\mbox{Mod}^{G}_{i}(\mathbb{A};\mathfrak{o}_{j})/\mbox{Mod}^{G}_{i+1}(\mathbb{A})\cong\mbox{Mod}^{G_{j}}_{i}(\mathbb{A};S_{j})/\mbox{Mod}^{G_{j}}_{i+1}(\mathbb{A})$ where $G_{j}$ is the stabilizer of the orbit representative $S_{j}$, and that further we have an equivalence of abelian categories $\mbox{Mod}^{G_{j}}_{i}(\mathbb{A},S_{j})/\mbox{Mod}_{i+1}^{G_{j}}(\mathbb{A})\cong\mbox{Nil}^{G_{j}}(\mathbb{A};{{S}_{j}})$ (7–7) Also, devissage gives us an equivalence of spectra $K(\mbox{Nil}^{G_{j}}(\mathbb{A};S_{j}))\cong K^{G_{j}}(\mbox{Mod}(\overline{\mathbb{A}}_{{S}_{j}}))$ (7–8) and therefore a decomposition of $K$-theory spectra $K(\mbox{Mod}^{G}_{i}(\mathbb{A}))/K(\mbox{Mod}^{G}_{i+1}(\mathbb{A}))\cong\bigvee_{j=1}^{t}K^{G_{j}}(\mbox{Nil}(\mathbb{A};{{S}_{l}}))\cong\bigvee_{j=1}^{t}K^{G_{j}}(\overline{\mathbb{A}}_{{S}_{j}})$ (7–9) The corresponding decomposition for the case of a product with a fixed affine Noetherian scheme with $G$-action $W$ $K(\mbox{Mod}^{G}_{i}(\mathbb{A},W))/K(\mbox{Mod}^{G}_{i+1}(\mathbb{A},W))\cong\bigvee_{j=1}^{t}K^{G_{j}}(\mbox{Nil}(\mathbb{A},W;S_{l}))\cong\bigvee_{j=1}^{t}K^{G_{j}}(\overline{\mathbb{A}}_{{S}_{j}}\times W)$ (7–10) follows similarly We now let $G$ be a profinite group, and let $\eta$ denote a fixed $l$-BAR of $G$. We also let $W$ denote a continuous Noetherian affine $G$-scheme. In practice, we will be concerned only with the case where $W=Spec(k)$, and the $G$-action is trivial, or the case where $G=G_{F}$ is the absolute Galois group of an extension $F$ of $k$, and $W=Spec(\overline{F})$. We wish to analyze $K^{G}({\cal A}_{\eta}\times W)$, $K^{G}({\cal T}_{\eta}\times W)$, and the map $K^{G}({\cal A}_{\eta}\times W)\rightarrow K^{G}({\cal T}_{\eta}\times W)$. We obtain $k$-linear representations $\rho_{s}\cdot\eta$ of $G$, and the corresponding varieties ${\cal A}_{\eta}^{s}$ and ${\cal T}_{\eta}^{s}$. We also denote the affine coordinate ring of ${\cal A}_{\eta}^{s}$ by $A_{s}$. We also defined the $l$-th power map $\theta:{\cal A}_{\eta}^{s+1}\rightarrow{\cal A}_{\eta}^{s}$, which is equivariant.Because the map $\theta$ respects the coordinate axes, it is clear that it carries coordinate ideals into coordinate ideals. From the definition of the subcategories $\mbox{Mod}^{G}_{i}$, it follows readily that the functor $A_{s+1}\mathrel{\mathop{\otimes}\limits_{A_{s}}}-$ carries $\mbox{Mod}_{i}^{G}({\cal A}_{\eta}^{s})\subseteq\mbox{Mod}^{G}({\cal A}_{\eta}^{s})$ to the subcategory $\mbox{Mod}_{i}^{G}({\cal A}_{\eta}^{s+1})\subseteq\mbox{Mod}^{G}({\cal A}_{\eta}^{s+1})$. Note that these tensoring functors are exact because ${A}_{s+1}$ is always a flat $A_{s}$-algebra. This permits us to define categories $\mbox{Mod}_{i}^{G}({\cal A}_{\eta})$ as the colimits of the systems $\cdots\rightarrow\mbox{Mod}_{i}^{G}({\cal A}_{\eta}^{s-1})\rightarrow\mbox{Mod}_{i}^{G}({\cal{A}}_{\eta}^{s})\rightarrow\mbox{Mod}_{i}^{G}({\cal A}_{\eta}^{s+1})\rightarrow\cdots$ A more intrinsic way to describe the subcategories of $\mbox{Mod}_{i}({\cal A_{\eta}})$ described above is as follows. For any $s\in\\{1,\ldots,n\\}$, we let $V(s)$ denote the subscheme defined by the ideal $(x_{s},x_{s}^{\frac{1}{l}},x_{s}^{\frac{1}{l^{2}}},\ldots)$. We can now construct $V(S)$ for $S\subseteq\\{1,\ldots,n\\}$ as before, and the corresponding module categories of finitely presented modules supported on the corresponding subschemes as well. We note that the analyses above apply equally well to a situation where we take products with a fixed affine Noetherian $G$-scheme. The arguments from above are adapted in a straightforward way. We now have the analogue of the Proposition 7.3. Let $\Phi^{G}({\cal A}_{\eta},{\cal T}_{\eta})$ denote the homotopy fiber of the map $K^{G}({\cal A}_{\eta})\rightarrow K^{G}({\cal T}_{\eta})$ ###### Proposition 7.4 There is a filtration of $\Phi^{G}({\cal A}_{\eta},{\cal T}_{\eta})$ by spectra $\Phi_{i}^{G}({\cal A}_{\eta},{\cal T}_{\eta})$, with $\simeq\Phi_{n+1}^{G}({\cal A}_{\eta},{\cal T}_{\eta})\subseteq\Phi_{n}^{G}({\cal A}_{\eta},{\cal T}_{\eta})\subseteq\Phi_{n-1}^{G}({\cal A}_{\eta},{\cal T}_{\eta})\subseteq\cdots\subseteq\Phi_{2}^{G}({\cal A}_{\eta},{\cal T}_{\eta})\subseteq\Phi_{1}^{G}({\cal A}_{\eta},{\cal T}_{\eta})=\Phi^{G}({\cal A}_{\eta},{\cal T}_{\eta})$ and so that $\Phi^{G}_{i}({\cal A}_{\eta},{\cal T}_{\eta})/\Phi^{G}_{i+1}({\cal A}_{\eta},{\cal T}_{\eta})\cong K^{G}(\mbox{\em Mod}_{i}({\cal A}_{\eta})/\mbox{\em Mod}_{i+1}({\cal A}_{\eta}))$ Further, for any orbit $\mathfrak{o}_{j}$ under the action of $G$ on ${\cal P}_{i}$, $A_{s+1}\mathrel{\mathop{\otimes}\limits_{A_{s}}}-$ carries the subcategory $\mbox{Mod}^{G}_{i}({\cal A}^{s}_{\eta},\mathfrak{o}_{j})$ into the subcategory $\mbox{Mod}^{G}_{i}({\cal A}_{\eta}^{s+1},\mathfrak{o}_{j})$. It follows that the decompositions 7–10 are respected by the map $\theta$. We can therefore define subcategories $\mbox{Mod}_{i}^{G}({\cal A}_{\eta};\mathfrak{o})\subseteq\mbox{Mod}_{i}^{G}({\cal A}_{\eta})$ as the colimits of the system of categories $\cdots\rightarrow\mbox{Mod}_{i}^{G}({\cal A}_{\eta}^{s-1};\mathfrak{o})\rightarrow\mbox{Mod}_{i}^{G}({\cal A}^{s}_{\eta};\mathfrak{o})\rightarrow\mbox{Mod}_{i}^{G}({\cal A}_{\eta}^{s+1};\mathfrak{o})\rightarrow\cdots$ and we obtain a decomposition $K^{G}(\mbox{Mod}_{i}({\cal A}_{\eta})/\mbox{Mod}_{i+1}({\cal A}_{\eta}))\cong\bigvee_{j=1,\ldots t}K^{G}(\mbox{Mod}_{i}({\cal A}_{\eta};\mathfrak{o_{j}})/\mbox{Mod}_{i+1}({\cal A}_{\eta}))$ (7–11) where $\\{\mathfrak{o}_{i},\ldots,\mathfrak{o}_{t}\\}$ is an orbit decomposition of the action of $G$ on the subsets of cardinality $i$ of $\\{1,\ldots,n\\}$. We will now focus on analyzing the effect of the map $K(\mbox{Mod}^{G}_{i}({\cal A}_{\eta}^{s};\mathfrak{o}_{j})/\mbox{Mod}^{G}_{i+1}({\cal A}_{\eta}^{s}))\rightarrow K(\mbox{Mod}^{G}_{i}({\cal A}_{\eta}^{s+1};\mathfrak{o}_{j})/\mbox{Mod}^{G}_{i+1}({\cal A}_{\eta}^{s+1}))$ induced by $\theta$ on the individual factors, so as to understand the groups $\pi_{}K^{G}(\mbox{Mod}_{i}({\cal A}_{\eta};\mathfrak{o_{j}})/\mbox{Mod}_{i+1}({\cal A}_{\eta}))$ We let $S_{j}$ denote a representative for the orbit $\mathfrak{o}_{j}$, and let $G_{j}\subseteq G$ denote the stabilizer of $S_{j}$. We also have the identifications in 7–7 and 7–8 above, and we wish to identify the effect of $\theta$ under this identification. The category $\mbox{Nil}^{G_{j}}({\cal A}_{\eta}^{s};{{S}_{j}})$ is described as follows. We suppose that an orbit representative for the set $S_{j}$ is the set $\\{1,\ldots,i\\}$. If not, perform a reordering of the indexing. The permutation representation of $G_{j}$ preserves the decomposition $\\{1,\ldots,i\\}\coprod\\{i+1,\ldots,n\\}$ (7–12) and yields decompositions $\rho_{s}\cdot\eta\simeq\sigma_{s}\oplus\tau_{s}$ and $\rho_{s+1}\cdot\eta\simeq\sigma_{s+1}\oplus\tau_{s+1}$, and corresponding decompositions of schemes with $G$-action ${\cal A}_{\eta}^{s}={\cal B}_{\eta}^{s}\times{\cal C}_{\eta}^{s}=\mathbb{A}(k,\sigma_{s})\times\mathbb{A}(k,\tau_{s})$ (7–13) for all $s$. We also remove the coordinate hyperplanes corresponding to the elements of the complement of $S_{j}$, and obtain the variety $\overline{{\cal A}}^{s}_{\eta}={\cal{B}}^{s}_{\eta}\times\overline{{\cal C}}^{s}_{\eta}=\mathbb{A}(k,\sigma_{s})\times\mathbb{T}(k,\tau_{s})$ Moreover, the power map $\theta=\theta_{\rho}$ preserves the decomposition, so that we have a commutative diagram $\begin{diagram}$ of $G_{j}$-equivariant schemes. The category $\mbox{Nil}^{G_{j}}({\cal A}_{\eta}^{s};{{S}_{j}})$ corresponds to the category of all finitely generated $\overline{{\cal A}}_{\eta}^{s}$-modules whose support is concentrated on the subscheme $\\{0\\}\times\overline{{\cal C}}_{\eta}^{s}$, and similarly for $\overline{{\cal A}}_{\eta}^{s+1}$. We let $A_{s}=k[x_{1},\ldots,x_{i},x_{i+1}^{\pm 1},\ldots,x_{n}^{\pm 1}],B_{s}=k[x_{1},\ldots,x_{i}]$, and $C_{s}=k[x_{i+1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ denote the corresponding affine coordinate rings, and similarly for $s+1$. ###### Proposition 7.5 Since the product decompositions 7–13 are equivariant, we have the following. 1. 1. The projections $\overline{\cal A}_{\eta}^{s}\rightarrow\overline{\cal C}_{\eta}^{s}$ and $\overline{\cal A}_{\eta}^{s+1}\rightarrow\overline{\cal C}_{\eta}^{s+1}$ induce equivalences $K^{G_{j}}(\overline{\cal A}_{\eta}^{s})\cong K^{G_{j}}(\overline{\cal C}_{\eta}^{s})$ and $K^{G_{j}}(\overline{\cal A}_{\eta}^{s+1})\cong K^{G_{j}}(\overline{\cal C}_{\eta}^{s+1})$ of weak ring spectra. 2. 2. There are weak equivalences $\varepsilon_{s}:K^{G_{j}}(\overline{\cal C}_{\eta}^{s})\xrightarrow{\sim}K(\mbox{\em Nil}^{G_{j}}({\cal A}_{\eta}^{s};S_{j}))$ and $\varepsilon_{s+1}:K^{G_{j}}(\overline{\cal C}_{\eta}^{s+1})\xrightarrow{\sim}K(\mbox{\em Nil}^{G_{j}}({\cal A}_{\eta}^{s+1};S_{j}))$ of modules over the weak ring spectra $K^{G_{j}}({\cal C}_{\eta}^{s})$ and $K^{G_{j}}({\cal C}_{\eta}^{s})$, which is given on $\pi_{0}$ by $K_{0}(\varepsilon_{s})(1)=[k\mathrel{\mathop{\otimes}\limits_{k}}C_{s}]$, where $k$ is the quotient of $B_{s}$ by the maximal ideal of the origin, where the group action on $k$ is by the identity, and on $C_{s}$ is given by the already prescribed action. The description of $\varepsilon_{s+1}$ is identical. Proof: The first result follows from the homotopy property in [31]. The second two are straightforward devissage arguments. $\square$ Next, we note that by the homotopy property of $K$-theory, we have equivalences $K^{G_{j}}(Spec(k))\rightarrow K^{G_{j}}({\cal B}_{\eta}^{s})$ and $K^{G_{j}}(Spec(k))\rightarrow K^{G_{j}}({\cal B}_{\eta}^{s+1})$ induced by the projections ${\cal B}_{\eta}^{s}\rightarrow Spec(k)$ and ${\cal B}_{\eta}^{s+1}\rightarrow Spec(k)$, respectively. These equivalences, taken together with the map $K^{G_{j}}({\cal B}_{\eta}^{s})\rightarrow K^{G_{j}}({\cal B}_{\eta}^{s+1})$ induced by $\theta_{\sigma}$, give a map $\overline{\theta}_{\sigma}:K^{G_{j}}(Spec(k))\rightarrow K^{G_{j}}(Spec(k))$, which we wish to identify. In order to do this, we consider a monomial $k$-linear representation $\nu$ of a finite group $G$, with the representation space $V_{\nu}$ of $G$ equipped with a basis $B_{\nu}$ in which the representation is monomial. Given any vector space $V$ equipped with a basis $B$, we may construct the symmetric algebra (over $k$) $S[V]$, and factor out the ideal generated by the set $\\{b^{n}\\}_{b\in B}$. We denote this vector space by $T_{n}(V,B)$. Note that the construction only depends on the set of lines spanned by the individual basis elements, and can hence be defined as a functor on the category of vector spaces equipped with a family of independent and spanning lines. This means that in our context, we can construct from the representation $\nu$ a new representation $T_{n}(\nu)$, which has $T_{n}(V_{\nu},B_{\nu})$ as its representation space. We can now describe the map $\overline{\theta}_{\sigma}$ as follows. ###### Proposition 7.6 The map $\overline{\theta}_{\sigma}:K^{G_{j}}(k)\rightarrow K^{G_{j}}(k)$ is characterized up to homotopy by the following two requirements. 1. 1. $\overline{\theta}_{\sigma}$ is a map of $K^{G_{j}}(k)$-modules. 2. 2. $\pi_{0}(\overline{\theta}_{\sigma})(1)=T_{l}(\sigma_{s+1})$ Proof: The first statement is immediate since the map $K^{G_{j}}({\cal B}_{\eta}^{s})\rightarrow K^{G_{j}}({\cal B}_{\eta}^{s+1})$ is a map of $K^{G_{j}}(k)$-modules. For the second, we pass to affine coordinate rings. We write ${\cal B}_{\eta}^{s+1}=Spec(B_{s+1})=Spec(k[x_{1},\ldots,x_{i}])$ with the $G_{j}$-action given by an action on the vector space spanned by $\\{x_{1},\ldots,x_{i}\\}$, which is monomial with respect to this basis. We have ${\cal B}_{\eta}^{s}=Spec(B_{s})=Spec(k[x_{1}^{l},\ldots,x_{i}^{l}])$ and the map $\theta_{\sigma}:{\cal B}_{\eta}^{s+1}\rightarrow{\cal B}_{\eta}^{s}$ is given by the evident inclusion $B_{s}\hookrightarrow B_{s+1}$. The restriction of the $G_{j}$-action to $B_{s}$ factors through the homomorphism $G_{j}\rightarrow G_{j}$. We are interested in the behavior of the functor $B_{s+1}\mathrel{\mathop{\otimes}\limits_{B_{s}}}-$ restricted to the category $\mbox{Nil}^{G_{j}^{s}}({\cal B}_{\eta}^{s})$, the category of modules whose support is the origin in the affine space ${\cal B}_{\eta}^{s}$. A devissage argument shows that a generator for $K_{0}(\mbox{Nil}^{G_{j}}({\cal B}_{\eta}^{s}))$ is the field $k$, regarded as a module over $B_{s}$ via the trivial (zero) action of each of the generators $x_{t}^{l}$ and as a $G_{j}$-representation by declaring the action to be the identity action. We now consider $B_{s+1}\mathrel{\mathop{\otimes}\limits_{B_{s}}}k$. It can clearly be identified with $B_{s+1}/(x_{1}^{l},\ldots,x_{i}^{l})$, which as a $G_{j}$-representation is isomorphic to $T_{l}(\sigma_{s+1})$. It now follows from the homotopy property that this element in $K_{0}^{G_{j}}({\cal B}_{\eta}^{s+1})$ is equal to the element $[T_{l}(\sigma_{s+1})]$. $\square$ ###### Corollary 7.1 There is a commutative diagram $\begin{diagram}$ for all $i$, where $\varepsilon_{s}$ and $\varepsilon_{s+1}$ are isomorphisms. Proof: This is a simple product calculation, noting that ${\cal A}^{s}\cong{\cal B}_{\eta}^{s}\times{\cal C}_{\eta}^{s}$. $\square$ ###### Corollary 7.2 We let $i$ be an integer between $1$ and $n$, that $\mathfrak{o}_{j}$ is an orbit of the action of $G$ on the collection of subsets of cardinality $i$ in $\\{1,\ldots,n\\}$ with orbit representative $S_{j}$, and that $G_{j}$ is the stabilizer of $S_{j}$. Then we have the following. 1. 1. The graded $R[G]$-module $K_{}^{G}(\mbox{ \em Mod}_{i}({\cal A}_{\eta};\mathfrak{o}_{j})/\mbox{\em Mod }_{i}({\cal A}_{\eta}))$ is isomorphic to the colimit of the system $K_{}^{G_{j}}(\overline{{\cal C}}^{0}_{\eta})\stackrel{{\scriptstyle T_{l}(\sigma_{1})\cdot K^{G_{j}}_{}(\theta_{\tau})}}{{\longrightarrow}}K_{}^{G_{j}}(\overline{{\cal C}}^{1}_{\eta})\stackrel{{\scriptstyle T_{l}(\sigma_{2})\cdot K^{G_{j}}_{}(\theta_{\tau})}}{{\longrightarrow}}K_{}^{G_{j}}(\overline{{\cal C}}^{2}_{\eta})\stackrel{{\scriptstyle T_{l}(\sigma_{3})\cdot K^{G_{j}}_{}(\theta_{\tau})}}{{\longrightarrow}}\cdots$ (7–14) where $\sigma_{s}$ is the $i$-dimensional representation of $G$ on the basis elements corresponding to the subset $S_{j}$ on the affine space $\mathbb{A}_{s}$. It follows that it is isomorphic to the colimit of the system $K_{}^{G_{j}}(\overline{{\cal C}}_{\eta})\stackrel{{\scriptstyle T_{l}(\sigma_{1})}}{{\longrightarrow}}K_{}^{G_{j}}(\overline{{\cal C}}_{\eta})\stackrel{{\scriptstyle T_{l}(\sigma_{2})}}{{\longrightarrow}}K_{}^{G_{j}}(\overline{{\cal C}}_{\eta})\stackrel{{\scriptstyle T_{l}(\sigma_{3})}}{{\longrightarrow}}K_{}^{G_{j}}(\overline{{\cal C}}_{\eta})\stackrel{{\scriptstyle T_{l}(\sigma_{4})}}{{\longrightarrow}}\cdots$ (7–15) 2. 2. Let $\overline{G}_{j}$ denote the image of $G_{j}$ under $\eta$. Then all of the elements $T_{l}(\sigma_{s})\in R[G_{j}]$ are in the image of the ring homomorphism $R[\overline{G}_{j}]\rightarrow R[G_{j}]$. Proof: For part (1), we observe that the colimit (7–14) above can be written as a two parameter colimit, of the following form. $\begin{diagram}$ This colimit may be computed in iterative fashion, first in the vertical direction and then in the horizontal. The diagram (7–15) above is obtained from this two parameter diagram by performing the vertical colimits. For part (2) above, it is clear that the representations $\sigma_{s}$ all factor through $\overline{G}_{j}$ by definition. $\square$ ## 8 Completed equivariant $K$-theory of ${\cal T}_{\eta}$ We suppose that we are given an $l$-BAR $\eta$, and construct the spectra ${\cal A}_{\eta}$ and ${\cal T}_{\eta}$. Also let $W$ be a Noetherian approximable continuous affine $G$-scheme, as in the previous section. We wish to prove that the map $\theta:\mathfrak{K}^{G}(k)\cong\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)$ induces an equivalence on derived completions at the homomorphism $\epsilon:\mathfrak{K}^{G}(k)\rightarrow\mathbb{H}_{l}$ where $\mathbb{H}_{l}$ denotes the mod $l$ Eilenberg spectrum. In order to do this, it will suffice to prove that the map $\mathbb{H}_{l}\cong\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}\xrightarrow{\theta\wedge id}\mathfrak{K}^{G}({\cal T}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}$ is an equivalence of spectra by Proposition 2.20. This analysis will require the filtrations discussed in Section 7. The computations there were exclusively using the weak ring spectra and modules constructed from Waldhausen’s $S_{.}$-construction. In order to apply those results, we have to construct the $\mathfrak{K}^{G}$-counterparts to the spectra $K^{G}(\mbox{Mod}_{i}(-))$ and $K^{G}(\mbox{Mod}^{G}_{i}(-)/\mbox{Mod}^{G}_{i+1}(-))$. To begin, we suppose that we are given a monomial representation $l$-BAR of $G$ , and the corresponding actions of $G$ on ${\cal A}_{\eta}^{s}$ and ${\cal T}_{\eta}^{s}$. In Section 7, we defined the abelian subcategories $\mbox{Mod}^{G}_{i}({\cal A}_{\eta}\times W)$ of the category $\mbox{Mod}^{G}({\cal A}_{\eta}\times W)$. We now define abelian categories $Q_{i}$ to be the quotient abelian categories $\mbox{Mod}^{G}({\cal A}_{\eta}\times W)/\mbox{Mod}^{G}_{i}({\cal A}_{\eta}\times W)$. Each of the categories $Q_{i}$ admits a strictly associative and coherently commutative tensor produce, and so is therefore suitable input to the Elmendorf-Mandell machine. Consequently, we may construct commutative $\mathfrak{K}^{G}(k)$-algebras $\mathfrak{K}^{G}(Q_{i})$, and we have natural homomorphisms $\mathfrak{K}^{G}(Q_{i})\rightarrow\mathfrak{K}^{G}(Q_{i-1})$ of such algebras. We now define the module $\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W)$ over $\mathfrak{K}^{G}(k)$ to be the homotopy fiber of the map $\mathfrak{K}^{G}(k)({\cal A}_{\eta}\times W)\rightarrow\mathfrak{K}^{G}(Q_{i})$ The localization theorem in Waldhausen theory shows that $\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W)$ has the same homotopy type as $K(\mbox{Mod}_{i}^{G}({\cal A}_{\eta}\times W))$ as a weak module spectrum over $K^{G}({\cal A}_{\eta}\times W)$. We now have natural homomorphisms of $\mathfrak{K}^{G}(k)$-modules $\mathfrak{M}^{G}_{i+1}({\cal A}_{\eta}\times W)\rightarrow\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W)$, and we can construct their cofibers $\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W)/\mathfrak{M}^{G}_{i+1}({\cal A}_{\eta}\times W)$ to obtain $\mathfrak{K}^{G}$ counterparts to the subquotients $K^{G}(\mbox{Mod}_{i}^{G}({\cal A}_{\eta}\times W)/\mbox{Mod}_{i+1}^{G}({\cal A}_{\eta}\times W))$. We also want to construct $\mathfrak{K}^{G}$-counterparts $\mathfrak{M}_{i}^{G}({\cal A}_{\eta}\times W;\mathfrak{o})$ to the spectra $K^{G}(\mbox{Mod}^{G}_{i}({\cal A}_{\eta}\times W;\mathfrak{o}))$ where $\mathfrak{o}$ is any orbit under the action of $G$ on the collection of subsets of cardinality $i$ of $\\{1,\ldots,n\\}$. In order to do this, we proceed as above by letting $Q_{i}(\mathfrak{o})$ denote the quotient abelian category $\mbox{Mod}^{G}({\cal A}_{\eta}\times W)/\mbox{Mod}^{G}_{i}({\cal A}_{\eta}\times W;\mathfrak{o})$, and defining $\mathfrak{M}_{i}^{G}({\cal A}_{\eta}\times W;\mathfrak{o})$ to be the homotopy fiber of the map $\mathfrak{K}^{G}(k)\rightarrow\mathfrak{K}^{G}(Q_{i}(\mathfrak{o}))$. We may apply the $K^{G}$ case to conclude that there is a decomposition $\mathfrak{M}_{i}^{G}({\cal A}_{\eta}\times W)/\mathfrak{M}_{i+1}^{G}({\cal A}_{\eta}\times W)\cong\bigvee_{\mathfrak{o}}\mathfrak{M}_{i}^{G}({\cal A}_{\eta}\times W;\mathfrak{o})/\mathfrak{M}^{G}_{i+1}({\cal A}_{\eta}\times W)$ (8–16) We now have the following Proposition concerning these modules. ###### Proposition 8.1 There are natural maps $\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\rightarrow Q_{\mbox{dim}(\eta)+1}\rightarrow Q_{1}\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)$. 1. 1. The map $\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\rightarrow Q_{\mbox{dim}(\eta)+1}$ is an equivalence of spectra, so we have $\mathfrak{M}^{G}_{\dim(\eta)+1}({\cal A}_{\eta}\times W)\simeq$. 2. 2. The map $Q_{1}\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)$ is an equivalence of spectra, so $\mathfrak{M}_{1}^{G}({\cal A}_{\eta}\times W)$ is equivalent to the homotopy fiber of the map $\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)$. Proof: For item (1), we note that the category $\mbox{Mod}^{G}_{\mbox{dim}(\eta)+1}({\cal A}_{\eta}\times W)$ consists only of the zero module, from which (1) follows. For (2), $Q_{0}$ consists of all modules on ${\cal A}_{\eta}\times W$ for which the support consists of coordinate subspaces, including the zero subspace. The localization is equivalent to the category of free modules over ${\cal T}_{\eta}\times W$. $\square$ ###### Proposition 8.2 In order to prove that the map $\mathbb{H}_{l}\cong\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}\xrightarrow{\theta\wedge id}\mathfrak{K}^{G}({\cal T}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}$ is an equivalence of spectra, it will suffice to prove that $\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W;\mathfrak{o})/\mathfrak{M}_{i+1}^{G}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}\cong$ for all $i=1,\ldots,n$ and all orbits $\mathfrak{o}$ of the action of $G$ on the subsets of cardinality $i$ of $\\{1,\ldots,n\\}$. Proof: Since by Proposition 8.1, $\mathfrak{M}^{G}_{1}({\cal A}_{\eta}\times W)$ is equivalent to the homotopy fiber of the map $\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)$ it will clearly suffice to prove that $\mathfrak{M}^{G}_{1}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}\simeq$ But we have a diagram of module spectra $\simeq\mathfrak{M}^{G}_{dim(\eta)+1}({\cal A}_{\eta}\times W)\rightarrow\mathfrak{M}^{G}_{dim(\eta)}({\cal A}_{\eta}\times W)\rightarrow\cdots\rightarrow\mathfrak{M}^{G}_{2}({\cal A}_{\eta}\times W)\rightarrow\mathfrak{M}^{G}_{1}({\cal A}_{\eta}\times W)$ from which it is clear that it suffices to prove that $\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W)/\mathfrak{M}^{G}_{i+1}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}\simeq$ for $i=1,2,\ldots,dim(\eta)$. The result now follows easily from the decomposition (8–16) above. $\square$ ###### Corollary 8.1 To prove that $\mathbb{H}_{l}\cong\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}\xrightarrow{\theta\wedge id}\mathfrak{K}^{G}({\cal T}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l}$ is an equivalence, it will suffice to prove that the groups $Tor^{R[G]}_{j}(\pi_{k}(\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W;\mathfrak{o})/\mathfrak{M}_{i+1}^{G}({\cal A}_{\eta}\times W)),\mathbb{F}_{l})$ vanish for all $i=1,\ldots,n$, and all $j,k$, and $\mathfrak{o}$. Proof: Follows immediately from Proposition 2.21. $\square$ Let $\eta$ denote an $n$-dimensional $l$-BAR of $G$, and consider an orbit $\mathfrak{o}$ under the action of $G$ on the subsets of order $i$ of $\\{1,\ldots,n\\}$. Let $H$ be the stabilizer of one orbit representative in $\mathfrak{o}$. We now have a commutative diagram $\begin{diagram}$ (8–17) and corresponding diagram of representation rings $\begin{diagram}$ (8–18) where the quotients $Q_{G}$ and $Q_{H}$ are the images of the composites $G\stackrel{{\scriptstyle k}}{{\longrightarrow}}\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n}\rightarrow\Sigma_{n}$ and $H\stackrel{{\scriptstyle j}}{{\longrightarrow}}\Sigma_{i}\times\Sigma_{n-i}\ltimes\mathbb{Z}_{l}^{n}\rightarrow\Sigma_{i}\times\Sigma_{n-i}$ The right hand horizontal maps are isomorphisms since the groups $Q_{G}$ and $Q_{H}$ are finite $l$-groups, by Proposition 6.3. We draw the following consequence of Proposition 7.2. ###### Proposition 8.3 The $R[G]$ -module $\pi_{k}(\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W;\mathfrak{o})/\mathfrak{M}_{i+1}^{G}({\cal A}_{\eta}\times W))$ is of the form $M\stackrel{{\scriptstyle\times r_{0}}}{{\longrightarrow}}M\stackrel{{\scriptstyle\times r_{1}}}{{\longrightarrow}}M\stackrel{{\scriptstyle\times r_{2}}}{{\longrightarrow}}M\stackrel{{\scriptstyle\times r_{3}}}{{\longrightarrow}}\cdots$ where 1. 1. $M$ is a module over $R[H]$. 2. 2. Each $r_{j}$ is a representation of dimension divisible by $l$ 3. 3. Each $r_{j}$ is the pullback along $j$ of a representation $\rho_{i}$ of $\Sigma_{i}\times\Sigma_{n-i}\ltimes\mathbb{Z}_{l}^{n}$. Proof: The first follows from Part (1) of Corollary 7.2, with $M=K^{G_{j}}_{i}(\overline{\cal C}_{\eta})$. The second follows because the construction $T_{l}$ used in Proposition 7.2 always yields dimension a multiple of $l$. The last statement follows since the representations $\sigma_{s}$ are pulled back along a homomorphism to $\Sigma_{i}\ltimes\mathbb{Z}_{l}^{i}$, which is a subgroup of $\Sigma_{i}\times\Sigma_{n-i}\ltimes\mathbb{Z}_{l}^{n}$. $\square$ ###### Corollary 8.2 There exists an $M$ so that for all $i$, the product $r_{i}\cdot r_{i+1}\cdot\cdots\cdot r_{i+M-1}\cdot r_{i+M}$ lies in the submodule $(l)+I_{G}\cdot R[H]$. Proof: We note that each $\rho_{i}$ has dimension divisible by $l$, and it readily follows that $\rho_{i}\in(l)+I_{\Sigma_{i}\times\Sigma_{n-i}\ltimes\mathbb{Z}_{l}^{n}}$ It follows from Corollary 6.4 that there is an $M$ so that $\rho_{i}\cdot\rho_{i+1}\cdot\cdots\cdot\rho_{i+M-1}\cdot\rho_{i+M}\in((l)+I_{\Sigma_{n}\ltimes\mathbb{Z}_{l}^{n}}+((l)+I_{\Sigma_{i}\times\Sigma_{n-i}})^{\infty})R[\Sigma_{i}\times\Sigma_{n-i}\ltimes\mathbb{Z}_{l}^{n}]$ Referring to the diagram 8–18 above, we have $j^{}(\rho_{i}\cdot\rho_{i+1}\cdot\cdots\cdot\rho_{i+M-1}\cdot\rho_{i+M})=r_{i}\cdot r_{i+1}\cdot\cdots\cdot r_{i+M-1}\cdot r_{i+M}$ and therefore that $r_{i}\cdot r_{i+1}\cdot\cdots r_{i+M-1}\cdot r_{i+M}\in((l)+I_{G}+((l)+I_{Q_{H}})^{\infty})R[H]$ But by Proposition 6.3, Part (5), the ideal $((l)+I_{Q_{H}})^{\infty}$ is equal to the zero ideal, since $Q_{H}$ an $l$-group. It now follows that $r_{i}\cdot r_{i+1}\cdot\cdots\cdot r_{i+M-1}\cdot r_{i+M}\in(l)+I_{G}R[H]$. $\square$ ###### Corollary 8.3 The groups $Tor^{R[G]}_{t}(\pi_{k}(\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W;\mathfrak{o})/\mathfrak{M}_{i+1}^{G}({\cal A}_{\eta}\times W)),\mathbb{F}_{l})$ vanish for all $t$. Proof: Multiplication by an element of $(l)+I_{G}$ annihilates $Tor^{R[G]}_{i}(M,\mathbb{F}_{l})$. $\square$ This corollary now yields the following. ###### Proposition 8.4 The groups $\pi_{}(\mathfrak{M}^{G}_{i}({\cal A}_{\eta}\times W;\mathfrak{o})/\mathfrak{M}_{i+1}^{G}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{\mathfrak{K}^{G}(k)}}\mathbb{H}_{l})$ are all trivial. Proof: This is an immediate consequence of Proposition 2.21. $\square$ ###### Corollary 8.4 The map $\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)$ induces an equivalence $\mathfrak{K}^{G}(W)\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}\rightarrow\mathfrak{K}^{G}({\cal A}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}$ Proof: Immediate from Proposition 7.4, and the decomposition 7–11. $\square$ ## 9 The main theorem We are now in position to prove our main theorem. ###### Proposition 9.1 Let $G$ be a totally torsion free profinite $l$-group, and let $W$ be any Noetherian approximable continuous affine $G$-scheme. Then the natural map of $\mathfrak{K}^{G}(k)$-module spectra $\mathfrak{K}^{G}(W)\rightarrow\mathfrak{K}^{G}({\cal E}G\times W)$ induces an equivalence $\mathfrak{K}^{G}(W)\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}\rightarrow\mathfrak{K}^{G}({\cal E}G\times W)\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}$ Proof: Since ${\cal E}G$ is an infinite product of continuous affine $G$-schemes of the form ${\cal T}_{\eta}$, the equivariant $\mathfrak{K}$-theory of ${\cal E}G\times W$ can be expressed as a colimit of spectra of the form $\mathfrak{K}^{G}({\cal T}_{\eta}\times W)$. Corollary 8.4 asserts that the map $\mathfrak{K}^{G}(W)\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}\rightarrow\mathfrak{K}^{G}({\cal T}_{\eta}\times W)\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}$ is an equivalence. The construction $-\mathrel{\mathop{\wedge}\limits_{K^{G}(k)}}\mathbb{H}_{l}$ commutes with colimits, and the result therefore follows. $\square$ ###### Theorem 9.1 Let $G$ and $W$ be as above, then there are equivalences of spectra $\mathfrak{K}^{G}(W)^{\wedge}_{\epsilon}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{K}^{G}({\cal E}G\times W)^{\wedge}_{\epsilon}\cong\mathfrak{K}({\cal E}G\mathrel{\mathop{\times}\limits_{G}}W)^{\wedge}_{l}$ well defined up to homotopy, and natural in $W$. Proof: Follows directly from Proposition 2.21, Corollary 5.3 and Proposition 2.16. $\square$ ###### Corollary 9.1 Let $G=G_{F}$ denote the absolute Galois group of a field $F$ containing $k$, so that $G_{F}$ is a pro-$l$ group and so that $l$ is prime to the characteristic of $k$. Let $\overline{F}$ denote the algebraic closure of $G$, equipped with the defining $G_{F}$-action. Then there are natural equivalences of spectra $\mathfrak{K}^{G}(k)^{\wedge}_{\epsilon}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{K}^{G}({\cal E}G)^{\wedge}_{\epsilon}\cong\mathfrak{K}({\cal E}G/G)^{\wedge}_{l}$ and spectra $\mathfrak{K}(F)^{\wedge}_{l}\cong\mathfrak{K}^{G}(\overline{F})^{\wedge}_{\epsilon}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{K}^{G}({\cal E}G\times Spec(\overline{F}))^{\wedge}_{\epsilon}\cong\mathfrak{K}({\cal E}G\mathrel{\mathop{\times}\limits_{G}}Spec(\overline{F}))^{\wedge}_{l}$ Proof: We simply apply Theorem 9.1 in the cases $W=Spec(k)$ and $W=Spec(\overline{F})$. The left hand equivalence in the case $W=Spec(\overline{F})$ is proved in Corollary 5.4. $\square$ ## References [1] M. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Ètudes Sci. Publ. Math. No. 9 1961 23–64. * [2] M. Atiyah and G. Segal, Equivariant $K$-theory and completion, J. Differential Geometry 3 1969 1–18. * [3] L. Auslander and M. 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Waterhouse, Introduction to Affine Group Schemes, Springer graduate texts in mathematics, vol. 2066, New York, 1979, ISBN 9780387904214.	arxiv-papers	2013-09-22T21:00:06	2024-09-04T02:49:51.322236	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Gunnar Carlsson", "submitter": "Gunnar Carlsson", "url": "https://arxiv.org/abs/1309.5649" }
1309.5699	# Distribution of Entropy of Bardeen Regular Black Hole with Corrected State Density Hai Huang Juhua Chen [email protected] Yongjiu Wang College of Physics and Information Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China ###### Abstract We consider corrections to all orders in the Planck length on the quantum state density, and calculate the statistical entropy of the scalar field on the background of the Bardeen regular black hole numerically. We obtain the distribution of entropy which is inside the horizon of black hole and the contribution of the vicinity of horizon takes a great part of the whole entropy. Keywords: entropy, Bardeen regular black hole ###### pacs: 04.70.Dy, 04.62.+v, 97.60.Lf In 1970’s, Bekenstein and Hawking found that the black hole(BH) entropy was proportional to the area of the event horizon by comparing BH physics with thermodynamics1 .’t Hooft statistically obtained the Bekenstein-Hawking entropy of a scalar field outside the horizon of the Schwarzschild BH by brick-wall model2 . The Bekenstein-Hawking entropy was identified with the statistical-mechanical entropy arising from a thermal bath of quantum fields propagating outside the event horizon,but there are some drawbacks like ultraviolet divergence near the horizon. An improved brick-wall method has been introduced by taking the thin-layer outside the event horizon of a BH as the integral region, but the artificial cutoffs is still unsolved3 . It’s commonly believes that there exists a minimal length which originates due to the quantum fluctuation of the gravitational field motivated by the generalized uncertainty principle (GUP) 4 ; 5 , the entropy integral of a radial component in the range near the horizon should be treated for a convergent entropy6 . The Feynman propagator displays an exponential ultraviolet cutoff of the form exp($-\lambda p^{2}$), where the parameter $\lambda$ actually plays the role of a minimal length7 . Quantum gravity phenomenology has been tackled with effective models based on GUPs and/or modified dispersion relations containing a minimal length as a natural ultraviolet cutoff 7.1 ; 7.2 ; 7.3 . At the quantum mechanical level, the essence of the ultraviolet finiteness of the Feynman propagator can be captured by a nonlinear relation $p=f(k)$, where $p$ and $k$ are the momentum and the wave vector of the particle, respectively. The commutator between the operators $x_{i}$ and $p_{j}$ is generalized to $[x_{i},p_{j}]=i\frac{\partial p_{i}}{\partial k_{j}}$; moreover, the usual momentum measure in $3+1$-dimensional spacetime $d^{3}p$ is modified to $d^{3}p\ \textrm{det}\left\|\frac{\partial k_{i}}{\partial p_{j}}\right\|=d^{3}p\prod_{i}\left\|\frac{\partial k_{i}}{\partial p_{j}}\right\|,$ (1) where $\frac{\partial k_{i}}{\partial p_{j}}=\delta_{ij}e^{\lambda p_{i}^{2}}$8 ; 9 ; 10 , the number of quantum states in a volume element in phase cell space based on the GUP is $dg_{+}(\omega)=\frac{d^{3}xd^{3}p}{(2\pi\hbar)^{3}}e^{-\lambda p^{2}},$ (2) where $p^{2}=p_{i}p^{i}$ is the square of momentum. Many authors calculated the entropy of BHs to the leading order in the Planck length by using this newly modified equation of states of density11 ; 12 ; 13 ; 14 . In their works, the entropy were concentrated on a small vicinity of the BH horizon, we do not know what’s the contribution of the interior of BH to the entropy. In this letter, we devote to consider the contribution of the whole interior of BH by numerical method. We study the entropy of a scalar field on a Bardeen regular black hole(BRBH) backgrounds. Contrary to our general expectation, this study of a regular case is not so trivial in contrast to the previous result of the non-regular case. By using the equation of states of density motivated by GUP in quantum gravity, we calculate the quantum entropy of a massive scalar field numerically. We obtain the desired Bekenstein-Hawking entropy through the whole event horizon and little mass approximation satisfying the asymptotic property of the wave vector $k$ in the modified dispersion relation. We take the units as $G=\hbar=c=k_{B}\equiv 1$. In the context of gravity coupled to some form of matter, regular solutions are obtained from a prototypical action of the form $S_{act}=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}(R-\pounds),$ (3) where $g$ is the determinant of the metric $g_{\mu\nu}$, $R$ is the scalar curvature, and $\pounds$ represents the Lagrangian of the matter fields. In the context of a specific nonlinear electrodynamics with Lagrangian given by $\pounds=(6m/(2\|\alpha\|\alpha^{2}))(\sqrt{2\alpha^{2}F}/(1+\sqrt{2\alpha^{2}F}))^{5/2}$, the BRBH line element takes the form $ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d\varphi^{2}),$ (4) note that the function $f(r)$ takes a particularly simple form $f(r)=1-\frac{2mr^{2}}{(r^{2}+\alpha^{2})^{3/2}},$ (5) where $(t,r,\theta,\phi)$ are the usual space-time spherical coordinates , $m$ and $\alpha$ stand for the mass and the monopole charge of a self-gravitating magnetic field of a non-linear electrodynamics source, respectively(seeing Fig.1). Depending on the value of $\alpha$, the above regular solution may present two distinct, one,or no horizons. It reduces to the Schwarzschild solution for $\alpha=0$. Figure 1: The figure illustrates the behavior of the metric function $f(r)$ given in (4) for different values of the parameter $\alpha$. In this BRBH background, let us consider massless scalar field which satisfies the Klein-Gordon equation $\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi)=0.$ (6) Substituting the line element(4) and using the leading order Wenzel-Kramers- Brillouin(WKB) approximation $\Phi=e^{-i\omega t}e^{iS(r,\theta,\phi)}$, Eq.(4) becomes $\frac{\omega^{2}}{f(r)}-f(r)\left(\frac{\partial S}{\partial r}\right)^{2}-\frac{1}{r^{2}}\left(\frac{\partial S}{\partial\theta}\right)^{2}-\frac{1}{r^{2}sin^{2}\theta}\left(\frac{\partial S}{\partial\varphi}\right)^{2}=0.$ (7) We have $p_{r}^{2}=\frac{1}{f(r)}\left(\frac{\omega^{2}}{f(r)}-\frac{1}{r^{2}}p_{\theta}^{2}-\frac{1}{r^{2}sin^{2}\theta}p_{\varphi}^{2}\right),$ (8) where $p_{r}=\frac{\partial S}{\partial r}$, $p_{\theta}=\frac{\partial S}{\partial\theta}$ and $p_{\varphi}=\frac{\partial S}{\partial\varphi}$, we also obtain the square module momentum $p^{2}=p_{i}p^{i}=g^{rr}p_{r}^{2}+g^{\theta\theta}p_{\theta}^{2}+g^{\varphi\varphi}p_{\varphi}^{2}=\frac{\omega^{2}}{f(r)}.$ (9) From Eq.(2), the number of quantum states with the energy is less than $\omega$ is given by $\displaystyle g_{+}(\omega)$ $\displaystyle=$ $\displaystyle\frac{1}{(2\pi)^{3}}\int e^{-\lambda p^{2}}drd\theta d\varphi dp_{r}dp_{\theta}dp_{\varphi}$ (10) $\displaystyle=$ $\displaystyle\frac{2}{(2\pi)^{3}}\int e^{-\lambda p^{2}}drd\theta d\varphi\int\frac{1}{\sqrt{f(r)}}\left(\frac{\omega^{2}}{f(r)}-\frac{1}{r^{2}}p_{\theta}^{2}-\frac{1}{r^{2}sin^{2}\theta}p_{\varphi}^{2}\right)^{\frac{1}{2}}dp_{\theta}dp_{\varphi}$ $\displaystyle=$ $\displaystyle\frac{2}{3\pi}\int\frac{r^{2}\omega^{3}}{f^{2}(r)}e^{-\lambda p^{2}}dr.$ By using Eq.(10), the free energy can be rewritten as $\displaystyle F_{+}$ $\displaystyle\approx$ $\displaystyle\frac{1}{\beta}\int dg_{+}(\omega)\ln(1-e^{-\beta\omega})$ (11) $\displaystyle=$ $\displaystyle-\int\frac{g_{+}(\omega)d\omega}{e^{\beta\omega}-1}$ $\displaystyle=$ $\displaystyle-\frac{2}{3\pi}\int_{0}^{r_{H}}e^{-\frac{\lambda\omega^{2}}{f(r)}}\frac{r^{2}}{f^{2}(r)}dr\int_{0}^{\infty}\frac{\omega^{3}d\omega}{e^{\beta\omega}-1}$ Here, we have taken the continuum limit of quantum numbers and integrated it. Furthermore, the contribution from the vacuum surrounding the system is ignored, we are not only concerned about the contribution from the just vicinity near the horizon, but also the whole region within the horizon $(0\rightarrow r_{H})$. We can obtain the entropy as follows: $\displaystyle S_{+}$ $\displaystyle=$ $\displaystyle\beta^{2}\frac{\partial F_{+}}{\partial\beta}$ (12) $\displaystyle=$ $\displaystyle\frac{2\beta^{2}}{3\pi}\int_{0}^{r_{H}}e^{-\frac{\lambda\omega^{2}}{f^{2}(r)}}\frac{r^{2}}{f^{2}(r)}dr\int_{0}^{\infty}\frac{e^{\beta\omega}\omega^{4}d\omega}{(e^{\beta\omega}-1)^{2}}$ Figure 2: The figure illustrates the behavior of entropy of massless scalar wave from BRBH with $\lambda$ from $-0.1$ to $0$ at inverse Hawking temperature $\beta=25.228$, $\alpha=0.1$, $k_{B}=1$. The bottom picture shows the entropy together at different Hawking temperature for $\lambda=-0.01$ ,$-0.03$, $-0.05$, respectively. Figure 3: The figure illustrates the behavior of entropy of massless scalar wave from BRBH with $\lambda$ from $-0.03$ to $0$ at inverse Hawking temperature $\beta=25.530$, $\alpha=0.2$, $k_{B}=1$. The bottom picture shows the entropy together at different Hawking temperature for $\lambda=-0.01$ ,$-0.03$, $-0.05$, respectively. Figure 4: The figure illustrates the behavior of entropy of massless scalar wave from BRBH with $\lambda$ from $-0.1$ to $0$ at inverse Hawking temperature $\beta=25.133$, $\alpha=0$, $k_{B}=1$. The bottom picture shows the entropy together at different Hawking temperature for $\lambda=-0.01$ ,$-0.03$, $-0.05$, respectively. Figs.2-4 illustrate the behavior of the BRBH entropy with respect to the monopole charge $\alpha=0,0.1,0.2$ and parameter $\lambda$, inverse temperature. We assume $\lambda=\lambda l_{p}^{2}$, where $l_{p}$ is the Planck length, and in the system of the Planck units $l_{p}=1$. At the same temperature, the larger the value of $\lambda$, the larger the corresponding value of entropy is. When $\lambda$ tend to zero, the entropy increase rapidly. The value of $\lambda$ corresponding to actual entropy is approximately -0.02. When we used fixed value of $\lambda$, we find the parameter $\lambda$ makes the entropy lower. In order to inspect the contribution of different parts of radius inside the external horizon, we maintain the entropy as $A/4$, and the radius is divided into 100 parts and integral. In Fig.5, we plot the integral of different parts for $\alpha=0$, we find the integral increases rapidly when the interval tends to external horizon. When the interval in the vicinity of zero, it makes little contribution to the entropy. We show the value of integral for different $\alpha$ in Table 1, we can see the last parts of integral constitute the main part of entropy, $\lambda$ approximately equivalent for different values of temperature. In Fig.6, we show the integral in logarithm forms to inspect more clearly. When $\alpha=0$(Schwarzschild case), there is no inner horizon, the entropy increases as integral parts tend to the external horizon, when $\alpha\neq 0$, there are two extremum. It’s obvious that the small vicinity of horizon radius show a great contribution than the parts around horizon radius and the contribution of the vicinity of external horizon almost takes the whole parts of the entropy. By comparing these curves, we can see that, the monopole charge $\alpha$ makes the entropy weaker and its distribution width narrower between two horizon. The inner horizon entropy increases while the external horizon entropy decrease. Here we adopted the proper value of $\lambda$, and no cut-off is adopted. In Fig.7, we try to discuss the situation outside horizon, it shows the contribution of vacuum outside the horizon. We plot the integral of entropy from external horizon $r=r_{H}$ to radius $r=100r_{H}$ for $\alpha=0$(Schwarzschild case),$\lambda=0.0175372$. When $r<10r_{H}$, the entropy maintains basically stable, when $r>20r_{H}$, the entropy increase as exponential form. This property indicates that if we make a cut-off between $r_{H}$ and $r_{H}+\varepsilon$($\varepsilon$ is a small quantity), we can get the Bekenstein-Hawking entropy. This agrees with the traditional brick-wall model12 and thin-layer model13 . In Fig.8, if we do not maintain $\lambda$ as a constant, the integral of entropy from $r_{H}$ to different upper limit can be adjusted to get the same Bekenstein-Hawking entropy. Figure 5: The integral of entropy of first 90 parts($0\rightarrow\frac{90}{100}r_{H}$) when $r_{H}$ is divided into 100 parts. The last 10 parts is showed in the table below. Table 1: Entropy Integral between Different Parts of Radius. \| $\alpha=0$ \| $\alpha=0.1$ \| $\alpha=0.2$ \| $\alpha=0.3$ ---\|---\|---\|---\|--- $\lambda$ \| -0.0175372 \| -0.0175376 \| -0.0174500 \| -0.0174500 $\beta$ \| 25.1327412 \| 25.2282438 \| 25.5307184 \| 26.0959399 $S_{+}$ \| 12.5664 \| 12.4719 \| 12.1855 \| 11.6975 91 \| 0.00205671 \| 0.00204622 \| 0.00201396 \| 0.00195699 92 \| 0.00268403 \| 0.00266955 \| 0.00262509 \| 0.00254671 93 \| 0.00360017 \| 0.00357973 \| 0.00351704 \| 0.00340665 94 \| 0.00500378 \| 0.00497400 \| 0.00488275 \| 0.00472224 95 \| 0.00729502 \| 0.00724965 \| 0.00711080 \| 0.00686681 96 \| 0.01137950 \| 0.01130580 \| 0.01108040 \| 0.01068470 97 \| 0.01967990 \| 0.01954740 \| 0.01914320 \| 0.01843340 98 \| 0.04065840 \| 0.04037460 \| 0.03951080 \| 0.03799340 99 \| 0.12414800 \| 0.12325100 \| 0.12053600 \| 0.11575000 100 \| 12.2699000 \| 12.1775000 \| 11.9590000 \| 11.481000 Figure 6: The logarithm of the integral of entropy for different interval within the external horizon $r_{H}$ for $\alpha=0$(Schwarzschild case),$0.1$, $0.2$, $0.3$ , respectively. Figure 7: The integral of entropy from external horizon $r=r_{H}$ to radius $r=100r_{H}$ for $\alpha=0$(Schwarzschild case),$\lambda=0.0175372$. It shows the contribution of vacuum outside the horizon. Figure 8: The integral upper limit corresponding to different value of $\lambda$ when entropy is maintain as $A/4$. In conclusion, we have investigated the entropy of scalar field on a BRBH backgrounds using analytic and numerical methods. We found the monopole charge $\alpha$ makes the entropy weaker, and the distribution width of entropy between two horizons become narrower. We gave the distribution of the entropy in the interior of BH, we also found that the contribution of the vicinity of external horizon almost taken the whole parts of entropy. This calculation did not need to introduce any cut-off while considered the exterior region of horizon we need to cut out a small region to get the Bekenstein-Hawking entropy, this agreed with the previous works list in the introduction. This project is supported by the National Natural Science Foundation of China under Grant No.10873004, the State Key Development Program for Basic Research Program of China under Grant No.2010CB832803 and the Program for Changjiang Scholars and Innovative Research Team in University, No. IRT0964. ## References * (1) J. D. Bekenstein, Lett. Nuovo Cimento 4 737(1972); J. D. Bekenstein, Phys. Rev. D 7 2333(1973); J. D. Bekenstein, Phys. Rev.D 9 3292(1974); S. W. Hawking, Commun. Math. Phys. 43 199(1975). * (2) G. ’t Hooft, Nucl. Phys. B256, 727 (1985). * (3) X. Li and Z. Zhao, Mod. Phys. Letts. A 15 1739(2000); X. Li and Z. Zhao, Int. J. Theor. Phys. 40 90(2001); T. P. Song, C. X. Hou and W. L. Shi, Acta Phys. Sin. 51 06(2002). * (4) D. J. Gross and P. F. Menda Nucl. Phys. B 303 407(1988). * (5) Y. W. Kim and Y. J. Park, Phys. Rev. D 77 067501(2008). * (6) M. Yoon, J. Ha and W. Kim, Phys. Rev. D 76 047501(2007). * (7) K. Nouicer, Phys. Lett. B 646, 63(2007). * (8) G. Amelino-Camelia, M. Arzano, Y. Ling, and G.Mandanici, Classical Quantum Gravity 23, 2585 (2006) * (9) S. Hossenfelder, Phys. Rev. D 73, 105013 (2006); S. Hossenfelder,Class. Quantum Grav. 23, 1815 (2006). * (10) M. Fontanini, E. Spallucci, and T. Padmanabhan, Phys.Lett. B 633, 627 (2006). * (11) L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi,Phys. Rev. D 65, 125028(2002). * (12) Y.W. Kim, Phys. Rev. D 77, 067501(2008). * (13) X. Li, Phys. Lett. B 540, 9(2002). * (14) X. Sun and W. Liu, Mod. Phys. Lett. A 19 677(2004). * (15) R. Zhao, Y. Q. Wu and L. C. Zhang Class. Quantum Grav. 20 4885(2003) C. Liu, X. Li and Z. Zhao, Gen. Rel. Grav. 36 1135(2004); C. Z. Liu, Int. J. Theor. Phys. 44 567(2005). * (16) W. Kim, Y. W. Kim and Y. J. Park Phys. Rev. D 74 104001(2006); W. Kim, Y. W. Kim and Y. J. Park, Phys. Rev. D 75 127501(2007) * (17) W. B. Liu, Chin. Phys. Lett. 20 440(2003)	arxiv-papers	2013-09-23T06:08:21	2024-09-04T02:49:51.346276	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hai Huang, Juhua Chen, Yongjiu Wang", "submitter": "Juhua Chen", "url": "https://arxiv.org/abs/1309.5699" }
1309.5717	# Ligand-induced stabilization of the aptamer terminal helix in the add adenine riboswitch Francesco Di Palma 1, Francesco Colizzi 1 and Giovanni Bussi 1∗ (SISSA - Scuola Internazionale Superiore di Studi Avanzati, via Bonomea 265, 34136 Trieste, Italy) Stabilization of the P1 stem in add riboswitch keywords: P1 stem, RNA aptamer, adenine riboswitch, molecular dynamics simulation, free energy calculation. 11footnotetext: To whom correspondence should be addressed. Email: [email protected]; Tel: +390403787407 ABSTRACT Riboswitches are structured mRNA elements that modulate gene expression. They undergo conformational changes triggered by highly-specific interactions with sensed metabolites. Among the structural rearrangements engaged by riboswitches, the forming and melting of the aptamer terminal helix, the so- called P1 stem, is essential for genetic control. The structural mechanisms by which this conformational change is modulated upon ligand binding mostly remain to be elucidated. Here we used pulling molecular dynamics simulations to study the thermodynamics of the P1 stem in the add adenine riboswitch. The P1 ligand-dependent stabilization was quantified in terms of free energy and compared with thermodynamic data. This comparison suggests a model for the aptamer folding in which direct P1-ligand interactions play a minor role on the conformational switch when compared with those related to the ligand- induced aptamer preorganization. ## INTRODUCTION Riboswitches are ligand-responsive regulatory elements located in untranslated regions of messenger RNAs (Serganov and Nudler 2013). They change their conformation in response to specific metabolite binding (Roth and Breaker 2009; Edwards and Batey 2010; Serganov and Patel 2012) and they have been proposed as modern descendants of an ancient sensory and regulatory system in the RNA world (Breaker 2012). Many pathogenic bacteria use riboswitches to control essential metabolic pathways and they are currently regarded as promising antibacterial drug targets (Blount and Breaker 2006; Mulhbacher et al. 2010; Deigan and Ferré-D’Amaré 2011). Riboswitches consist of an aptamer domain that binds the effector ligand and an expression platform that transduces the ligand-induced conformational “switch” into a modulation of gene expression (Barrick and Breaker 2007; Garst and Batey 2009). Among more than 20 natural aptamer classes (Breaker 2012), purine-sensing riboswitches have the peculiarity to recognize the targeted purine by utilizing a conserved pyrimidine (Kim and Breaker 2008; Batey 2012). One of the most characterized members of this class is the adenine sensing riboswitch (A-riboswitch) _cis_ -regulating the _add_ gene in _Vibrio vulnificus_ (Mandal and Breaker 2003). The ligand-bound structure of its aptamer is a three-way junction composed of three stems (P1, P2, P3) with the ligand completely encapsulated into the structure (Fig. 1) (Serganov et al. 2004; Mandal and Breaker 2004). The specificity for adenine is ensured by canonical Watson-Crick (WC) base pairing established between a uracil in conserved position and the ligand (Noeske et al. 2005; Gilbert et al. 2006). The A-riboswitch acts as a translational regulator (Serganov et al. 2004; Lemay et al. 2011). In the absence of adenine the ribosome binding site and the initiation codon, which are portions of the expression platform, are sequestered by pairing with a portion of the aptamer (OFF-state, Fig. 2B). The presence of adenine stabilizes an aptamer conformation in which the terminal P1 helix is well structured and both the regulatory sequences are available for ribosomal binding thus enabling mRNA translation (ON-state, Fig. 2A) (Rieder et al. 2007; Lee et al. 2010; Leipply and Draper 2011). The structural mechanism regulating the switch between the ON- and the OFF-state upon ligand binding mostly remains to be elucidated. The P1 stem is formed in the ON-state and it is disrupted in the OFF-state (Mandal and Breaker 2003; Serganov et al. 2004). It has been proposed that P1 is stabilized by the ligand (Batey et al. 2004) and that this could be a common feature in many riboswitch classes (Montange and Batey 2006). The role of ligand binding in the structural organization of the aptamer has been investigated with single-molecule spectroscopy providing an insightful overview on the folding dynamics (Neupane et al. 2011), yet lacking the critical atomistic details needed for an accurate structural characterization of the process as extensively discussed by Lin et al. (2012). Although _in silico_ techniques have been used to investigate the ligand role (Lin and Thirumalai 2008; Sharma et al. 2009; Priyakumar and MacKerell 2010; Gong et al. 2011; Allnér et al. 2013), a quantitative estimation of the energetic contributions associated to ligand binding, in particular regarding the role of direct P1-ligand interactions, has not yet been provided. In this context, state-of-the-art free-energy methods combined with atomistic simulations can bridge the gap providing an unparalleled perspective on the mechanism and dynamics of the biomolecular process of interest (Dellago and Bolhuis 2009). In this work we used steered molecular dynamics (SMD) (Grubmuller et al. 1996; Sotomayor and Schulten 2007) simulations to study the thermodynamics of the P1 stem formation in the presence and in the absence of the cognate ligand. We enforced the breaking of the P1 stem base pairs (bp) and then using a recently developed reweighting scheme (Colizzi and Bussi 2012) we quantitatively estimated the ligand-induced stabilization of the helix. The A9-U63 bp which directly stacks with adenine was used as a proxy for the P1 stability. Our non-equilibrium simulations provide measurements of the stability of the A9-U63 bp and quantify the direct ligand-dependent stabilization of the pairing. In the following our results are presented and compared with melting and single-molecule experiments. A structural model for the conformational switch emerging from the combination of our results and previous experimental data is also discussed. ## RESULTS We carried out the simulations of the aptamer domain of the _add_ A-riboswitch in different forms, namely the entire aptamer (PDB id 1Y26) (Serganov et al. 2004) has been simulated in the presence (Holo) and in the absence (Apo) of the cognate ligand, the adenine; additionally, to better estimate the ligand- induced stabilization, we also simulated a truncated aptamer ($\triangle$1-8/64-71), both in the Apo and Holo forms. Long unbiased molecular dynamics (MD) for all the four systems were performed to test the stability of the aptamer in different conditions. In the truncated systems the terminal bp was restrained in its initial configuration to mimic the presence of the rest of the stem. Furthermore, the full-length systems were pulled from the terminal bases to disrupt the entire P1 stem (Fig. 3) thus allowing its different stability between in the Holo and in the Apo forms to be qualitatively inferred. At last, to quantify this difference, SMD simulations of both the $\triangle$1-8/64-71 systems were done enforcing the breaking of the A9-U63 bp that directly stacks with the ligand (Fig. 4). The stability of both the Apo and Holo systems was evaluated monitoring the root mean square deviation (RMSD) from the native structure along 48 ns MD runs (Fig. 5A-B). Ligand removal (see Materials and Methods for details) did not affect the overall stability of the Apo aptamer in this time-scale, and secondary and tertiary structures were substantially unchanged. The analysis of the trajectories obtained by pulling the P1 stem showed that the secondary and tertiary structure elements of the rest of the aptamer were not affected by the opening of the helix (data not shown). Focusing our attention on the P1 stem we observed that in the Apo form the A9-U63 bp (Fig. 3) was broken when the distance between the centers of mass of the terminal bases reached a value of $\approx$9.8 nm. Differently, in presence of the ligand a longer pulling was needed and the rupture only happened at a distance of $\approx$11.5 nm (Fig. 6). This behavior is compatible with the picture in which the ligand stabilizes the P1 stem (Montange and Batey 2006). It was however difficult to extract quantitative information on the ligand-P1 interaction from these simulations because the rupture is a stochastic event and extensive sampling would be required. Moreover, as pointed out in a recent paper (Lin et al. 2012), the end-to-end distance could be a non-optimal CV for pulling experiments or simulations since local bp formation plays an important role in global stem folding. The quantitative analysis of the P1-ligand interaction was better obtained from the simulation of both the $\triangle$1-8/64-71 systems. We verified that, when the P1 stem is replaced with the A9-U63 bp restrained to be in canonical WC pairing, the aptamer remains stable (Fig. 5C-D). Remarkably, fluorescence experiments have shown that the aptamer can also fold and bind adenine when large fractions of the P1 stem are removed (Lemay and Lafontaine 2007). This validates the possibility of using the two structures, $\triangle$1-8/64-71 Holo and $\triangle$1-8/64-71 Apo, to investigate the direct P1 stabilization given by the adenine. In the following we focus on the SMD simulations performed on these truncated forms. Typical initial and final conformations from the SMD are shown in Fig. 4. ### Analysis of work profiles The unbinding event of the A9-U63 bp is described as a function of the value of the steered RMSD in Fig. 7. The initial value corresponds to the configuration with the WC pairing formed, whereas at the final value (0.35 nm) the pairing is completely broken. Even if the ensembles of work profiles for the two forms are broadly spread and superposed, the free-energy profiles computed using the Jarzynski equality (Jarzynski 1997 _a_) as the exponential average of the two sets of data are clearly distinguishable (Fig. 7A). Qualitatively it is worth highlighting that during the breaking of the A9-U63 bp the Apo form (red line) is always lower in free energy than the Holo form (blue line). It follows that the breaking of the monitored bp in the Apo form was unambiguously more probable than in the Holo one (Fig. 7A). However, such an approach was still a long way off from quantitatively accounting for the energetic stabilization of the A9-U63 bp related to the presence of adenine in the binding site. Within this framework there was no way to automatically detect when the nucleobases reached the unbound configuration. It was thus difficult to avoid systematic errors in the comparison of the two systems. Furthermore, few low-work realizations occurred during the unpairing in the presence of adenine. In these low-work realizations the number of hydrogen bonds was non-zero at large RMSD values and the structural analysis of the trajectories revealed the transient formation of a cis-sugar edge pair (data not shown) (Leontis et al. 2002). Due to the exponential nature of the Jarzynski average, these low-work realizations dominated the free energy profile for the Holo form further compromising the possibility of a quantitative comparison with the Apo form. ### Energetics of hydrogen bond breaking We thus analyzed the trajectories in terms of number of hydrogen bonds formed between A9 and U63, a discrete variable that more strictly reported on the breaking of the pairing. In this metrics, the bound (1 or 2 hydrogen bonds) and unbound (0 hydrogen bond) ensembles could be clearly and unambiguously identified thus allowing a quantitative comparison between the Apo and the Holo system. Additionally, the configurations from the outlier trajectories could be assigned properly to one or the other ensemble in spite of their atypical RMSD value. The differences in free energy ($\Delta F$) between the ensembles, with and without hydrogen bonds, was computed using a reweighting scheme (Colizzi and Bussi 2012). The values and the associated standard errors were estimated for both systems. For the Apo form $\Delta F=-2.5\pm 1.4\ kJ/mol$ suggesting that the bp could spontaneously break in the absence of adenine. For the Holo form $\Delta F=1.9\pm 1.7\ kJ/mol$ implying that the presence of the ligand and its pairing with U63 stabilized the stacked bp. The $\Delta\Delta F$ between the two forms is equal to $-4.4\pm 2\ kJ/mol$. This value quantifies the thermodynamic stabilization to the formation of the base pair which directly interacts with adenine in the P1 stem. ## DISCUSSION AND CONCLUSIONS Our simulations at atomistic detail provide for the first time the free-energy contribution of ligand stacking to the formation of the P1 stem in a riboswitch. In particular, the presented _in silico_ approach allows the energetics involved in the aptamer stabilization upon ligand binding to be dissected in detail. Below we compare our results with single-molecule manipulation, both _in vitro_ and _in silico_ , and thermodynamic data from dsRNA melting experiments. We also provide a model for ligand-modulated co- transcriptional folding of the _add_ riboswitch. ### Comparison with related works Our results are in nice agreement with thermodynamic data based on dsRNA melting experiments (Mathews et al. 2004; Turner and Mathews 2010). The comparison between our simulations and those experiments can be straightforwardly achieved by considering the pairing between U62 and the sensed adenine as an additional terminal bp of the P1 stem. The direct stabilization of the P1 stem due to the cognate-ligand binding should be thus equivalent to that given by adding one further AU bp to the P1 helix. Using the most recent nearest neighbor energy parameters for the comparison of RNA secondary structures (Mathews et al. 2004; Turner and Mathews 2010), the free- energy difference between the sequence of the P1 stem with and without the additional AU base pair, ${}_{\textrm{5'-CGCUUCAUA\emph{A}-3'}}$ ${}^{\textrm{3'-GUGAAGUAU\emph{U}-5'}}$ and ${}_{\textrm{5'-CGCUUCAUA-3'}}$ ${}^{\textrm{3'-GUGAAGUAU-5'}}$ , can be computed (Hofacker et al. 1994; Lorenz et al. 2011) as $\Delta\Delta F=-3.7\ kJ/mol$, consistently with our results. Our free-energy estimates complement previously reported investigations in which the role of the ligand in the folding process of the A-riboswitch has always been referred to the whole aptamer (Lin and Thirumalai 2008; Neupane et al. 2011) and never specifically to the P1 stem. Using a one-bead-per- nucleotide coarse-grained model, the $\Delta\Delta F$ has been computed as approximately $-15\ kJ/mol$ (Lin and Thirumalai 2008). Notably also this calculation has been done using a shortened P1 stem, possibly affecting the $\Delta F$ estimation. Single-molecule force spectroscopy experiments have been also performed to characterize the folding pathway of the aptamer with an estimated $\Delta\Delta F\simeq-33\ kJ/mol$ (Neupane et al. 2011). However in both these works the separated contributions of the P1-ligand stacking, of the interaction between the ligand and the junctions J1-2, J2-3 and J3-1, and of the interaction between loops L2-L3 could not be discerned (secondary structure elements labeled as in Fig. 8). From the comparison of our data with the above mentioned experimental and computational works, a twofold modular role for the ligand emerges. On the one hand, the binding of adenine can contribute to the aptamer preorganization and it could allow the long-range induction of the tight hydrogen-bonding and base-stacking networks observed in the native state (Rieder et al. 2007; Lee et al. 2010). This preorganization would reduce the distance between A9 and U63, thus increasing the probability of their pairing. A similar mechanism has been proposed also for the SAM-I riboswitch (Whitford et al. 2009). On the other hand, adenine binding enhances the P1 formation by direct stacking interaction, mimicking the extension of the helix by an additional bp. Notably, the energetic contribution of the direct stacking is smaller than that involved in the aptamer preorganization. The latter can be estimated as the difference between the global ligand-induced aptamer stabilization (Lin and Thirumalai 2008; Neupane et al. 2011) and the stacking contribution dissected in our work. ### Folding model Our work provides atomistic details and energetic estimates to the currently accepted model for the folding of the _add_ riboswitch upon ligand binding (Rieder et al. 2007; Lee et al. 2010; Leipply and Draper 2011). Altogether, our data and the related experimental works suggest a folding model as depicted in Fig. 8. Initially, only the P2 and P3 stems and the corresponding loops (L2, L3, still not interacting each other) are formed and not fully stable (Fig. 8A). Then, adenine binding allows for a preorganization of the aptamer where the three junctions arrange around the ligand (Fig. 8$\textrm{B}_{\textrm{1}}$), stabilizing also the previously formed helices (Rieder et al. 2007). It has not been established clear if the loop-loop interaction is formed before or after ligand binding (Leipply and Draper 2011). Thus, an alternative pathway, the junctions and the P1 could acquire a partially folded conformation also in the absence of adenine (Fig. 8$\textrm{B}_{\textrm{2}}$) (Lee et al. 2010). Finally, the P1 helix becomes fully structured and stabilized by the ligand (Fig. 8C), to the detriment of the expression platform (see Fig. 2) (Lemay et al. 2011). This step is mandatory for translation to be initiated. We quantified the ligand contribution to the P1 stem formation due to direct interactions to be approximately -4 kJ/mol. Our result is compatible with both the folding pathways (Figs. 8$\textrm{B}_{\textrm{1}}$ and 8$\textrm{B}_{\textrm{2}}$) irrespectively of their relative population and cannot discriminate among them. The relative probability between the two paths can be modulated by the ligand concentration and its binding affinity. On the one hand, the intermediate shown in Fig. 8$\textrm{B}_{\textrm{1}}$ could be relevant for ligand-RNA binding in an early transcriptional context in which the last 9 nucleobases (i.e those allowing P1 formation) of the aptamer have not yet been synthesized. Indeed, it has been shown that also an aptamer missing a large portion of the P1 stem is able to bind adenine (Lemay and Lafontaine 2007). On the other hand, the intermediate shown in Fig. 8$\textrm{B}_{\textrm{2}}$ could be populated at low-ligand concentration once the nucleobases allowing P1 formation are synthesized. Later on, after the synthesis of the expression platform, ligand binding could shift the thermodynamic equilibrium towards one of two competing riboswitch conformations (P1 formed and non formed). ### Conclusion Ligand-induced stabilization of the P1 stem is crucial for A-riboswitch regulation and function. Here we quantified the direct interaction between adenine and P1 stem and analyzed it at atomistic detail. Our results suggest a model for the aptamer folding in which the direct P1-ligand interactions play a minor role on the conformational switch when compared with those related to the ligand-induced aptamer preorganization. Because the structural/functional role of the aptamer terminal helix is a common feature in the “straight junctional” riboswitches (Serganov and Nudler 2013), we foresee a wider validity of the model presented herein. ## MATERIALS AND METHODS ### System description and set-up We simulated the Holo and the Apo form of the A-riboswitch aptamer domain both composed of 71 nucleotides. The Apo form was generated by adenine removal from the ligand-bound (Holo) crystal structure (PDB id: 1Y26) (Serganov et al. 2004). This deletion is justified by the fact that the Apo and Holo form have been shown experimentally to share an overall similar secondary structure (Lemay et al. 2011). This is at variance with e.g. the _pbuE_ adenine riboswitch in which the two structures are different. The generation of the Apo form by simply removing the ligand has been adopted also in a recent work (Allnér et al. 2013). Molecular dynamics (MD) simulations were performed using the Amber99 force field (Wang et al. 2000) refined with the _parmbsc0_ corrections (Pérez et al. 2007). From the analysis of the SMD trajectories we do not expect the refinement on the $\chi$ torsional parameters (Zgarbová et al. 2011) to affect the results. Adenine was parametrized using the general Amber force field (gaff) (Wang et al. 2004). Partial atomic charges were assigned using the restricted electrostatic potential fit method (Bayly et al. 1993) based on an electronic structure calculation at the HF/6-31G* level of theory performed with Gaussian03 (Frisch et al. 2004). Bond-lengths were constrained with LINCS (Hess et al. 1997) and the electrostatic interactions were calculated using the particle-mesh Ewald method (Darden et al. 1993). For both forms, the following protocol was used to prepare the systems (Table 1 for details) for MD simulations: steepest descent minimization (200 steps) starting from the X-ray structure. Solvation with $\approx$13000 TIP3P water molecules (Jorgensen et al. 1983) and NaCl at 0.15 M concentration (plus extra Na+ counter-ions to neutralize the charges of the systems) in a hexagonal prism (lattice vectors in nm [(10,0,0), (0,7,0), (0,$\frac{7}{2}$,$\frac{7\cdot\sqrt{3}}{2}$)]) that was created orienting the major length of the aptamer along the X axis. Steepest descent minimization (200 steps) for ions and solvent; the systems were thermalized at 300 K, initially for 200 ps with frozen solute positions and then for 5 ns in NPT ensemble (1 atm) with stochastic velocity rescaling (Bussi et al. 2007) and Berendsen barostat (Berendsen et al. 1984); to maintain the systems oriented along the largest lattice vector (X) a restraint was imposed with a force constant of 4$\cdot$103$\nicefrac{\mathrm{(\nicefrac{\mathrm{kJ}}{\mathrm{mol}})}}{\mathrm{nm^{2}}}$ on the Y and Z components of the distance between phosphate atoms of A52 and G71. Each system was simulated for 48 ns in NVT ensemble to assess the stability of the aptamer. ### Steered molecular dynamics To perform SMD simulations inducing the opening of the whole P1 stem the systems were solvated again with $\approx$39500 water molecules in a larger rhombic dodecahedral box with distance between periodic images equal to 12 nm, adding ions to maintain the same ionic strengh (P1-SMD systems in Table 1). The same protocol described above was applied for the minimization, thermalization and equilibration of this larger Holo and Apo systems for the pulling simulations. An incremental separation between the centers of mass of the terminal nucleotides (C1 and G71) was imposed from an initial value of 1.05 nm to a value sufficient to completely unfold the 9 bp of the P1 helix (Apo 10.05 nm, Holo 11.75 nm) at a speed of 0.56 nm/ns (see Fig. 3). The spring constant was set to 3.9$\cdot$104$\nicefrac{\mathrm{(\nicefrac{\mathrm{kJ}}{\mathrm{mol}})}}{\mathrm{nm^{2}}}$. The first eight bp of the P1 stem (i.e. whole stem from C1 to U8 and from A64 to G71, except for A9-U63 bp) were then cut in both systems creating the $\triangle$1-8/64-71 Holo and $\triangle$1-8/64-71 Apo structures (Table 1 for details). Water molecules were allowed to relax filling the space left by the 16 removed bases through an additional 1 ns equilibration in which the positions of aptamer atoms were frozen followed by 5 ns of unrestrained NPT simulation. Then the systems were simulated for 48 ns in the NVT ensemble restraining the terminal bases in the initial state to avoid any spontaneous flipping. The pairs deletion is not biologically meaningless, because it has been shown experimentally that a series of aptamer variants with shorter P1 helix are still able to bind the ligand (Lemay and Lafontaine 2007). The deletion reduced the noise during the pulling allowing to focus the calculation on the influence of the ligand on the A9-U63 pairing. This bp rupture was here enforced by pulling on the RMSD between the heavy atoms of A9 and U63 with reference to the crystal structure. This collective variable (CV) was chosen as it identifies the native conformation (RMSD $\approx$ 0) of the A9-U63 bp, which is necessary for the initiation of the P1 stem formation. The steered CV was pulled at constant velocity of 0.175 $\nicefrac{{nm}}{{ns}}$ from 0 to 0.35 nm in 2 ns. This pulling induced the complete opening of the A9-U63 bp in presence and absence of the ligand (Fig. 4). The spring constant was set to 3.9$\cdot$104$\nicefrac{\mathrm{(\nicefrac{\mathrm{kJ}}{\mathrm{mol}})}}{\mathrm{nm^{2}}}$. The starting points were extracted equidistantly (one every 16 ps) from a 8.192 ns run NVT ensemble restraining the RMSD value of those atoms at 0. For the two system 512 independent SMD simulations were performed, corresponding to an aggregate time of approximately 1$\mu$s each. Simulations were carried out with the Gromacs 4.0.7 program package (Hess et al. 2008) combined with the PLUMED 1.3 plug-in (Bonomi et al. 2009). ### Analysis The Jarzynski equality (Jarzynski 1997 _a_) was used to estimate the equilibrium free-energy landscape of the process from the collected work profiles. The simulations were then analyzed using a recently proposed scheme (Colizzi and Bussi 2012) which combines an identity by Jarzynski (Jarzynski 1997 _b_) with the weighted-histogram analysis method (Kumar et al. 1992). The algorithm allows the free-energy profiles to be projected onto any _a posteriori_ chosen CV. It is well known that free-energy calculations using Jarzynski-based relationships are difficult to converge. Statistical errors were thus estimated by the bootstrapping procedure described in Do et al. (2013) indicating that our results were converged within $\approx k_{B}T$. The VIENNA RNA package (Hofacker et al. 1994; Lorenz et al. 2011) was used to compare our results with the thermodynamic data based on dsRNA melting experiments (Mathews et al. 2004; Turner and Mathews 2010). ## ACKNOWLEDGMENTS We thank Daniel Lafontaine and Gabriele Varani for reading the manuscript and providing critical comments. Sandro Bottaro is also acknowledged for carefully reading the manuscript and suggesting several improvements. We acknowledge the CINECA award no. HP10B2G6OF, 2012 under the ISCRA initiative for the availability of high performance computing resources. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 306662, S-RNA-S. ## References * Allnér et al. (2013) Allnér O, Nilsson L, Villa A. 2013. 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(2011) Zgarbová M, Otyepka M, Šponer J, Mládek A, Banáš P, Cheatham III TE, Jurečka P. 2011. Refinement of the Cornell et al. nucleic acids force field based on reference quantum chemical calculations of glycosidic torsion profiles. _J Chem Theory Comput_ 7:2886–2902. Table 1: Specifications for the simulated systems System \| Total atoms \| Water molecules \| aptamer atoms \| Ions (Na+Cl) ---\|---\|---\|---\|--- Holo form \| 41628 \| 13078 \| 2257 \| 122 (96+26) Apo form \| 41676 \| 13099 \| 2257 \| 122 (96+26) Holo (P1-SMD) \| 120654 \| 39364 \| 2257 \| 290 (180+110) Apo (P1-SMD) \| 120588 \| 39347 \| 2257 \| 290 (180+110) Holo $\Delta$1-8/64-71 \| 41101 \| 13078 \| 1746 \| 106 (80+26) Apo $\Delta$1-8/64-71 \| 41149 \| 13099 \| 1746 \| 106 (80+26) Figure 1: Adenine riboswitch aptamer and binding site. A) Secondary structure elements and B) 3-dimensional structure with bound adenine. The P1 stem is grey, the other stems and loops are black. C) Cartoon representation of the binding site; the two dotted lines represent the hydrogen bonds of the WC pairing between the U62 and the ligand. Figure 2: Secondary structure representation of the _add_ riboswitch in the ON (A) and OFF (B) states. The ligand, the initiation codon and the Shine- Dalgarno sequence are labeled. Figure 3: Initial (A) and final (B) configuration of the SMD simulation opening the P1 stem here shown for the Holo form. The backbone of the aptamer is in blue except for the P1 stem, in light blue. The ligand and the 18 bases forming the helix are shown. The P1 stem is formed in A and disrupted in B. Figure 4: Representative structures of the Holo binding pocket at the beginning (A, RMSD = 0 nm) and at the end (B, RMSD = 0.35 nm) of the SMD. The portion of the P1 stem removed in our simulations is in light blue. Bases forming the binding pocket are labeled, ligand is shown in red. A9-U63 pair is formed in A and disrupted in B. Figure 5: Root mean square deviation (RMSD) from native structure. A) Holo form during 48 ns equilibration, computed on the whole aptamer (black) and on the bases from 9 to 63 (grey). B) Same as A) done on the Apo form (whole aptamer, black; bases from 9 to 63, grey). The difference between black and grey profiles in both panels, indicates that the P1 stem is less stable than the rest of the aptamer. C) $\triangle$1-8/64-71 Holo RMSD along the NVT 48 ns equilibration. D) Same as C) for the $\triangle$1-8/64-71 Apo form. Figure 6: Base-pair ruptures during P1 pulling. In the pulling simulations the 9 bp forming the P1 stem were unpaired. We here monitored the RMSD of each bp (grey-scale) from their native conformation (A, Holo; B, Apo). A9-U63 bp (in black) was disrupted (RMSD $\approx$0.5 nm, arrows) later in the Holo form ($\approx$19 ns) than in the Apo one ($\approx$16 ns). Figure 7: Unbinding process of the A9-U63 bp. (A) Mechanical work performed as a function of the value of the steered RMSD, or equivalently of time, for 512 simulations for Apo (pink) and Holo form (light-blue). The respective free energies resulting from the Jarzynski equality are shown in thicker red and blue lines. The initial free-energy decrease related to the entropy gain induced by the restraint movement has no consequence on the final result. (B) Hydrogen bonds occurrence for both the systems Apo (red) and Holo (blue). (C) Snapshots of the Holo form (ligand in black) with 2, 1 or 0 hydrogen bond (dotted lines) formed between A9-U63. Figure 8: Schematic representation of the aptamer secondary structure in its folding intermediates with and without the ligand. A) Stems P2 and P3, loops L2 and L3 are folded but not stable. $\textrm{B}_{\textrm{1}})$ The junctions (J1-2, J2-3, J3-1) arrange around the ligand (ADE) and the inter-loop pairings occur (L2-L3) stabilizing also the stems. $\textrm{B}_{\textrm{2}})$ Alternative possible intermediate in which all the 3 stems are not completely and stably folded before ligand binding. The junctions and the tertiary interaction between the loops are not stable. C) P1 stem is fully folded and stabilized by the ligand.	arxiv-papers	2013-09-23T07:27:52	2024-09-04T02:49:51.353122	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Francesco Di Palma, Francesco Colizzi, and Giovanni Bussi", "submitter": "Giovanni Bussi", "url": "https://arxiv.org/abs/1309.5717" }
1309.5767	Hugo Herbelin and Arnaud Spiwack Billy Editor, Bill Editors2Conference title on which this volume is based on111 10.4230/LIPIcs.xxx.yyy.p # The Rooster and the Syntactic Bracket111This research has received funding from the European Research Council under the FP7 grant agreement 278673, Project MemCAD Hugo Herbelin Inria Paris-Rocquencourt Paris, France [email protected] Arnaud Spiwack Inria Paris-Rocquencourt Paris, France [email protected] ###### Abstract. We propose an extension of pure type systems with an algebraic presentation of inductive and co-inductive type families with proper indices. This type theory supports coercions toward from smaller sorts to bigger sorts via explicit type construction, as well as impredicative sorts. Type families in impredicative sorts are constructed with a bracketing operation. The necessary restrictions of pattern-matching from impredicative sorts to types are confined to the bracketing construct. This type theory gives an alternative presentation to the calculus of inductive constructions on which the Coq proof assistant is an implementation. ###### Key words and phrases: Coq, Calculus of inductive constructions, Impredicativity, Strictly positive type families, Inductive type families ###### 1991 Mathematics Subject Classification: F.3.3 Studies of Program Constructs ## 1\. Introduction In the Coq proof assistant [14] inductive types are treated as toplevel definitions. If it makes sense from a convenience or an efficiency point of view, the monolithic nature of the definitions make it hard to describe what they precisely mean. As a matter of fact, inductive definitions mean different things depending on the type they are defined in: specifically, some types are interpreted differently in impredicative sorts like Prop or the impredicative variant of Set. In this article, we present a calculus of inductive and co-inductive constructions where inductive and co-inductive types are presented algebraically. The algebraic presentation is an extension of a PTS [3] with inductive and co-inductive type families. Thanks to its modularity, it is meant to serve as a description which is simpler to expose and more mathematically amenable than the monolithic scheme which is found in a practical system such as Coq. For the sake of clarity, the system is given with a single universe and explicit subtyping, although Coq has an unbounded cumulative hierarchy of universes and implicit subtyping. Apart from these technicalities, it is believed that our calculus of algebraic inductive and co-inductive constructions expresses all the features of the Set-impredicative Calculus of Inductive Constructions that Coq implements, _e.g._ in its version 8.4 when launched with option -impredicative-set. This work draws most of its inspiration from Morris & al [9, 10] for the algebraic presentation of inductive type families in a predicative sort, and Awodey & Bauer [2] for the treatment of impredicative sorts. We use examples from Coq to illustrate the algebraic presentation. To differentiate expressions in Coq from expressions in the algebraic presentation, the former are typeset in a sans-serif font and the latter in a roman font. ## 2\. Pure type systems To model the type system of Coq, we start with the classic presentation of pure type systems (pts) of Barendregt [3], which we will then extend to model type families. A pts is characterised by a _single_ syntactic category of terms which are used both as ${\lambda}$-terms and as types. It has a single form of typing judgment $\Gamma\vdash u:A$, where $u$ and $A$ are terms, and $\Gamma$ a context assigning terms to variables. A pts has a set of _sorts_ , which we shall denote schematically by the symbol $s$. Every sort is an atomic term. A pts has a conversion relation $u\equiv v$. Here we diverge from the presentation of [3] which always uses ${\beta}$-conversion. Coq, on the other hand, uses ${\beta}$${\eta}$-conversion on the fragment described in this section. In this article we will take the conversion rule as abstract, not even requiring it to be decidable. We will only require that it contains all the reduction rules which are given in the form $u\leadsto v$ (in this section, we only have ${\beta}$-reduction). $\Gamma,x{:}A\vdash x:A\lx@proof@logical@and\Gamma\vdash A:sx\mbox{{ is fresh in }}\Gamma$ $\Gamma,x{:}B\vdash u:A\lx@proof@logical@and\Gamma\vdash u:A\Gamma\vdash B:sx\mbox{{ is fresh in }}\Gamma$ $\Gamma\vdash\lambda{x}^{A}.\,u:\prod_{x{:}A}B\lx@proof@logical@and\Gamma\vdash\prod_{x{:}A}B:s\Gamma,x{:}A\vdash u:B$ $\Gamma\vdash{u}\,{v}:{B}{\left[\,x\,{\setminus}\,v\,\right]}\lx@proof@logical@and\Gamma\vdash u:\prod_{x{:}A}B\Gamma\vdash v:A$ $\Gamma\vdash u:B\lx@proof@logical@and\Gamma\vdash u:A\Gamma\vdash B:sA\equiv B$ ${\left(\lambda{x}^{A}.\,u\right)}\,{v}\leadsto{u}{\left[\,x\,{\setminus}\,v\,\right]}$ Figure 1. Generic rules of pts The typing rules of a pts comprise of a set of generic rules given in Figure 1, together with a number of rules of the form $\vdash{s}_{1}:{s}_{2}$ called axioms, and rules of the form $\Gamma\vdash\prod_{x{:}A}B:{s}_{3}\lx@proof@logical@and\Gamma\vdash A:{s}_{1}\Gamma,x{:}A\vdash B:{s}_{2}$ of product formation rules. As usual we write $A\rightarrow B$ for $\prod_{x{:}A}B$ when $x$ does not bind a variable in $B$. As a starting point of the algebraic presentation, we shall use a pts with two sorts, $\mathrm{Type}$ and $\Box$, together with the following axiom: $\Gamma\vdash\mathrm{Type}:\Box$ and the following product formation rule: $\Gamma\vdash\prod_{x{:}A}B:{\mathrm{max}}\,{{s}_{1}}\,{{s}_{2}}\lx@proof@logical@and\Gamma\vdash A:{s}_{1}\Gamma,x{:}A\vdash B:{s}_{2}$ where ${\mathrm{max}}\,{s}\,{s}=s$ and ${\mathrm{max}}\,{\mathrm{Type}}\,{\Box}={\mathrm{max}}\,{\Box}\,{\mathrm{Type}}=\Box$ The sorts $\mathrm{Type}$ and $\Box$ are predicative. The sort $\Box$ plays a technical role in allowing type variable and the formation of type-level functions; it cannot, however, be referenced in terms. In the following sections, $\Box$ will also be used to be able to define types by pattern- matching (_strong elimination_). To model the entire Coq system, $\mathrm{Type}$ and $\Box$ would be replaced with a hierarchy of predicative sorts ${\mathrm{Type}}_{i}$, such that $\Gamma\vdash{\mathrm{Type}}_{i}:{\mathrm{Type}}_{i+1}$ are axioms. Adapting the presentation to a hierarchy of sorts is straightforward, but in the interest of keeping to the heart of the matter we give a presentation with two sorts. ## 3\. Inductive type families We shall now extend the algebraic presentation with a notion of _inductive type families_ to model (dependent) datatypes. In this section we will stay in the predicative fragment of Coq. Contrary to the inductive types of Coq, where inductive definitions must be _named_ at _toplevel_ , like in: Inductive Even : Type := ${\mid}$ eo : Even ${\mid}$ es : Odd ${\rightarrow}$ Even with Odd : Type := ${\mid}$ os : Even ${\rightarrow}$ Odd. the presentation given here is essentially anonymous, and inductive families need not be defined at toplevel prior to use. Mutually inductive types such as Even and Odd are not modelled directly, rather they are encoded using an index: Inductive EvenOdd : bool ${\rightarrow}$ Type := ${\mid}$ eo : EvenOdd true ${\mid}$ es : EvenOdd false ${\rightarrow}$ EvenOdd true ${\mid}$ os : EvenOdd true ${\rightarrow}$ EvenOdd false. Definition Even := EvenOdd true. Definition Odd := EvenOdd false. This encoding works as long as all the mutual definitions are all in the same sort. A variant for mutual definition involving Type and Prop is demonstrated in Section 4.3. When the types being defined are in different predicative sorts, however, we have to resort to another encoding which involves nested datatypes [4, Section 8.6]. We will not explore the latter kind of mutual definition. However, nested datatypes – where recursive calls occur as arguments of another type – such as: Inductive List (A:Type) : Type := ${\mid}$ nil : List A ${\mid}$ cons : A ${\rightarrow}$ List A ${\rightarrow}$ List A. Inductive Tree : Type := ${\mid}$ node : List Tree ${\rightarrow}$ Tree. are indeed modelled in the algebraic presentation. ### 3.1. Regular types To be able to traverse terms of inductive type, the core pts constructions is extended with a _recursive fixed point_ on functions: $\Gamma\vdash\mathrm{fix}\leavevmode\nobreak\ f\leavevmode\nobreak\ {x}_{1}{:}{A}_{1}\leavevmode\nobreak\ \mbox{\ldots}\leavevmode\nobreak\ {x}_{n-1}{:}{A}_{n-1}\leavevmode\nobreak\ {x}_{n}{:}{A}_{n}\Rightarrow u:\prod_{{x}_{1}{:}{A}_{1},\mbox{\ldots},{x}_{n-1}{:}{A}_{n-1},{x}_{n}{:}{A}_{n}}B\begin{array}[]{c}\eqmakebox[tag1]{$\Gamma\vdash\prod_{{x}_{1}{:}{A}_{1},\mbox{\ldots},{x}_{n-1}{:}{A}_{n-1},{x}_{n}{:}{A}_{n}}B : s$${\qquad}$${\mathrm{guarded}}\,{f}\,{{x}_{1}}\,{\mbox{\ldots}}\,{{x}_{n-1}}\,{{x}_{n}}\,{u}$}\\\ \eqmakebox[tag1]{$\Gamma, f{:}\prod_{{x}_{1}{:}{A}_{1},\mbox{\ldots},{x}_{n-1}{:}{A}_{n-1},{x}_{n}{:}{A}_{n}}B, {x}_{1}{:}{A}_{1}, \mbox{\ldots}, {x}_{n-1}{:}{A}_{n-1}, {x}_{n}{:}{A}_{n}\vdash u : B$}\\\ \end{array}$ Recursive fixed points are unfolded when fully applied ${\left(\mathrm{fix}\leavevmode\nobreak\ f\leavevmode\nobreak\ {x}_{1}{:}{A}_{1}\leavevmode\nobreak\ \mbox{\ldots}\leavevmode\nobreak\ {x}_{n}{:}{A}_{n}\Rightarrow u\right)}\,{{v}_{1}}\,{\mbox{\ldots}}\,{{v}_{n}}\leadsto{u}{\left[\,{x}_{i}\,{\setminus}\,{v}_{i}\,\right]}$ To ensure strong normalisation, this reduction rule is limited, and a guard condition is imposed on the recursive calls to $f$. It is not, however, the object of this article to discuss these restriction or the guard condition. Briefly, Coq currently relies on a single _structural_ argument in the block ${x}_{1}$, …, ${x}_{n}$: fixed points are not unfolded until their structural argument starts with a constructor, and the guard condition ensures that each recursive call is performed on a subterm of said structural argument, for a relaxed notion of subterm. Other possibilities exist: Agda2 [11] uses any number of arguments as structural, and tries to find a lexicographic ordering. Yet another possibilities is to use sized types [1]. We shall simply assume that an adequate guard condition is given. We now extend the grammar of type constructors. The presentation of this article is largely inspired by the synthetic definition of _strictly positive families_ by Morris & al [9, 10], but is adapted to _intensional type theory_. The presentation of [9, 10] is designed for generic programming inside a type theory, they give codes for strictly positive families which are then decoded into an actual type of the ambient theory. No elimination principle needs to be given for the strictly positive families, as they are implicit in their decoding. Here, we are defining the syntax of inductive type families, including their elimination rules. Sum type $\Gamma\vdash A+B:\mathrm{Type}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Type}\Gamma\vdash B:\mathrm{Type}$ $\Gamma\vdash{\mathrm{inl}}\,{t}:A+B\lx@proof@logical@and\Gamma\vdash A+B:s\Gamma\vdash t:A$ $\Gamma\vdash{\mathrm{inr}}\,{u}:A+B\lx@proof@logical@and\Gamma\vdash A+B:s\Gamma\vdash u:B$ $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{inl}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ {\mathrm{inr}}\,{z}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &w\\\ \end{array}\\\ \end{array}:{P}{\left[\,x\,{\setminus}\,u\,\right]}\lx@proof@logical@and\Gamma\vdash u:A+B\Gamma,x{:}A+B\vdash P:s\Gamma,y{:}A\vdash v:{P}{\left[\,x\,{\setminus}\,{\mathrm{inl}}\,{y}\,\right]}\Gamma,z{:}A\vdash w:{P}{\left[\,x\,{\setminus}\,{\mathrm{inr}}\,{z}\,\right]}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\mathrm{inl}}\,{u}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{inl}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ {\mathrm{inr}}\,{z}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &w\\\ \end{array}\\\ \end{array}\leadsto{v}{\left[\,y\,{\setminus}\,u\,\right]}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\mathrm{inr}}\,{u}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{inl}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ {\mathrm{inr}}\,{z}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &w\\\ \end{array}\\\ \end{array}\leadsto{w}{\left[\,z\,{\setminus}\,u\,\right]}$ Unit type $\Gamma\vdash 1:\mathrm{Type}$ $\Gamma\vdash():1$ $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}()\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:{P}{\left[\,x\,{\setminus}\,u\,\right]}\lx@proof@logical@and\Gamma\vdash u:1\Gamma,x{:}1\vdash P:s\Gamma\vdash v:{P}{\left[\,x\,{\setminus}\,()\,\right]}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ ()\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}()\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\leadsto v$ Empty type $\Gamma\vdash 0:\mathrm{Type}$ $\Gamma\vdash\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ A\leavevmode\nobreak\ \mathrm{with}\leavevmode\nobreak\ \cdot:A\lx@proof@logical@and\Gamma\vdash u:0\Gamma\vdash A:\mathrm{Type}$ Figure 2. Regular type constructors The regular type constructors, whose typing rules are given in Figure 2, are the empty type $0$, the unit type $1$, and the sum of two types. The elimination rules are given in the form of _dependent pattern-matching_ with a syntax made to remind of that of Coq. We shall often omit the typing predicate when it is clear from the context, especially when it does not depend on the branch. With this material we can define a first example type, namely the booleans: $\left\\{\begin{aligned} \mathbb{B}&=1+1\\\ \mathrm{true}&={\mathrm{inl}}\,{()}\\\ \mathrm{false}&={\mathrm{inr}}\,{()}\end{aligned}\right.$ ### 3.2. Inductive type families Inductive families differ from regular inductive types in that they are parametrised by _indices_ , that is they are functions $F:A\rightarrow\mathrm{Type}$ for some $A$. An inductive family of the form $\lambda{x}^{A}.\,R$, is said to be _uniformly parametrised_ by $A$. In general, inductive families are not uniformly parametrised: the value of the index is allowed to vary in recursive calls, and constructors may build values of ${F}\,{x}$ for certain $x$ only. Remember, for instance, the EvenOdd family: Inductive EvenOdd : bool ${\rightarrow}$ Type := ${\mid}$ eo : EvenOdd true ${\mid}$ es : EvenOdd false ${\rightarrow}$ EvenOdd true ${\mid}$ os : EvenOdd true ${\rightarrow}$ EvenOdd false. The inductive family constructors, presented in Figure 3, warrant individual discussion. First, notice that as a simplifying hypothesis, inductive families have exactly one index. This is, of course, not a limitation in expressive power as multiple indices can be encoded as a dependent sum, and the unit type allows us to write families without an index. Inductive fixed point $\Gamma\vdash\mu{X}^{A\rightarrow\mathrm{Type}}.\,F:A\rightarrow\mathrm{Type}\lx@proof@logical@and\Gamma\vdash A:s\Gamma,X{:}A\rightarrow\mathrm{Type}\vdash F:A\rightarrow\mathrm{Type}{{\mbox{{sp}}}_{X}}\,{F}$ $\mu{X}^{A\rightarrow s}.\,F\equiv{F}{\left[\,X\,{\setminus}\,\mu{X}^{A\rightarrow s}.\,F\,\right]}$ Proper indices $\Gamma\vdash{\sum}_{x{:}A}^{f}T:B\rightarrow\mathrm{Type}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Type}\Gamma\vdash B:\mathrm{Type}\Gamma,x{:}A\vdash T:\mathrm{Type}\Gamma,x{:}A\vdash f:B$ $\Gamma\vdash{\left(u,v\right)}_{x{:}A.T}^{f}:{\left({\sum}_{x{:}A}^{f}T\right)}\,{\left({f}{\left[\,x\,{\setminus}\,u\,\right]}\right)}\lx@proof@logical@and\Gamma\vdash{\sum}_{x{:}A}^{f}T:B\rightarrow s\Gamma\vdash u:A\Gamma\vdash v:{T}{\left[\,x\,{\setminus}\,u\,\right]}$ $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ z\leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\ y\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\left(i,j\right)}_{x{:}A.T}^{f}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:{P}{\left[\,y\,{\setminus}\,b\,,\,z\,{\setminus}\,u\,\right]}\begin{array}[]{c}\eqmakebox[tag2]{$\Gamma\vdash u : {\left( {\sum}_{x{:}A}^{f}T\right) }\,{b}$${\qquad}$$\Gamma, y{:}B, z{:}{\left( {\sum}_{x{:}A}^{f}T\right) }\,{y}\vdash P : s$}\\\ \eqmakebox[tag2]{$\Gamma, i{:}A, j{:}{T}\,{i}\vdash v : {P}{\left[ \,y\,{\setminus}\,{f}\,{i}\,,\,z\,{\setminus}\,{\left( i, j\right) }_{x{:}A.T}^{f}\,\right] }$}\\\ \end{array}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\left(u,v\right)}_{x{:}A.T}^{f}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ z\leavevmode\nobreak\ \mathrm{in}\leavevmode\nobreak\ y\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\left(i,j\right)}_{x{:}A.T}^{f}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &w\\\ \end{array}\\\ \end{array}\leadsto{w}{\left[\,i\,{\setminus}\,u\,,\,j\,{\setminus}\,v\,\right]}$ Figure 3. Inductive type families ${{\mbox{{sp}}}_{X}}\,{y}$ ${{\mbox{{sp}}}_{X}}\,{0}$ ${{\mbox{{sp}}}_{X}}\,{1}$ ${{\mbox{{sp}}}_{X}}\,{\left(\prod_{x{:}A}B\right)}\lx@proof@logical@and X\mbox{{ is fresh in }}A{{\mbox{{sp}}}_{X}}\,{B}$ ${{\mbox{{sp}}}_{X}}\,{\left(A+B\right)}\lx@proof@logical@and{{\mbox{{sp}}}_{X}}\,{A}{{\mbox{{sp}}}_{X}}\,{B}$ ${{\mbox{{sp}}}_{X}}\,{\left(\mu{Y}^{A}.\,F\right)}\lx@proof@logical@and X\mbox{{ is fresh in }}A{{\mbox{{sp}}}_{X}}\,{F}$ ${{\mbox{{sp}}}_{X}}\,{\left(\lambda{x}^{A}.\,T\right)}\lx@proof@logical@and X\mbox{{ is fresh in }}A{{\mbox{{sp}}}_{X}}\,{T}$ ${{\mbox{{sp}}}_{X}}\,{\left({U}\,{t}\right)}\lx@proof@logical@and{{\mbox{{sp}}}_{X}}\,{U}X\mbox{{ is fresh in }}t$ ${{\mbox{{sp}}}_{X}}\,{A}\lx@proof@logical@and{{\mbox{{sp}}}_{X}}\,{B}B\equiv A$ ${{\mbox{{sp}}}_{X}}\,{\left({\sum}_{x{:}A}^{f}T\right)}\lx@proof@logical@and X\mbox{{ is fresh in }}f{{\mbox{{sp}}}_{X}}\,{A}{{\mbox{{sp}}}_{X}}\,{T}$ Figure 4. Strict positivity condition The construction $\mu{X}^{A\rightarrow\mathrm{Type}}.\,F$ constructs the _inductive fixed point_ of $F$. It acts on type families, because indices vary through recursive calls to $X$. To be able to form an inductive fixed point, occurrences of $X$ must be strictly positive in $F$, rules for strict positivity are given in Figure 4. The rules of Figure 4 are a simple set which suits the needs of this article, however in practice, we may want to consider strict positivity rules involving elimination rules and a finer treatment of application. Strict positivity ensures that no non-terminating term can be written without recursive fixed points, so that the guard condition suffices to enforce termination. Paradoxes which can be derived from non-positive or non-strictly positive inductive fixed points can be found in [13, Chapter 4, Section 4.2][8, Chapter 3][4, Chapter 8]. To avoid clutter, we give a presentation where inductive fixed points can be freely rolled and unrolled thanks to the conversion. An alternative can be to give an explicit term constructor for fixed points, see Section 3.4. We will also use an inductive fixed point on nullary families, defined as: $\displaystyle\mu{X}^{\mathrm{Type}}.\,F$ $\displaystyle={\left(\mu{Y}^{1\rightarrow\mathrm{Type}}.\,{F}{\left[\,X\,{\setminus}\,{Y}\,{()}\,\right]}\right)}\,{()}$ from which we have that $\mu{X}^{\mathrm{Type}}.\,F$ can be freely rolled from or unrolled to ${F}{\left[\,X\,{\setminus}\,\mu{X}^{\mathrm{Type}}.\,F\,\right]}$. With inductive fixed points, we can, for instance, define the accessibility predicate. In Coq: Inductive Acc (A:Type) (R:A${\rightarrow}$A${\rightarrow}$Type) (x:A) : Type := ${\mid}$ acc_intro : (forall y:A, R y x ${\rightarrow}$ Acc A R y) ${\rightarrow}$ Acc A R x. This type represents the generic form of termination proofs: any terminating recursive fixed point can be made structural over a proof of accessibility. In the algebraic presentation, it is defined as: $\left\\{\begin{aligned} \mathrm{Acc}&=\lambda{A}^{\mathrm{Type}}\,{R}^{A\rightarrow A\rightarrow\mathrm{Type}}.\,\mu{\mathrm{Acc}}^{A\rightarrow\mathrm{Type}}.\,\lambda{x}^{A}.\,\prod_{y{:}A}{R}\,{y}\,{x}\rightarrow{\mathrm{Acc}}\,{y}\\\ {\mathrm{acc}}_{\mathrm{intro}}&=\lambda{A}^{\mathrm{Type}}\,{R}^{A\rightarrow A\rightarrow\mathrm{Type}}\,{x}^{A}\,{f}^{\prod_{y{:}A}{R}\,{y}\,{x}\rightarrow{\mathrm{Acc}}\,{A}\,{R}\,{y}}.\,f\end{aligned}\right.$ Because inductive fixed points are treated transparently, the constructor is rather trivial. However, notice how, in the definition of $\mathrm{Acc}$, the parameter $x$ is treated differently from $A$ and $R$. The reason is that $A$ and $R$ are _uniform parameters_ , in that they do not vary through recursive calls, whereas $x$ does: it is a _non-uniform parameter_. The parameter $x$ is, hence, represented as an index. However, such an index is not sufficient to encode types like EvenOdd. Representing proper indices requires a new type construction, which we write ${\sum}_{x{:}A}^{f}T$. This construction comes from [9, 10], where it is inspired by a categorical point of view: in a sufficiently extensional setting, ${\sum}_{x{:}A}^{f}T$ is the right adjoint to a pullback functor. The similarity with the usual notation of dependent sum is not fortuitous, indeed we can define dependent sum as a special case of proper indexing: $\left\\{\begin{aligned} \sum_{x{:}A}B&={\left({\sum}_{x{:}A}^{()}B\right)}\,{()}\\\ {\left(u,v\right)}_{{\mbox{\scriptsize{$x${:}$A$.$B$}}}}&={\left(u,v\right)}_{x{:}A.B}^{()}\end{aligned}\right.$ We also write $A\times B$ and $\left(u,v\right)$ for $\sum_{x{:}A}B$ and ${\left(u,v\right)}_{{\mbox{\scriptsize{$x${:}$A$.$B$}}}}$, respectively, when $x$ is not free in $B$. In the case of dependent sums, the index is trivial. When it is not, however, the pattern matching return clause $P$ is allowed to depend on the value of the index. This is the purpose of Coq’s in-pattern. With the algebraic presentation, the in-pattern has the pleasant property of being confined to the proper indexing construction, hopefully making its meaning more explicit. The syntax differs a little from that of Coq, however: Coq renders the in clause as a pattern with the type name at the head: match n as n’ in EvenOdd b return P n’ b with … end. In the algebraic presentation, types not having a name, the in clause simply consists of a name for the index. The prototype of proper indexing is the identity type, which we name $\mathrm{Eq}$. In Coq: Inductive Eq (A:Type) (x:A) : A ${\rightarrow}$ Type := ${\mid}$ eq_refl : Eq A x x in the algebraic presentation: $\left\\{\begin{aligned} \mathrm{Eq}&=\lambda{A}^{\mathrm{Type}}\,{x}^{A}.\,{\sum}_{\\_{:}1}^{x}1\\\ {\mathrm{eq}}_{\mathrm{refl}}&=\lambda{A}^{\mathrm{Type}}\,{x}^{A}.\,{\left((),()\right)}_{\\_{:}1.1}^{x}\end{aligned}\right.$ In fact, dependent sums and identity types are sufficient to define proper indexing. Indeed ${\sum}_{x{:}A}^{f}T$ can be redefined as: $\left\\{\begin{aligned} {\sum}_{x{:}A}^{f}T&=\lambda{y}^{B}.\,\sum_{x{:}A}\left({\mathrm{Eq}}\,{B}\,{y}\,{f}\right)\times T\\\ {\left(u,v\right)}_{x{:}A.T}^{f}&={\left(u,\left({{\mathrm{eq}}_{\mathrm{refl}}}\,{B}\,{{f}{\left[\,x\,{\setminus}\,u\,\right]}},v\right)\right)}_{{\mbox{\scriptsize{$x${:}$A$.$\left({\mathrm{Eq}}\,{B}\,{\left({f}{\left[\,x\,{\setminus}\,u\,\right]}\right)}\,{f}\right)$${\times}$$T$}}}}\end{aligned}\right.$ It is closer to the spirit of Coq, but in no way essential, to take a proper indexing construction rather than equality as primitive. In Morris & al [9, 10], the dependent sum and equality of the ambient type theory is used to define ${\sum}_{x{:}A}^{f}T$ which is then taken as primitive. An other choice lies in the use of $A+B$ as primitive. It is the only type construction which allows to define a type with distinct elements. However, a common alternative is to take $\mathbb{B}$ as primitive, in which case we can define $A+B$ as: $\left\\{\begin{aligned} A+B&=\sum_{b{:}\mathbb{B}}\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ b\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}\mathrm{true}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &A\\\ \mathrm{false}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &B\\\ \end{array}\\\ \end{array}\\\ \mathrm{inl}&=\lambda{x}^{A}.\,{\left(\mathrm{true},x\right)}_{{\mbox{\scriptsize{$b${:}$\mathbb{B}$.$\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ b\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}\mathrm{true}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &A\\\ \mathrm{false}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &B\\\ \end{array}\\\ \end{array}$}}}}\\\ \mathrm{inr}&=\lambda{y}^{B}.\,{\left(\mathrm{false},y\right)}_{{\mbox{\scriptsize{$b${:}$\mathbb{B}$.$\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ b\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}\mathrm{true}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &A\\\ \mathrm{false}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &B\\\ \end{array}\\\ \end{array}$}}}}\end{aligned}\right.$ ### 3.3. Examples The previous section concludes the description of the predicative fragment of the algebraic presentation. We can now give definitions of the remaining inductive families we have encountered in terms of the algebraic presentation. Starting with EvenOdd: Inductive EvenOdd : bool ${\rightarrow}$ Type := ${\mid}$ eo : EvenOdd true ${\mid}$ es : EvenOdd false ${\rightarrow}$ EvenOdd true ${\mid}$ os : EvenOdd true ${\rightarrow}$ EvenOdd false. translates, in the algebraic presentation, to: $\left\\{\begin{aligned} \mathrm{EvenOdd}&=\mu{\mathrm{EvenOdd}}^{\mathbb{B}\rightarrow\mathrm{Type}}.\,\left({\sum}_{\\_{:}1}^{\mathrm{true}}1+{\mathrm{EvenOdd}}\,{\mathrm{false}}\right)+\left({\sum}_{\\_{:}1}^{\mathrm{false}}{\mathrm{EvenOdd}}\,{\mathrm{true}}\right)\\\ \mathrm{eo}&={\mathrm{inl}}\,{{\left((),{\mathrm{inl}}\,{()}\right)}_{\\_{:}1.1}^{\mathrm{true}}}\\\ \mathrm{es}&=\lambda{x}^{{\mathrm{EvenOdd}}\,{\mathrm{false}}}.\,{\mathrm{inl}}\,{{\left((),{\mathrm{inr}}\,{x}\right)}_{\\_{:}1.1}^{\mathrm{true}}}\\\ \mathrm{os}&=\lambda{x}^{{\mathrm{EvenOdd}}\,{\mathrm{true}}}.\,{\mathrm{inr}}\,{{\left((),x\right)}_{\\_{:}1.1}^{\mathrm{false}}}\end{aligned}\right.$ Here is the definition of List in Coq: Inductive List (A:Type) : Type := ${\mid}$ nil : List A ${\mid}$ cons : A ${\rightarrow}$ List A ${\rightarrow}$ List A. and in the algebraic presentation: $\left\\{\begin{aligned} \mathrm{List}&=\lambda{A}^{\mathrm{Type}}.\,\mu{\mathrm{List}}^{\mathrm{Type}}.\,1+A\times\mathrm{List}\\\ \mathrm{nil}&=\lambda{A}^{\mathrm{Type}}.\,{\mathrm{inl}}\,{()}\\\ \mathrm{cons}&=\lambda{A}^{\mathrm{Type}}\,{x}^{A}\,{l}^{{\mathrm{List}}\,{A}}.\,{\mathrm{inr}}\,{\left(x,l\right)}\end{aligned}\right.$ Finally the definition of Tree: Inductive Tree : Type := ${\mid}$ node : List Tree ${\rightarrow}$ Tree. translates to: $\left\\{\begin{aligned} \mathrm{Tree}&=\mu{\mathrm{Tree}}^{\mathrm{Type}}.\,{\mathrm{List}}\,{\mathrm{Tree}}\\\ \mathrm{node}&=\lambda{f}^{{\mathrm{List}}\,{\mathrm{Tree}}}.\,f\end{aligned}\right.$ Note that in the definition of $\mathrm{Tree}$, we must ${\beta}$-reduce ${\mathrm{List}}\,{\mathrm{Tree}}$ for the recursive definition to be strictly positive. A more complex example is given by the type of lists indexed by their length, often called _vectors_ : Inductive Nat : Type := ${\mid}$ o : Nat ${\mid}$ s : Nat ${\rightarrow}$ Nat. Inductive Vector (A:Type) : Nat ${\rightarrow}$ Type := ${\mid}$ vnil : Vector A o ${\mid}$ vcons : forall n, A ${\rightarrow}$ Vector A n ${\rightarrow}$ Vector A (s n). It is encoded in the algebraic presentation as: $\left\\{\begin{aligned} \mathrm{Nat}&=\mu{\mathrm{Nat}}^{\mathrm{Type}}.\,1+\mathrm{Nat}\\\ o&={\mathrm{inl}}\,{()}\\\ s&=\lambda{n}^{\mathrm{Nat}}.\,{\mathrm{inr}}\,{n}\\\ \mathrm{Vector}&=\lambda{A}^{\mathrm{Type}}.\,\mu{V}^{\mathrm{Nat}\rightarrow\mathrm{Type}}.\,\lambda{n}^{\mathrm{Nat}}.\,{\left({\sum}_{\\_{:}1}^{o}1\right)}\,{n}+{\left({\sum}_{n^{\prime}{:}\mathrm{Nat}}^{{s}\,{n^{\prime}}}A\times{V}\,{n^{\prime}}\right)}\,{n}\\\ \mathrm{vnil}&=\lambda{A}^{\mathrm{Type}}.\,{\mathrm{inl}}\,{{\left((),()\right)}_{\\_{:}1.1}^{o}}\\\ \mathrm{vcons}&=\lambda{A}^{\mathrm{Type}}\,{n}^{\mathrm{Nat}}\,{a}^{A}\,{v}^{{\mathrm{Vector}}\,{n}\,{a}}.\,{\mathrm{inr}}\,{{\left(n,\left(a,v\right)\right)}_{n^{\prime}{:}\mathrm{Nat}.A\times{V}\,{n^{\prime}}}^{{s}\,{n^{\prime}}}}\end{aligned}\right.$ Contrary to proper indices, the types of non-uniform parameters are allowed to be in $\Box$, this allows the definition of types such as the binary lists [12]: Inductive BList (A:Type) : Type := ${\mid}$ one : A ${\rightarrow}$ BList A ${\mid}$ twice : BList (A$$A) ${\rightarrow}$ BList A ${\mid}$ stwice : A ${\rightarrow}$ BList (A$$A) ${\rightarrow}$ BList A which are rendered in the algebraic presentation as: $\left\\{\begin{aligned} \mathrm{BList}&=\mu{\mathrm{BList}}^{\mathrm{Type}\rightarrow\mathrm{Type}}.\,\lambda{A}^{\mathrm{Type}}.\,A+\left({\mathrm{BList}}\,{\left(A\times A\right)}+A\times\left({\mathrm{BList}}\,{\left(A\times A\right)}\right)\right)\\\ \mathrm{one}&=\lambda{A}^{\mathrm{Type}}\,{x}^{A}.\,{\mathrm{inl}}\,{x}\\\ \mathrm{twice}&=\lambda{A}^{\mathrm{Type}}\,{l}^{{\mathrm{BList}}\,{\left(A\times A\right)}}.\,{\mathrm{inr}}\,{\left({\mathrm{inl}}\,{l}\right)}\\\ \mathrm{stwice}&=\lambda{A}^{\mathrm{Type}}\,{x}^{A}\,{l}^{{\mathrm{BList}}\,{\left(A\times A\right)}}.\,{\mathrm{inr}}\,{\left({\mathrm{inr}}\,{\left(a,l\right)}\right)}\end{aligned}\right.$ In Coq, where there is a hierarchy of universe, types of proper indices can be in any sort. However, a proper index whose type is in ${\mathrm{Type}}_{i}$ constrains the final type to be in ${\mathrm{Type}}_{i+1}$ or higher. Uniform parameters, of any type, do not constrain the type they parametrise. Inductive types are consumed by recursive fixed points. Using the implicit unfolding of inductive fixed points, we can pattern-match over the top constructor directly. The Coq function Fixpoint add (x y:Nat) : Nat := match y with ${\mid}$ o ${\Rightarrow}$ x ${\mid}$ s y’ ${\Rightarrow}$ s (add x y’) end is rendered in the algebraic presentation as $\displaystyle\mathrm{add}$ $\displaystyle=\mathrm{fix}\leavevmode\nobreak\ \mathrm{add}\leavevmode\nobreak\ x{:}\mathrm{Nat}\leavevmode\nobreak\ y{:}\mathrm{Nat}\Rightarrow\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ y\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ \\_\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ \mathrm{Nat}\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{inl}}\,{\\_}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &x\\\ {\mathrm{inr}}\,{y^{\prime}}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &{s}\,{\left({\mathrm{add}}\,{x}\,{y^{\prime}}\right)}\\\ \end{array}\\\ \end{array}$ ### 3.4. Co-induction In addition to inductive fixed points, Coq also has support for co-inductive fixed points. Co-inductive fixed points are required to be strictly positive, like inductive fixed points. We choose in this section, a presentation of co- inductive data where fixed points are explicitly introduced with a constructor. Below we will use this explicit presentation to give a variation on Coq’s co-inductive fixed points. $\parbox{433.62pt}{ \begin{center} \makebox[433.62pt]{\parbox{433.62pt}{ \begin{center} $\Gamma\vdash\nu{X}^{A\rightarrow\mathrm{Type}}.\,F:A\rightarrow\mathrm{Type}\lx@proof@logical@and\Gamma\vdash A:s\Gamma,X{:}A\rightarrow\mathrm{Type}\vdash F:A\rightarrow\mathrm{Type}{{\mbox{{sp}}}_{X}}\,{F}$ \end{center} }}\\\ \makebox[433.62pt]{\parbox{433.62pt}{ \begin{center} $\Gamma\vdash{\mathrm{forced}}\,{u}:{\left(\nu{X}^{A\rightarrow s}.\,F\right)}\,{i}\lx@proof@logical@and\Gamma\vdash\nu{X}^{A\rightarrow s}.\,F:A\rightarrow s\Gamma\vdash i:A\Gamma\vdash u:{{F}{\left[\,X\,{\setminus}\,\nu{X}^{A\rightarrow s}.\,F\,\right]}}\,{i}$ \end{center} }}\\\ \makebox[433.62pt]{\parbox{433.62pt}{ \begin{center} $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:{P}{\left[\,x\,{\setminus}\,u\,\right]}\begin{array}[]{c}\eqmakebox[tag3]{$\Gamma\vdash\nu{X}^{A\rightarrow s}.\,F : A\rightarrow s$${\qquad}$$\Gamma\vdash i : A$${\qquad}$$\Gamma\vdash u : {\left( \nu{X}^{A\rightarrow s}.\,F\right) }\,{i}$}\\\ \eqmakebox[tag3]{$\Gamma, x{:}{\left( \nu{X}^{A\rightarrow s}.\,F\right) }\,{i}\vdash P : s'$${\qquad}$$\Gamma, y{:}{{F}{\left[ \,X\,{\setminus}\,\nu{X}^{A\rightarrow s}.\,F\,\right] }}\,{i}\vdash v : {P}{\left[ \,x\,{\setminus}\,{\mathrm{forced}}\,{y}\,\right] }$}\\\ \end{array}$ \end{center} }}\\\ \makebox[433.62pt]{\parbox{433.62pt}{ \begin{center} $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\mathrm{forced}}\,{u}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\leadsto{v}{\left[\,y\,{\setminus}\,u\,\right]}$ \end{center} }}\\\ \end{center} }$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\mathrm{forced}}\,{u}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\leadsto{v}{\left[\,y\,{\setminus}\,u\,\right]}$ Just like inductive data is _destructed_ by a recursive fixed point operation, co-inductive data is _constructed_ by a co-recursive fixed point operation, allowing co-inductive data to be infinite. The guard condition on co-recursive fixed points ensures that a finite number of unfolding will eventually produce a forced value. $\Gamma\vdash\mathrm{cofix}\leavevmode\nobreak\ f\leavevmode\nobreak\ {x}_{1}{:}{A}_{1}\leavevmode\nobreak\ \mbox{\ldots}\leavevmode\nobreak\ {x}_{n}{:}{A}_{n}\Rightarrow u:\prod_{{x}_{1}{:}{A}_{1},\mbox{\ldots},{x}_{n}{:}{A}_{n}}{\left(\nu{X}^{A\rightarrow s}.\,F\right)}\,{i}\begin{array}[]{c}\eqmakebox[tag4]{$\Gamma\vdash\prod_{{x}_{1}{:}{A}_{1},\mbox{\ldots},{x}_{n}{:}{A}_{n}}{\left( \nu{X}^{A\rightarrow s}.\,F\right) }\,{i} : s$${\qquad}$${\mathrm{coguarded}}\,{f}\,{{x}_{1}}\,{\mbox{\ldots}}\,{{x}_{n}}\,{u}$}\\\ \eqmakebox[tag4]{$\Gamma, f{:}\prod_{{x}_{1}{:}{A}_{1},\mbox{\ldots},{x}_{n}{:}{A}_{n}}{\left( \nu{X}^{A\rightarrow s}.\,F\right) }\,{i}, {x}_{1}{:}{A}_{1}, \mbox{\ldots}, {x}_{n}{:}{A}_{n}\vdash u : {\left( \nu{X}^{A\rightarrow s}.\,F\right) }\,{i}$}\\\ \end{array}$ Co-recursive fixed-point are meant to represent infinite data: they cannot be unfolded eagerly, lest they would fail to terminate. They are unfolded only when they appear at the head of a pattern-matching expression: $\begin{array}[]{ll}&\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\left(\mathrm{cofix}\leavevmode\nobreak\ f\leavevmode\nobreak\ {x}_{1}{:}{A}_{1}\leavevmode\nobreak\ \mbox{\ldots}\leavevmode\nobreak\ {x}_{n}{:}{A}_{n}\Rightarrow u\right)}\,{{v}_{1}}\,{\mbox{\ldots}}\,{{v}_{n}}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\\\ \leadsto&\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {u}{\left[\,f\,{\setminus}\,\left(\mathrm{cofix}\leavevmode\nobreak\ f\leavevmode\nobreak\ {x}_{1}{:}{A}_{1}\leavevmode\nobreak\ \mbox{\ldots}\leavevmode\nobreak\ {x}_{n}{:}{A}_{n}\Rightarrow u\right)\,,\,{x}_{i}\,{\setminus}\,{v}_{i}\,\right]}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\\\ \end{array}$ The dependent elimination rule for co-inductive fixed points asserts, in essence, that every co-inductive data is of the form ${\mathrm{forced}}\,{u}$. Even though it would be fine for inductive fixed points – this is why we could leave the unrolling to the conversion – this does not reflect well the computational aspects of co-inductive data: suspended co-recursive fixed points are values, and won’t be evaluated until the context demands it. The fact that the elimination for co-inductive data claims that all values are forced gives rise to undesirable behaviour. Take for instance the following simple co-inductive type, and data: $\left\\{\begin{aligned} T&=\nu X.\,X\\\ i&=\mathrm{cofix}\leavevmode\nobreak\ i\Rightarrow{\mathrm{forced}}\,{i}\end{aligned}\right.$ So that $i$ is effectively an infinite sequence of $\mathrm{forced}$. Using the elimination principle above, it is possible to give a _closed_ proof that ${\mathrm{Eq}}\,{T}\,{i}\,{\left({\mathrm{forced}}\,{i}\right)}$: $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ i\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ {\mathrm{Eq}}\,{T}\,{x}\,{\left({\mathrm{forced}}\,{\left(\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &y\\\ \end{array}\\\ \end{array}\right)}\right)}\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &{{\mathrm{eq}}_{\mathrm{refl}}}\,{T}\,{y}\\\ \end{array}\\\ \end{array}$ However, $i$ and $\left({\mathrm{forced}}\,{i}\right)$ are not convertible, yet, as every closed proof of equality does, this proof reduces to ${\mathrm{eq}}_{\mathrm{refl}}$, hence should relate convertible terms. The dependent elimination rule of co-inductive fixed points compromises the type safety of the logic. Coq uses the above dependent elimination rule for co-inductive fixed points. It was a deliberate decision made for practical purposes. Nonetheless, one may want to weaken it to avoid the incompatibility between equality and conversion. To do so, it suffices to erase the dependency of the return predicate over the matched term: $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{forced}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:P\lx@proof@logical@and\Gamma\vdash\nu{X}^{A\rightarrow s}.\,F:A\rightarrow s\Gamma\vdash i:A\Gamma\vdash u:{\left(\nu{X}^{A\rightarrow s}.\,F\right)}\,{i}\Gamma\vdash P:s^{\prime}\Gamma\vdash v:P$ ## 4\. Prop With all the common baggage for predicative sorts set in place, we can add impredicative sorts to the algebraic presentation. The main such sort in Coq is the sort Prop of propositions. The design of Prop is guided by _proof irrelevance_ : even if it is not actually provable in Coq, different proofs of a proposition are thought of as being equal. This property is useful for program extraction: only the _computationally relevant_ parts of a program need to be executed to get the final result. In other words: propositions are considered as _static_ data. It is why, with disjunction and existential defined as: Inductive Or (A B:Prop) : Prop := ${\mid}$ or_introl : A ${\rightarrow}$ A ${\lor}$ B ${\mid}$ or_intror : B ${\rightarrow}$ A ${\lor}$ B. Inductive Ex (A:Type) (P:A${\rightarrow}$Prop) : Prop := ${\mid}$ ex_intro : forall x:A, P x ${\rightarrow}$ Ex A P. the following terms are refused by type-checking: match x with ${\mid}$ or_introl $\\_$ ${\Rightarrow}$ true ${\mid}$ or_intror $\\_$ ${\Rightarrow}$ false end. and match x with ${\mid}$ ex_intro x $\\_$ ${\Rightarrow}$ x end. On the other hand, it is not the case of every inductive type defined in Prop, that they cannot be eliminated into Type. Conjunction and falsity are two counter-examples: Inductive False : Prop := . Inductive And (A B:Prop) : Prop := ${\mid}$ conj : A ${\rightarrow}$ B ${\rightarrow}$ A ${\land}$ B. Coq allows elimination over these two propositions into Type, and both following terms are well-typed: match x return Bool with end. and match x with ${\mid}$ conj $\\_$ $\\_$ ${\Rightarrow}$ true end. The object of this section is to make syntactically explicit what happens when an inductive type of Coq is declared to be of sort Prop. The description elaborated in this section has strong similarities with the system of bracket- types proposed by Awodey & Bauer [2]. They describe the propositions as the subset of types with at most one element, and introduce a left adjoint, written as brackets, to the inclusion of propositions into types. We will reuse their notation, even though, in our intensional setting, T:Prop does not enforce that T has a most one element, and the bracketing operation does not properly form an adjunction with the inclusion from Prop to Type. ### 4.1. Impredicativity Let us start by introducing the new sort $\mathrm{Prop}$ in the algebraic presentation: $\Gamma\vdash\mathrm{Prop}:\mathrm{Type}$ As in [2], propositions form a subset of types. Coq has a subtyping rule (also known as _cumulativity_) to make the inclusion transparent. We will, however, render it with a syntactic construct: $\Gamma\vdash\left\\{A\right\\}:\mathrm{Type}\Gamma\vdash A:\mathrm{Prop}$ $\Gamma\vdash{\mathrm{prf}}\,{u}:\left\\{A\right\\}\Gamma\vdash u:A$ $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{prf}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:{P}{\left[\,x\,{\setminus}\,u\,\right]}\lx@proof@logical@and\Gamma\vdash u:\left\\{A\right\\}\Gamma,x{:}\left\\{A\right\\}\vdash P:s\Gamma,y{:}A\vdash v:{P}{\left[\,x\,{\setminus}\,{\mathrm{prf}}\,{y}\,\right]}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\mathrm{prf}}\,{u}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{prf}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\leadsto{v}{\left[\,y\,{\setminus}\,u\,\right]}$ This definition simply makes $\left\\{A\right\\}$ a synonym of $A$, except of sort $\mathrm{Type}$. It is strictly positive in $A$: ${{\mbox{{sp}}}_{X}}\,{\left\\{A\right\\}}{{\mbox{{sp}}}_{X}}\,{A}$ The fact that $\mathrm{Prop}$ is impredicative – _i.e._ supports the following product formation rules: $\Gamma\vdash\prod_{x{:}A}B:\mathrm{Prop}\lx@proof@logical@and\Gamma\vdash A:s\Gamma,x{:}A\vdash B:\mathrm{Prop}$ is easily understood in terms of proof irrelevance. Indeed, if for all $x$, $B$ has at most one element, so has the product over $x$. Even though it uses functional extensionality, which is not provable. ### 4.2. Singleton rules The types which (ideally) preserve the proof irrelevance property are sometimes called singleton types in the setting of Coq. In our algebraic presentation, they correspond to inductive type family constructors with extra formation rules to make them preserve propositions. The rules are shown in Figure 5. $\Gamma\vdash 0:\mathrm{Prop}$ $\Gamma\vdash 1:\mathrm{Prop}$ $\Gamma\vdash{\sum}_{x{:}A}^{f}T:B\rightarrow\mathrm{Prop}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Prop}\Gamma\vdash B:\mathrm{Prop}\Gamma,x{:}A\vdash T:\mathrm{Prop}\Gamma,x{:}A\vdash f:B$ Figure 5. Singleton rules This elucidates why Coq allows elimination over False and And into arbitrary type: False is implemented as $0$ and And A B and $A\times B$. The elimination rules being unchanged, pattern-matching over proofs of False and And are unrestricted. Because restricted pattern-matching is often seen as the default, singleton types are said to enjoy _singleton elimination_. Remark that, proofs of propositions being uninformative, there is essentially nothing to be gained from depending on, or being indexed over a proposition. In consequence, the type formation rule for proper indexing in Figure 5 is only useful, in practice, for the subcase of cartesian product. Coq actually implements two other singleton rules. The first one is for inductive fixed points. In our algebraic presentation: $\Gamma\vdash\mu{X}^{A\rightarrow\mathrm{Prop}}.\,F:A\rightarrow\mathrm{Prop}\lx@proof@logical@and\Gamma\vdash A:s\Gamma,X{:}A\rightarrow\mathrm{Prop}\vdash F:A\rightarrow\mathrm{Prop}{{\mbox{{sp}}}_{X}}\,{F}$ It allows to type the accessibility predicate $\mathrm{Acc}$ in $\mathrm{Prop}$. This rule is sound in that fixed points indeed preserve proof irrelevance in presence of functional extensionality. It is also very useful for extraction: structural recursion over $\mathrm{Acc}$ allows the definition of functions whose termination cannot be proved automatically by the guard condition. However, the proof is no longer needed to ensure termination in the target languages of extraction. In this sense, at least, it is static data. The last singleton rule allows properly indexed families in $\mathrm{Prop}$ (not how it is stronger than the rule dependent sum of Figure 5): $\Gamma\vdash{\sum}_{x{:}A}^{f}T:B\rightarrow\mathrm{Prop}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Prop}\Gamma\vdash B:\mathrm{Type}\Gamma,x{:}A\vdash T:\mathrm{Prop}\Gamma,x{:}A\vdash f:B$ It turns the identity type $\mathrm{Eq}$ into a proposition. It is known to be sound to accept that $\mathrm{Eq}$ is proof irrelevant [6]. It is also useful for extraction, as equal types, in a closed environment, are extracted to the same type. Hence a program may safely eliminate over $\mathrm{Eq}$ knowing that it will not affect the performances of the extracted code. In Coq, the index $B$ in the rule above can be of any sort ${\mathrm{Type}}_{i}$, however, this wisdom has been challenged in recent years with the formulation of the _univalence principle_ [15], of which a simple consequence is that $\mathrm{Eq}$ is _not_ proof irrelevant at every type. Indeed, some extracted Coq programs written assuming the univalence principle crash. To correct for the univalence principle, the singleton rule for proper indices can be simply dropped; but it can also be restricted to the lowest sort: $B:{\mathrm{Type}}_{0}$. More precisely the conjunction of the univalence principle and the proof irrelevance principle is consistent as long as the singleton rule of proper indices is restricted to sorts $s$ such that there is no sort $s^{\prime}$ other than $\mathrm{Prop}$ such that $s^{\prime}:s$. Because, if such a sort $s^{\prime}$ exists, $\mathbb{B}:s^{\prime}$ and by univalence, ${\mathrm{Eq}}\,{\mathbb{B}}$ has two distinct elements contradicting proof irrelevance. For types which do not enjoy singleton elimination, turning them into propositions means restricting their elimination. We achieve this effect by adding a single type construction coercing from $\mathrm{Type}$ to $\mathrm{Prop}$: $\Gamma\vdash\left[A\right]:\mathrm{Prop}\Gamma\vdash A:\mathrm{Type}$ $\Gamma\vdash\left\langle u\right\rangle:\left[A\right]\Gamma\vdash u:A$ $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}\left\langle y\right\rangle\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:{P}{\left[\,x\,{\setminus}\,u\,\right]}\lx@proof@logical@and\Gamma\vdash u:\left[A\right]\mbox{${\quad}$$\Gamma,x{:}\left[A\right]\vdash P:\mathrm{Prop}$${\quad}$}\Gamma,y{:}A\vdash v:{P}{\left[\,x\,{\setminus}\,\left\langle y\right\rangle\,\right]}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ \left\langle u\right\rangle\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}\left\langle y\right\rangle\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\leadsto{v}{\left[\,y\,{\setminus}\,u\,\right]}$ The important rule is the elimination rule, where the return clause is limited to be of sort $\mathrm{Prop}$, whereas every other type construction can be eliminated to any sort. Apart from this restriction $\left[A\right]$ is a synonym of $A$, except in $\mathrm{Prop}$. In [2], the type theory is extensional, in that the identity type and the conversion relation coincide. The elimination rules for bracket is much finer and reflects precisely the fact that propositions are proof-irrelevant. In an intensional type theory, restricting with respect to sorts approximates this behaviour: even if we constrained propositions to be proof-irrelevant, not every proof irrelevant type will have type $\mathrm{Prop}$. The bracketing construction is also strictly positive: ${{\mbox{{sp}}}_{X}}\,{\left[A\right]}{{\mbox{{sp}}}_{X}}\,{A}$ It is actually possible, using only the impredicative dependent product to define a bracketing operation: $\prod_{P{:}\mathrm{Prop}}\left(A\rightarrow P\right)\rightarrow P$. Like $\left[A\right]$ it behaves as $A$ except it can only be used to form a proposition. However, the impredicative encoding is positive but not strictly, which motivates the introduction of the extra construction. ### 4.3. Examples The logical connectives can be defined as follows: $\left\\{\begin{aligned} \mathrm{False}&=0\\\ \mathrm{And}&=\lambda{A}^{\mathrm{Prop}}\,{B}^{\mathrm{Prop}}.\,A\times B\\\ \mathrm{pair}&=\lambda{A}^{\mathrm{Prop}}\,{B}^{\mathrm{Prop}}\,{x}^{A}\,{y}^{B}.\,\left(x,y\right)\\\ \mathrm{Or}&=\lambda{A}^{\mathrm{Prop}}\,{B}^{\mathrm{Prop}}.\,\left[A+B\right]\\\ {\mathrm{or}}_{\mathrm{introl}}&=\lambda{A}^{\mathrm{Prop}}\,{B}^{\mathrm{Prop}}\,{x}^{A}.\,\left\langle{\mathrm{inl}}\,{x}\right\rangle\\\ {\mathrm{or}}_{\mathrm{intror}}&=\lambda{A}^{\mathrm{Prop}}\,{B}^{\mathrm{Prop}}\,{y}^{B}.\,\left\langle{\mathrm{inr}}\,{y}\right\rangle\\\ \mathrm{Ex}&=\lambda{A}^{\mathrm{Type}}\,{P}^{A\rightarrow\mathrm{Prop}}.\,\left[\sum_{x{:}A}{P}\,{x}\right]\\\ {\mathrm{ex}}_{\mathrm{intro}}&=\lambda{A}^{\mathrm{Type}}\,{P}^{A\rightarrow\mathrm{Prop}}\,{x}^{A}\,{p}^{{P}\,{x}}.\,\left\langle{\left(x,p\right)}_{{\mbox{\scriptsize{$x${:}$A$.$P$}}}}\right\rangle\end{aligned}\right.$ Note how, because of the brackets, existentials and disjunctions are prohibited from being eliminated to non-propositional types. Thanks to the singleton rules, however, conjunction and falsity do not require brackets. As a final example, consider the type Ascending n p of ascending sequences of integers between p and n defined by mutual recursion with the proposition Ge m p which stands from m is greater than or equal to p: Inductive Ascending : Nat ${\rightarrow}$ Nat ${\rightarrow}$ Type := ${\mid}$ top : forall n, Ascending n n ${\mid}$ up : forall n p m, Ge m (s p) ${\rightarrow}$ Ascending n m ${\rightarrow}$ Ascending n p with Ge : Nat ${\rightarrow}$ Nat ${\rightarrow}$ Prop := ${\mid}$ ascend : forall m p, Ascending m p ${\rightarrow}$ Ge m p. As Ascending has type Type, whereas Ge has type Prop, the translation to a single inductive type is not as straightforward as Even and Odd. The translation requires the use of brackets around the recursive calls: $\left\\{\begin{aligned} \mathrm{AscendingGe}&=\begin{array}[]{l}\mu{X}^{\left(\mathrm{Nat}\times\mathrm{Nat}\right)+\left(\mathrm{Nat}\times\mathrm{Nat}\right)\rightarrow\mathrm{Type}}.\,\lambda{i}^{\left(\mathrm{Nat}\times\mathrm{Nat}\right)+\left(\mathrm{Nat}\times\mathrm{Nat}\right)}.\,\mbox{\phantom{a}}\\\ \begin{array}[]{cl}&{\left({\sum}_{n{:}\mathrm{Nat}}^{{\mathrm{inl}}\,{\left(n,n\right)}}1\right)}\,{i}\\\ +&{\left({\sum}_{j{:}\mathrm{Nat}\times\mathrm{Nat}}^{{\mathrm{inl}}\,{j}}\sum_{m{:}\mathrm{Nat}}\left[{X}\,{\left({\mathrm{inr}}\,{\left(m,{s}\,{\left({{\pi}_{2}}\,{j}\right)}\right)}\right)}\right]\times{X}\,{\left({\mathrm{inl}}\,{\left({{\pi}_{1}}\,{j},m\right)}\right)}\right)}\,{i}\\\ +&{\left({\sum}_{j{:}\mathrm{Nat}\times\mathrm{Nat}}^{{\mathrm{inr}}\,{j}}{X}\,{\left({\mathrm{inl}}\,{j}\right)}\right)}\,{i}\\\ \end{array}\\\ \end{array}\\\ \mathrm{Ascending}&=\lambda{n}\,{p}.\,{\mathrm{AscendingGe}}\,{\left({\mathrm{inl}}\,{\left(n,p\right)}\right)}\\\ \mathrm{Ge}&=\lambda{m}\,{p}.\,\left[{\mathrm{AscendingGe}}\,{\left({\mathrm{inr}}\,{\left(m,p\right)}\right)}\right]\end{aligned}\right.$ ## 5\. Impredicative Set In addition to the impredicative sort $\mathrm{Prop}$, Coq has a sort $\mathrm{Set}$ which is predicative by default but can be turned impredicative with a flag. Where $\mathrm{Prop}$ is meant to be used in the context of separating static and dynamic information, the spirit of the impredicative sort $\mathrm{Set}$ is to be as powerful as possible without being inconsistent. In the algebraic presentation, that means being stable by every construction except dependent sums with the first projection in an arbitrary sort (_strong sums_). To mirror the optional nature of the impredicativity of $\mathrm{Set}$, the rules for a predicative sort $\mathrm{Set}$ are given in Figure 6; to turn impredicativity on, the rules of Figure 7 must be used _in addition_ to those of predicative $\mathrm{Set}$. This presentation makes immediately apparent that impredicative $\mathrm{Set}$ is an extension of predicative $\mathrm{Set}$, in that every program of the latter typechecks in the former. The rules of $\mathrm{Set}$ are the same as those of $\mathrm{Prop}$, with the exception of $A+B$ which is in $\mathrm{Set}$ when both $A$ and $B$ are – even with predicative $\mathrm{Set}$. Hence, there are types in $\mathrm{Set}$ with several elements – _e.g._ $\mathbb{B}$. As a consequence, the bracketing operation which coerces types in $\mathrm{Type}$ to $\mathrm{Set}$ does not enjoy an explanation in terms of proof irrelevance, as was the case in $\mathrm{Prop}$. As a matter of fact, there is no clear set-theoretical description at all. A close cousin of $\mathrm{Set}$ bracketing, however, can be found in homotopy type theory [15], where, roughly, groupoids are _truncated_ to sets through a quotient of their homsets by the total relation. $\Gamma\vdash\mathrm{Set}:\mathrm{Type}$ $\Gamma\vdash\prod_{x{:}A}B:\mathrm{Set}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Set}\Gamma,x{:}A\vdash B:\mathrm{Set}$ $\Gamma\vdash\sum_{x{:}A}P:\mathrm{Set}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Set}\Gamma,x{:}A\vdash P:\mathrm{Set}$ $\Gamma\vdash A+B:\mathrm{Set}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Set}\Gamma\vdash B:\mathrm{Set}$ $\Gamma\vdash 1:\mathrm{Set}$ $\Gamma\vdash 0:\mathrm{Set}$ $\Gamma\vdash\mu{X}^{A\rightarrow\mathrm{Set}}.\,F:A\rightarrow\mathrm{Set}\lx@proof@logical@and\Gamma\vdash A:s\Gamma,X{:}A\rightarrow\mathrm{Set}\vdash F:A\rightarrow\mathrm{Set}{{\mbox{{sp}}}_{X}}\,{F}$ $\Gamma\vdash{\sum}_{x{:}A}^{f}T:B\rightarrow\mathrm{Set}\lx@proof@logical@and\Gamma\vdash A:\mathrm{Set}\Gamma\vdash B:\mathrm{Set}\Gamma,x{:}A\vdash T:\mathrm{Set}\Gamma,x{:}A\vdash f:B$ $\Gamma\vdash\nu{X}^{A\rightarrow\mathrm{Set}}.\,F:A\rightarrow\mathrm{Set}\lx@proof@logical@and\Gamma\vdash A:s\Gamma,X{:}A\rightarrow\mathrm{Set}\vdash F:A\rightarrow\mathrm{Set}{{\mbox{{sp}}}_{X}}\,{F}$ $\Gamma\vdash{\left\\{A\right\\}}_{\mathrm{Set}}:\mathrm{Type}\Gamma\vdash A:\mathrm{Set}$ $\Gamma\vdash{\mathrm{elt}}\,{u}:{\left\\{A\right\\}}_{\mathrm{Set}}\Gamma\vdash u:A$ $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{elt}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:{P}{\left[\,x\,{\setminus}\,u\,\right]}\lx@proof@logical@and\Gamma\vdash u:{\left\\{A\right\\}}_{\mathrm{Set}}\Gamma,x{:}{\left\\{A\right\\}}_{\mathrm{Set}}\vdash P:s\Gamma,y{:}A\vdash v:{P}{\left[\,x\,{\setminus}\,{\mathrm{elt}}\,{y}\,\right]}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\mathrm{elt}}\,{u}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\mathrm{elt}}\,{y}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\leadsto{v}{\left[\,y\,{\setminus}\,u\,\right]}$ ${{\mbox{{sp}}}_{X}}\,{{\left\\{A\right\\}}_{\mathrm{Set}}}{{\mbox{{sp}}}_{X}}\,{A}$ Figure 6. Rules for predicative $\mathrm{Set}$ $\Gamma\vdash\prod_{x{:}A}B:\mathrm{Set}\lx@proof@logical@and\Gamma\vdash A:s\Gamma,x{:}A\vdash B:\mathrm{Set}$ ${{\mbox{{sp}}}_{X}}\,{{\left[A\right]}_{\mathrm{Set}}}{{\mbox{{sp}}}_{X}}\,{A}$ $\Gamma\vdash{\left[A\right]}_{\mathrm{Set}}:\mathrm{Set}\Gamma\vdash A:\mathrm{Type}$ $\Gamma\vdash{\left\langle u\right\rangle}_{\mathrm{Set}}:{\left[A\right]}_{\mathrm{Set}}\Gamma\vdash u:A$ $\Gamma\vdash\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ u\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\left\langle y\right\rangle}_{\mathrm{Set}}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}:{P}{\left[\,x\,{\setminus}\,u\,\right]}\lx@proof@logical@and\Gamma\vdash u:{\left[A\right]}_{\mathrm{Set}}\Gamma,x{:}{\left[A\right]}_{\mathrm{Set}}\vdash P:\mathrm{Set}\Gamma,y{:}A\vdash v:{P}{\left[\,x\,{\setminus}\,{\left\langle y\right\rangle}_{\mathrm{Set}}\,\right]}$ $\begin{array}[]{@{}l}\mathrm{match}\leavevmode\nobreak\ {\left\langle u\right\rangle}_{\mathrm{Set}}\leavevmode\nobreak\ \mathrm{as}\leavevmode\nobreak\ x\leavevmode\nobreak\ \mathrm{return}\leavevmode\nobreak\ P\leavevmode\nobreak\ \mathrm{with}\\\ \begin{array}[]{\|l@{~\Rightarrow~}l}{\left\langle y\right\rangle}_{\mathrm{Set}}\hfil\leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ &v\\\ \end{array}\\\ \end{array}\leadsto{v}{\left[\,y\,{\setminus}\,u\,\right]}$ Figure 7. Rules for impredicative $\mathrm{Set}$ ## 6\. Conclusion The algebraic presentation of Coq makes the conversion between sorts explicit. The toplevel inductive definitions of Coq can be understood as implicitly inserting canonical bracketing operations when an inductive type is declared inside an impredicative sort but should be of a different sort due to its form; and inserting type coercion from a smaller sort to a bigger sort when applying a cumulativity rule. Monolithic type definitions like in Coq have a number of advantages over the algebraic presentation, they boil down to better type errors due to naming, better type inference and better memory representation due to $n$-ary sums and products. However, the value of the implicit coercions between sorts is less clear. In particular, the bracketing operation to impredicative sorts is probably a better guide for program extraction than the current method of figuring whether or not a given type is a proposition, which interacts badly with universe polymorphism [7]. Explicit coercions for extraction are also in the spirit of [5]. All of the algebraic type constructors can actually be defined in Coq, except the two fixed-points because there is no way to abstract over strictly positive type families. So is it clear that expressions of the algebraic presentation which do not use inductive or co-inductive fixed points can be translated into Coq. Occurrences of the fixed points in a type must be ${\lambda}$-lifted and given a toplevel name. Some care must be given to avoiding the duplication of such definitions otherwise types which must be convertible for the expression to typecheck, might be seen as different in the Coq translation. Apart from this technicality, translation from the algebraic presentation to Coq is straightforward. We claim that, at least if we extend the algebraic presentation to a hierarchy of universes and the strict positivity condition is made a bit more fine-grained, Coq terms can be, conversely, translated into the algebraic presentation. ## References * [1] Andreas Abel. A polymorphic lambda-calculus with sized higher-order types. PhD thesis, 2006. * [2] Steve Awodey and Andrej Bauer. Propositions as [Types]. Journal of Logic and Computation, 14(4):447–471, August 2004. * [3] Henk Barendregt. Lambda calculus with types. Handbook of logic in computer science, 1992. * [4] Bruno Barras. Semantical Investigations in Intuitionistic Set Theory and Type Theories with Inductive Families. Thèse d’Habilitation, 2013. * [5] Bruno Barras and Bruno Bernardo. The implicit calculus of constructions as a programming language with dependent types. Foundations of Software Science and Computational Structures, 4962:365–379, 2008. * [6] Gyesik Lee and Benjamin Werner. Proof-irrelevant model of CC with predicative induction and judgmental equality. 2011\. * [7] Pierre Letouzey and Bas Spitters. Implicit and noncomputational arguments using monads. pages 1–16, 2005. * [8] Peter Morris. Constructing Universes for Generic Programming. PhD thesis, 2007. * [9] Peter Morris and Thorsten Altenkirch. Constructing strictly positive families. CATS ’07 Proceedings of the thirteenth Australasian symposium on Theory of computing, pages 111–121, 2007. * [10] Peter Morris, Thorsten Altenkirch, and Neil Ghani. A universe of strictly positive families. IInternational journal of foundations of computer science, pages 83–107, 2009. * [11] Ulf Norell. Towards a practical programming language based on dependent type theory. PhD thesis, Chalmers University of Technology, 2007. * [12] Chris Okasaki. Purely functional data structures. 1999\. * [13] Christine Paulin-Mohring. Définitions inductives en théorie des types d’ordre supérieur. PhD thesis, Université Claude Bernard-Lyon I, 1996. * [14] The Coq development team. The Coq Proof Assistant. * [15] The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Mathematics.	arxiv-papers	2013-09-23T11:14:12	2024-09-04T02:49:51.363899	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hugo Herbelin and Arnaud Spiwack", "submitter": "Arnaud Spiwack", "url": "https://arxiv.org/abs/1309.5767" }
1309.5853	11institutetext: Space Weather Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD, USA 11email: [email protected]; [email protected] 22institutetext: Max-Planck-Institut für Sonnensystemforschung, Max-Planck- Strasse 2, 37191 Katlenburg-Lindau, Germany 22email: [email protected] 33institutetext: National Solar Observatory, Sunspot, NM 88349, USA. 33email: [email protected]; [email protected] # First use of synoptic vector magnetograms for global nonlinear force free coronal magnetic field models Tilaye Tadesse 11 T. Wiegelmann 22 S. Gosain 33 P. MacNeice 11 Alexei A. Pevtsov 33 (Received ******** / Accepted *****) ###### Abstract Context. The magnetic field permeating the solar atmosphere is generally thought to provide the energy for much of the activity seen in the solar corona, such as flares, coronal mass ejections (CMEs), etc. To overcome the unavailability of coronal magnetic field measurements, photospheric magnetic field vector data can be used to reconstruct the coronal field. Currently there are several modelling techniques being used to calculate three-dimension of the field lines into the solar atmosphere. Aims. For the first time, synoptic maps of photospheric vector magnetic field synthesized from Vector Spectromagnetograph (VSM) on Synoptic Optical Long- term Investigations of the Sun (SOLIS) are used to model the coronal magnetic field and estimate free magnetic energy in the global scale. The free energy (i.e., the energy in excess of the potential field energy) is one of the main indicators used in space weather forecasts to predict the eruptivity of active regions. Methods. We solve the nonlinear force-free field equations using optimization principle in spherical geometry. The resulting three-dimensional magnetic fields are used to estimate the magnetic free energy content $E_{\mathrm{free}}=E_{\mathrm{nlfff}}-E_{\mathrm{pot}}$, i.e., the difference of the magnetic energies between the nonpotential field and the potential field in the global solar corona. For comparison, we overlay the extrapolated magnetic field lines with the extreme ultraviolet (EUV) observations by the Atmospheric Imaging Assembly on board SDO. Results. For a single Carrington rotation 2121, we find that the global NLFFF magnetic energy density is $10.3\%$ higher than the potential one. Most of this free energy is located in active regions. Conclusions. ###### Key Words.: Magnetic fields – Sun: corona – Sun: photosphere – methods: numerical ## 1 Introduction In the solar coronal plasma, magnetic energy is the prime energy reservoir that fuels the dynamical evolution of eruptive events, but it remains an open question how the magnetic energy is released. The amount of energy associated with the magnetic field is much larger than other energy sources, and the dynamics of the coronal configuration is determined by the evolution of its magnetic field (Forbes 2000; Low 2001). Free energy is defined as excess of energy as compared with potential field. In fact, one can in principle release more energy than free energy by, for example, annihilating magnetic field. Also, numerically, one can get pre-post flare difference in energy larger than free magnetic energy if magnetic field at the photosphere changes (e.g., some flux elements disappear). It quantifies the energy deviation of the coronal magnetic field from its potential state (Metcalf et al. 2005; Régnier 2007; Aschwanden 2012). The magnetic free energy is stored in the form of electric currents flowing along the magnetic field. Free magnetic energy of solar solar magnetic fields can be affected by several processes such as, e.g., photospheric shearing flows, magnetic flux emergence and magnetic reconnection (e.g., Welsch 2006; Fang et al. 2012). To understand the role that the magnetic field plays in energizing the solar corona, it is important to calculate the amount of free energy in order to quantify the energy budget in a catastrophic energy release event, as well as for estimating upper limits in forecasting individual events in real-time. Using various extrapolation techniques for the coronal magnetic field under the assumption of force-free fields, the spatial and temporal evolution of the coronal magnetic free energy during solar flares has been extensively studied (Régnier & Canfield 2006; Guo et al. 2008; Jing et al. 2010; Tadesse et al. 2012a; Meyer et al. 2013). From the 3D coronal magnetic configurations, we can derive the magnetic energy in the corona as: $E_{\mathrm{M}}=\frac{1}{8\pi}\int_{V}\textbf{B}\cdot\textbf{B}\,\,r^{2}\sin\theta drd\theta d\phi$ (1) The free magnetic energy in spherical geometry is calculated by computing the nonpotential field $B_{\mathrm{nlfff}}(r,\theta,\phi)$ with a numerical nonlinear force-free field (NLFFF) code and a potential field $B_{\mathrm{pot}}(r,\theta,\phi)$ for the same photospheric boundary data, so that the difference of the magnetic field energy density in the coronal volume V encompassing the active regions of interest can be quantified as $E_{\mathrm{M_{free}}}=E_{\mathrm{M_{nlfff}}}-E_{\mathrm{M_{pot}}}$. Since the corona is optically thin, direct measurements of the 3-D magnetic field are very difficult to implement and interpret. Therefore, the present observations for the magnetic fields based on the spectropolarimetric method (the Zeeman and the Hanle effects) are limited to low layers of solar atmosphere (photosphere and chromosphere). Even if direct measurement techniques for the 3-D magnetic fields in the chromosphere and the corona have considerably improved in recent decades (Lin et al. 2000, 2004; Liu & Lin 2000), further developments are needed before accurate data are routinely available. Thus, the problem of measuring the coronal field and its embedded electrical currents leads us to use numerical modelling to infer the field strength in the higher layers of the solar atmosphere from the measured photospheric field. Force-free extrapolation of photospheric magnetic fields is currently used as the primary tool for the modeling of coronal magnetic fields. In this model assumption, the corona magnetic forces are dominant so that all non-magnetic forces like plasma pressure gradient and gravity can be neglected in the lowest order (Gary 2001). This implies that, if appreciable currents are present, these must be aligned with the magnetic field, since otherwise the resulting Lorentz forces could not be balanced by the nonmagnetic forces. The equilibrium structure of the static coronal magnetic field can be described using the force-free assumption as: $(\nabla\times\textbf{B})=\alpha\textbf{B}\Rightarrow(\nabla\times\textbf{B})\times\textbf{B}=0$ (2) $\nabla\cdot\textbf{B}=0$ (3) where B is the magnetic field. The force-free parameter $\alpha$ of Equation (2) can be a function of position, but the combination of Equation (2) and (3) ($\textbf{B}\cdot\nabla\alpha=0$) requires that $\alpha$ be constant along a given field line. Potential ($\alpha=0$) and linear force-free fields (whenever $\alpha$ is constant everywhere in the volume under consideration) can be used as a first step to model the general structure of magnetic fields in the solar corona. Practically the pre-eruptive magnetic fields are nonlinear force-free fields ($\alpha$ being a function of position) as supported by both observational and theoretical reasons. For details of those models we direct the readers to Wiegelmann & Sakurai (2012). Nonlinear force-free field codes have been routinely applied to the reconstruction of the coronal field of a single active region using the Cartesian geometry. In that case, the curvature of the solar surface does not play an important role. Solar Dynamics Observatory (SDO) mission has made repeated observations of large scale events in which connections between widely separated active regions may play fundamental role (Martens et al. 2012). Even before SDO, it was known that large-scale connectivity is important for solar eruptive and non-eruptive activity (e.g., studies of sympathetic flares, transequatorial loops (Pevtsov 2000), effects of distant active regions on large-scale coronal brightness (Pevtsov & Acton 2001), eruption of filaments triggered by remote flux emergence and evolution (e.g., Balasubramaniam et al. 2011). Therefore, this needs motivate us to implement a NLFFF procedure in spherical geometry (Wiegelmann 2007; Tadesse et al. 2009, 2011, 2012a, 2012b; Guo et al. 2012; Tadesse et al. 2013a; Amari et al. 2013) In this study, we estimate the free magnetic energy for global corona using data from SOLIS/VSM. We compare the extrapolated potential and NLFFF magnetic loops with extreme ultraviolet (EUV) observations by the Atmospheric Imaging Assembly (AIA) on board SDO. This comparison helps to identify whether the NLFFF model reconstructs the magnetic configuration better than the potential field model in the global scale. In this paper, we present some descriptions of the dataset used for analysis in Section 2. The spherical optimization procedure used for modeling 3-D magnetic field in global corona is presented in Section 3. Then, we present results of our studies in Section 4. A summary and discussions are finally presented in Section 5. ## 2 Instrumentation and data set In this paper, we use the first ever synoptic vector magnetograms, i.e., Carrington maps of the three components of the magnetic field vector, the radial $B_{r}$, the poloidal $B_{\theta}$, and the toroidal $B_{\phi}$, to perform first global nonlinear force free field extrapolation based on optimization algorithm (Wiegelmann 2007; Tadesse et al. 2009). The synoptic vector field maps are derived using daily full disk photospheric vector magnetograms from Vector Spectromagnetograph (VSM) instrument of Synoptic Optical Long-term Investigations of the Sun (SOLIS), a synoptic observing facility (Keller et al. 2003; Balasubramaniam & Pevtsov 2011). Further detail about these synoptic maps and their properties can be found in Gosain et al. (2013). Here we briefly describe the instrument characteristics and the full disk vector field observations. VSM routinely obtains full disk magnetograms in photospheric and chromospheric lines as a part of the synoptic program of National Solar Observatory i.e., NSO Integrated Synoptic Program (NISP). In order to obtain full disk photospheric vector magnetograms SOLIS/VSM measures Stokes $(S=I,Q,U,V)$ profiles in photospheric Fe I $630.15-630.25$ nm line pair. A single full disk scan (2048 scan lines) takes only about 20 minutes, thanks to the long slit of the spectrograph which intersects the solar disk from one limb to another in one shot. The spatial sampling is 1 arc-sec per pixel with square pixels. The spectral sampling is 2.4pm per pixel. A single Stokes set $(I,Q,U,V)$ per slit position is obtained typically in about 0.6 second. The telescope itself is designed to be free of instrumental polarization by employing symmetric optical configuration and performing polarization modulation just after the prime focus, after the slit. Dual beam analysis of polarization using polarizing beam splitter avoids seeing induced cross talks in the signal. The polarimeter calibration is done routinely to calibrate the Stokes vector for cross-talks. The signal to noise ratio (SNR) in the continuum of the Stokes profiles is typically $>1000$. The magnetic field vector is inferred from the calibrated Stokes profiles by performing inversion in the framework of Milne- Eddington model for stellar atmosphere following Unno-Rachkovsky formalism (Skumanich & Lites 1987). Only pixels with polarization signal above the threshold of $0.1\%$ of continuum intensity, Ic, are inverted to obtain the magnetic (field strength, inclination angle, and azimuth angle) and thermodynamic (e.g., Doppler width, Doppler velocity, source function, temperature) parameters. The threshold of $0.1\%$ of Ic corresponds to typical noise level in the continuum. Using this threshold avoids fitting profiles buried in the noise. Further details about instrument and pipeline reduction steps can be found elsewhere (e.g., Jones et al. 2002; Henney et al. 2006; Balasubramaniam & Pevtsov 2011). The noise in SOLIS magnetograms is estimated to be few Gauss in longitudinal and 70 G in the transverse field measurements (Tadesse et al. 2013a). The 180 degree azimuth ambiguity is resolved using a different (faster) ambiguity resolution method developed by Rudenko & Anfinogentov (2011). Figure 1: Synoptic Carrington maps of the vector magnetic field components are shown for CR-2121. The panels from top to bottom show the distribution of the $B(r)$, $B(\phi)$ and $B(\theta)$ components, respectively. The $B_{r}$ map is scaled between $\pm$ 100 G, and the $B_{\phi}$ and $B_{\theta}$ maps are scaled to $\pm$ 20 G. The positive values of $B_{r}$, $B_{\phi}$ and $B_{\theta}$ point, respectively, upward, to the right (westward) and southward. ## 3 Magnetic field modeling For modelling the coronal magnetic field in a global scale, we use the variational principle originally proposed by Wheatland et al. (2000). Wiegelmann (2007) has developed an optimization method to reconstruct the NLFFF for global solar corona by minimizing an objective functional $L$ that combines Lorentz forces and the divergence of the magnetic field in spherical geometry. The code has been tested with semi-analytic force-free solutions (Low & Lou 1990). If the functional is minimized to zero, Equations (2) and (3) are satisfied simultaneously. The optimization procedure in the spherical geometry has been implemented by Tadesse et al. (2009), for restricted area with large field of views. Later Tadesse et al. (2011) modified the objective functional of the optimization method (Wiegelmann & Inhester 2010) for spherical geometry as $L=L_{f}+L_{d}+\nu L_{\mathrm{photo}}$ (4) $L_{\mathrm{f}}=\int_{V}B^{-2}\big{\|}(\nabla\times{\textbf{B}})\times{\textbf{B}}\big{\|}^{2}r^{2}\sin\theta drd\theta d\phi$ $L_{\mathrm{d}}=\int_{V}\big{\|}\nabla\cdot{\textbf{B}}\big{\|}^{2}r^{2}\sin\theta drd\theta d\phi$ $L_{\mathrm{photo}}=\int_{S}\big{(}\textbf{B}-\textbf{B}_{\mathrm{obs}}\big{)}\cdot\textbf{W}(\theta,\phi)\cdot\big{(}\textbf{B}-\textbf{B}_{\mathrm{obs}}\big{)}r^{2}\sin\theta d\theta d\phi$ where $L_{f}$ and $L_{d}$ measure how well the force-free Eqs. (2) and divergence-free (3) conditions are fulfilled, respectively. The main reason for modification of the code was that we need to deal with boundary data of different noise levels and qualities or even miss some data points completely. Hence, the third integral, $L_{\mathrm{photo}}$, is the surface integral over the photosphere which allows us to relax the field on the photosphere towards force-free solution without too much deviation from the original surface field data. Figure 2: Global field lines of (a) the potential field model and (b) the NLFFF model overlaid on the AIA 171Å image. Green and red lines represent open and closed magnetic field lines, respectively. SOLIS/VSM provides full disk vector magnetograms from which Synoptic Carrington maps of the vector magnetic field components are synthesized. However, for pixels below the polarization threshold the inversion was not performed and field data there will be missing for these pixels (see Figure 1). Typically, the field is missing where its magnitude is small; thus these pixels would have a small impact on the model even if they were measured correctly. Within the error margin of a measured field value, any value is just as good as any other, and from this range of values we take the value that fits the force-free field best. In order to treat those pixels with missing data, we used the diagonal matrix, $\textbf{W}(\theta,\phi)$, which gives different weights to the observed surface field components depending on the relative accuracy in measurement. In this sense, missing data is considered most inaccurate and is taken into account by setting $\textbf{W}(\theta,\phi)$ to zero in all elements of the matrix. Photospheric magnetic field has a plasma-$\beta$ of order unity, which does not satisfy the force-free condition (Gary 2001). Therefore, the vector magnetogram data are inconsistent with the force-free assumption, which is absolutely essential condition for NLFFF extrapolation. To find suitable boundary conditions for the NLFFF field modeling, we have to preprocess the measured synoptic vector magnetograms by using a preprocessing scheme developed by Tadesse et al. (2009) in spherical geometry. This preprocessing scheme removes forces and torques from the boundary and approximates the photospheric magnetic field to the low plasma-$\beta$ force-free chromosphere. For a detailed description of the current code implementation, we refer to Wiegelmann (2007) and Tadesse et al. (2011). ## 4 Results This study requires extrapolating the three-dimensional potential and NLFFF coronal fields from the photospheric boundary in global scale. We use synoptic maps of photospheric vector magnetic field observed during 4-31 March, 2012. During this observation there were about 25 active regions all over the solar globe. In order to use our spherical optimization code for global corona, we adopt a uniform spherical grid $r$, $\theta$, $\phi$ with $n_{r}=300$, $n_{\theta}=450$, and $n_{\phi}=900$ grid points in the direction of radius, latitude, and longitude, respectively, with the field of view of $[r_{\rm{min}}=1R_{\odot}:r_{\rm{max}}=2.5R_{\odot}]\times[\theta_{\rm{min}}=-85^{\circ}:\theta_{\rm{max}}=85^{\circ}]\times[\phi_{\rm{min}}=0^{\circ}:\phi_{\rm{max}}=360^{\circ}]$. The code solves the NLFFF equations in the bounded domain between $1R_{\odot}$ and the source surface at $2.5R_{\odot}$. The outer boundary is kept fixed using the initial potential field values. All current-carrying field lines have to close inside the volume. The domain outside $2.5R_{\odot}$ is not included in the model, because the force-free approach is no longer justified here. The magnetic field extrapolation has been carried out almost for global corona by excluding the polar regions where the magnetic field measurements are highly being influenced by noise. Before performing NLFFF extrapolations, we use the preprocessed radial magnetic field component ${\bf{B}}_{r}$ to compute the corresponding potential field using a spherical harmonic expansion for initializing our spherical NLFFF code during relaxation towards a force- free state in the computational volume. Figure 3: Field lines of (a) the potential field model and (b) the NLFFF model around ARs 11429 and 11430 overlaid on the AIA 171Å image. Green and red lines represent open and closed magnetic field lines, respectively. Figure 4: The magnetic field line skeletons (a) of the entire solar globe from the NLFFF model and image of the sun observed by SOHO/LASCO C2 coronagraph at 16:33UT. The main purpose of this work is to study the structures of the global potential and NLFFF magnetic fields and to estimate free magnetic energy available to power solar eruptions during Carrington rotation 2121. In addition, we compare which of those two models is best agree with observation in global environment. To do this, we plot the selected fieldlines of the potential and NLFFF models in Figure 2. We overlay the field lines with an AIA 171 Å image. The field lines of the potential and NLFFF models are reconstructed from the same footpoints. The potential field lines in Figure 3(a) have an obvious deviation from the observed EUV loops, since the projection of the field lines indicated by blue and black arrows divate from EUV loops. However, those loops are best overlaid by NLFFF lines than potential ones (see Figure 3(b)). Therefore, the qualitative comparison between the model magnetic field lines and the observed EUV loops indicates that the NLFFF model provides a more consistent field for global corona magnetic field reconstruction. Figure 4 shows that there is over all similarity between the corresponding NLFFF model field lines and image of the sun observed by SOHO/LASCO C2 coronagraph. However, one can see that NLFFF does not represent well a linear structure above coronal helmet in low-left corner in Figure 3(b). This could be due to the missing data from the polar region. Figure 5: Evolution of the entire functional L (blue line) and its three term in Eq. 4 during the optimization process. The black line corresponds to $L_{\rm photo}$, the red line to $L_{\rm f}$, and the green line to $L_{\rm d}$. In addition to the above comparison to quantify the degree of disagreement between the two model vector field solutions in the global corona volume that are specified on the identical sets of grid points, we use the vector correlation metric ($C_{\rm vec}$) which is also used analogous to the standard correlation coefficient for scalar functions. The correlation was calculated (Schrijver et al. 2006) from $C_{\mathrm{vec}}=\frac{\sum_{i}\textbf{v}_{i}\cdot\textbf{u}_{i}}{\Big{(}\sum_{i}\|\textbf{v}_{i}\|^{2}\sum_{i}\|\textbf{u}_{i}\|^{2}\Big{)}^{1/2}},$ (5) where $\textbf{v}_{i}$ and $\textbf{u}_{i}$ are the vectors at each grid point $i$. If the vector fields are identical, then $C_{\rm vec}=1$; if $\textbf{v}_{i}\perp\textbf{u}_{i}$ , then $C_{\rm vec}=0$. The degree of convergence towards a force-free and divergence-free model solution can be quantified by the integral measures of the Lorentz force and divergence terms in the minimization functional in Equation (4), computed over the entire solar globe. $L_{\rm f}$ and $L_{\rm d}$ of Equation (4) measure how well the force- free and divergence-free conditions are fulfilled, respectively. In Table 1, we provide some quantitative measures to rate the quality of our reconstruction. Column $1$ names the corresponding models. Columns $2-3$ show how well the force-balance and solenoidal conditions are fulfilled for both models. Figure 5 shows how well the functional $L$ converge to zero during iteration process. In the last column, the vector correlation shows that there is disagreement between the two model field solutions. The energy stored in the magnetic field as a result of field line stressing into a nonpotential configuration has been identified as the source of flare energy. Therefore, to understand the physics of solar flares, including the local reorganisation of the magnetic field and the acceleration of energetic particles, we have first to estimate the free magnetic energy available for such phenomena. This free magnetic energy can be converted into kinetic and thermal energy. We estimate the free magnetic energy, the difference between the extrapolated NLFFF and the potential field with the same normal boundary conditions in the photosphere. We therefore estimate the upper limit to the free magnetic energy associated with coronal currents of the form $E_{\mathrm{free}}=\frac{1}{8\pi}\int_{V}\Big{(}{B^{2}}_{\mbox{nlff}}-{B^{2}}_{\mbox{pot}}\Big{)}dV\,\,$ (6) ${B}_{\mbox{pot}}$ and ${B}_{\mbox{nlff}}$ represent the potential and NLFFF magnetic field, respectively. Our result for the estimation of free-magnetic energy in Table 2 shows that the NLFFF model has $10.3\%$ more energy than the corresponding potential field model. Figure 6 shows iso-surface plot of free magnetic energy density in the volume above synoptic map. There are strong free energy concentrations above each active region over the solar globe. Figure 6: Iso-surfaces (ISs) of free magnetic energy density ($8.5\times 10^{20}$ erg) computed within the entire computational domain. Table 1: Evaluation of the reconstruction quality for the potential field and NLFFF models. We have used spherical grids of $300\times 450\times 900$. Model \| $L_{\rm f}$ \| $L_{\rm d}$ \| $L_{\rm photo}$ \| $C_{\rm vec}$ ---\|---\|---\|---\|--- Potential \| $0.000$ \| $0.000$ \| $0.001$ \| $1$ NLFFF \| $0.391$ \| $0.697$ \| $0.302$ \| $0.893$ Table 2: The magnetic energy associated with the 3-D potential and NLFFF field configurations calculated from synoptic vector magnetogram. Model \| $E_{\rm{total}}(10^{33}{\rm{erg}})$ \| $E_{\rm{free}}(10^{33}{\rm{erg}})$ ---\|---\|--- Potential \| $41.6$ \| $0$ NLFFF \| $46.4$ \| $4.8$ ## 5 Conclusion and outlook Most of the NLFFF procedures are implemented in the Cartesian coordinates. Therefore, both potential and nonlinear force-free field (NLFFF) codes in Cartesian geometry are not well suited for larger domains, since the spherical nature of the solar surface cannot be neglected when the field of view is large. Therefore, it is necessary to implement a NLFFF procedure in spherical geometry for use when large-scale boundary data are in use. In this study, we have investigated the coronal magnetic field and free magnetic energy associated with global corona by analyzing Carrington synoptic maps of photospheric vector magnetic field synthesized from Vector Spectromagnetograph (VSM) on Synoptic Optical Long-term Investigations of the Sun (SOLIS) which has been observed during 4-31 March, 2012. The Carrington rotation number for this observation is 2121. During this particular observation, there were about ten active regions distributed across the globe. We have used our spherical NLFFF and potential codes to compute the magnetic field solutions over global corona. This is the first NLFFF magnetic field extrapolation for the global corona using real data. We have compared the magnetic field solutions from both potential and NLFFF models. The qualitative comparison between the model magnetic field lines and the observed EUV loops indicates that the NLFFF model provides a more consistent field for global corona magnetic field reconstruction. For this particular Carrington rotation we find that the global NLFFF magnetic energy density is $10.3\%$ higher than the potential one. For future, we have a plan to study the evolution of global free magnetic energy. In this study, most of this free energy is located in active regions. ###### Acknowledgements. This work utilizes SOLIS data obtained by the NSO Integrated Synoptic Program (NISP), managed by the National Solar Observatory, which is operated by the Association of Universities for Research in Astronomy(AURA), Inc. under a cooperative agreement with the National Science Foundation. This research was supported by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center (GSFC), administered by Oak Ridge Associated Universities through a contract with NASA. The work of T. Wiegelmann was supported by DLR- grant $50$ OC $453$ $0501$. ## References Amari et al. (2013) Amari, T., Aly, J. J., Canou, A. et al., 2013, A&A, 553, A43 * Aschwanden (2012) Aschwanden, M. J., 2012, Sol. Phys., preprint doi:10.1007/s11207-012-0203-6 * Balasubramaniam et al. (2011) Balasubramaniam, K. S., Pevtsov, A. A. & Cliver, E. W., et al., 2011, ApJ, 743, 202 * Balasubramaniam & Pevtsov (2011) Balasubramaniam, K. 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1309.5886	# Generalized three-intensity decoy state method for measurement device independent quantum key distribution Zong-Wen Yu1,2, Yi-Heng Zhou1, and Xiang-Bin Wang1,3111Email Address: [email protected] 1State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, People s Republic of China 2Data Communication Science and Technology Research Institute, Beijing 100191, China 3 Shandong Academy of Information and Communication Technology, Jinan 250101, People s Republic of China ###### Abstract We present improved results and more general results for the decoy-state measurement device independent quantum key distribution (MDI-QKD) where the intensities of all 3 pulses can be non-zero. We present a more tightened explicit formula for the lower bound of the yield of two single-photon pulses for this generalized 3-intensity protocol, which can be applied to the recently proposed MDI-QKD with imperfect single-photon sources such as the coherent states or the heralded states from the parametric down-conversion. We strictly prove that our result is better than the prior art result. With this result, the final key rate rises drastically. ###### pacs: 03.67.Dd, 42.81.Gs, 03.67.Hk ## I Introduction Recently, much attention has been spent on the security of real set-ups in quantum key distribution (QKD) BB84 ; GRTZ02 . After the proposal of the decoy state method ILM ; H03 ; wang05 ; LMC05 ; AYKI ; haya ; peng ; wangyang ; rep ; njp , the security of QKD with an imperfect single photon source is guaranteed given whatever Eve’s attack to the channel, including the famous photon-number splitting attack PNS1 ; PNS . However, the guaranteed security assumes that Eve cannot attack device inside Alice or Bob’s lab. As was demonstrated lyderson , given a lossy channel and limited detection efficiency, Eve can attack the gated-mode detectors inside Alice or Bob’s lab if there is no additional device to detect the abnormally strong light at Alice or Bob’s lab. To reach the goal of ultimate security, one can, in principle use the device independent protocol ind1 . However, these protocols seem to be technically demanding at the moment. Recently, measurement device independent QKD (MDI-QKD) was proposed based on the idea of entanglement swapping ind3 ; ind2 . If we want to obtain a higher key rate, we can choose to directly use an imperfect single-photon source such as the coherent state ind2 with decoy-state method for this, say the MDI decoy-state method. Calculation formulas for the practical decoy-state implementation with only a few different states has been studied in, e.g., Refs. wangPRA2013 ; qing3 , and then further studied both experimentally tittel1 ; tittel2 ; liuyang and theoretically lopa ; han ; curty ; Wang2013 . Theoretical study for the decoy-state MDI-QKD with only a few intensities are of particularly interesting. The first formula for 3-intensity decoy-state MDI-QKD was proposed in wangPRA2013 . It assumes that each user uses three different states, one vacuum state and two non-vacuum states. The results can apply to sources such as the coherent states, the parametric down conversion, and so on. It was then extended to the case that the 3 intensities are all non-vacuum coherent states lopa . One can see that the major formula there is identical with the one in Ref.wangPRA2013 in the case when Alice and Bob use coherent pulses and the weakest pulse is vacuum. In Ref.Wang2013 ; Wang201309 , we presented an improved explicit formulas of 3-state decoy-state method for the MDI-QKD with the weakest pulse being vacuum. Here in this work, we shall present an improved results for the decoy-state MDI-QKD where all 3 intensities are non-vacuum. Compared with the prior-art result lopa , our improved formulas here make a more tightened faithful estimation of the yield and the phase flip rate of the single-photon pulse pairs and hence can raise the final key rate drastically. Also, our result is for a general type of sources with a simple condition, it does not limit to the coherent-state source. In the second section, we present the improved explicit formulas for 3-intensity decoy-state MDI-QKD and prove analytically that our result is better than the previous lopa . In the third section, we give the numerical simulations. The article is ended with a concluding remark. ## II Generalized Three-intensity decoy state method for MDI-QKD In the protocol, each time a pulse-pair (two-pulse state) is sent to the relay for detection. The relay is controlled by an un-trusted third party (UTP). The UTP will announce whether the pulse-pair has caused a successful event. Those bits corresponding to successful events will be post-selected and further processed for the final key. Since real set-ups only use imperfect single- photon sources, we need the decoy-state method for security. We assume Alice (Bob) has three sources, $v_{A},d_{A},s_{A}$ ($v_{B},d_{B},s_{B}$) which can only emit three different states $\rho_{v_{A}},\rho_{d_{A}},\rho_{s_{A}}$ ($\rho_{v_{B}},\rho_{d_{B}},\rho_{s_{B}}$), respectively, in photon number space. It’s worth pointing out that our protocol is applicable to the weakest pulse being vacuum or not. Suppose $\left\\{\begin{array}[]{lcr}\rho_{v_{A}}=\sum_{k}a^{v}_{k}\|k\rangle\langle k\|,&&\rho_{v_{B}}=\sum_{k}b^{v}_{k}\|k\rangle\langle k\|,\\\ \rho_{d_{A}}=\sum_{k}a^{d}_{k}\|k\rangle\langle k\|,&&\rho_{d_{B}}=\sum_{k}b^{d}_{k}\|k\rangle\langle k\|,\\\ \rho_{s_{A}}=\sum_{k}a^{s}_{k}\|k\rangle\langle k\|,&&\rho_{s_{B}}=\sum_{k}b^{s}_{k}\|k\rangle\langle k\|.\end{array}\right.$ (1) At each time, Alice will randomly select one of her 3 sources to emit a pulse, and so does Bob. The pulse form Alice and the pulse from Bob form a pulse pair and are sent to the UTP. We regard equivalently that each time a two-pulse source is selected and a pulse pair (one pulse from Alice, one pulse from Bob) is emitted. There are many different two-pulse sources used in the protocol. We denote $\alpha\beta$ for the two pulse source when the pulse-pair is produced by source $\alpha$ at Alice’s side and source $\beta$ at Bob’s side, $\alpha$ can be one of $\\{v_{A},d_{A},s_{A}\\}$ and $\beta$ can be one of $\\{v_{B},d_{B},s_{B}\\}$. For example, at a certain time $j$ Alice uses source $v_{A}$ and Bob uses source $s_{B}$, we say the pulse pair is emitted by source $v_{A}s_{B}$. In the protocol, two different bases (e.g., $Z$ basis consisting of horizontal polarization $\|H\rangle\langle H\|$ and vertical polarization $\|V\rangle\langle V\|$, and $X$ basis consisting of $\pi/4$ and $3\pi/4$ polarizations) are used. The density operator in photon number space alone does not describe the state in the composite space. We shall apply the decoy-state method analysis in the same basis (e.g., $Z$ basis or $X$ basis) for pulses from sources $\alpha\beta(\alpha\in\\{v_{A},d_{A},s_{A}\\},\beta\in\\{v_{B},d_{B},s_{B}\\})$. Therefore we only need consider the density operators in the photon number space. According to the decoy-state theory, the yield of a certain set of pulse pairs is defined as the happening rate of a successful event (announced by the UTP) corresponding to pulse pairs out of the set. We use capital letter $Y_{\alpha\beta}$ to denote the nine yields for sources $\alpha\beta(\alpha\in\\{v_{A},d_{A},s_{A}\\},\beta\in\\{v_{B},d_{B},s_{B}\\})$ and these values can be directly calculated from the observed experimental data. We use $y_{mn}$ to denotes the yield given that Alice and Bob send respectively an $m$-photon and $n$-photon pulses. Our first major task is to deduce $y_{11}$ from the known values, i.e., to formulate $y_{11}$, the yield of state $\|1\rangle\langle 1\|\otimes\|1\rangle\langle 1\|$ in the nine known values $\\{Y_{\alpha\beta}\|\alpha\in\\{v_{A},d_{A},s_{A}\\},\beta\in\\{v_{B},d_{B},s_{B}\\}\\}$. Using the convex proposition, we have the following relations $Y_{\alpha\beta}=\sum_{k,l\geq 0}a^{\alpha}_{k}b^{\beta}_{l}y_{kl},$ (2) where $\alpha\in\\{v_{A},d_{A},s_{A}\\}$ and $\beta\in\\{v_{B},d_{B},s_{B}\\}$. In practice, the imperfect single-photon sources have the property that the probability of emitting an empty pulse is greater than the probability of the no-empty pulse. With this fact, in order to estimate the lower bound of $y_{11}$, we need to eliminate the unknown values $y_{0l}(l\geq 0)$ and $y_{k0}(k\geq 0)$ firstly. Without causing any ambiguity, we omit the subscripts A and B in the following of this paper. Cancel out the terms $y_{0k}$ and $y_{j0}$ using Gaussian elimination, we obtain the following four relations $\displaystyle\left\\{\begin{array}[]{c}a_{0}^{v}b_{0}^{v}\hat{Y}_{dd}-a_{0}^{v}b_{0}^{d}\hat{Y}_{dv}-a_{0}^{d}b_{0}^{v}\hat{Y}_{vd}+a_{0}^{d}b_{0}^{d}\hat{Y}_{vv}=0,\\\ a_{0}^{v}b_{0}^{v}\hat{Y}_{ds}-a_{0}^{v}b_{0}^{s}\hat{Y}_{dv}-a_{0}^{d}b_{0}^{v}\hat{Y}_{vs}+a_{0}^{d}b_{0}^{s}\hat{Y}_{vv}=0,\\\ a_{0}^{v}b_{0}^{v}\hat{Y}_{sd}-a_{0}^{v}b_{0}^{d}\hat{Y}_{sv}-a_{0}^{s}b_{0}^{v}\hat{Y}_{vd}+a_{0}^{s}b_{0}^{d}\hat{Y}_{vv}=0,\\\ a_{0}^{v}b_{0}^{v}\hat{Y}_{ss}-a_{0}^{v}b_{0}^{s}\hat{Y}_{sv}-a_{0}^{s}b_{0}^{v}\hat{Y}_{vs}+a_{0}^{s}b_{0}^{s}\hat{Y}_{vv}=0,\end{array}\right.$ (7) where $\hat{Y}_{ij}=Y_{ij}-\sum_{k,l\geq 1}a_{k}^{i}b_{l}^{j}y_{kl},(i,j\in\\{v,d,s\\})$. For the sake of convenience, we denote $\displaystyle\left\\{\begin{array}[]{c}\tilde{Y}_{dd}^{\prime}=a_{0}^{v}b_{0}^{v}{Y}_{dd}-a_{0}^{v}b_{0}^{d}{Y}_{dv}-a_{0}^{d}b_{0}^{v}{Y}_{vd}+a_{0}^{d}b_{0}^{d}{Y}_{vv},\\\ \tilde{Y}_{ds}^{\prime}=a_{0}^{v}b_{0}^{v}{Y}_{ds}-a_{0}^{v}b_{0}^{s}{Y}_{dv}-a_{0}^{d}b_{0}^{v}{Y}_{vs}+a_{0}^{d}b_{0}^{s}{Y}_{vv},\\\ \tilde{Y}_{sd}^{\prime}=a_{0}^{v}b_{0}^{v}{Y}_{sd}-a_{0}^{v}b_{0}^{d}{Y}_{sv}-a_{0}^{s}b_{0}^{v}{Y}_{vd}+a_{0}^{s}b_{0}^{d}{Y}_{vv},\\\ \tilde{Y}_{ss}^{\prime}=a_{0}^{v}b_{0}^{v}{Y}_{ss}-a_{0}^{v}b_{0}^{s}{Y}_{sv}-a_{0}^{s}b_{0}^{v}{Y}_{vs}+a_{0}^{s}b_{0}^{s}{Y}_{vv},\end{array}\right.$ (12) and introduce the following notations $h_{x_{k}}^{i}=\frac{x_{k}^{i}}{x_{0}^{i}},\quad(k\geq 0;x\in\\{a,b\\};i\in\\{v,d,s\\}).$ (13) Divide the four relations in Eq.(7) by the positive factors $a_{0}^{v}a_{0}^{d}b_{0}^{v}b_{0}^{d}$, $a_{0}^{v}a_{0}^{d}b_{0}^{v}b_{0}^{s}$, $a_{0}^{v}a_{0}^{s}b_{0}^{v}b_{0}^{d}$ and $a_{0}^{v}a_{0}^{s}b_{0}^{v}b_{0}^{s}$ respectively, we obtain the following simplified equation $\displaystyle\left\\{\begin{array}[]{c}\tilde{Y}_{dd}=\sum_{k,l\geq 1}(h_{a_{k}}^{d}-h_{a_{k}}^{v})(h_{b_{l}}^{d}-h_{b_{l}}^{v})y_{kl},\\\ \tilde{Y}_{ds}=\sum_{k,l\geq 1}(h_{a_{k}}^{d}-h_{a_{k}}^{v})(h_{b_{l}}^{s}-h_{b_{l}}^{v})y_{kl},\\\ \tilde{Y}_{sd}=\sum_{k,l\geq 1}(h_{a_{k}}^{s}-h_{a_{k}}^{v})(h_{b_{l}}^{d}-h_{b_{l}}^{v})y_{kl},\\\ \tilde{Y}_{ss}=\sum_{k,l\geq 1}(h_{a_{k}}^{s}-h_{a_{k}}^{v})(h_{b_{l}}^{s}-h_{b_{l}}^{v})y_{kl}.\end{array}\right.$ (18) where $\tilde{Y}_{ij}$ are defined as follows $\tilde{Y}_{ij}=\frac{\tilde{Y}_{ij}^{\prime}}{a_{0}^{v}a_{0}^{i}b_{0}^{v}b_{0}^{j}},\quad(i,j\in\\{d,s\\}),$ (19) with $\tilde{Y}_{ij}^{\prime}$ being presented in Eq.(12). Furthermore, if we introduce $\tilde{h}_{x_{k}}^{i}=h_{x_{k}}^{i}-h_{x_{k}}^{v},\quad(k\geq 0;x\in\\{a,b\\};i\in\\{v,d,s\\}),$ (20) with $h_{x_{k}}^{i}$ defined in Eq.(13), we can write the relations about variables $y_{kl}$ given by Eq.(18) into the following simplified form $\displaystyle\tilde{Y}_{dd}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{d}y_{11}+\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{2}}^{d}y_{12}+\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}y_{21}+J_{dd},$ (21) $\displaystyle\tilde{Y}_{ds}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{s}y_{11}+\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{2}}^{s}y_{12}+\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{s}y_{21}+J_{ds},$ (22) $\displaystyle\tilde{Y}_{sd}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{1}}^{d}y_{11}+\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{2}}^{d}y_{12}+\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{d}y_{21}+J_{sd},$ (23) $\displaystyle\tilde{Y}_{ss}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{1}}^{s}y_{11}+\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{2}}^{s}y_{12}+\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}y_{21}+J_{ss},$ (24) where $\tilde{Y}_{ij}$ is defined in Eq.(12), $\tilde{h}_{x_{k}}^{i}$ is given by Eq.(20), and $J_{ij}=\sum_{(k,l)\in J_{1}}{\tilde{h}_{a_{k}}^{i}\tilde{h}_{b_{l}}^{j}y_{kl}},(i,j\in\\{d,s\\})$ with $J_{1}=\\{(k,l)\|k,l\geq 1;k+l\geq 4\\}$. With these preparations, now we are going to introduce the following very important condition: $\frac{\tilde{h}_{x_{k}}^{s}}{\tilde{h}_{x_{k}}^{d}}\geq\frac{\tilde{h}_{x_{2}}^{s}}{\tilde{h}_{x_{2}}^{d}}\geq\frac{\tilde{h}_{x_{1}}^{s}}{\tilde{h}_{x_{1}}^{d}},\quad(x\in\\{a,b\\})$ (25) for $k\geq 2$. We can easily prove that the imperfect sources used in practice such as the coherent state source, the heralded source out of the parametric- down conversion, satisfy the above restriction. In order to get a lower bound of $y_{11}$, we should derive the expression of $y_{11}$ with Eqs.(21-24) firstly. Combining Eqs.(21-23), we obtain the expression of $y_{11}$ by eliminating $y_{12}$ and $y_{21}$ such that $y_{11}=y_{11}^{(123)}+\sum_{(m,n)\in J_{1}}f_{11}^{(123)}(m,n)y_{mn},$ (26) where $J_{1}=\\{(m,n)\|m\geq 1,n\geq 1,m+n\geq 4\\}$, $y_{11}^{(123)}=\frac{(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})\tilde{Y}_{dd}-\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})\tilde{Y}_{ds}-\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{d}(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})\tilde{Y}_{sd}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})},$ (27) and $f_{11}^{(123)}(m,n)=\frac{\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{n}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{m}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{m}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})+\tilde{h}_{a_{m}}^{d}\tilde{h}_{b_{1}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})(\tilde{h}_{b_{2}}^{d}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{b_{2}}^{s}\tilde{h}_{b_{n}}^{d})}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})}.$ (28) In these expressions, we use the superscript $^{(123)}$ to denote the result obtained with the first three equations from Eqs.(21-24). Under the conditions presented in Eq.(25), we can easily find out that $(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})\geq 0$, $(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})\geq 0$, $(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{m}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{m}}^{d})\geq 0$ for all $m\geq 1$ and $(\tilde{h}_{b_{2}}^{d}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{b_{2}}^{s}\tilde{h}_{b_{n}}^{d})\geq 0$ for all $n\geq 2$. Then we know that $f_{11}^{(123)}(m,n)\geq 0$ hold for all $(m,n)\in J_{1}$. With this fact, we obtain a lower bound from Eq.(26) by setting $y_{mn}=0,(m,n)\in J_{1}$ such that $\underline{y_{11}}=y_{11}^{(123)}\leq y_{11},$ (29) where $y_{11}^{(123)}$ is defined by Eq.(27). This and Eq.(27) are our major formulas for the decoy-state method implementation for MDI-QKD in this section. Similarly, we can get other expressions with choosing any other three equations from Eqs.(21-24). For example, we choose Eqs.(21-22,24). By eliminating $y_{12}$ and $y_{21}$, we get another expression of $y_{11}$ such that $y_{11}=y_{11}^{(124)}+\sum_{(m,n)\in J_{1}}f_{11}^{(124)}(m,n)s_{mn},$ (30) where $y_{11}^{(124)}=\frac{\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})\tilde{Y}_{dd}+(\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})\tilde{Y}_{ds}-\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{d}(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})\tilde{Y}_{ss}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})},$ (31) and $f_{11}^{(124)}(m,n)=\frac{\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{n}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{m}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{m}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})+\tilde{h}_{a_{m}}^{d}\tilde{h}_{b_{1}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})(\tilde{h}_{b_{2}}^{d}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{b_{2}}^{s}\tilde{h}_{b_{n}}^{d})}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})}.$ (32) Under the conditions presented in Eq.(25), we can also find out that $f_{11}^{(124)}(m,n)\geq 0$ for all $(m,n)\in J_{1}$. Then we know that $y_{11}^{(124)}$ is alos a lower bound of $y_{11}$. On the other hand, by comparing $f_{11}^{(123)}(m,n)$ and $f_{11}^{(124)}(m,n)$, we have $\displaystyle f_{11}^{(123)}(m,n)-f_{11}^{(124)}(m,n)$ $\displaystyle=$ $\displaystyle\frac{\tilde{h}_{a_{2}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{m}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{m}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{n}}^{d})}{-\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{1}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})}\leq 0,$ for all $(m,n)\in J_{1}$. Then we know that $y_{11}^{(123)}\geq y_{11}^{(124)},$ (33) with Eq.(26) and Eq.(30). With the relation presented in Eq.(33) we know that the lower bound $y_{11}^{(123)}$ is tighter than the lower bound $y_{11}^{(124)}$. In the same way, we can get another two lower bounds $y_{11}^{(134)}$ and $y_{11}^{(234)}$ of $y_{11}$ with Eqs.(21,23-24) and Eqs.(22-24) respectively. Furthermore, we can also prove that $y_{11}^{(123)}\geq y_{11}^{(134)},\quad y_{11}^{(123)}\geq y_{11}^{(234)}.$ (34) Now we only consider Eq.(21) and Eq.(24). By eliminating $y_{12}$ or $y_{21}$ respectively, we get two expressions of $y_{11}$ such that $\displaystyle y_{11}$ $\displaystyle=$ $\displaystyle y_{11}^{(14a)}+\sum_{(m,n)\in J_{2}}f_{11}^{(14a)}(m,n)y_{mn},$ (35) $\displaystyle y_{11}$ $\displaystyle=$ $\displaystyle y_{11}^{(14b)}+\sum_{(m,n)\in J_{2}}f_{11}^{(14a)}(m,n)y_{mn},$ (36) where $J_{2}=\\{(m,n)\|m\geq 1,n\geq 1,m+n\geq 3\\}$, $\displaystyle y_{11}^{(14a)}$ $\displaystyle=$ $\displaystyle\frac{\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{2}}^{s}\tilde{Y}_{dd}-\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{2}}^{d}\tilde{Y}_{ss}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{1}}^{s}(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})},$ (37) $\displaystyle y_{11}^{(14b)}$ $\displaystyle=$ $\displaystyle\frac{\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}\tilde{Y}_{dd}-\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{Y}_{ss}}{\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{1}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})},$ (38) and $\displaystyle f_{11}^{(14a)}(m,n)$ $\displaystyle=$ $\displaystyle\frac{\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{2}}^{d}\tilde{h}_{a_{m}}^{s}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{2}}^{s}\tilde{h}_{a_{m}}^{d}\tilde{h}_{b_{n}}^{d}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{1}}^{s}(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})},$ (39) $\displaystyle f_{11}^{(14b)}(m,n)$ $\displaystyle=$ $\displaystyle\frac{\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{a_{m}}^{s}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}\tilde{h}_{a_{m}}^{d}\tilde{h}_{b_{n}}^{d}}{\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{1}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})}.$ (40) For any sources used in the protocol, we must have either $K_{a}=\frac{\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{2}}^{s}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{2}}^{d}}\leq\frac{\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}}{\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}}=K_{b}\quad\textrm{or}\quad K_{a}>K_{b}.$ Suppose the former one holds, we can easily find out that $f_{11}^{(14a)}(m,n)\geq 0$ for all $(m,n)\in J_{2}$ and $y_{11}^{(14a)}$ is a lower bound of $y_{11}$. On the other hand, if $K_{a}\geq K_{b}$ holds, we have $f_{11}^{(14b)}(m,n)\geq 0$ for all $(m,n)\in J_{2}$ and $y_{11}^{(14b)}$ is a lower bound of $y_{11}$. Considering the following two relations $K_{a}-K_{b}=\frac{\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{d}},$ (41) and $\displaystyle f_{11}^{(14a)}(m,n)-f_{11}^{(14b)}(m,n)$ $\displaystyle=$ $\displaystyle\frac{\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{2}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{m}}^{s}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{m}}^{d}\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{n}}^{d})(K_{a}-K_{b})}{\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{1}}^{s}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})},$ we know that $K_{a}-K_{b}$ and $f_{11}^{(14a)}-f_{11}^{(14b)}$ have the same sign which means that they are both positive or negative simultaneously. Then we can write the lower bound of $y_{11}$ with Eq.(21) and Eq.(24) into the following compact form $y_{11}^{(14)}=\min\\{y_{11}^{(14a)},y_{11}^{(14b)}\\}.$ (42) If both Alice and Bob use coherent pulses, the lower bound given by Eq.(42) is just the result presented in Refs. lopa ; curty . In the coming, we will prove that the lower bound $y_{11}^{(123)}$ given in Eq.(27) is more tightly than $y_{11}^{(14)}$. Firstly, if we suppose $K_{a}\leq K_{b}$ holds, then we know that $\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}\leq\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d}$ and $y_{11}^{(14)}=y_{11}^{(14a)}$. For any $(m,n)\in J_{1}$ we have $\displaystyle f_{11}^{(123)}(m,n)-f_{11}^{(14a)}(m,n)$ $\displaystyle=$ $\displaystyle-\frac{(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{m}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{m}})D_{a}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{1}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})},$ where $D_{a}=(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{d}\tilde{h}_{b_{n}}^{s}+\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d}\tilde{h}_{b_{n}}^{d}+\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}\tilde{h}_{b_{n}}^{d}+\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{d}\tilde{h}_{b_{n}}^{s})\geq\tilde{h}_{b_{2}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{2}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{2}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{n}}^{d})$. Then we know that $\displaystyle f_{11}^{(123)}(m,n)-f_{11}^{(14a)}(m,n)$ $\displaystyle\leq$ $\displaystyle-\frac{\tilde{h}_{b_{2}}^{d}(\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{m}}^{s}-\tilde{h}_{a_{1}}^{s}\tilde{h}_{a_{m}}^{d})(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{n}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{n}}^{d})}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{1}}^{d}(\tilde{h}_{b_{1}}^{d}\tilde{h}_{b_{2}}^{s}-\tilde{h}_{b_{1}}^{s}\tilde{h}_{b_{2}}^{d})}\leq 0.$ We can easily know that $y_{11}^{(123)}\geq y_{11}^{(14)}$ when $K_{a}\leq K_{b}$ with this equation. Secondly, if we suppose $K_{a}>K_{b}$ holds, we can easily prove that $f_{11}^{(123)}(m,n)-f_{11}^{(14b)}(m,n)\leq 0$ for all $(m,n)\in J_{1}$ within the same way. Then we get $y_{11}^{(123)}\geq y_{11}^{(14)}$ when $K_{a}>K_{b}$. This completes the proof that $y_{11}^{(123)}\geq y_{11}^{(14)}$. In order to estimate the final key rate, we also need the upper bound of error rate caused by the two single-photon pulses, say $e_{11}$. Similar to the total gain, the total error rate with source $\alpha\beta$ chosen by Alice and Bob can be written as ind2 $\displaystyle\tilde{T}_{dd}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{d}t_{11}+\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{2}}^{d}t_{12}+\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{d}t_{21}+K_{dd},$ (43) $\displaystyle\tilde{T}_{ds}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{s}t_{11}+\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{2}}^{s}t_{12}+\tilde{h}_{a_{2}}^{d}\tilde{h}_{b_{1}}^{s}t_{21}+K_{ds},$ (44) $\displaystyle\tilde{T}_{sd}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{1}}^{d}t_{11}+\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{2}}^{d}t_{12}+\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{d}t_{21}+K_{sd},$ (45) $\displaystyle\tilde{T}_{ss}$ $\displaystyle=$ $\displaystyle\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{1}}^{s}t_{11}+\tilde{h}_{a_{1}}^{s}\tilde{h}_{b_{2}}^{s}t_{12}+\tilde{h}_{a_{2}}^{s}\tilde{h}_{b_{1}}^{s}t_{21}+K_{ss},$ (46) where $T_{ij}=Y_{ij}E_{ij},(i,j\in\\{v,d,s\\})$, $t_{mn}=y_{mn}e_{mn}$, and $\displaystyle\tilde{T}_{dd}$ $\displaystyle=$ $\displaystyle\frac{T_{dd}}{a_{0}^{d}b_{0}^{d}}-\frac{T_{dv}}{a_{0}^{d}b_{0}^{v}}-\frac{T_{vd}}{a_{0}^{v}b_{0}^{d}}+\frac{T_{vv}}{a_{0}^{v}b_{0}^{v}},$ (47) $\displaystyle\tilde{T}_{ds}$ $\displaystyle=$ $\displaystyle\frac{T_{ds}}{a_{0}^{d}b_{0}^{s}}-\frac{T_{dv}}{a_{0}^{d}b_{0}^{v}}-\frac{T_{vs}}{a_{0}^{v}b_{0}^{s}}+\frac{T_{vv}}{a_{0}^{v}b_{0}^{v}},$ (48) $\displaystyle\tilde{T}_{sd}$ $\displaystyle=$ $\displaystyle\frac{T_{sd}}{a_{0}^{s}b_{0}^{d}}-\frac{T_{sv}}{a_{0}^{s}b_{0}^{v}}-\frac{T_{vd}}{a_{0}^{v}b_{0}^{d}}+\frac{T_{vv}}{a_{0}^{v}b_{0}^{v}},$ (49) $\displaystyle\tilde{T}_{ss}$ $\displaystyle=$ $\displaystyle\frac{T_{ss}}{a_{0}^{s}b_{0}^{s}}-\frac{T_{sv}}{a_{0}^{s}b_{0}^{v}}-\frac{T_{vs}}{a_{0}^{v}b_{0}^{s}}+\frac{T_{vv}}{a_{0}^{v}b_{0}^{v}},$ (50) and $K_{ij}=\sum_{(k,l)\in J_{1}}{\tilde{h}_{a_{k}}^{i}\tilde{h}_{b_{l}}^{j}t_{kl}},(i,j\in\\{d,s\\})$ with $J_{1}=\\{(k,l)\|k,l\geq 1,k+l\geq 4\\}$. According to Eq.(43), we can find out the upper bound of $e_{11}$ such that $e_{11}\leq e_{11}^{(1)}=\frac{\tilde{T}_{dd}}{\tilde{h}_{a_{1}}^{d}\tilde{h}_{b_{1}}^{d}y_{11}}=\overline{e_{11}}.$ (51) In the protocol, there are two different bases. We denote $y_{11}^{Z}$ and $y_{11}^{X}$ for yields of single-photon pulse pairs in the $Z$ and $X$ bases, respectively. Consider those post-selected bits cased by source $d_{A}d_{B}$ in the $Z$ basis. After an error test, we know the bit-flip error rate of this set, say $T_{d_{A}d_{B}}^{Z}=Y_{d_{A}d_{B}}^{Z}E_{d_{A}d_{B}}^{Z}$. We also need the phase-flip rate for the subset of bits which are caused by the two single-photon pulse, say $e_{11}^{ph}$, which is equal to the flip rate of post-selected bits caused by a single photon in the $X$ basis, say $e_{11}^{X}$. Given this, we can now calculate the key rate by the well-know formula. For example, for those post-selected bits caused by source $s_{A}s_{B}$, it is $R=a_{1}^{s}b_{1}^{s}y_{11}^{Z}[1-H(e_{11}^{X})]-Y_{s_{A}s_{B}}^{Z}fH(E_{s_{A}s_{B}}^{Z}),$ (52) where $f$ is the efficiency factor of the error correction method used. ## III Numerical Simulation Figure 1: (Color online) The relative value between estimated parameter of $y_{11}$ and the asymptotic limit of the infinite decoy-state method versus the total channel transmission loss using 3-intensity decoy state MDI-QKD. In the simulations, we assume that $\mu_{v}=\nu_{v}=0.01$, $\mu_{d}=\nu_{d}=0.1$, $\mu_{s}=\nu_{s}=0.5$ and the value of other parameters are presented in Table 1. Figure 2: (Color online) The estimated parameter of $e_{11}$ versus the total channel transmission loss using 3-intensity decoy state MDI-QKD. In the simulations, we assume that $\mu_{v}=\nu_{v}=0.01$, $\mu_{d}=\nu_{d}=0.1$, $\mu_{s}=\nu_{s}=0.5$ and the value of other parameters are presented in Table 1. Figure 3: (Color online) The estimated key rate $R$ versus channel transmission using 3-intensity decoy state MDI-QKD. In the simulations, we assume that $\mu_{v}=\nu_{v}=0.01$, $\mu_{d}=\nu_{d}=0.1$, $\mu_{s}=\nu_{s}=0.5$ and the value of other parameters are presented in Table 1. Figure 4: (Color online) The relative value between the optimal key rate obtained with different methods and the asymptotic limit of the infinite decoy-state method versus the total channel transmission loss using 3-intensity decoy state MDI-QKD. In the simulations, we assume that $\mu_{v}=\nu_{v}=0.01$, $\mu_{d}=\nu_{d}=0.1$, $\mu_{s}=\nu_{s}$ and the value of other parameters are presented in Table 1. The optimal key rate is just the value with maximizing the key rate with $\mu_{s}=\nu_{s}\in(\mu_{d},1)$. Figure 5: (Color online) The optimal intensities versus the total channel transmission loss using 3-intensity decoy state MDI-QKD. In the simulations, we assume that $\mu_{v}=\nu_{v}=0.01$, $\mu_{d}=\nu_{d}=0.1$, $\mu_{s}=\nu_{s}$ and the value of other parameters are presented in Table 1. The optimal densities of the signal-state pulses are obtained by maximizing the key rate. Table 1: List of experimental parameters used in numerical simulations: $e_{0}$ is the error rate of background, $e_{d}$ is the misalignment-error probability; $p_{d}$ is the dark count rate per detector; $f$ is the error correction inefficiency. $e_{0}$ \| $e_{d}$ \| $p_{d}$ \| $f$ ---\|---\|---\|--- 0.5 \| 1.5% \| $3.0\times 10^{-6}$ \| 1.16 Now, we present some numerical simulations to comparing our results with the existing results lopa ; curty . Below for simplicity, we suppose that Alice and Bob use the coherent-state sources. Here, we denote Alice’s sources $\\{v_{A},d_{A},s_{A}\\}$ by their intensities $\\{\mu_{v},\mu_{d},\mu_{s}\\}$ and Bob’s sources $\\{v_{B},d_{B},s_{B}\\}$ by their intensities $\\{\nu_{v},\nu_{d},\nu_{s}\\}$ respectively. The UTP locates in the middle of Alice and Bob, and the UTP’s detectors are identical, i.e., they have the same dark count rate and detection efficiency, and their detection efficiency does not depend on the incoming signals. We shall estimate what values would be probably observed for the gains and error rates in the normal cases by the linear models as in ind2 ; Wang2013 ; Wang201309 : $\displaystyle\|n\rangle\langle n\|=\sum_{k=0}^{n}C_{n}^{k}\xi^{k}(1-\xi)^{n-k}\|k\rangle\langle k\|$ where $\xi^{k}$ is the transmittance for a distance from Alice to the UTB. For fair comparison, we use the same parameter values used in ind2 ; Wang2013 ; Wang201309 for our numerical evaluation, which follow the experiment reported in UrsinNP2007 . For simplicity, we shall put the detection efficiency to the overall transmittance $\eta=\xi^{2}\zeta$. We assume all detectors have the same detection efficiency $\zeta$ and dark count rate $p_{d}$. The values of these parameters are presented in Table 1. With this, the total gains $Y_{\mu_{i},\nu_{j}}^{\omega},(\omega=X,Z)$ and error rates $Y_{\mu_{i},\nu_{j}}^{\omega}E_{\mu_{i},\nu_{j}}^{\omega},(\omega=X,Z)$ of Alice’s intensity $\mu_{i}(i\in\\{v,d,s\\})$ and Bob’s intensity $\nu_{j}(j\in\\{v,d,s\\})$ can be calculated. By using these values, we can estimate the lower bounds of yield $y_{11}^{Z}$ with Eq.(29) and Eq.(42). Also, we can estimate the upper bounds of error rate $e_{11}^{X}$ with Eq.(51) where the lower bound of $y_{11}^{X}$ being estimated by Eq.(29) and Eq.(42). The estimated parameters of $y_{11}$ and $e_{11}$ are shown in Fig.1 and Fig.2, respectively, which clearly shows that our methods are more tightly than the pre-existed results. In order to see more clearly, in Fig.1, we plot the relative value of $y_{11}$ to the result obtained with the infinite decoy- state method. We can observe that our results are more close to the asymptotic limit of the infinite decoy-state method than the pre-existed results lopa ; curty . Furthermore, with these parameters, we can estimate the final key rate $R$ of this protocol with Eq.(52) which is shown in Fig.3. In these figures, the blue dashed line is obtained by Eq.(42) which is just the results presented in Refs. lopa ; curty , the red solid line is obtained by the method presented in section 2 with Eq.(29), the black dash-dot line is the result obtained by the infinite decoy-state method. In the simulation, the densities used by Alice and Bob are assigned to $\mu_{v}=\nu_{v}=0.01$, $\mu_{d}=\nu_{d}=0.1$, $\mu_{s}=\nu_{s}=0.5$. Furthermore, if we fix the densities of the decoy-state pulses used by Alice and Bob, the final key rate will change with Alice and Bob taking different densities for their signal-state pulses. Then we can find out the optimal densities of their signal-state pulses with maximizing the final key rate. Here, we also take $\mu_{v}=\nu_{v}=0.01$, $\mu_{d}=\nu_{d}=0.1$ and assume that $\mu_{s}=\nu_{s}\in(\mu_{d},1)$. In Fig.4, we plot the relative value of the optimal key rate to the result obtained with the infinite decoy-state method. We can observe that our result is better than the pre-existed results. The optimal densities with the optimal key rate versus the total channel transmission loss is given in Fig.5. ## IV Conclusion We study the MDI-QKD in practice where the intensities of all 3 different states can be nonzero. Our result here is the most general result for the 3-intensity method: Setting the weakest state to be zero, we obtain the result in the special case where a vacuum state and two non-vacuum states are usedWang2013 . The result here is not limited to the coherent states. It applies to any states that satisfy Eq.(25), e.g., the heralded states from the parametric down conversion. Our result is most efficient: It offers a more tightened bound for $y_{11}$ and therefore a higher key rate than the prior artlopa , as has been strictly proven and also numerically demonstrated. The key rate can be further improved when it is combined with the method as shown in Ref.Wang201309 . Acknowledgement: We acknowledge the support from the 10000-Plan of Shandong province, the National High-Tech Program of China Grants No. 2011AA010800 and No. 2011AA010803 and NSFC Grants No. 11174177 and No. 60725416. ## References (1) C.H. Bennett and G. Brassard, in Proc. of IEEE Int. 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Phys. 3, 481 (2007).	arxiv-papers	2013-09-23T17:24:50	2024-09-04T02:49:51.384117	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Z.-W. Yu, Y.-H. Zhou, and Xiang-Bin Wang", "submitter": "Xiang-Bin Wang", "url": "https://arxiv.org/abs/1309.5886" }
1309.5983	# Remarks on the tensor degree of finite groups Ahmad M.A. Alghamdi Department of Mathematical Sciences, Faculty of Applied Sciences Umm Alqura University, P.O. Box 14035, Makkah, 21955, Saudi Arabia [email protected] and Francesco G. Russo DEIM, Universitá degli Studi di Palermo Viale Delle Scienze, Edificio 8, 90128, Palermo, Italy [email protected] ###### Abstract. The present paper is a note on the tensor degree of finite groups, introduced recently in literature. This numerical invariant generalizes the commutativity degree through the notion of nonabelian tensor square. We show two inequalities, which correlate the tensor and the commutativity degree of finite groups, and, indirectly, structural properties will be discussed. ###### Key words and phrases: Tensor degree, commutativity degree, exterior degree ###### 2010 Mathematics Subject Classification: Primary: 20J99, 20D15; Secondary: 20D60; 20C25 ## 1\. The relative tensor degree All the groups of the present paper are supposed to be finite. Having in mind the exponential notation for the conjugation of two elements $x$ and $y$ in a group $G$, that is, the notation $x^{y}=y^{-1}xy$, we may follow [3, 4, 13] in saying that two normal subgroups $H$ and $K$ of $G$ act compatibly upon each other, if $\left(h_{2}^{k_{1}}\right)^{h_{1}}=\left(\left({h_{2}}^{h_{1}^{-1}}\right)^{k_{1}}\right)^{h_{1}}\ \mathrm{and}\ \left(k_{2}^{h_{1}}\right)^{k_{1}}=\left(\left({k_{2}}^{k_{1}^{-1}}\right)^{h_{1}}\right)^{k_{1}}$ for all $h_{1},h_{2}\in H$ and $k_{1},k_{2}\in K$, and if $H$ and $K$ act upon themselves by conjugation. Given $h\in H$ and $k\in K$, the nonabelian tensor product $H\otimes K$ is the group generated by the symbols $h\otimes k$ satisfying the relations $h_{1}h_{2}\otimes k_{1}=(h_{2}^{h_{1}}\otimes{k_{1}}^{h_{1}})\ (h_{1}\otimes k_{1})$ and $k_{1}k_{2}\otimes h_{1}=(k_{1}\otimes h_{1})\ (h_{1}^{k_{1}}\otimes k_{2}^{k_{1}})$ for all $h_{1},h_{2}\in H$ and $k_{1},k_{2}\in K$. The map $\kappa_{H,K}:h\otimes k\in H\otimes K\mapsto[h,k]\in[H,K]$ turns out to be an epimorphism, whose kernel $\ker\kappa_{H,K}=J(G,H,K)$ is central in $H\otimes K$. The reader may find more details and a topological approach to $J(G,H,K)$ in [4, 5, 11, 13]. The short exact sequence $\begin{CD}1@>{}>{}>J(G,H,K)@>{}>{}>H\otimes K@>{\kappa_{H,K}}>{}>[H,K]@>{}>{}>1\end{CD}$ is a central extension. In the special case $G=H=K$, we have that $J(G)=J(G,G,G)=\ker\kappa_{G,G}=\ker\kappa$ and $H\otimes K=G\otimes G$ is called nonabelian tensor square of $G$. The fundamental properties of $G\otimes G$ have been described in the classical paper [3], in which it is noted that $\kappa:x\otimes y\in G\otimes G\mapsto\kappa(x\otimes y)=[x,y]\in G^{\prime}$ is an epimorphism of groups with $\ker\kappa=J(G)$ and $1\rightarrow J(G)\rightarrow G\otimes G{\overset{\kappa}{\rightarrow}}G^{\prime}\rightarrow 1$ is a central extension. The group $J(G)$ is important from the perspective of the algebraic topology, in fact $J(G)\cong\pi_{3}(SK(G,1))$ is the third homotopy group of the suspension of an Eilenberg–MacLane space $K(G,1)$ (see [4] for more details). As done in [12], we may consider the tensor centralizer $C^{\otimes}_{K}(H)=\\{k\in K\ \|\ h\otimes k=1,\ \forall h\in H\\}=\bigcap_{h\in H}C^{\otimes}_{K}(h)$ and the tensor center $C^{\otimes}_{G}(G)=Z^{\otimes}(G)={\underset{x\in G}{\bigcap}}C_{G}^{\otimes}(x)$ and one can check that $C^{\otimes}_{G}(x)$ and $Z^{\otimes}(G)$ are subgroups of $G$ such that $C^{\otimes}_{G}(x)\subseteq C_{G}(x)$ and $Z^{\otimes}(G)\subseteq Z(G)$. Generalizing what has been done in [12], we may define the relative tensor degree $d^{\otimes}(H,K)=\frac{\|\\{(h,k)\in H\times K\ \|\ h\otimes k=1\\}\|}{\|H\|\|K\|}=\frac{1}{\|H\|\ \|K\|}\sum_{h\in H}\|C^{\otimes}_{K}(h)\|$ of $H$ and $K$. Notice that $d^{\otimes}(G)=1$ if and only if $Z^{\otimes}(G)=G$. Unfortunately, few results are available on the relative tensor degree at the moment and these are contained mainly in [12], where it is discussed the tensor degree $d^{\otimes}(G)=d^{\otimes}(G,G)$ of $G$. On the other hand, there is a rich literature (see for instance [1, 2, 7, 8, 9]) on the relative commutativity degree $d(H,K)=\frac{\|\\{(h,k)\in H\times K\ \|\ [h,k]=1\\}\|}{\|H\|\|K\|}=\frac{1}{\|H\|\ \|K\|}\sum_{h\in H}\|C_{K}(h)\|=\frac{k_{K}(H)}{\|H\|}$ of $H$ and $K$ (not necessarily normal this time) of $G$. Here $k_{K}(H)$ is the number of $K$–conjugacy classes that constitute $H$. In particular, if $G=H=K$, we find the well known commutativity degree $d(G)=d(G,G)=k_{G}(G)/\|G\|$. Our fist main contribution is the following. ###### Theorem 1.1. Let $H,K$ be two normal subgroups of a group $G$ and $p$ the smallest prime divisor of $\|G\|$. Then the following inequalities are true: $None$ $\frac{d(H,K)}{\|J(G,H,K)\|}+\frac{\|C^{\otimes}_{K}(H)\|}{\|H\|}\left(1-\frac{1}{\|J(G,H,K)\|}\right)\leq d^{\otimes}(H,K)$ $None$ $d^{\otimes}(H,K)\leq d(H,K)-\left(1-\frac{1}{p}\right)\left(\frac{\|C_{K}(H)\|-\|C^{\otimes}_{K}(H)\|}{\|H\|}\right).$ On the other hand, we may correlate the relative tensor degree, the relative commutativity degree and another notion, studied recently in [11]. In order to proceed in this direction, we recall from [3, 5, 10] that the nonabelian exterior product $H\wedge K$ of $H$ and $K$ is the quotient the nonabelian tensor produc $H\otimes K$, defined by $H\wedge K=(H\otimes K)/\nabla(H\cap K)=\langle(x\otimes y)\nabla(H\cap K)\ \|\ x,y\in H\cap K\rangle=\langle x\wedge y\ \|\ x,y\in H\cap K\rangle$, where $\nabla(H\cap K)=\langle x\otimes x\|\ x\in H\cap K\rangle$. From [3, 4], we may note that $\kappa^{\prime}_{H,K}:h\wedge k\in H\wedge K\mapsto\kappa^{\prime}_{H,K}(h\wedge k)=[h,k]\in[H,K]$ is an epimorphism of groups such that $\begin{CD}1@>{}>{}>M(G,H,K)@>{}>{}>H\wedge K@>{\kappa^{\prime}_{H,K}}>{}>[H,K]@>{}>{}>1\end{CD}$ is a central extension, where $M(G,H,K)=\ker\kappa^{\prime}_{H,K}$ is the so–called Schur multiplier of the triple $(G,H,K)$. We inform the reader that several references on the theory of the Schur multipliers of triples can be found in [4, 11]. In particular, $M(G,G,G)=M(G)=H_{2}(G,\mathbb{Z})$ is the Schur multiplier of $G$, that is, the second integral homology group over $G$. In our situation, it is possible to consider the set $C^{\wedge}_{K}(H)=\\{k\in K\ \|\ h\wedge k=1,\ \forall h\in H\\}=\bigcap_{h\in H}C^{\wedge}_{K}(h),$ called exterior centralizer of $H$ with respect to $K$ and it is actually a subgroup of $K$ (see [10] for details). In particular, $C^{\wedge}_{G}(G)=Z^{\wedge}(G)=\bigcap_{x\in G}C^{\wedge}_{G}(x)$ is called exterior center of $G$. It is easy to check that $C^{\wedge}_{G}(x)\subseteq C_{G}(x)$ and $Z^{\wedge}(G)\subseteq Z(G)$. Some recent papers as [11] show that it is possible to have a combinatorial approach for measuring how far a group $G$ is from $Z^{\wedge}(G)$ and this is interesting, because a result of Ellis [5] characterize a capable groups by the triviality of its exterior center (i.e.: a group $G$ is capable if $G\simeq E/Z(E)$ for a given group $E$). This aspect has motivated the notion of relative exterior degree $d^{\wedge}(H,K)=\frac{\|\\{(h,k)\in H\times K\ \|\ h\wedge k=1\\}\|}{\|H\|\|K\|}=\frac{1}{\|H\|\ \|K\|}\sum_{h\in H}\|C^{\wedge}_{K}(h)\|$ of $H$ and $K$. When $G=H=K$, we find the exterior degree $d^{\wedge}(G,G)=d^{\wedge}(G)$ of $G$ in [11]. It is easy to prove that $d^{\wedge}(G)=1$ if and only if $G=Z^{\wedge}(G)$. Hence the exterior degree represents the probability that two randomly chosen elements commute with respect to the operator $\wedge$. Roughly speaking, this means that there are many chances of finding capable groups for small values of exterior degree. From [12, Theorem 2.8], we may correlate the above notions via the inequality $d^{\otimes}(G)\leq d^{\wedge}(G)\leq d(G)$ and our second main theorem shows that something of more general holds. ###### Theorem 1.2. Let $H,K$ be normal subgroups of a group $G$. Then $d^{\otimes}(H,K)\leq d^{\wedge}(H,K)\leq d(H,K).$ Moreover, if $J(G,H,K)$ is trivial, then $d^{\otimes}(H,K)=d^{\wedge}(H,K)=d(H,K)$. ## 2\. Proofs of the main results We begin with a technical lemma, whose proof uses an argument which appears in [11, Lemma 2.1] and [12, Lemma 2.2] in different ways. ###### Lemma 2.1. Let $H,K$ be normal subgroups of a group $G$. Then $d^{\otimes}(H,K)=\frac{1}{\|H\|}\sum^{k_{K}(H)}_{i=1}\frac{\|C^{\otimes}_{K}(h_{i})\|}{\|C_{K}(h_{i})\|}.$ In particular, if $G=HK$, then $C_{K}(h_{i})/C^{\otimes}_{K}(h_{i})$ is isomorphic to a subgroup of $J(G,H,K)$ and $\|C_{G}(h_{i}):C^{\otimes}_{G}(h_{i})\|\leq\|J(G,H,K)\|$ for all $i=1,2,\ldots,k_{K}(H)$. ###### Proof. Since $H$ is normal in $G$, we consider the $K$–conjugacy classes $C_{1},\ldots,C_{k_{K}(H)}$ that constitute $H$. It follows that $\|H\|\ \|K\|\ d^{\otimes}(H,K)=\sum_{h\in H}\|C^{\otimes}_{K}(h)\|=\sum^{k_{K}(H)}_{i=1}\sum_{h\in C_{i}}\|C^{\otimes}_{K}(h)\|$ $=\sum^{k_{K}(H)}_{i=1}\|K:C_{K}(h_{i})\|\ \|C^{\otimes}_{K}(h_{i})\|=\|K\|\ \sum^{k_{K}(H)}_{i=1}\frac{\|C^{\otimes}_{K}(h_{i})\|}{\|C_{K}(h_{i})\|}.$ Now assume that $G=HK$. For all $i=1,\ldots,k_{K}(H)$, the map $\varphi:kC^{\otimes}_{K}(h_{i})\in C_{K}(h_{i})/C^{\otimes}_{K}(h_{i})\longmapsto k\otimes h_{i}\in J(G,H,K)$ satisfies the condition $\varphi(k_{1}k_{2}C^{\otimes}_{K}(h_{i}))=k_{1}k_{2}\otimes h_{i}=(k_{1}\otimes h_{i})^{k_{2}}\ (k_{2}\otimes h_{i})$ $=(k_{1}\otimes h_{i})\ (k_{2}\otimes h_{i})=\varphi(k_{1}C^{\otimes}_{K}(h_{i}))\ \varphi(k_{2}C^{\otimes}_{K}(h_{i}))$ for all $k_{1},k_{2}\in C_{K}(h_{i})$. Furthermore, $\ker\varphi=\\{kC^{\otimes}_{K}(h_{i})\ \|\ k\otimes h_{i}=1\\}=C^{\otimes}_{K}(h_{i})$. Then $\varphi$ is a monomorphism and $C_{K}(h_{i})/C^{\otimes}_{K}(h_{i})$ is isomorphic to a subgroup of $J(G,H,K)$. We conclude that $\|C_{K}(h_{i}):C^{\otimes}_{K}(h_{i})\|\leq\|J(G,H,K)\|$. ∎ Now we may prove Theorem 1.1. It is an interesting bound, which connects the notion of relative tensor degree with that of relative commutativity degree. ###### Proof of Theorem 1.1. We begin to prove $(a)$. From Lemma 2.1, we have $\|C^{\otimes}_{K}(H)\|/\|C_{K}(H)\|\geq 1/\|J(G,H,K)\|$ and, together with the equality $d(H,K)=\frac{k_{K}(H)}{\|H\|}$, we deduce $d^{\otimes}(H,K)=\frac{1}{\|H\|}\sum^{k_{K}(H)}_{i=1}\left\|\frac{C^{\otimes}_{K}(h_{i})}{C_{K}(h_{i})}\right\|\geq\frac{1}{\|H\|}\left(\|C^{\otimes}_{K}(H)\|+\frac{k_{K}(H)-\|C^{\otimes}_{K}(H)\|}{\|J(G,H,K)\|}\right)$ $=\frac{k_{K}(H)}{\|H\|\ \|J(G,H,K)\|}+\frac{\|C^{\otimes}_{K}(H)\|}{\|G\|}\left(1-\frac{1}{\|J(G,H,K)\|}\right)$ $=\frac{d(H,K)}{\|J(G,H,K)\|}+\frac{\|C^{\otimes}_{K}(H)\|}{\|H\|}\left(1-\frac{1}{\|J(G,H,K)\|}\right)$ Conversely, $\|K:C^{\otimes}_{K}(h_{i})\|\geq p$ implies that $d^{\otimes}(H,K)=\frac{1}{\|H\|}\sum^{k_{K}(H)}_{i=1}\left\|\frac{C^{\otimes}_{K}(h_{i})}{C_{K}(h_{i})}\right\|$ $\leq\frac{\|C^{\otimes}_{K}(H)\|}{\|H\|}+\frac{1}{p}\left(\frac{\|C_{K}(H)\|-\|C^{\otimes}_{K}(H)\|}{\|H\|}\right)+\frac{k_{K}(H)-\|C_{K}(H)\|}{\|H\|}$ $=d(H,K)-\frac{p-1}{p}\left(\frac{\|C_{K}(H)\|-\|C^{\otimes}_{K}(H)\|}{\|H\|}\right).$ ∎ Immediately, we note that [12, Theorem 2.3] describes a special case of Theorem 1.1. The following consequence of Theorem 1.1 is interesting, too. ###### Corollary 2.2. Let $G=HK$ for two normal subgroups $H$ and $K$. Then $\frac{d(H,K)}{\|J(G,H,K)\|}\leq d^{\otimes}(H,K)\leq\ d(H,K).$ In particular, if $J(G,H,K)$ is trivial, then $d^{\otimes}(H,K)=d(H,K).$ The second main theorem is a result of comparison. Its proof is the following. ###### Proof of Theorem 1.2. We have $d^{\wedge}(H,K)=\frac{1}{\|H\|}\sum^{k_{K}(H)}_{i=1}\left\|\frac{C^{\wedge}_{K}(h_{i})}{C_{K}(h_{i})}\right\|\leq\frac{1}{\|H\|}\sum^{k_{K}(H)}_{i=1}\left\|\frac{C_{K}(h_{i})}{C_{K}(h_{i})}\right\|=d(H,K)$ and the upper bound follows. Now $k\in C^{\wedge}_{K}(H)$ if and only if $k\wedge h=1$ for all $h\in H$ if and only if $(k\otimes h)\nabla(H\cap K)=\nabla(H\cap K)$ if and only if $k\otimes h\in\nabla(H\cap K)$. This condition is weaker than the condition $k\otimes h=1$, characterizing the elements of $C^{\otimes}_{K}(H)$. Then $C^{\otimes}_{K}(H)\subseteq C^{\wedge}_{K}(H)\subseteq C_{K}(H)$. This and Lemma 2.1 imply the lower bound $d^{\otimes}(H,K)=\frac{1}{\|H\|}\sum^{k_{K}(H)}_{i=1}\left\|\frac{C^{\otimes}_{K}(h_{i})}{C_{K}(h_{i})}\right\|\leq\frac{1}{\|H\|}\sum^{k_{K}(H)}_{i=1}\left\|\frac{C^{\wedge}_{K}(h_{i})}{C_{K}(h_{i})}\right\|=d^{\wedge}(H,K).$ The rest follows from Corollary 2.2. ∎ ## References * [1] A.M.A. Alghamdi and F.G. Russo, A generalization of the probability that the commutator of two group elements is equal to a given element, Bull. Iranian Math. Soc. 38 (2012), 973–986. * [2] A. Amit and U. Vishne , Characters and solutions to equations in finite groups, J. Algebra Appl. 10 (2011), 675–686. * [3] R. Brown, D. L. Johnson and E. F. Robertson, Some computations of non-abelian tensor products of groups, J. Algebra 111 (1987), 177–202. * [4] R. Brown, P.J. Higgins and R. Sivera, Nonabelian algebraic topology, EMS Tracts in Mathematics, EMS Publishing, Zürich, 2011. * [5] G. Ellis, Tensor products and $q$–crossed modules, J. London Math. Soc. 51 (1995), 243–258. * [6] R.M. Guralnick and G.R. Robinson, On the commuting probability in finite groups, J. Algebra 300 (2006), 509–528. * [7] W.H. Gustafson, What is the probability that two groups elements commute? Amer. Math. Monthly 80 (1973), 1031–1304. * [8] P. Lescot, Isoclinism classes and commutativity degrees of finite groups, J. Algebra 177 (1995), 847–869. * [9] P. Lescot, Central extensions and commutativity degree, Comm. Algebra 29 (2001), 4451–4460. * [10] P. Niroomand and F.G. Russo, A note on the exterior centralizer, Arch. Math. (Basel) 93 (2009), 505–512. * [11] P. Niroomand, R. Rezaei and F.G. Russo, Commuting powers and exterior degree of finite groups, J. Korean Math. Soc. 49 (2012), 855–865. * [12] P. Niroomand and F.G. Russo, On the tensor degree of finite groups, Ars Comb., to appear. Available as preprint at: http://arxiv.org/abs/1303.1364. * [13] D.E. Otera, F.G. Russo and C. Tanasi, Some algebraic and topological properties of nonabelian tensor product, Bull. Korean Math. Soc. 50 (2013),1069–1077.	arxiv-papers	2013-09-23T21:13:56	2024-09-04T02:49:51.396576	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ahmad M.A. Alghamdi (Umm Alqura University, Mekkah, Saudi Arabia) and\n Francesco G. Russo (Universita' degli Studi di Palermo, Palermo, Italy)", "submitter": "Francesco G. Russo", "url": "https://arxiv.org/abs/1309.5983" }
1309.5984	# An evolutionary approach to Function Phillip Lord School of Computing Science Newcastle University Newcastle-Upon-Tyne United Kingdom NE1 7RU [email protected] ###### Abstract Background: Understanding the distinction between function and role is vexing and difficult. While it appears to be useful, in practice this distinction is hard to apply, particularly within biology. Results: I take an evolutionary approach, considering a series of examples, to develop and generate definitions for these concepts. I test them in practice against the Ontology for Biomedical Investigations (OBI). Finally, I give an axiomatisation and discuss methods for applying these definitions in practice. Conclusions: The definitions in this paper are applicable, formalizing current practice. As such, they make a significant contribution to the use of these concepts within biomedical ontologies. ## 1 Background Large parts of modern biology are aimed at answering questions about function. For example, much of the Gene Ontology deals with molecular function(Ashburner et al., 2000). In dealing with the social aspects of science, roles are similarly important. It is clear, therefore, that function and role are important concepts in biomedical ontologies and are prime candidates for inclusion in an upper ontology. A coherent, consistent and shared definition for function and role is likely to decrease the effort required to integrate independently-developed ontologies. One upper ontology, the Basic Formal Ontology (BFO)(bfo, ), currently in use by a number of groups, includes a definition of function. However, this definition is not naturally applicable to biology; it is not clear, for instance, that Gene Ontology molecular functions are also BFO functions, as discussed later (Section 2.1). An alternative is available within the General Formal Ontology (GFO), which provides an ontology of function(Burek et al., 2006). This ontology provides a more extensive framework for describing functions but, in itself, does not define biological function. Despite this, there is a reasonable degree of informal agreement among biologists as to the meaning of the word. Conversely, while formal and informal definitions for role seems clear, many people have difficulties in applying it in practice. In this paper, I address two key issues relating to the modeling of function and role coherently for biomedical ontologies: firstly, how I unify the definitions of function as they apply to artifacts and to life; and, secondly, how do I differentiate between roles and functions? I consider illustrative examples where the answers are reasonably clear and evolve a definition from them. I then consider examples of the application of current definitions from BFO in a practical biomedical use. Finally, I offer an axiomatisation in OWL stemming from my definitions and consider how they could be applied in practice. ## 2 Results ### 2.1 Biological Function First, consider the current definition of function provided by BFO (see Def: 1). ###### Definition 1 Function is a realizable entity the manifestation of which is an essentially end-directed activity of a continuant entity being a specific kind of entity in the kind or kinds of context that it is made for As a simple example, a hammer (_the continuant entity_) was made to hammer nails (_the function_) in a hammering process (_the end-directed activity_). This definition is problematic for biological systems. The problem here is simple: most biological systems were not made or designed for any purpose. Although, it has not been incorporated into BFO yet, there is a potential definition for “biological function” which would become a new child of function (Def: 2)(Arp and Smith, 2008). ###### Definition 2 A biological function is a function which inheres in an independent continuant that is i) part of an organism and ii) exists and has the physical structure as a result of the coordinated expression of that organism’s structural genes. As an example, a foot (_the independent continuant_) is part of an organism, exists, and develops in a controlled way as a result of gene expression. There are some difficulties with this definition. Consider the following examples: a differentiated tumour has its structure through coordinated expression of its genes, it exists and it engages in end-directed activity; a male ant has its structure as a result of gene expression, and engages in end-directed activity, however, it is not part of an organism. From this, I conclude that a differentiated tumour does, indeed, have a function (growing). A male ant, however does not have a function. Also consider molecular function: the physical structure of a protein is independent of the expression of an organisms structural genes – only its presence depends on this. A protein, therefore, does not have a function, by this definition. So, there are two key problems: the definition does not work for entities above or below a certain size; and most biological entities have their structure as a result of coordinated expression. I offer the following alternate definition (Def: 3). ###### Definition 3 A biological function is a realizable entity that inheres in a continuant which is realized in an activity, and where the homologous structure(s) of individuals of closely related and the same species bear this same biological function. The definition given uses the notion of homology; evolution is key to our understanding of biology and it is appropriate that it should be used to define biological function. If a biological structure has a function, then this function will have evolved along with the structure; so, other structures with a common evolutionary descent will display the same behaviour. This definition also mirrors closely normal biological practice; the most common way to determine the function of an unknown structure is to look for function of a homologous structure. It should be noted that this definition of biological function is _not_ circular, although it has itself as part of its definition; rather it is recursive; a chimp hand and a human hand can have the same function because of each other. It does require that a structure must have a homolog for it to have a function. It does not require that these homologous structures be extant. Applying this to our examples: the tumour now has no function because it has no homologs (different tumours arise as independent events and share no common ancestor). Likewise, the activity of the male ant now clearly is a function, as many different, related organisms behave in a similar way. Finally, a protein may have a function depending on the activity of its homologs. In short, this definition results in the same conclusions as our biological understanding. ### 2.2 Relating the Functions As well as biological function, another subclass, artifactual function has been suggested(Arp and Smith, 2008). Next, I consider the relationship between function and these biological and artifactual subclasses. Taking an illustrative example, consider the sole of my foot and the sole of my shoe. They appear to operate to: provide a frictional surface to enable motion; provide padding to reduce shock to everything above; be tough enough to resist abrasion. They would appear to have the same function; indeed, like many artifacts, we would guess that the shoe owes much of its design to mimicry of biology. It would seem, therefore, sensible that instances of Shock Resistance could be either a biological or artifactual function. The alternative would be to duplicate many functions under both subclasses (“biological shock resistance”, and “artifactual shock resistance”) with very similar definitions. Some functions such as reproduction have to be a biological function, while others liquifying iron can never be. In short, whether an instance of function is biological or artifactual should be determined from the nature of the entity in which it inheres, rather than the process by which it is realized. I therefore offer a simple definition of function which reflects this (Def: 4). ###### Definition 4 A function is a realizable entity which is a biological function or an artifactual function. For the purposes of this paper, I note that the definition given for function earlier (Def: 1), can serve as a reasonable definition for artifactual function. It is also interesting that this definition covers some unusual but nonpathological examples. Take a bacterium whose colonies change colour depending on the presence of a toxin and which was produced using synthetic biology techniques. The components have all evolved, but the organisation has not. Is the detection of the toxin then a biological or artifactual function? This is clearly a difficult, if uninspiring, question but given Def: 4, one that can avoid by simply describing it as a function; perhaps more intuitively, it can be described as both a biological and an artifactual function, suggesting strongly that these two classes should not be disjoint. ### 2.3 Roles Next, having considered the definition of function and its applicability to biology, I consider the issue of roles. The current definition (Def: 5) is complex; put more simply, it suggests that the entity having that role can be involved in an activity but that it was not necessarily intended for, nor necessarily has the structure for this. ###### Definition 5 A realizable entity the manifestation of which brings about some result or end that is not essential to a continuant in virtue of the kind of thing that it is but that can be served or participated in by that kind of continuant in some kinds of natural, social or institutional contexts. Consider the relationship between role and function. Again, I shall use a simple biological example, in this case of a man walking on his hands. By our earlier definition (Def: 3), “to walk on hands” is _not_ a function. While the homologous structure is, indeed, used for walking on in all closely related species, most humans do not walk on their hands. It would, therefore appear to be a role. In this context the hand has a role of Shock Resistance. This realizable entity also appears in the hands of many other primates; in this case, however, it would appear to be a function of the primate hand, as it is a function of their feet. I am left with a similar conclusion as previously. Just as Shock Resistance maybe either a biological or artifactual function, I must also conclude that the individuals of the same class can be a role, depending on the nature of the relationship between the independent continuant and realizable entity. ### 2.4 OBI as a case study So far in this paper, I have considered a number of illustrative examples and used these to draw conclusions about definitions for functions and roles. This methodology is appropriate, but has the limitation that the choice of other examples may have led us to different conclusions. In this section, therefore, I will consider the use that OBI (Ontology for Biomedical Investigations)(The OBI consortium, 2009) has made of function and role (Analysis was performed on OBI rc-1, (release 2009-11-06)). I choose to use OBI as it was built after BFO and with knowledge of it; many of the ontologies available from OBO were started without its use or knowledge. Function Bearer \| OBI Function ---\|--- Human \| Perturb, Measure, Separation (3), Sort Computer \| Information Processor (3), Consume Data Highly Generic \| Freeze, Heat, Environment Control, Mechanical, Record, Contain, Transfer, Cool, Connection, Synthesizing, Excitation (2), Ionization, Energy Supply (1) Distant Galaxy \| Magnify Out of Scope \| Molecular Function (3) Table 1: OBI Functions considered as Roles. I provide suggestions for entities that could engage in the same end-directed activity, but without being designed for the purpose. Function has been omitted from the OBI term names. Numbers indicate child terms which have been omitted for brevity. I do not consider Information Processor to be the function of a computer, as the definition is more specific than the term suggests. Considering first the functions of which OBI has 38. Can these functions be fulfilled by an entity which was not designed for the purpose? As shown in Table 1 most of them can, often by considering a highly generic device (like a computer) or organism (like a human). Some are highly generic in themselves and can be performed by many things (heat for example). I find no cases where a function could not be fulfilled by an entity which was not designed for the purpose. I consider “Molecular Function” to be out-of-scope for this section as it is not clear whether it fulfils the BFO definition of function. Next, I consider OBI roles. There are many more roles than functions – around 90 in fact. Due to the size, here I consider whole branches of the role hierarchy. There are a set of roles relating to reagents and their states. While label role (defined as a reagent role realized in a detection of label assay) seems sensible, it would appear that S35 CTP, used to label DNA, is manufactured specifically for this purpose. It certainly does not occur in nature. It would appear to fulfil the requirements for a function. Next, let us consider reference substance roles (a role which can support the observation of similarities, differences, relative magnitude or change). In many cases, biological assays use a reference which is not manufactured. However, consider $\lambda$-HindIII fragments or a calibration standard. These would all appear to function as an reference and have been produced specifically for this purpose. Finally, there are a number of roles for molecules or organisms: antigen role, pathogen role and primer role. In an age where we can engineer the production of DNA, protein and organisms, it is not clear that these can only ever be roles and not functions. As a result of this analysis, I suggest a modification to the current definition of role (Def: 5). Both roles and functions can become apparent (_realized_) in natural, social or institutional contexts. That such a context exists does not provide a clear differentation between role and function; the critical distinction relates to whether the entity in question was designed to be or has homologs that are engaged in a given process. I suggest, therefore, this alternate and simpler definition for role (Def: 6). ###### Definition 6 A realizable entity the manifestation of which brings about some result or end that is not essential to a continuant because of its kind. In short, from this case study, I conclude that the role/function distinction is not clear. While OBI has a specific intent in mind with its application of the distinction (broadly and not exhaustively, social or experimental roles, device or instrument functions), this distinction is not the distinction made in the current definitions of role and function in BFO; further given that most functions could also appear to be roles, and many roles appear also to be functions, I suggest that the distinction made in the current definitions is not useful in the context of OBI. The earlier theoretical analysis seems to be confirmed in practice within OBI. This suggests that the limitations in the definitions drawn from the earlier illustrative examples are general and not simply as a result of the specific examples chosen. ### 2.5 An axiomatisation for function and role I have produced an axiomatisation in OWL of functions and roles as defined in this paper, available as described in the abstract; due to space considerations I report here key differences between this and the BFO axiomatisation. * • There is an explicit relationship between RealizableEntity and Process. Subclasses use a more specific relationship. So, a ToAbsorbShock function may only be realized by a AbsorbtionOfShock process, if it is realized at all. * • Function and Role are defined classes. Stating that an instance of ToAbsorbShock is_function_of instance of FootSole implies, therefore, that the former is a function. * • Most leaves of RealizableEntity are direct children of RealizableEntity, with a few exceptions (ToReproduce is a child of BiologicalFunction). The definitions could be extended further; for simplicity, I have not added classes to differentiate between organisms and artifacts. These could be added to automate the population of BiologicalFunction and ArtifactualFunction. At the current time, it remains an open question whether Role, Function and its children should be disjoint. The key example of the function of a synthetic biological organism suggests that Biological and ArtifactualFunction function should not be disjoint (as it appears to be both), but I have no example which suggests whether Role and Function should be disjoint. In axiomatisating the examples given, these disjoint statements make no practical difference. Many ontologies are built using the OBO format; while this has a slightly weaker semantics than OWL it is possible to represent much of OWL in OBO format(Golbreich et al., 2007). The axiomatisation presented here can be represented using OBO union and intersection of to describe classes, which is usually translated as a definition. The universal link between RealizableEntity and Process has no natural equivalent. However, as this link has its own specific relationship which is restricted to this use, problems caused by the lack of an inexact semantic equivalent are likely to be relatively minor. The axiomatisation presented here is related to that produced by others; Dumontier(Dumontier, 2008) focuses more on roles, while Burek et al.(Burek et al., 2006) provides for a more complex representation, covering issues such as preconditions. ### 2.6 Applying the Definitions in Practice Finally, I consider whether these definitions are _applicable_ ; for a given set of entities how do we decide whether we have a function (of either subclass) or a role. The definition of an artifactual function easily allows its application: first, we determine whether the entity in question is an organism or part of one (which it should not be); second, we could ask whoever produced the entity what it was designed for. Of course, the second may not always be possible, in which case, we can guess from its design what its purpose is. In most cases, these questions will provide a clear answer. For biological function, the situation is less clear. Whether an entity is an organism or part of one is, in practice, likely to be straightforward for extant entities; otherwise, we can apply palentological techniques. Likewise, identification of closely related species and homologous structures is well known as it forms the basis of taxonomy. While developing an exact definition for “closely related” is outside the scope of this paper, it is possible. The definition that I introduce for Function in this paper (Defn 4) is conjunctive; it is either biological or artifactual. Here I have given little evidence that these are the only kind of function. Fundamentally, these two arise from very different mechanisms. There could be other appropriate subclasses of function; the most obvious possibility would be Chemical Function. However, artifacts are designed by humans who understand, mimic and improve on biology by building tools. It is this mimicry that we wish to reflect with a common definition joining biological and artifactual function; this is not true for Chemical Function. To determine whether something is a role, it is possible to make a determination on the basis of whether the context is optional(Arp and Smith, 2008). However, this optionality is a difficult criterion; firstly, all RealizableEntity’s are optional in the sense that they might never be realized and, secondly, the optionality can depend on how specifically we define the bearer. A hammer is not designed to hammer nails, as claimed earlier, it is designed to hit things; a nail hammer is designed to hit nails, a toffee hammer to hit toffee, a warhammer to hit anyone who irritates you. In practice, a role can be considered to be a negative definition; if there is a continuant and an end-directed activity that the continuant can be involved in, and this involvement is known not to fulfil the definition of either function, then we have a role. In this paper, I have considered OBI and found that the distinction between role and function is hard to apply; this is not true for all ontologies. For example, consider the Gene Ontology. In many cases, the homology will be considered as a standard part of the operating procedure(goe, ) in determining the function of a gene product; regardless, the evidence codes would allow us to make the distinction. We can conclude, therefore, that when the Gene Ontology is used to annotate a protein, this describes a biological function rather than a role. ### 2.7 Life is hard So far, I have considered a set of examples and how the definitions might be applied, including examples from OBI which have not been preselected. However, categorising life is hard; here, I consider some examples which present difficulties for the definitions I have given and the implications of these examples. In the first example, which I term a _drop out_ species, consider a human walking on their hands. Earlier (Section 2.3), I have suggested that this should be considered a role. Most of the primates do, however, walk on their hands. However, given that the homologous structure of closely related species use the structure for the same purpose, the definition of function (Def: 3) would appear to apply. It is for this reason that the definition specifies that (most) individuals of the same species must also demonstrate this behaviour (This definition differs slightly from that given at Bio-Ontologies 2009). In short, in the absence of most individuals in a species using a structure in a specific process, we should not use consider this structure to have a function. The second example, I term a _drop in_ species. Again, using a human example, I use my larynx for vocalisation and talking. Most primates, likewise, vocalise with their larynx; therefore, according to the given definition, this is a function of the larynx. However, speech is considered unique to humans, and therefore, their larynx; given that homologous structures in closely related organisms do not bear this realizable entity, which is part of the definition for biological function (Def: 3), I am forced to conclude that this is a role of the larynx and not a function. In short, while sharing a realizable entity within a species is NOT sufficient to allow the conclusion that this entity IS a function, NOT sharing a realizable entity within a species is sufficient to conclude that this entity is NOT a function. One solution to this difficulty is to state that where most individuals in a single species use a structure within a given process, this alone is sufficient to conclude that the structure has a function. Simply, most humans talk with their larynx, therefore this would be a function. I counter this, however, with the example that most humans use their fingers to operate their mobile phones, so we would be forced to conclude that this would also be a function. As this seems opposed to normal biological intuition and usage, I conclude, the presence of most individuals in a species using a structure in a specific process, is _not_ sufficient to conclude that this structure has a function. It is also possible that this difficulty could be resolved with greater knowledge or changes in biology. Def: 3 does not require species be extant; if a close, but extinct, relative of humans were shown to speak with their larynx, or if humans speciated while maintaining their speech, again, I would conclude that this represented a function. While human speciation seems unlikely, it is much more relevant to other taxa. Bacteria, in particular, evolve rapidly. There are many genes and proteins in bacteria which are unique to a species, family or lineage(Siew et al., 2004). In this case, the requirement for closely-related species seems to rule out the presence of a function. Again, this seems opposed to normal biological intuition and usage. I would counter this with two arguments. First, unlike primates, our knowledge of the extant bacterial species is very limited. The lack of knowledge of another speaking primate species is good evidence that no such species exists; the lack of knowledge of a close relative for a given bacterial species is not. Second, any definition which relies on a notion of a species is only as good as the definition of species; for bacteria, there is considerable debate about the utility of a species classification(Rosselló- Mora, 2003); my definition of function will need to evolve along with our understanding of bacterial ecology and gene flow; it may be necessary, as has been suggested with definitions for bacterial species(Rosselló-Mora, 2003), to have different definitions of BiologicalFunction tailored to different parts of the taxonomy. The evolution of a definition of role and function for proteins is difficult. At the level of the protein, I side with Dumontier(Dumontier, 2008), who suggests that the role/function distinction may be redundant; broadly, proteins can do anything their structure allows, and only do things their structure allows. The definitions given in this paper have a consistent interpretation at the level of the protein; this avoids the necessity of deciding at which level of granularity to stop making the role/function distinction. We can make the distinction at all levels if we choose, but we are not forced to do so, at those levels of granularity where it is not useful. It is important to note that arguing against a role/function distinction for proteins is not to dismiss the experience of biologists in the analysis of function assignment for genes. In this sense, the word “function” is being used to describe an association between a protein and a process that a protein molecule may be involved in; in short, the word “function”, in this case, can be considered to be a synonym for “realizable entity”. ## 3 Conclusions Here, I have taken an evolutionary approach to function and role by considering examples and using this to derive definitions which are as consistent as possible with current use within biomedical sciences. These definitions have been encoded in an axiomatisation which should enable the use of these definitions in a machine-interpretable way. The applicability of these definitions is a key advantage; the current distinction being made between function and role is a hard one to understand and apply. My definition distinguishes between the two based on the nature of the relationship to the independent continuant in which they inhere. I suggest that it is very hard to make the distinction at the class level; my study of OBI shows that very few of the functions and roles clearly fall into one category or another. For an individual continuant bearing a realizable entity, this distinction appears to be much more straightforward. I also provide a definition of biological function, something that is currently lacking in BFO. I have paid close attention to current biological usage; the definition is close to the process used to determine function. Moreover, it is highly applicable; all parts of the definition are measurable. The desire for an applicable and measurable definition is also the reason that I have avoided a definition based on the outcome of selective pressure; this is hard to test in most circumstances, requiring expensive evolutionary studies, and impossible for extinct species. Selective pressure can also be transient. Consider industrial melanism(Majerus, 1998); should melanic coloring be considered to gain its function during periods of pollution and lose it in post-industrial periods? By way of analogy, should a spanner measured in inches be considered to lose its function following metrication? Serendipitously, it also avoids difficult questions about artificial selection; we can state clearly that cows do not have a function of producing beef, though this is the outcome of selection. Importantly, my definition of biological function works across multiple levels of granularity: from organisms and organism parts through to genes and molecules; this is not true of previous definitions(Arp and Smith, 2008), which cover only anatomy. It is, however, not clear how useful the role/function distinction is for proteins and genes, as discussed earlier (Section 2.7); It is for this reason that I have used homology rather than orthology as the basis for the definition, as the latter is limited to the genetic scale, where the distinction is least useful. Finally, my definitions also do not allow distinctions that may often be made between different types of function. For example, most biologists would consider motion the most important function of muscle, while heat production a byproduct; or, for a more pathological example after Hoehndorf et al. (2009), most biologists would consider blood circulation to be a function of the heart, but “making loud thumping noises” not to be. This is a concern which could be best addressed by incorporating a degree of social ascription into the categorisation of realizable entities within biology; although it is outside the scope of this paper, this would provide a valuable and useful addition to the current ontological practice. In summary, I believe that the definitions and axiomatisation given in this paper make a significant contribution to the use of role and function in biomedical ontologies. They should enable a consistent use of these classes, because they consider current usage of the terms and the applicability of these definitions. I seek not to change current use but to formalize it. ## 4 Competing Interests The author has no competing interests. ## 5 Acknowledgements Thanks to Frank Gibson, Allyson Lister, James Malone, Helen Parkinson, Matt Pocock and Robert Stevens for many useful discussions on the contents of this paper. Thanks to the audience at Bio-Ontologies 2009 for the example of rapidly-evolving bacteria. To the memory of Mike Majerus who showed me how fundamentally strange evolution can be. ## References * (1) Basic Formal Ontology. URL http://www.ifomis.org/bfo. * (2) Guide to GO Evidence Codes. URL http://www.geneontology.org/GO.evidence.shtml. * Arp and Smith (2008) R. Arp and B. Smith. Function, role and disposition in basic formal ontology. In _The BIo-Ontologies Workshop (at ISMB 2008)_ , 2008. * Ashburner et al. (2000) M. Ashburner, C. Ball, J. Blake, D. Botstein, H. Butler, J. Cherry, A. Davis, K. Dolinski, S. Dwight, J. Eppig, et al. Gene ontology: tool for the unification of biology. The Gene Ontology Consortium. _Nat Genet_ , 25(1):25–9, 2000. * Burek et al. (2006) P. Burek, R. Hoehndorf, F. Loebe, J. Visagie, H. Herre, and J. Kelso. 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Opinion: the species problem, can we achieve a universal concept? _Syst Appl Microbiol_ , 26(3):323–326, Sep 2003\. * Siew et al. (2004) N. Siew, Y. Azaria, and D. Fischer. The orfanage: an orfan database. _Nucleic Acids Res_ , 32(Database issue):D281–D283, Jan 2004. doi: 10.1093/nar/gkh116. URL http://dx.doi.org/10.1093/nar/gkh116. * The OBI consortium (2009) The OBI consortium. Modeling biomedical experimental processes with obi. In _Bio-Ontologies 2009: Knowledge in Biology_ , 2009.	arxiv-papers	2013-09-23T21:15:10	2024-09-04T02:49:51.402166	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Phillip Lord", "submitter": "Phillip Lord Dr", "url": "https://arxiv.org/abs/1309.5984" }
1309.6089	# Hysteretic phenomena in a 2DEG in quantum Hall effect regime studied in a transport experiment M. V. Budantsev [email protected] A. V. Rzhanov Institute of Semiconductor Physics of SB RAS, Novosibirsk, Russia Novosibirsk State University, Russia D. A. Pokhabov A. G. Pogosov E. Yu. Zhdanov A. V. Rzhanov Institute of Semiconductor Physics of SB RAS, Novosibirsk, Russia Novosibirsk State University, Russia A. K. Bakarov A. I. Toropov A. V. Rzhanov Institute of Semiconductor Physics of SB RAS, Novosibirsk, Russia ###### Abstract We investigated experimentally non-equilibrium state of a two-dimensional electron gas (2DEG) in the quantum Hall effect (QHE) regime, studying the hysteresis of magnetoresistance of a 2DEG with a constriction. The large amplitude of the hysteresis enabled us to make the consistent phenomenological description of the hysteresis. We studied the dependence on the magnetic field sweep prehistory (minor loop measurements), recovered the anhysteretic curve, and studied the time dependence of the magnetoresistance. We showed that the hysteresis of magnetoresistance of a 2DEG in the QHE regime has significant phenomenological similarities with the hysteresis of magnetization of ferromagnetic materials, showing multistability, jumps of relaxation, and having the anhysteretic curve. Nevertheless, we revealed the crucial difference, manifested itself in an unusual inverted (anti-coercive) behavior of the magnetoresistance hysteresis. The time relaxation of the hysteresis has fast and slow regimes, similar to that of non-equilibrium magnetization of a 2DEG in QHE regime pointing to their common origin. We studied the dependence of the hysteresis loop area on the lithographic width of the constriction and found the threshold value of width $\sim$1.35 $\mu$m beyond which the hysteresis is not observed. This points to the edge nature of the non- equilibrium currents (NECs) and allows us to determine the width of the NECs area ($\sim$0.5 $\mu$m). We suggest the qualitative picture of the observed hysteresis, based on non-equilibrium redistribution of the electrons among the Landau level states and assuming huge imbalance between the population of bulk and edge electronic states. ###### pacs: 71.10.Ca, 71.70.Di, 73.43.Qt ## I Introduction A two-dimensional electron gas (2DEG) in the quantum Hall effect (QHE) regime demonstrates a number of intriguing phenomena that are not fully explained up to date. Among them are non-equilibrium phenomena manifested in hysteresis of sheet electron density, Pudalov1984 ; Dolgopolov1992 ; Dolgopolov1993 magnetization (see for review Ref.[Usher2009, ]) and electrochemical potential Huels2004 ; Klaffs2004 as a function of magnetic field and gate voltage (for gated structures). The hysteresis was observed in an ordinary single-layer 2DEG in nonmagnetic materials such as Si MOSFETs and AlGaAs/GaAs heterostructures at integer and fractional filling factors corresponding to a longitudinal resistance ($R_{\mathrm{L}}$) vanishing. It was found that the hysteretic variations of magnetization Usher2009 and electrochemical potential Huels2004 ; Klaffs2004 exceed equilibrium values by a factor of 10-60. In contrast the conventional electron transport measurements in QHE regime with vanishing $R_{\mathrm{L}}$ are not sensitive to the hysteresis, though, the question of a probable indirect influence of the non-equilibrium on the transport properties of 2DEG still remains open. Recently in a number of papers it was demonstrated that creation of a constriction in a macroscopic bath of 2DEG allows to reveal the hysteretic behavior in magnetoresistance Budantsev2007 ; Budantsev2009 ; Budantsev22009 ; Budantsev2011; Pioro-Ladriere2006 ; Smith2011 . It was shown that at low temperatures (lower than 100 mK) the hysteresis relaxes over a day Pioro-Ladriere2006 which is far beyond the time usually spent on a conventional measurement (a magnetic field sweep) in the QHE regime. In terms of usual magnetic hysteresis description Bertotti1998 this type of hysteresis is considered as time-independent. The most common explanation of the hysteretic phenomenon is based on non- equilibrium currents (NECs) slowly relaxing providing $R_{\mathrm{L}}$ vanishing. But there are no unambiguous models in the literature, adequately describing this phenomenon. The detailed consideration of NECs induction is proposed in the Ref. [Shikin2002, ], but it concerns only the case of weak non-equilibrium. Ruhe et al. Ruhe2009 studied the case of time-dependent hysteresis. The most developed model up to now Matthews2004 is based on the assumption of redistribution of “frozen” charges over the whole bulk area of the 2DEG Dyakonov1991 . These charges give rise to the radial (for the disk geometry) electric field that in turn gives rise to azimuthal drift magnetization current. The magnetization maximum is determined by the QHE breakdown current, calculated in the frames of quasi-elastic inter-Landau level scattering (QUILLS) approach.Eaves1986 The mentioned model correctly describes experimentally observed linear temperature suppression of the hysteresis amplitude observed both in magnetization Matthews2004 and constriction conductance.Budantsev2007 ; Budantsev2009 But the magnetization maximum calculated in the model is less than experimentally obtained value by a factor of 4. Moreover, the model predicts bulk NECs distribution, which, to our opinion, contradicts with experimental data of Klaffs et. al. Klaffs2004 In the present work we study the hysteresis of the magnetoresistance of the conventional 2DEG with constriction, placed in it. It was found that at certain critical width of the constriction the hysteresis of the magnetoresistance vanishes. This observation strongly suggests that NECs are localized near the edge of the sample. A phenomenological comparison of the 2DEG magnetoresistance hysteresis in the QHE regime with the hysteresis of magnetization of ordinary ferromagnetic materials is performed. Particularly, we discuss the dependence on the magnetic field sweep prehistory for both cases, studying minor loops, anhysteretic curves and time relaxations. Mostly we observe a similar behavior. However significant difference has been found manifested itself in unusual advancing (anti-coercive) response (2DEG magnetoresistance) to external parameter (magnetic field) change, which is just opposite to the behavior of ferromagnetic materials. We compare our results with the hysteresis of magnetoresistance in systems with pseudo-spin degree of freedom — so-called “quantum Hall ferromagnets” Piazza1999 ; DePoortere2000 ; Jungwirth2001 . In some cases (see e.g. Ref. [Piazza1999, ]) analogical anti-coercive behavior is found. This phenomenological similarity suggests that there are similar mechanisms underlying the origin of the hysteresis both in ordinary 2DEG and in QH ferromagnets. In addition to the phenomenological description, we provide a qualitative picture of a spatial distribution of NECs and electrochemical potential in the sample. The paper is organized in the following way. In Sec. II we describe the experimental detail. In Sec. III we present the experimental data of following measurements: the magnetoresistance of the 2DEG with the constriction in the QHE regime, minor loop, anhysteretic curve, time relaxation dependence of the magnetoresistance, and the dependence of the magnetoresistance hysteresis loop area on the lithographic width of the constriction and discuss them. In Sec. IV we develop a detailed qualitative picture of spatial distribution of NECs in a sample, explaining the hysteresis of the magnetoresistance of the constriction as well as the non-equilibrium magnetization of a 2DEG in the QHE regime. We discuss inverted (anti-coercive) behavior of the hysteresis and compare the hysteresis of the magnetoresistance with the hysteresis of the ferromagnetic materials magnetization. We conclude in Sec. V. ## II Experimental details Experimental samples were fabricated from GaAs/AlGaAs heterojunctions with 2DEG of two types (type I and type II) grown in different cycles of molecular beam epitaxy. The electron mobility $\mu$ and spacer layer thickness $d$ are $\mu$=0.6$\div$0.8$\times$106 cm2/V$\cdot$s and $d=400$ Å in the type I sample and $\mu$=0.8$\div$1.0$\times$106 cm2/V$\cdot$s and $d=300$ Å in the type II sample. The electron densities of the macroscopic 2DEGs of both types are ns=3$\times$1011 cm-2 at the temperature of 4.2 K. Hall bars of the dimensions $W$$\times$$L$=50$\times$100 $\mu$m2 have been created on the surface of the heterostructures by means of photolithography. In the middle part of the Hall bars constrictions of the effective width 0.6$\div$0.8 $\mu$m and the length 3 $\mu$m have been created by means of electron beam lithography (see the inset of Fig. 1). The samples of the both type allow to measure longitudinal magnetoresistance $R_{\mathrm{L}}=U_{35}/I_{12}$ and Hall magnetoresistance of macroscopic 2DEG $R_{\mathrm{H}}=U_{36}/I_{12}$. The measurements are carried out in the linear response regime on the alternating current of the magnitude 1$\div$10 nA and the frequency 7 Hz at the temperature of 60 mK. The magnetic field covering a range of 0$\div$15 T was oriented perpendicular to the 2DEG. Figure 1: The magnetoresistance of the samples of the type I. At the centers of the plateaux of macroscopic 2DEG Hall resistance $R_{\mathrm{H}}~{}=~{}U_{36}/I_{12}$ at odd and even integer filling factors a giant hysteresis of longitudinal magnetoresistance $R_{\mathrm{L}}~{}=~{}U_{35}/I_{12}$ is observed. The geometry of the sample is shown in the inset. ## III Experimental results First, we have studied the type I samples. A giant hysteresis of longitudinal magnetoresistance $R_{\mathrm{L}}$ of the constriction with respect to magnetic field sweep direction is observed in magnetic fields corresponding to centers of the plateau of macroscopic 2DEG Hall resistance $R_{\mathrm{H}}$ at both odd and even integer filling factors (Fig. 1). The effect is independent on a direction of magnetic field. The most pronounced hysteresis loop is observed near the filling factor $\nu=1$ in the range of filling factors between $\nu_{\mathrm{h1}}=0.95$ and $\nu_{\mathrm{h2}}=1.07$ on the background of macroscopic 2DEG Hall plateau observed in the range of filling factors between $\nu_{1}=0.85$ and $\nu_{2}=1.22$ (Fig. 1). The hysteresis has the amplitude 10 k$\Omega$ amounting to 100$\%$ of measuring signal and the width width of the hysteresis is 1.2 T at filling factor $\nu=1$. Drastic step-like rise of the magnetoresistance from zero up to 10 $k\Omega$ at up- sweep of the magnetic field and sharp drop down to zero at down-sweep of the magnetic field allows us to consider this behavior as the magneto-induced breakdown of QHE. The large amplitude of the hysteresis made it possible to carry out a detailed study. It has been found that the response of the system (magnetoresistance) outpaces the external excitation (magnetic field change) rather than retards from it, unlike the case of ferromagnetic magnetization. In mathematical terms the hysteresis loop is negatively oriented curve, while in ferromagnetic materials the hysteresis loop is positively oriented. One can see that the magnetoresistance dramatically changes at the entrance to the area of hysteresis, that is $\|dR/dB\|$ tends to a huge value, and it remains almost unchanged when leaving the area. In the case of retarded behavior the magnetoresistance would practically not be changed at the entrance to the area of hysteresis, that is $\|dR/dB\|$ would be close to zero. In this sense the observed hysteresis is abnormally inverted having a negative coercivity in contrast to the ferromagnetic magnetization. Figure 2: The hysteresis of the magnetoresistance of the II type sample at the filling factor $\nu=1$. (a) Major and minor loops. (b) Anhysteretic curve. Arrows indicate the magnetic field sweep direction. The sweep-rate is 0.05 T/min. The geometry of the sample and relaxation are shown in the insets. One can see the fast relaxation regime with $\tau$=2.4 s. The relaxation is occurred by jumps simultaneously in both structures. The magnetoresistance of the type II sample also demonstrates the hysteresis of magnetoresistance (Fig. 2) in the range of filling factors between $\nu_{\mathrm{h1}}=0.96$ and $\nu_{\mathrm{h2}}=1.06$ on the background of macroscopic 2DEG Hall plateau observed in the range of filling factors between $\nu_{1}=0.88$ and $\nu_{2}=1.16$. One can see that the hysteresis of magnetoresistance of the type II samples is observed at the same range of filling factors as in the type I samples but it does not demonstrate a giant amplitude and drastic jumps. This distinction is probably resulted from the differences in heterostructures parameters and needs to be discussed separately. Nevertheless, we succeeded to reveal inverted (anti-coercive) behavior of the magnetoresistance hysteresis of the type II sample studying the form of minor hysteretic loops. One can see that a change of the magnetic field sweep direction inside the major hysteretic loop leads to dramatic change of the magnetoresistance, i.e. the derivative $\|dR/dB\|$ tends to a huge value (Fig. 2(a)), while the retarding behavior would correspond to zero $\|dR/dB\|$ (as in the case of minor loops of ferromagnetic materials, where $\|dM/dH\|=0$). Moreover the analysis of the minor loops allows us to conclude that there is a large number of sample states inside the major hysteresis loop, and therefore the sample state is multistable. It is generally accepted that long-lived NECs are induced on the background of Hall plateaus where $\sigma_{xx}$ is vanishing. Our measurements shows that the range of existence of the hysteresis $\Delta\nu_{\mathrm{h}}$ is approximately 3 times less than the Hall plateau width $\Delta\nu$: $\Delta\nu_{\mathrm{h}}$/$\Delta\nu$$\approx$1/3. That is the condition $\sigma_{xx}\approx 0$ is necessary but not sufficient to observe the hysteresis. To perform consistent phenomenological comparison of the 2DEG magnetoresistance hysteresis in QHE regime with the hysteresis of magnetization of ferromagnetic materials we have measured the anhysteretic curves (Fig. 2(b)). Each experimental point $R_{i}(B_{i})$ has been obtained by cycling the magnetic field around $B_{i}$ with a decreasing amplitude. The form of the anhysteretic curve and its position inside the loop are phenomenologically similar to that of ferromagnetic materials. We have studied the time relaxation of the hysteresis, measuring the time dependence of the magnetoresistances of two conducting channels situated in the same Hall bar and separated by macroscopic 2DEG reservoir of the length of 250 $\mu$m and the width of 50 $\mu$m (see the top inset of Fig. 2(a)). The dependencies are shown in the insets of Fig. 2(a) and (b). The curves have been obtained after stopping the up-sweep of the magnetic field at the value 12.25 T, corresponding to the filling factor $\nu=1$ where the hysteresis has the maximum amplitude. It has been found that time relaxation has two phases: the fast initial one followed by slow phase. After stopping the magnetic field sweep we have observed the fast exponential relaxation by relatively small value (less than 25$\%$) with relaxation time 2.4 s (see the bottom inset of Fig. 2(a)). Such relaxation time is too big to be explained by spin-orbital coupling mechanism of relaxation Muller1992 ; Khaetskii1992 and resembles the nuclear relaxation time in GaAs which is about 30 s.Dixon1997 ; Devyatov2004 It should be noted that the non-equilibrium magnetization relaxes in the similar way.Kershaw2007 It also has regime of initial exponential decay followed by a much slower power-law decay, however the relaxation time in the first regime is longer ($\sim$20 s) than obtained in our work. In Ref. [Matthews2004, ] it has been suggested that magnetic field change results in QHE breakdown accompanied by electron transitions between adjacent Landau levels with opposite spin. The observed fast relaxation can be explained by such inter-Landau levels scattering and brings the system to a local (not global) minimum of energy after stopping the magnetic field sweep. According to Ref. [Matthews2004, ] such scattering results from QUILLS process Eaves1986 and takes place in a part of the sample where the maximum electric field is reached. In Refs. [Matthews2004, , Dyakonov1991, ] it is suggested that maximum electric field is reached in the bulk of a sample. However our results are sensitive to the sample edge (see Section IV) since the magnetoresistance is defined by the edge states filling. This allows us to conclude that QHE breakdown takes place at the sample edge. Further relaxation is much slower. During the next 40 min the resistances of both structures relax by a small value ($\sim$20$\%$). Moreover the most notable changes are occurred by sudden jumps simultaneously in both remote structures (see the inset of Fig. 2(b)). Taking into account that the structures are separated by macroscopic 2DEG reservoir of the length of 250 $\mu$m and the width of 50 $\mu$m it can be concluded that these jumps are caused by relaxation process in the macroscopic 2DEG reservoir. This observation once again confirms the conclusion that the narrow conductive channel is a tool for the study of the non-equilibrium phenomena taking place in a macroscopic 2DEG.Budantsev2007 The observed stepwise relaxation is phenologically similar to Barkhausen jumps observed in the ferromagnetic materials and originating from spin domain structure transformations.Bertotti1998 A formation of a spatial spin polarization in a single quantum wire has been considered in Ref. [Ihnatsenka2007, ] and it has been shown that this would result in hysteresis in electron transport measurements. However, all the calculated characteristics demonstrate the conventional coercive behavior. Consequently, the spin mechanism is not relevant to our results. Moreover, earlier it have been shown that the hysteresis of magnetoresistance is independent on the in-plain component of magnetic field.Budantsev2009 It allowed us to conclude that the hysteretic effect is not related to the electron spin. The measurements of non-equilibrium magnetization of a 2DEG also show stepwise relaxation.Smith2011 It should be noted that transport measurements give a valuable complementary information about the non-equilibrium state of a 2DEG. Fact is that the magnetization is an integral characteristic determined by the sum of all the magnetic moments, which are caused by all magnetization currents in a sample, while the magnetoresistance is sensitive to a local electrochemical potential of edge states. Particularly in this paper we succeed to observe the correlation between the relaxation of magnetoresistance of two remote constrictions and have established that NECs are localized near the edge. Figure 3: The dependence of the hysteresis loop area on the constriction width at the filling factors $\nu$=1 and 2 at temperature 0.48 K. To determine the width of spatial localization of NECs the dependence of magnetoresistance hysteresis loop area on the lithographic width of the constriction has been experimentally studied. For this purpose we used the series of samples fabricated from the same heterostructure. The samples were the Hall bars with constrictions of different widths. The constrictions of widths from 0.8 $\mu$m to 1.3 $\mu$m have been fabricated by means of electron beam lithography in the central part of Hall bars. All the constrictions have low resistance in zero magnetic field (0.2$\div$0.8 k$\Omega$) and demonstrate QHE in high magnetic fields. At the temperature of 0.48 K the hysteresis of magnetoresistance have been observed at the filling factors $\nu$=1, 2 and 4 for all the samples. Fig. 3 shows the hysteresis loop area as a function of the constriction width at the filling factors $\nu$=1 and 2. Experimental points are well fitted by linear functions. One can see that the hysteresis loop area decreases with the constriction width and vanishes at certain critical width $W_{0}$. Moreover these critical widths coincide for both filling factors and are about $W_{0}\approx$1.35$\mu$m. Assuming that the critical lithographic width comprises of two counter-propagating NECs of the width $W_{\mathrm{NEC}}$ and two depletion regions of the width $W_{\mathrm{depl}}\approx$0.2$\mu$m: $W_{0}=2W_{\mathrm{NEC}}+2W_{\mathrm{depl}}$, we obtain $W_{\mathrm{NEC}}\approx$0.5$\mu$m is comparable with the depletion width. The edge character of NECs has been also reported in the Ref. [Klaffs2004, ] devoted to the study of spatial distribution of the non-equilibrium electrostatic potential on a 2DEG surface, where it has been shown that the most significant changes of potential take place close to the edge. The observed hysteresis of magnetoresistance exhibits the following properties: (i) it has two phases of relaxation, fast and slow; (ii) the slow relaxation is occurred by jumps resembling the Barkhausen jumps in ferromagnetics; (iii) it demonstrates inverted anti-coercive behavior; (iv) it shows multistability and has an anhysteretic curve; (v) it can be observed in sufficiently narrow channel thus pointing out its edge character. ## IV Discussion The existing physical models are able to interpret only separate experiments. Unfortunately there is no any model which would not contradict all the set of experimental data on the non-equilibrium state in a 2DEG in the QHE regime. Analyzing the experimental data obtained in our and other studiesBudantsev2007 ; Klaffs2004 ; Matthews2004 ; Usher2009 ; Ruhe2009 we suggest such a qualitative picture. We propose the picture of spatial distribution of NECs in a 2DEG and discuss its relation to the experimentally observed non-equilibrium magnetization of the 2DEG. For simplicity, we consider a 2DEG at the filling factor near $\nu=1$, when electrons form the incompressible liquid on the first Landau level and inter Landau level scattering is assumed to be suppressed. ### IV.1 Magnetoresistance of constriction The obtained experimental data on the magnetoresistance can be interpreted in the frame of the following physical picture. Down-sweep of the magnetic field gives rise to the vortex electric field that induces outflow of the electrons from the bulk to the edge.Dolgopolov1992 ; Laughlin1981 In this case the area of the incompressible liquid increases and the edge channels are shifted closer to the lithographic borders of the sample. It means that the counter-propagating edge currents in the constriction become more distant from each other (Fig. 4(a-c)) than in the case of equilibrium state and the backscattering is suppressed. In the type II samples the magnetoresistance is indeed lower than its equilibrium value on the anhysteretic curve. In the type I samples at filling factor $\nu=1$ the decrease of magnetic field leads to complete suppression of the backscattering in the constriction, manifested in the vanishing of the longitudinal magnetoresistance and the establishment of the QHE regime. Up-sweep of the magnetic field leads to vortex electric field, which in turn induces outflow of the electrons from the edge to the bulk. In this case the area of the incompressible liquid decreases and the edge channels are shifted away from the lithographic border of the sample. The counter-propagating edge currents in the constriction come closer together (Fig. 4(d-f)) than in the case of equilibrium state and the backscattering increases. This results in magnetoresistance rise as we have observed in both types of the samples. The magnetoresistance of the type II samples is indeed higher than its equilibrium value on the anhysteretic curve. In the type I samples at the filling factor $\nu=1$ the increase of magnetic field switches the constriction from the QHE regime to the resistive state. In other words, the magnetic field sweep leads to magnetic field induced QHE breakdown. Figure 4: Non-equilibrium electron redistribution in the macroscopic 2DEG (a), (d) and in the constriction (b), (e) and top view of the 2DEG with narrowing (c), (f) at up- and down-sweep of the magnetic field correspondingly at the filling factor $\nu$=1. Bulk states in the constriction are occupied (c) or not occupied (f) with incompressible liquid in different magnetic field sweep direction. Arrows designate nonequilibrium currents. ### IV.2 Non-equilibrium magnetization Let us consider how the proposed picture corresponds to the non-equilibrium magnetization measurements. The spatial distribution of the NECs have been discussed in Refs. [Usher2009, , Ruhe2009, , Matthews2004, ]. However, the details of the mentioned models are not fully consistent with the experimental data obtained up to date. In particular, the model proposed in Refs. [Usher2009, , Matthews2004, ] assumes that the non-equilibrium charge is redistributed over the whole plane of a 2DEG and consequently the electric field is formed in the bulk. This contradicts experimental data obtained in the present study and discussed in Ref. [Klaffs2004, ] pointing to the fact that the non-equilibrium charge redistribution takes place only near the edge. In Ref. [Ruhe2009, ] the spatial distribution of the magnetization currents at the 2DEG edge is qualitatively described (Fig.8 in [Ruhe2009, ]), but the provided picture corresponds rather to equilibrium state because the common Fermi level for the bulk and the edge is introduced, up to which all the electronic states are occupied. Besides in this case the observed magnetization should not exceed the amplitude of the equilibrium dHvA oscillations while the experimental non-equilibrium magnetization is about 20$\div$60 times larger. Figure 5: The structure of the non-equilibrium edge currents at down- (a) and up-sweep (b) of the magnetic field. The decrease of magnetic field at the filling factor $\nu=1$ shifts the electrons from the bulk to the edge within the first Landau level. Due to the fact that the area occupied by the bulk states is much larger than that occupied by the edge states one can neglect the changes of the electrostatic potential in the bulk and consider that it remains flat. In this case the most significant changes are occurred near the edge, where a dipolar strip decreasing the electrostatic potential in the bulk and increasing it near the edge is formed. The electrons on the first Landau level populate the edge states up to the level of non-equilibrium electrochemical potential $\mu_{down}$ exceeding the potential of the empty bulk states by the value $\Delta\mu$ (Fig. 5(a)). In the crossed electric and magnetic fields the electrons drift along the edge forming the magnetization current approximately equal to $e^{2}/h\cdot\Delta\mu$. This current corresponds to the experimentally observed non-equilibrium paramagnetic magnetization.Matthews2004 ; Ruhe2009 The increase of magnetic field at the filling factor $\nu=1$ leads to more complicated distribution of the magnetization currents. The outflow of the electrons from the edge to the bulk leads to the formation of the dipolar strip near the edge that rises the electrostatic potential in the bulk and forms the potential well near the edge followed by the depletion region. It means that the electrostatic potential is non-monotonic and has a minimum near the edge, caused by interplay of two oppositely directed electric fields: the field of overpopulated bulk and the depletion field (Fig. 5(b)). On either side of the potential minimum the incompressible electron liquid forms counter-propagating drifting currents — giant diamagnetic current flowing closer to the bulk and low paramagnetic current flowing in close proximity to the edge. The presence of the paramagnetic current is confirmed by the experimental fact that the sign of the Hall resistance $R_{H}$ does not depend on the magnetic field sweep direction in a given orientation of the magnetic field. Hence the direction of the transport current, flowing along the edge of a sample, coincides with the direction of the paramagnetic current as in case of equilibrium state. Edge states are populated up to the non-equilibrium electrochemical potential level $\mu_{up}$ so that the empty edge states on the first Landau level are lower than populated bulk states by the value of $\Delta\mu$. The total magnetization current is dominated by the giant diamagnetic current $e^{2}/h\cdot\Delta\mu$. To explain the experimentally observed giant non- equilibrium magnetization of a 2DEG Usher2009 it is necessary that $\Delta\mu\geq(20\div 60)\hbar\omega_{c}$, where $\omega_{c}$ is a cyclotron frequency. Here we do not discuss the finite magnetic field range of the hysteresis loop as it requires detailed consideration of disorder in nonlinear screening conditions.Chklovskii1992 ### IV.3 Anti-coercive behavior There are two possible reasons for anti-coercive behavior: (i) an abrupt movement of the boundary of the incompressible liquid at small change of the magnetic field, (ii) topological transitions in quantum Hall liquid. Discuss them in more detail. (i) at the filling factor $\nu=1$ the magnetic field change $\Delta B$ causes the shift of the boundary of the incompressible liquid relative to the lithographic edge. Besides, the larger the sample the smaller the change of the magnetic field $\Delta B$ is required to shift this boundary by an amount of the order of the depletion width. For example for disk-shaped 2DEG of radius $R$: $\Delta B\approx\frac{W_{depl}}{2R}\cdot B.$ (1) (ii) As seen from Figs. 4(c) and 4(f) the shift of the boundary of the incompressible liquid leads to topological transitions in quantum Hall liquid. Fig. 4(c) corresponds to the total transmission of the edge channels through the constriction, while Fig. 4(f) corresponds to the intense backscattering. Thus, a small change of the magnetic field leads to a drastic change of the magnetoresistance. Topological transition has been observed in samples of type I: the vanishing of the longitudinal magnetoresistance has been observed indicating total suppression of backscattering in the constriction at down-sweep of the magnetic field, while up-sweep of the magnetic field causes sharp jump of the magnetoresistance from zero to 10 k$\Omega$. Abrupt movement of the boundary of the incompressible electron liquid at small change of the magnetic field is inevitably realized in the both types samples. This is manifested itself in the sharp changes of the magnetoresistance in minor loops measurements. Figure 6: Hysteresis of 2DEG magnetization (large loop) in QHE regime against the de Haas - van Alphen oscillations (thick line) is inverted (anti-coercive behavior) that is the response advances the magnetic field sweep. Small loop demonstrates the retarding (coercive) behavior. The discovered unusual anti-coercive behavior is in accordance with the law of energy conservation. Consider the behavior of the non-equilibrium magnetization of a 2DEG in the QHE regime. In Fig. 6 a typical hysteresis of the magnetization on the background of dHvA oscillations at the filling factor $\nu=1$ is shown. Matthews2004 The system response (magnetization) also advances the external parameter (magnetic field) change as in the case discussed. However the total work performed by the system $-\oint{Md\,B}$ (shadow area in Fig. 6) is negative, that is in accordance with energy conservation law. Retarding response behavior would correspond to the positive total system work (dashed area in Fig. 6) per cycle contrary to the law of conservation of energy. ### IV.4 Comparison with ferromagnetics Let us compare the observed hysteresis of magnetoresistance with the hysteresis of magnetization of ferromagnetics. It is well known that ferromagnetism is of spin origin. The exchange interaction in ferromagnetics causes the spins align. At the same time, the requirement of a minimum of the total energy of the system leads to its splitting into domains with opposite spin orientations. A change of the magnetic field results in spin flip process near the domain walls causing their “motion”. However, due to a disorder the transformations of domain structure is retarded with respect to the external magnetic field change. This is the reason of hysteresis with positive coercivity in ferromagnetics. While in a 2DEG a small changes of the external magnetic field lead to sharp motion of the boundary of the incompressible liquid and causes the hysteresis with negative coercivity. Earlier it has been found Budantsev2007 that the hysteresis of magnetoresistance is suppressed by temperature. It has been shown that the amplitude of hysteresis linearly increases as the temperature decreases and saturates at certain low temperature ($\sim$400 mK). The non-equilibrium magnetization of a 2DEG has similar temperature dependence.Matthews2004 The hysteresis of magnetization of ferromagnetic materials is also suppressed by temperature but it has different functional dependence described by Weiss theory of ferromagnetism. Bertotti1998 It should be noted that NECs in the QHE regime are induced by sweep of the magnetic field and can be considered as eddy currents as they are often called. However, in ferromagnetics it is enough to reduce the sweep rate to eliminate the influence of eddy currents. In this sense eddy currents in ferromagnetics are time-dependent because of finite longitudinal conductivity $\sigma_{xx}$. While in a 2DEG in the QHE regime when $\sigma_{xx}\approx 0$ the generation of NECs is inescapable at any reasonable sweep rate. According to our results NECs relaxation lasts for hours after stopping the magnetic field sweep. In Ref. [Kershaw2007, ] NECs have been shown to circulate for many hours. In terms of Ref. [Bertotti1998, ] under conventional experimental conditions NECs can be considered as time-independent. ## V Conclusions We have shown that sweep of the magnetic field in the QHE regime results in drastic changes of electrostatic potential in the bulk with respect to the edge if the inter Landau levels scattering is suppressed. This leads to giant magnetization currents flowing along the edge and topological transitions in a 2DEG with a constriction. The latter results in changes of magnetoresistance. The hysteresis of magnetoresistance of a 2DEG in the QHE regime has significant phenomenological similarities with the hysteresis of magnetization of ferromagnetic materials, showing multistability, jumps of relaxation, temperature suppression and having the anhysteretic curve. At the same time a fundamental difference expressed in anti-coercive hysteresis behavior of the magnetoresistance has been found. The possible causes of the limitations of the inter Landau level scattering as well as the possible limiting mechanisms of NECs are beyond the scope of this article and require further investigation. However we have experimentally shown that NECs are induced in a narrow ($\sim$0.5$\mu$m) area along the edge. Despite the large number of studies devoted to the non-equilibrium state of a 2DEG induced by the sweep of the magnetic field in the QHE regime there is no clear understanding of the phenomena at a microscopic level as well as a systematic phenomenological study of the dependence of the phenomena on parameters of heterostructures and conditions of epitaxial growth up to date. Unfortunately the NECs are insensible in a conventional transport measurement in a linear response regime due to zero longitudinal resistance. In such measurements the condition of the 2DEG is not clear in advance. Such uncertainty can bring unexpected difficulties in studies of a 2DEG in the QHE regime, such as the study of QHE breakdown by the electric current, that has no full and consistent model up to date (see for a review Ref. [Nachtwei1999, ]). The present study raises the issue of the need to search critical parameters that define the phenomena and explain the phenomena at a microscopic level. Therefore, some existing concepts on the phenomena in 2DEG in the QHE regime can be extended or revised taking into account the found non-equilibrium. ## Acknowledgments The reported study was supported by RFBR (research project No.14-02-31740-mol-a and No.12-02-00532-a), Program of fundamental scientific research DNIT RAS (project 3.2). ## References * (1) V. M. Pudalov, S. G. Semenchinsky, and V. S. Edelman, Sol. St. Commun.51, 713 (1984). * (2) V. T. Dolgopolov, A. A. Shashkin, N. B. Zhitenev, S. I. Dorozhkin, and K. von Klitzing, Phys. Rev. B 46, 12560 (1992). * (3) V. T. Dolgopolov, A. A. Shashkin, G. V. Kravchenko, S. I. Dorozhkin, and K. von Klitzing, Phys. Rev. B 48, 8480 (1993). * (4) A. Usher and M. Elliott, J. Phys.: Condens. Matter 21, 103202 (2009). * (5) J. Huels, J. Weis, J. Smet, K. v. Klitzing, and Z. R. Wasilewski, Phys. Rev. B 69, 085319 (2004). * (6) T. Klaffs, V. A. Krupenin, J. Weis, and F. J. Ahlers, Physica E 22, 737 (2004). * (7) M. V. Budantsev, A. G. 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Dolgopolov, D. Reuter, and A. D. Wieck, Phys. Rev. B 69, 115330 (2004). * (25) T. J. Kershaw, A. Usher, A. S. Sachrajda, J. Gupta, Z. R. Wasilewski, M. Elliott, D. A. Ritchie, and M. Y. Simmons, New J. of Phys. 9, 71 (2007). * (26) S. Ihnatsenka, I. V. Zozoulenko, Phys. Rev. B 75, 035318 (2007). * (27) R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). * (28) D. B. Chklovskii, B. I. Shklovskii, and L. I. Glazman, Phys. Rev. B 46, 4026 (1992). * (29) G. Nachtwei, Physica E 4, 79 (1999).	arxiv-papers	2013-09-24T09:09:46	2024-09-04T02:49:51.423544	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. V. Budantsev, D. A. Pokhabov, A. G. Pogosov, E. Yu. Zhdanov, A. K.\n Bakarov, and A. I. Toropov", "submitter": "Dmitry Pokhabov A", "url": "https://arxiv.org/abs/1309.6089" }
1309.6117	# Efficient Gate-tunable light-emitting device made of defective boron nitride nanotubes: from ultraviolet to the visible Claudio Attaccalite1, Ludger Wirtz2,3, Andrea Marini4, Angel Rubio5,6111To whom correspondence should be addressed; email: [email protected] 1 Institut Néel, CNRS, 25 rue des Martyrs BP 166, 38042 Grenoble cedex 9 France 2 Institute for Electronics, Microelectronics, and Nanotechnology (IEMN), CNRS UMR 8520, Dept. ISEN, 59652 Villeneuve d’Ascq Cedex, France 3 Physics and Material Sciences Research Unit, University of Luxembourg, 162a ave. de la Faïencerie, L-1511 Luxembourg 4 Istituto di Struttura della Materia (ISM), Consiglio Nazionale delle Ricerche, Via Salaria Km 29.5, CP 10, 00016 Monterotondo Stazione, Italy 5 Nano-Bio Spectroscopy Group and ETSF Scientific Development Centre, Departamento de Física de Materiales, Centro de Física de Materiales CSIC- UPV/EHU-MPC and DIPC, Universidad del País Vasco UPV/EHU, Av. Tolosa 72, E-20018 San Sebastián, Spain 6 Fritz Haber Institut der Max Planck Gesellschaft, Faradayweg 4-6, 14195, Berlin,Germany Boron nitride is a promising material for nanotechnology applications due to its two-dimensional graphene-like insulating and highly-resistant structure.Pakdel2012256 ; wirtz_review ; RevModPhys.82.1843 Recently it has received a lot of attention as a substrate to grow and isolate graphenebngraphene as well as for its intrinsic UV lasing response.watanabe1 ; watanabe2 Similar to carbon, one-dimensional boron nitride nanotubes (BNNTs) have been theoretically predictedrubiobn and later synthesised.tubesynthesis Here we use first principles simulations to unambiguously demonstrate that i) BN nanotubes inherit the highly efficient UV luminescence of hexagonal BN; ii) the application of an external perpendicular field closes the electronic gap keeping the UV lasing with lower yield; iii) defects in BNNTS are responsible for tunable light emission from the UV to the visible controlled by an transverse electric field (TEF). Our present findings pave the road towards optoelectronic applications of BN-nanotube-based devices that are simple to implement because they do not require any special doping or complex growth. Scientists have worked hard in the last decades to grow defect free nano- structures. The near-perfect atomic arrangement at the nano-scale has been employed to create new efficient devices as light-emitters, transistors and sensors. While many applications benefit from using defect free materials, the presence of particular impurities can generate new and fascinating properties. For example F-centre in ionic crystals have been widely used in luminescence applications.RevModPhys.18.384 More recently nitrogen vacancies in diamond have been proposed for quantum computation.PhysRevA.71.060310 Defects play an important role also for the luminescence properties of hexagonal boron nitride and related nanostructures.PhysRevLett.100.189701 ; attaccalite2011 ; PhysRevB.75.085205 ; PhysRevB.77.235422 ; PhysRevB.78.155204 Figure 1: Panel $a)$ left(right): lowest conduction band (highest valence band) in presence of an external field E=0.206 V/Å. In panel $b)$ we report the band gap of BN $8\times 8$ nanotube versus the TEF in LDA and $G_{0}W_{0}$ approximations; in panel $c)$ the corresponding optical absorption computed at the many-body Bethe-Salpeter level. Similarly to graphene, a single BN layer can be rolled up to form new structures ranging from single and multi-wall nanotubes to BN- fullerenswirtz_review . In contrast to graphite, the ionic character of the BN bond results in a wide band-gap of about 6 eV for bulk hexagonal- BNPhysRevLett.96.126104 ; PhysRevLett.100.189701 ; arnoud2006 . The combination of a such large gap with a strong electron-hole attraction makes the optical properties of hexagonal-BN based nanostructures largely independent of the layer arrangement and dimensionality.PhysRevLett.96.126104 ; wirtz_review Although bulk h-BN has been shown to exhibit a strong luminescence, it cannot be used for optical applications in the visible range because the emission frequency is fixed to about 5.75 eV in the UV. However the presence of impurities can drastically modify this scenario, as it has been shown, theoreticallyattaccalite2011 and experimentally.PhysRevB.75.085205 ; PhysRevB.78.155204 ; PhysRevB.77.235422 . The defects electronic structure and formation energies have been widely studied in bulk h-BN and BN nanotubes.PhysRevB.67.113407 ; p911623m ; jp056941l ; berzina ; jp800096s ; zobelli Luminescence in the visible was attributed to the presence of deep levels in the sample,ADMA200700493 ; nl0726151 ; Chen2010S353 whereas the UV emission is an intrinsic response (Frenkel-exciton) of the structurally perfect hexagonal BN.pierret2013excitonic Here we propose to engineer BNNTs with particular defects in order to generate light-emission in a wide range of frequencies, that can be tuned by means of an external electric field. The range of tunability of the proposed nanotube-based light emitting device depends from defect location and type. For pure BNNTs, it has been shown that the application of a transverse electric field generates a Stark effect leading to a strong reduction of the band gapPhysRevB.69.201401 ; PhysRevLett.94.056804 . The external field leads to a localisation of the conduction-band minimum/valence-band maximum on opposite sides of the tube(see Fig. 1). The corresponding energy shift of the band edges is thus proportional to the nanotube diameter. Experimentally as- grown nanotubes contain defects that lead to both deep or shallow levels in the gap. The wave-functions of these levels are, a priori, only slightly affected by the presence of an external electric fieldattaccalite2007 ; jp800096s because they are associated with localised orbitals centred on the impurity. However, their level position with respect to the bands edges changesjp800096s ; attaccalite2007 because valence(conduction) bands are modified by the external field. Here we will show that this property can be employed to produce tunable and highly efficient bright light-emission devices based on defective BN nanotubes. ## Results We model the electronic and structural properties of the pure and defected BN nanotubes under a TEF using state-of-the-art first-principles methods based on density functional theory(DFT) combined with many-body perturbation- theory(MBPT) approaches. These methods allow calculation of quasiparticle-band structure and optical properties with a high degree of accuracy (see Methods section for details). In the past this theoretical framework has been shown to be very efficient in predicting the electronic properties on BN nanotubes that were later on confirmed in the experiments.wirtz_review ; rubiobn ; PhysRevLett.96.126104 ; RevModPhys.82.1843 We start our study by analysing the case of pure isolated BN nanotubes immersed in a static transverse electric field. A transverse electric field reduces the band gap, as shown in panel $b)$ of Fig. 1. The gap reduction, induced by the TEF, is directly proportional to the electric field strength and to the tube diameter.PhysRevB.78.085423 Surprisingly the shrinking of the band-gap slightly modifies the optical response of the tube.attaccalite2007 The main exciton remains in the same position, while a small fraction of its spectral weight is redistributed to higher excitons (see panel $c)$ in Fig. 1). In fact the conduction and valence orbitals contributing to the gap reduction are localised on the opposite side of the tube, they have a very little overlap (see panel $a)$ of Fig. 1) and therefore their contribution to the optical response is negligible. Therefore we conclude that while the giant Stark effect present in BNNTs can modify their transport propertiestransport , it leaves the optical response mainly unchanged. The light emission spectra of pure BNNTs is thus not tunable by an external electric field. The presence of defects drastically modifies this picture. Different experiments have shown that impurities induce light emission below 5 eV in BN nanostructures,PhysRevB.75.085205 ; PhysRevB.78.155204 ; PhysRevB.77.235422 and modify the luminescence arising from the main bulk exciton.attaccalite2011 These effects can be explained by the presence of deep levels in the BN band gap.attaccalite2011 The low frequency emissions are due to transitions from and to these levels. Moreover when the impurity levels are close to the top valence band or bottom conduction band they mix with the bulk excitations giving rise to a splitting of the main excitonic peak.attaccalite2011 Among the different impurities responsible for light emission we can distinguish two families: defect complexes and single defect centres. The first family is formed by multiple defects as for instance di-vacancies, defect lines and so on. The second family consists in a single defect centre as for instance boron (nitrogen) vacancies or a substitution of a boron(nitrogen) atom with a carbon one. The main difference between these two families lies in the different kind of transitions involved in light absorption/emission processes. In the case of defect complexes both donor and acceptor states are present in the band gap while in the other case there is only a single donor or acceptor state. Therefore in the case of defect complexes the optical response is dictated by the quasi-donor-acceptor transitionsMuseur2008 , while in the simple defect centres light absorption/emission is due to transition between bulk states and deep defect levels (see Fig. 2). Figure 2: In the top panel we report the oscillator strength versus the angle between the defect and the bottom conduction band for $V_{B}$. The optical matrix elements are averaged on the first conduction bands within an energy range of $0.15$ eV. The straight lines are the dipole matrix elements at zero TEF. In the same figure it is present also a schematic representation of the a BN tube with a defect in presence of a TEF. In the bottom panels we show the band structure of the same tube versus the angle between the defect and the electric field. The red arrow represents the transition responsible for the luminescence in presence of $V_{B}$.attaccalite2011 The intensity of the TEF is 0.206 V/Å. The electronic structure of defects in BN nanotubes is similar to the one of defects in a single BN-sheet.PhysRevB.67.113407 In fact due to the large band gap curvature effects play a minor role on the optical properties of pure BNNTs, where the strong localisation of excitons renders the optical spectra almost independent from the tube diameter and chirality.louietube ; PhysRevLett.96.126104 In order to simulate a tube with defects we use the same methodology employed for the pure tube but with larger supercells in such a way to reduce the defect-defect interaction. Although large part of the tubes produced in the experiments are multi-wall and possess a zig-zag chirality, we chose a $12\times 12$ armchair one as prototype for our study. This choice is motivated by two reasons: first the primitive cell of an armchair tube, radius being equal, contains less atoms than a chiral one, second we expect only small differences with respect to the optical response of multi-wall or chiral nanotubes for the reasons discussed above. When we turn on a TEF, the band gap of the tube shrinks and consequently, the defect levels change position with respect to the band edges.jp800096s ; attaccalite2007 The orbitals associated with defects levels are strongly localised on the impurities (see right panel of Fig. 3) and therefore they are slightly deformed by the presence of the external field. To first order, the shift of the defects levels is thus given by the potential generated by the TEF and depends therefore on the position of the defect respect to the direction of the electric field (see inset in Fig. 2). This is visualised in the bottom panel of Fig. 2 for three different defect positions. We will show in the following, how this property gives rise to a tunable and efficient light emission. In order to predict the emission frequencies of BN nanotubes in presence of defects, we used a simplified approach. The first necessary ingredient to get light emission is non-vanishing optical matrix elements between the discrete donor(acceptor) state and the continuum states of the bottom conduction(top valence) bands. In the upper panel of Fig. 2 we show the strength of the optical matrix elements between the defect level and the bottom conduction bands for the case of a Boron vacancy, $V_{B}$. The optical matrix element displays a strong dependence on the angle between the defect position and the electric field (see inset in the top panel of Fig. 2). The same phenomena occurs for other simple acceptor or donor defects like substitution of a Nitrogen or a Boron atom by a carbon one, $C_{B}$ and $C_{N}$ respectively. Furthermore, we note that the optical matrix elements for polarisation along the tube axis (z-axis) dominate, which also holds for the optical response of pure nanotubeswirtz_review . The presence of the external electric field localises valence and conduction bands on opposite sides of the tube (see Fig. 1), therefore transitions are maximal only when the defect is aligned with the bottom(top) of the conduction(valence) band. The dipole element decreases to almost zero as the defect is turned to the opposite side of the tube. Consequently one can expect that luminescence, which is generated by transitions from and to the defects levels, will be efficient only when the defects are positioned on the side of ”localised” conduction(valence) band edge. This focusing effect increases with the tube size and field intensity. Now that we are sure that transitions from and to simple defect centres are not zero in presence of a transverse electric field, we investigate how the field modifies the emission frequencies. In order to predict light emission we start from the quasi-particle(QP) band structure in presence of defects. We consider the energy differences for transitions between defect states and the top valence(bottom conduction) states. This allows us to investigate light- emission versus transverse electric field, without including electron-hole interaction or lattice relaxation (see Methods section). We found that also in presence of defects the GW renormalisation for conduction(valence) bands and defects levels is almost a constant respect to the external electric field (see also Fig. 1(b)). --- Figure 3: Schematic representation of light emission process for an acceptor impurity in a BN nanotube. On the left a simplified band structure picture in presence of a TEF. On the right conduction and defect orbitals responsible for the emission process. Luminescence can be estimated from the QP band structure in presence of TEF as the sum of independent transitions between conduction bands and defect states. However light emission originates from electron-hole recombination, a two- particle process that cannot be described by means of the quasi-particle band structure only. In fact electrons and holes attract each other and this attraction modifies the transition energies. These processes can be naturally treated within a two-particle Green’s functions formalismstrinati and it has been shown that transition energies from and to defect states are strongly renormalisedattaccalite2011 by the electron-electron correlation. In order to model this correction, we calculate the exchange and electron-hole attraction between the defect level and the bottom conduction(top valence) bands only. In the past this approximation has been successfully employed to predict excitation energies of F-colour centres.louie In addition to the corrections originating from the electronic correlation, we have to consider the contributions due to the lattice relaxation induced by the excited carriers. These are the so called Stokes and anti-Stokes shifts. The Stokes shifts can be estimated by means of a constrained DFT calculation with different defect occupation. We investigated three different defects, an acceptor the Boron vacancy $V_{B}$, a donor the Carbon substitution of a Boron atom $C_{B}$, and the Boron-Nitrogen di-vacancy $V_{BN}$. In order to estimate the Stokes shift we considered the case of a completely empty acceptor state(or a completely filled donor state). In principle one should consider also the correction coming from the partial filling (emptying) of the conduction(valence) bands, but this is supposed to be a minor effect because these bands are delocalised along the $\vec{z}$ direction. In this way we obtain a rough estimation of the Stokes shift of $\Delta E_{s}\simeq 0.19~{}eV$ for $C_{B}$ and $\Delta E_{s}\simeq 0.03~{}eV$ for $V_{B}$. We did not calculated any Stokes shift for $V_{BN}$ because in this case it is irrelevant as it will be clear in the following. Vested with this theoretical approach we proceed in the study of light emission versus the external electric field. Figure 4: Predicted light emission for different defects as a function of the transverse electric field. All the defects are taken in the position of maximum emission, according to their optical matrix elements, see also Fig. 2. In the $V_{BN}$ we did not include any Stokes shift. We report our predicted light emission for $BN(12,12)$ tube versus the transverse electric field in Fig. 4. As one can see from the figure an external electric field allows to vary the emission frequency in a large spectral range for the CB and VB cases. Notice that in presence of defect complexes, as for instance the BN di-vacancy $V_{BN}$, the emission frequency does not change with the external field. In fact in this case the emission is dominated by transitions between donor and acceptor states in the band- gap.Museur2008 ; attaccalite2011 Since the wave-functions associated to these states are localised on the impurity, the effect of external electric field is irrelevant. In the left panel of Fig. 3 we present a schematic representation of the light emission process from BN nanotubes in presence of defects. We want to underline that this process happens only when the defect is aligned with the conduction(valence) maximum, otherwise the emission will be inefficient due to the small dipole matrix elements. Although the results of Fig. 4 can be theoretically extended to larger tubes, calculations become soon prohibitive due to the large number of atoms, the vacuum in the super-cell and the number of conduction bands that enter in the many-body operators. Therefore in order to predict light emission in larger (more realistic) tubes we assume many-body corrections to be a constant with respect to the tube size and we fit the emission energy with a simple linear curve $E_{emission}=E_{0}+\alpha\xi$ (1) where $\xi$ is the external electric field. This relation was already employed to describe the band gap closing of h-BN nanotubes under the effect of a TEF in simple tight-binding models and _ab-initio_ calculations.PhysRevB.78.085423 ; Chegel2012154 In principle the linear coefficient $\alpha$ depends on the tube size. In order to estimate this dependence we performed different calculations at the DFT level, varying the tube size. We found that $\alpha$ changes linearly with the tube radius $R$, $\alpha(R)=\alpha_{0}+R\beta$. A similar behaviour has been found for the pure BNNTs gap versus the electric- field and tube radius.PhysRevB.78.085423 Combining the previous two equations we can predict the electric field $\xi$ necessary to produce light emission at a given frequency $E_{1}$: $\xi=\frac{E_{1}-E_{0}}{\alpha_{0}+R\beta}.$ (2) Now we use Eq. 2 to estimate the intensity of the TEF that will induce emission in the visible range ($1.65-3.1$ eV). In Fig. 5 we report the visible emission range for the $V_{B}$ and $C_{B}$ cases versus the TEF intensity and tube radius. Figure 5: Visible light emission range as function of the tube radius and the external electric field for $V_{B}$ and $C_{B}$ cases (rainbow colours are just a guide to the eyes). In general an increase of the tube size reduces the strength of the transverse electric field necessary to obtain emission in the visible range, For sufficient large tubes the TEF intensity is of the same order of the one available in small devices. Notice that an TEF produces an electrostatic potential inside the tube that is proportional to the TEF intensity and the tube radius.jp800096s Therefore Eq. 2 breaks down for too large nanotubes or too strong fields. However a giant Stark effects has been experimentally measured in pure BN nanotubes with a radius of about $23~{}\AA$ and a TEF of $0.08~{}V/\AA$. In the same experiment a gap reduction of more than $1~{}eV$ has been obtained.PhysRevLett.94.056804 Comparing these values with our extrapolation in Fig. 5 it is clear that there is a large margin to produce visible light with experimentally accessible nanotubes and electric fields. Finally we consider defects formation and their charge state. Recent experimentsjin2009fabrication have shown that it is possible to introduce defects in h-BN structures by means of electron irradiation. This process is mainly dominated by boron mono-vacancies even if other larger vacancies are present. These vacancies can also be transformed in substitutional defects by introducing C atoms in the experiment,krivanek2010atom ; risto and the final process can be controlled by charging the system during the irradiation.risto These advances make possible the realisation of the device that we are going to discuss in the following. Regarding the charge state of the defects, in the present paper we investigated only neutral ones. Charged defects posses different relaxation energies and electronic structure. This fact influences also their optical properties, as it has been recently shown in the case of vacancies in SiC.bockstedte2010many The present results can easily be extended to charged defects and we expect that the main findings will remain valid. In fact the tunability of the light emission it is related to the localisation of defect states versus the delocalised bulk ones. Therefore a different charge state will modify the emission at zero field but not its behaviour in presence of a TEF. Now that we have shown how to produce tunable light emission with defective BN nanotubes and discussed the feasibility of our idea, we briefly present the possible configurations of a device based on BN nanotubes. The generic configuration of the device (see Fig. 6) comprises depositing as-grown BN nanotubes on an insulating surface (for example silicon oxide) acting as a dielectric to enable the application of the gated electric field that controls the light emission. The configurations is very much similar to the one of a field effect transistor (FET). The activation of the BN-defected optoelectronic device could be done by one of the following three processes: i) using UV light, ii) introducing an ambipolar current that recombines in the defect and emits light dictated by the applied gate voltageChen18112005 iii) using tunnelling current through an STM tip close to the nanotube. The excited electrons would inelastically decay very fast into the lowest energy state (the defect-liked Frenkel exciton) that would further decay by emitting light, again with a frequency dictated by the applied voltage, a process similar to the one leading to light emission in electronically excited semiconductors and fluorescent materials. A schematic set-up of those devices is illustrated in Fig. 6. ## Discussion In conclusion, we have shown that light emission from BNNTs with simple defect centres can be tuned by the presence of TEF. This opens the possibility to use these systems as light emitting devices. The use of (non-tunable) UV-light emitting devices based on crystalline hexagonal BN has already been suggested beforepatent . Here, we move one important step further by showing how nanotube based devices could operate in the UV and visible range by varying the external field. The external electric field, necessary to tune the emission, can be applied using a field-effect transistor configurationtransport . The intensity necessary to produce visible light decreases with the tube size. The present results have been patented by some of the authors.tubepatent Figure 6: Schematic set-up for the suggested three possibilities to activate the optoelectronic device based on defective BN nanotubes (i) light-induced luminescence (ii) ambipolar transistor configuration (iii) electron induced electron-hole pairs by means of STM tip. Finally we envision that the present findings can be applied to other two- dimensional semiconducting or insulating layered materials that form tubes, as it is the case for transition metal dichalcogenides.cogenides ## Methods BNNTs are simulated by using a supercell approach,rubiobn where the tube is oriented along the $z$ direction, and a large empty space is left in the other two directions between tube replica in order to reduce the tube-tube interactions. Subsequently a sawtooth electric field (see inset in Fig. 1(a)) with the cell periodicity is added along the $x$ direction. In order to simulate light emission in BNNTs we employed a combination of Density- Functional Theory (DFT) plus Many Body Perturbation Theory (MBPT). DFT is an exact theory for ground state properties and it is known to describe very well the structural properties of boron-nitride nanostructures within Local Density Approximation (LDA). All DFT calculations have been performed using a $1\times 1\times 5$ supercell containing 240 atoms. The distance between the tube replica was 29 a.u. , we used a $1\times 1\times 2$ k-point sampling, LDA for the exchange correlation functionalceperley , a plane waves cutoff of 45 Ry for the wave-function and norm-conserving pseudo-potentials.troullier All DFT calculations have been performed with the PWSCF codepwscf and the atomic structures have been relaxed using a BFGS quasi-Newton algorithm. Excited state and optical properties have been studied by means of MBPT. We calculated quasi-particle properties solving a Dyson equation within the so-called $G_{0}W_{0}$ approximationaryasetiawan1998gw ; gw-strinati , where all the Green’s functions and the self-energy operator are constructed with eigenvalues and eigenvectors of the Kohn-Sham(KS) Hamiltonian. Non-self consistent GW calculations have been performed with the code YAMBOyambo using a plasmon pole approximation for the dielectric constant. We used 30.000 G-vectors for the wave-function, 2 Ha for the response block size and 3000 bands for the screening. A cylindrical cutoff has been applied to the Coulomb potential in order to reduce the tube-tube interaction. Neutral excitations, responsible for the absorption spectra were obtained from a two-particle Green’s function equation, the Bethe-Salpeter equation, that is solved in the static ladder approximationstrinati , including excitonic effects. We excluded quasi-free electron statesPhysRevB.69.201401 in the Bethe-Salpeter equation, because they are not supposed to be responsible for luminescence. We performed all calculations without including spin-polarisation effects. Even if we know that exchange-splitting slightly modifies the defects levels positionsattaccalite2011 ; PhysRevB.76.014405 , this effect does not modify the main results of the paper. For the large tubes employed to get the results in Fig. 4 we estimated the GW and electron-hole interaction from the one of a BN-sheet with the same defects and a distance between the periodic replica equal to the inter-tube distance.attaccalite2011 ## Acknowledgements We acknowledge financial support from the European Research Council Advanced Grant DYNamo (ERC-2010-AdG - 267374) Spanish Grant FIS2011-65702-C02-01, Grupo Consolidado UPV/EHU del Gobierno Vasco (IT-319-07) and European Commission project CRONOS (280879-2). Computational time was granted by i2basque and BSC ”Red Espanola de Supercomputacion” and GENCI-IDRIS (Nos. 100063 and No. 091827). A. 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Ab initio studies of possible magnetism in a $bn$ sheet by nonmagnetic impurities and vacancies. _Phys. Rev. B_ 76, 014405 (2007).	arxiv-papers	2013-09-24T11:38:06	2024-09-04T02:49:51.433552	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Claudio Attaccalite, Ludger Wirtz, Andrea Marini and Angel Rubio", "submitter": "Claudio Attaccalite CA", "url": "https://arxiv.org/abs/1309.6117" }
1309.6176	# Feature Learning with Gaussian Restricted Boltzmann Machine for Robust Speech Recognition ###### Abstract In this paper, we first present a new variant of Gaussian restricted Boltzmann machine (GRBM) called multivariate Gaussian restricted Boltzmann machine (MGRBM), with its definition and learning algorithm. Then we propose using a learned GRBM or MGRBM to extract better features for robust speech recognition. Our experiments on Aurora2 show that both GRBM-extracted and MGRBM-extracted feature performs much better than Mel-frequency cepstral coefficient (MFCC) with either HMM-GMM or hybrid HMM-deep neural network (DNN) acoustic model, and MGRBM-extracted feature is slightly better. Index Terms: restricted Boltzmann machine, robust speech recognition, feature learning ## 1 Introduction Since hybrid hidden markov model (HMM)-deep neural network (DNN) was introduced to large-vocabulary continuous speech recognition (LVCSR), the accuracy of speech recognition system has made significant performance improvements in idealized environments [1][2]. Such progression urges the development of speech recognition systems that are robust to background noise and channel distortion, as more and more speech applications are deploying on mobile devices. State-of-the-art robust automatic speech recognition system usually involves intensive specialized domain knowledge [3]. But we are more interested in the transformation of feature. Feature learning (representation learning) [4] is a developing field that grows alongside with deep learning. The aim of feature learning is to learn a certain kind of transformation through which we are able to extract information that makes discrimination much easier for classifiers. Feature learning needs as little feature engineering as possible, and transformed feature is much closer to real underlying factors that generate the original features that we observe. In this paper, we made our first attempt to apply the idea of feature learning to robust speech recognition. Although dozens of alternatives have been proposed over the past few decades, MFCC is still the default choice of feature for many speech applications. Instead of trying to propose another alternative, we are more interested in learning a better representation of MFCC feature with restricted Boltzmann machine (RBM) and its variants. We will explain why RBM may be able to learn a more suitable representation of MFCC for robust speech recognition. We will also be proposing a new variant of RBM called multivariate Gaussian restricted Boltzmann machine (MGRBM) which is specially designed for modeling the distribution of speech data. MGRBM is able to capture the evolving characteristic of speech within a context of several frames which is difficult to model with a Gaussian restricted Boltzmann machine (GRBM). We perform our experiments on the Aurora2 corpus and our results show that the learned features are better than the original feature for robust speech recognition. ## 2 Model ### 2.1 Restricted Boltzmann Machine The Boltzman machine is a special kind of Markov random field which models the joint probability distribution of the visible variable and hidden variable. Visible and hidden variable are both defined to be multidimensional Bernoulli variables. The distribution can be written as: $\displaystyle p(v,h)=\frac{1}{Z}e^{-E(v,h;\theta)}$ (1) and $\displaystyle E(v,h;\theta)=-\frac{1}{2}v^{T}Uv-\frac{1}{2}h^{T}Vh-v^{T}Wh-a^{T}v-b^{T}h$ (2) $E(v,h;\theta)$ in (1) is called energy function. $\theta=\\{U,V,W,a,b\\}$. $U,V,W$ models the visible-visible, hidden-hidden, and visible-hidden interaction respectively. $a$ and $b$ are bias vectors. The Restricted Boltzmann Machine (RBM) [5] is perhaps the most widely-used variant of Boltzmann machine. The energy function of RBM is the simplified version of that in the Boltzmann machine by making $U=0$ and $V=0$. That is, the energy function of an RBM is: $\displaystyle E(v,h;\theta)=-a^{T}v-b^{T}h-v^{T}Wh$ (3) An RBM is typically trained with maximum likelihood estimation. Taking the derivative with respect to the logarithm of the product of all the probability of training cases, we can derive the learning algorithm of RBM as follows: $\displaystyle\Delta W$ $\displaystyle=$ $\displaystyle\epsilon(<vh^{T}>_{data}-<vh^{T}>_{model})$ (4) $\displaystyle\Delta a$ $\displaystyle=$ $\displaystyle\epsilon(<v>_{data}-<v>_{model})$ (5) $\displaystyle\Delta b$ $\displaystyle=$ $\displaystyle\epsilon(<h>_{data}-<h>_{model})$ (6) The symbol $<\cdot>_{data}$ in (4)(5)(6) represents an average with respect to the conditional distribution $p(v\|h)$ and $<\cdot>_{model}$ represents an average with respect to the joint distribution $p(v,h)$. The $<\cdot>_{data}$ for RBM is generally easy to train because: $\displaystyle p(h_{j}=1\|v)=sigmoid(b_{j}+W_{j}v)$ (7) $\displaystyle p(v_{i}=1\|h)=sigmoid(a_{i}+W_{\cdot i}^{T}h)$ (8) (7)(8) can be derived from the definition of RBM and $sigmoid(x)=\frac{1}{1+e^{-x}}$. However, $<\cdot>_{model}$ is much harder to obtain. To address this problem, $<\cdot>_{model}$ is usually approximated with $<\cdot>_{recon}$ as the following: $\displaystyle\Delta W$ $\displaystyle=$ $\displaystyle\epsilon(<vh^{T}>_{data}-<vh^{T}>_{recon})$ (9) $\displaystyle\Delta a$ $\displaystyle=$ $\displaystyle\epsilon(<v>_{data}-<v>_{recon})$ (10) $\displaystyle\Delta b$ $\displaystyle=$ $\displaystyle\epsilon(<h>_{data}-<h>_{recon})$ (11) The $<\cdot>_{recon}$ represents an average with respect to the reconstruction of the visible data. The reconstruction of visible data is obtained by setting each node in hidden layer value 1 with probability (7), followed by setting each node in visible layer value 1 with probability (8). This is the contrastive divergence algorithm (CD) [6] for training of RBM. CD has been empirically showed to be adequate for many applications. ### 2.2 Gaussian Restricted Boltzmann Machine To model real-valued data, the Gaussian restricted Boltzmann machine (GRBM) has been proposed [7][8]. The energy function of GRBM is typically defined with: $\displaystyle E(v,h;\theta)=\sum_{i}\frac{(v_{i}-a_{i})^{2}}{2\sigma_{i}^{2}}-\sum_{ij}W_{ij}\frac{v_{i}}{\sigma_{i}}h_{j}-\sum_{j}b_{j}h_{j}$ (12) in which $\theta=\\{a,b,W,\sigma\\}$ and $\sigma_{i}$ models the standard deviation of each visible units. Conveniently, the learning algorithm of GRBM is the same with RBM (9)(10)(11). The conditional probabilities necessary for CD of a GRBM are: $\displaystyle p(v_{i}\|h)=\mathcal{N}(a_{i}+\sigma_{i}\sum_{j}W_{ij}h_{j},\sigma_{i}^{2})$ (13) $\displaystyle p(h_{j}=1\|v)=sigmoid(b_{j}+\sum_{i}W_{ij}\frac{v_{i}}{\sigma_{i}})$ (14) Generally speaking, $\sigma_{i}$ can be learned from data, but it’s difficult with CD. The training data to be modeled with a GRBM are always pre-processed mean 0 and variance 1 and thus $\sigma_{i}$ can be fixed with 1 and not trained. The reason why $\sigma_{i}$ is difficult to train with CD can be explained as follows: When $\sigma_{i}$ is much smaller than 1, the visible- hidden effects (13) tends to be large and hidden-visible effects (14) tends to be small. The result of such effect is that hidden units always tend to be firmly 1 or 0, and thus undermine the whole training process. One disadvantage of GRBM is its conditional independence assumption. That is, conditioned on the hidden layer, each visible unit is assumed to follow a Gaussian distribution and independent with each other. However, for natural data such as speech and image, they tend to have local similarity property. Take speech data for example, the smoothness of speech always makes one frame of acoustic feature similar to the frames next to it. Such local similarity property is difficult to capture with GRBM and yet contains certain amount of information. To offset this problem, we propose a variant of GRBM called multivariate Gaussian restricted Boltzmann machine. ### 2.3 Multivariate Gaussian Restricted Boltzmann Machine Figure 1: Graphical model of a multivariate Gaussian restricted Boltzmann machine. Links between sub-units in visible layer and hidden units are fully connected. The interaction of the $k$th sub-unit in the $i$th visible unit with $j$th hidden unit is modeled with $W_{i}^{kj}$. The multivariate Gaussian restricted Boltzmann machine (MGRBM) is a natural generalization of GRBM. The graphical model of a MGRBM is illustrated in Figure 1. Compared with GRBM, in which each unit in visible layer is modeled with a Gaussian distribution given the hidden layer, a MGRBM assumes that each unit in visible layer is modeled with a multivariate Gaussian distribution given the hidden layer. Similar to what $\sigma_{i}$ models in a GRBM, we denote the covariance matrix of each unit in visible layer of a MGRBM with $\Sigma_{i}$. Consider only the non-degenerate case, $\Sigma_{i}$ is a positive-definite matrix and thus Cholesky decomposition can be applied: $\Sigma_{i}=AA^{T}$. Since matrix $A$ is full-rank, we can denote $B=A^{-1}$. With these notation, we can define the energy function of an MGRBM as: $\displaystyle\begin{split}E(v,h;\theta)=\frac{1}{2}\sum_{i}(v_{i}-\mu_{i})^{T}B_{i}B_{i}^{T}(v_{i}-\mu_{i})\\\ -\sum_{i}v_{i}^{T}B_{i}W_{i}h-b^{T}h\end{split}$ (15) Suppose the number of units in visible layer and hidden layer is $N_{v}$ and $N_{h}$ respectively, and each unit in visible layer has $d$ dimension. Then $v_{i},\mu_{i}$ each is a $d\times 1$ vector; $B_{i}$ each is a $d\times d$ matrix; $W_{i}$ each is a $d\times N_{h}$ matrix; $b$ and $h$ are both $N_{h}\times 1$ vectors. Similar to GRBM, we can also prove that: $\displaystyle p(v_{i}\|h)=\mathcal{N}(\mu_{i}+(B_{i}^{T})^{-1}W_{i}h,(B_{i}^{T})^{-1}B_{i}^{-1})$ (16) $\displaystyle p(h_{j}=1\|v)=sigmoid(b_{j}+\sum_{i}W_{ij}^{T}B_{i}v_{i})$ (17) and learning algorithm is: $\displaystyle\begin{split}\Delta\mu_{i}=\epsilon(<B_{i}B_{i}^{T}(v_{i}-\mu_{i})>_{data}\\\ -<B_{i}B_{i}^{T}(v_{i}-\mu_{i})>_{model})\end{split}$ (18) $\displaystyle\Delta b=\epsilon(<h>_{data}-<h>_{model})$ (19) $\displaystyle\Delta W_{i}=\epsilon(<B_{i}^{T}v_{i}h^{T}>_{data}-<B_{i}^{T}v_{i}h^{T}>_{model})$ (20) $\displaystyle\begin{split}\Delta B_{i}=\epsilon(<(v_{i}-\mu_{i})(v_{i}-\mu_{i})^{T}B_{i}-v_{i}h^{T}W_{i}^{T}>_{data}\\\ -<(v_{i}-\mu_{i})(v_{i}-\mu_{i})^{T}B_{i}-v_{i}h^{T}W_{i}^{T}>_{model})\end{split}$ (21) In our experiment, we use persistent contrastive divergence (PCD) [9] to train MGRBM. The algorithm can be written as: $\displaystyle\begin{split}\Delta\mu_{i}=\epsilon(<B_{i}B_{i}^{T}(v_{i}-\mu_{i})>_{data}\\\ -<B_{i}B_{i}^{T}(v_{i}-\mu_{i})>_{fanta})\end{split}$ (22) $\displaystyle\Delta b=\epsilon(<h>_{data}-<h>_{fanta})$ (23) $\displaystyle\Delta W_{i}=\epsilon(<B_{i}^{T}v_{i}h^{T}>_{data}-<B_{i}^{T}v_{i}h^{T}>_{fanta})$ (24) $\displaystyle\begin{split}\Delta B_{i}=\epsilon(<(v_{i}-\mu_{i})(v_{i}-\mu_{i})^{T}B_{i}-v_{i}h^{T}W_{i}^{T}>_{data}\\\ -<(v_{i}-\mu_{i})(v_{i}-\mu_{i})^{T}B_{i}-v_{i}h^{T}W_{i}^{T}>_{fanta})\end{split}$ (25) with $<\cdot>_{fanta}$ denotes average with respect to fantasy particles. Notice that the problem for updating variances in a GRBM which we described in section 2.2 still exists in MGRBM. In section 3.1 we will explain how we address this problem in our experiments. MGRBM is specially designed for speech data to address the problem of GRBM described above. Typically, for the task of speech recognition with hybrid HMM-neural network (NN) method, a context of several frames of acoustic feature is used for each training case. Unlike GRBM, MGRBM can explicitly model the evolving characteristics in each context. How a MGRBM can be used for robust feature extraction is illustrated in figure 2. Concretely, suppose the original acoustic feature for each frame has $D$ dimensions and each context is chosen to be $C$ frames. Then the visible layer of MGRBM has $D$ units, each has dimension $C$; the $d$th dimension in the $c$th frame acoustic feature corresponds to the $c$th dimension of $d$th unit. Figure 2: Extract new feature from acoustic feature with multivariate Gaussian restricted Boltzmann machine. There are two reasons for the above setting. First, the correlation modeled across each dimension of acoustic feature would act as a strong regularization in temporal perspective if training data is much different from testing data. Second, as is the case with GRBM, MGRBM also has conditional independence assumption. Fortunately, this assumption is indeed satisfied if we use MFCC as acoustic feature, for the step of discrete cosine transform already has the effect of decorrelation. ### 2.4 Feature Learning for Robust Speech Recognition Despite of its prevalence and huge success in phoneme recognition [10][11] and LVCSR [1], deep neural network (DNN) stand alone is rarely used as a acoustic model for robust speech recognition. We believe that this is due to the fact that neural network is a discriminative model, whose optimization objective is better discriminative power and lower classification error. However, for the task of robust speech recognition, especially in mismatched scenario where the training data is clean and the testing data is noisy, the power of such model is greatly degraded due to its poor generalization over highly distorted data. We assert that DNN performs significantly poorly in very noisy conditions than GMM and we will prove it in our experiments later. Generative models focus on modeling how the data is “generated”. Natural data such as speech and image are usually high-dimensional, but all that make sense occupy only a subspace (or lower-dimensional manifold). The learning process of a generative model is essentially to find out this manifold by tuning all its parameters. Background noise from real-life environment, we believe, is substantially different from white noise, because the former kind of signal contains certain characteristic shared with sounds that spread through the air. For this reason, we consider it more appropriate to model speech with generative models. So, we would like to find a model that can leverage such advantage of generative models and yet escape from a model with strong assumption like GMM, and RBM is indeed a such model. As a generative model, RBM (and its variants) makes little assumption of data. What is more important, it belongs to a family called product of experts (as opposed to mixture of experts which GMM belongs to) [12]. This makes it exponentially more powerful and less prone to over-fitting. Other than modeling the distribution of data, it provides a natural way to transform feature by applying (7)(14)(17). Since such transformation takes the whole distribution of data into consideration, the learned feature tends to be much more abstract and expressive. ## 3 Experiments In this paper, we used the Aurora2 data set [13] for our experiments. In all the experiments described below, we used only clean data set as training data and whole test set as testing data. We intentionally use acoustic models that are simple and comparable. Two different kinds of acoustic model is utilized in our experiments : HMM-GMM and hybrid HMM-DNN. With each kind of acoustic model, we compare the word error rate (WER) of three kinds of feature : MFCC (12 coefficients + energy + delta + acceleration, 39-dimension), GRBM-extracted feature (G-feature) and MGRBM- extracted feature (M-feature). We used standard Aurora2 setup described in [13] for baseline system (GMM + MFCC). We use DNN simply because it is a good classifier and it is much more natural to employ RBM-extracted feature. ### 3.1 RBM Training We used two kinds of RBM in our experiment, one for feature extraction and the other for pre-training of DNN. GRBM and MGRBM are both trained to extract feature for comparison. For these GRBMs and MGRBMs, PCD with stochastic gradient descent (SGD) was used. The size of a mini-batch is 128 and no momentum was applied. The number of fantasy particles was the same with the size of a mini-batch, and one full Gibbs update was performed for each gradient estimate. For all the weights and biases, learning rate was 0.001. For the updating of $B_{i}$ in (15) of an MGRBM, learning rate was 0.0001. To avoid the problem of updating variances described in section 2.2, we divide each $B_{i}$ by its trace after each updating. This step makes all diagonal elements of $B_{i}$ fixed with one and thus make the learning stable. All GRBM and MGRBM were trained for 400 epochs and each with 1024 hidden units, which made G-feature and M-feature both has 1024 dimensions. The visible layer corresponds to a context of 9 frame of MFCC feature, which makes the number of visible units of GRBM 351 and MGRBM 39$\times$9. For all the GRBMs and RBMs that were used for pre-training DNNs, CD algorithm (CD-1), SGD with batch size 128 and a momentum of 0.9 were used. We trained all the RBMs with learning rate of 0.01 for 50 epochs and all the GRBMs with learning rate of 0.001 for 100 epochs. ### 3.2 DNN Training All DNNs in our experiments have 4 hidden layers, each hidden layer with 1024 units. DNNs were pre-trained with stacked RBM (section 3.1) as described in [14] and fine-tuned with back-propagation algorithm with SGD as described in [11]. The learning rate for back-propagation started from 1.0 and was halved if an increase of substitution error on development set was observed during the end of each epoch and all the weights roll-back to the end of last epoch. When using MFCC feature, the input layer of DNN corresponds to 9 frame of MFCC, which is 351 units. The output layer has 180 units, with each unit corresponding to each state in HMM. All data to be trained with DNN are normalized to have mean 0 and variance 1 with respect to each dimension. ### 3.3 Results All results from our experiments are shown in Table 1. We averaged WER in all test set across different noises. The first thing we should notice is that in the lowest SNR condition, HMM-DNN performs consistently poorer than HMM-GMM, which confirms our assertion in section 2.4. Those WER that exceeded 100% was due to many substitution errors. From the last three columns, we can clearly see the advantage of G-feature over MFCC and M-feature over G-feature. Although the improvements does not seems to be obvious, the trend that the gap between M-feature and G-feature increases with the decrease of SNR is still distinguishable. The reason why the improvements of M-feature over G-feature is marginal, we believe, is that the model which we trained is still not good enough. All the learning rates, number of epochs and number of hidden units are chosen heuristically, and 9 frame of context might be not sufficiently long. So there is no reason to believe we have exploited the full potential of MGRBM. SNR \| GMM \| DNN ---\|---\|--- MFCC \| G-feat (PCA) \| M-feat (PCA) \| MFCC \| G-feat \| M-feat Clean \| 1.05 \| 1.94 \| 2.75 \| 1.27 \| 0.76 \| 0.82 20dB \| 6.10 \| 4.28 \| 6.21 \| 3.76 \| 2.93 \| 2.49 15dB \| 15.18 \| 8.50 \| 13.67 \| 9.09 \| 6.47 \| 5.64 10dB \| 34.82 \| 21.64 \| 32.06 \| 25.85 \| 17.37 \| 15.79 5dB \| 61.27 \| 49.80 \| 59.75 \| 58.05 \| 41.87 \| 40.99 0dB \| 82.56 \| 75.15 \| 79.89 \| 93.16 \| 78.32 \| 75.28 -5dB \| 91.20 \| 87.59 \| 90.07 \| 108.79 \| 106.20 \| 94.68 Table 1: Comparison of different features, all numbers are percentage of WER. G-feat and M-feat here represents G-feature and M-feature respectively. With HMM-GMM as acoustic model, the comparison is not so straightforward, because training a 1024-dimensional GMM would leads to severe over-fitting. So we reduced the dimension of G-feature and M-feature to 39 dimensions with principal component analysis (PCA). Notice that this is actually not the right thing to do, because the sum of top 39 eigenvalues only consists 91.8% and 69.7% of sum of all eigenvalues for G-feature and M-feature respectively. Despite of such great loss of information, both G-feature and M-feature performs better than MFCC for all SNR levels except the clean speech. ## 4 Conclusion and Discussion In this paper, we briefly reviewed the definition and learning algorithm of RBM and GRBM. We then propose a new variant of RBM called MGRBM by which we would like to model the covariance of adjacent frames within each context. After that we offered an explanation of why a feature learned with RBM (and its variants) might be able to enhance the performance of robust speech recognition over original feature. Finally we performed our experiments on Aurora2 and showed that feature that learned with GRBM and MGRBM would indeed improve the average accuracy across environment in every SNR condition over original MFCC feature. Throughout the process of feature learning with GRBM, virtually nothing is presupposed. This makes it adaptable with any feature generated from a front- end. Aside from training a GRBM, which can be done off-line, the extra cost of the feature transformation is merely a multiplication with a matrix. From this perspective, feature learning with GRBM is similar to the TANDEM system [15], but the latter is purely supervised. Hence many advantages can be gained if lots of data is accessible but little is labeled. What is more important, GRBM-extracted feature can be used for the TANDEM System seamlessly, which makes TANDEM system semi-supervised and thus further enhancement of performance can be achieved. ## 5 Acknowledgements This work is partially supported by the National Basic Research Program (973 Program) of China(2012CB316401), the National Natural Science Foundation of China (60928005, 60805008, 60931160443 and 61003094), the Ph.D. Programs Foundation of Ministry of Education of China (200800031015), the Upgrading Plan Project of Shenzhen Key Laboratory and the Science and Technology R&D Funding of the Shenzhen Municipal. ## References * [1] G. Dahl, D. Yu, L. Deng, , and A. Acero, “Large vocabulary continuous speech recognition with context-dependent dbn-hmms,” in _Proc. ICASSP_ , 2011. * [2] G. E. Hinton, L. Deng, D. Yu, G. Dahl, A. Mohamed, N. Jaitly, A. Senior, V. Vanhoucke, P. Nguyen, T. Sainathand, and B. Kingsbury, “Deep neural networks for acoustic modeling in speech recognition,” _IEEE Signal Processing Magazine_ , 2012. * [3] ETSI, “Advanced front-end feature extraction algorithm,” ETSI ES 202 050, Tech. Rep., 2007. * [4] Y. Bengio, A. Courville, and P. Vincent, “Representation learning: A review and new perspectives,” arXiv:1206.5538 [cs.LG]. * [5] P. Smolensky, “Information processing in dynamical systems: Foundations of harmony theory,” _Parallel Distributed Processing: Explorations in the Microstructure of Cognition_ , vol. 1, 1986. * [6] G. E. Hinton, “Training products of experts by minimizingc ontrastive divergence,” _Neural Computation_ , vol. 14, pp. 1771–1800, 2002. * [7] G. E. Hinton and R. Salakhutdinov, “Reducing the dimensionality of data with neural networks,” _Science_ , vol. 313, pp. 504–507, 2006. * [8] G. W. Taylor, G. E. Hinton, and S. T. Roweis, “Two distributed-state models for generating high-dimensional time series,” _The Journal of Machine Learning Research_ , vol. 12, pp. 1025–1068, 2011. * [9] T. Tieleman, “Training restricted boltzmann machines using approximations to the likelihood gradient,” in _Proc. of ICML_ , 2008. * [10] A. Mohamed, G. E. Dahl, and G. E. Hinton, “Deep belief networks for phone recognition,” in _NIPS Workshop on Deep Learning for Speech Recognition and Related Applications_ , 2009. * [11] A. Mohamed, G. Dahl, and G. Hinton, “Acoustic modeling using deep belief networks,” _IEEE Trans. on Audio, Speech and Language Processing_ , 2011\. * [12] G. E. Hinton, “Products of experts,” _ICANN_ , vol. 1, pp. 1–6, 1999. * [13] H. G. Hirsch and D. Pierce, “The aurora experimental framework for the performance evaluation of speech recognition systems under noise conditions,” in _Proc. ISCA ITRW ASR2000_ , 2000. * [14] G. E. Hinton, S. Osindero, and Y. Teh, “A fast learning algorithm for deep belief nets,” _Neural Computation_ , vol. 18, pp. 1527–1554, 2006. * [15] H. Hermansky, D. P. W. Ellis, and S. Sharma, “Tandem connectionist feature extraction for conventional hmm systems,” in _Proc. ICASSP_ , 2000, pp. 1635–1638.	arxiv-papers	2013-09-23T13:51:28	2024-09-04T02:49:51.446285	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xin Zheng, Zhiyong Wu, Helen Meng, Weifeng Li, Lianhong Cai", "submitter": "Xin Zheng", "url": "https://arxiv.org/abs/1309.6176" }
1309.6266	# ENERGY OF SIGNED DIGRAPHS S. Pirzada and Mushtaq A. Bhat Department of Mathematics University of Kashmir Srinagar, Hazratbal 190006 India [email protected]; [email protected] [email protected] ###### Abstract. In this paper we extend the concept of energy to signed digraphs. We obtain Coulson’s integral formula for energy of signed digraphs. Formulae for energies of signed directed cycles are computed and it is shown that energy of non cycle balanced signed directed cycles increases monotonically with respect to number of vertices. Characterization of signed digraphs having energy equal to zero is given. We extend the concept of non complete extended $p$ sum (or briefly, NEPS) to signed digraphs. An infinite family of equienergetic signed digraphs is constructed. Moreover, we extend McClelland’s inequality to signed digraphs and also obtain sharp upper bound for energy of signed digraph in terms the number of arcs. Some open problems are also given at the end. ###### Key words and phrases: Energy of signed digraph, NEPS,McClelland’s inequality. ###### 2000 Mathematics Subject Classification: fill it, ###### 2010 Mathematics Subject Classification: 05C50, 05C22,05C76. ## 1\. introduction A signed digraph (or briefly sidigraph) is defined to be a pair $S=(D,\sigma)$ where $D=(V,\mathscr{A})$ is the underlying digraph and $\sigma:\mathscr{A}\rightarrow\\{-1,1\\}$ is the signing function. The sets of positive and negative arcs of $S$ are respectively denoted by $\mathscr{A}^{+}(S)$ and $\mathscr{A}^{-}(S)$. Thus $\mathscr{A}(S)=\mathscr{A}^{+}(S)\cup\mathscr{A}^{-}(S)$. A sidigraph is said to be homogeneous if all of its arcs have either positive sign or negative sign, otherwise heterogeneous. A sidigraph without orientations of its arcs (when it is regarded as a simple undirected sigraph) is known as its underlying sigraph and is denoted by $S^{u}$. Two vertices are adjacent if they are connected by an arc. A path of length $n-1$ $(n\geq 2)$, denoted by $P_{n}$, is a sidigraph on $n$ vertices $v_{1},v_{2},\cdots,v_{n}$ with $n-1$ signed arcs $(v_{i},v_{i}+1)$. A cycle of length $n$, is a sidigraph having vertices $v_{1},v_{2},\cdots,v_{n}$ and signed arcs $(v_{i},v_{i}+1)$ $i=1,2,\cdots,n-1$ and $(v_{n},v_{1})$. A sidigraph is linear if each vertex has both indegree and outdegree equal to one. The sign of a sidigraph is defined as the product of signs of its arcs. A sidigraph is said to be positive (negative) if its sign is positive (negative) i.e., it contains an even (odd) number of negative arcs. A signed digraph is said to be all-positive (respectively, all negative) if all its arcs are positive (negative). A signed digraph is said to be cycle balanced if each of its cycles is positive, otherwise non cycle balanced. Throughout this paper we call cycle balanced cycle a positive cycle and non cycle balanced cycle a negative cycle and respectively denote them by $C_{n}$ and ${\bf C}_{n}$, where $n$ is number of vertices. A sidigraph is symmetric if $(u,v)\in\mathscr{A}^{+}(S)~{}~{}(or~{}~{}\mathscr{A}^{-}(S))$ then $(v,u)\in\mathscr{A}^{+}(S)~{}~{}(or~{}~{}\mathscr{A}^{-}(S))$, where $u,v\in V(S)$. A one to one correspondence between sigraphs and symmetric sidigraphs is given by $S\rightsquigarrow\overleftrightarrow{S}$, where $\overleftrightarrow{S}$ has the same vertex set as that of sigraph $S$, and each signed edge $(u,v)$ is replaced by a pair of symmetric arcs $(u,v)$ and $(v,u)$ both with same sign as that of edge $(u,v)$. Under this correspondence a sigraph can be identified with a symmetric sidigraph. A sidigraph is said to be skew symmetric if its adjacency matrix is skew symmetric. A skew symmetric sidigraph is obtained by replacing each edge of sigraph by a pair of opposite oriented and opposite signed arcs. We denote a skew symmetic digraph of order $n$ by ${\bf S}_{n}$. Two sidigraphs (sigraphs) on same number of vertices are said to be cospectral if they have same spectrum. Acharya, Gill, Patwardhan [2] introduced the concept of Quasicospectrality in digraphs (graphs). Two digraphs (graphs) $G$ and $H$ of same order are said to be quasicospectral if there exist sidigraphs (sigraphs) $S$ and $S^{\prime}$ on $G$ and $H$ respectively which are cospectral. The weighted directed graph $S$ of an $n\times n$ matrix $M=(m_{ij})$ of reals consists of $n$ vertices, with vertex $i$ joined by a directed arc with weight $m_{ij}$ to vertex $j$ if and only if $m_{ij}$ is non-zero. In case the matrix consists of entries $-1,0$ and $1$, then we get a sidigraph. Thus there is a one to one correspondence between the set of integral $(-1,0,1)$-matrices of order $n$ and set of sidigraphs of order $n$. The cartesian product (or sum) of two sidigraphs $S_{1}=(V_{1},A_{1},\sigma_{1})$ and $S_{2}=(V_{2},A_{2},\sigma_{2})$ denoted by $S_{1}\times S_{2}$ is the sidigraph $(V_{1}\times V_{2},A,\sigma)$, where the arc set is that of the cartesian product of unsigned digraphs and the sign function is defined by $\sigma((u_{i},v_{j}),(u_{k},v_{l}))=\left\\{\begin{array}[]{lr}\sigma_{1}(u_{i},u_{k}),&\mbox{if $j=l$},\\\ \sigma_{2}(v_{j},v_{l}),&\mbox{if $i=k.$}\end{array}\right.$ The Kronecker product (strong product or conjunction) of two sidigraphs $S_{1}=(V_{1},A_{1},\sigma_{1})$ and $S_{2}=(V_{2},A_{2},\sigma_{2})$ denoted by $S_{1}\otimes S_{2}$ is the sidigraph $(V_{1}\times V_{2},A,\sigma)$, where arc set is the arc set of underlying unsigned digraphs and the sign function is defined by $\sigma((u_{i},v_{j}),(u_{k},v_{l}))=\sigma_{1}(u_{i},u_{k})\sigma_{2}(v_{j},v_{l}).$ The adjacency matrix of a sidigraph $S$ whose vertices are ${v_{1},v_{2},\cdots,v_{n}}$ is the $n\times n$ matrix $A(S)=(a_{ij})$, where $a_{ij}=\left\\{\begin{array}[]{lr}\sigma(v_{i},v_{j}),&\mbox{if there is an arc from $v_{i}$ to $v_{j},$}\\\ 0,&\mbox{otherwise.}\end{array}\right.$ The characteristic polynomial $\|xI-A(S)\|$ of the adjacency matrix $A(S)$ of sidigraph $S$ is called the characteristic polynomial of $S$ and is denoted by $\phi_{S}(x)$. The eigenvalues of $A(S)$ are called eigenvalues of $S$. The following is the coefficient Theorem for sidigraphs [2] Theorem 1.1. If $S$ be a sidigraph with characteristic polynomial $\phi_{S}(x)=x^{n}+c_{1}x^{n-1}+\cdots+c_{n-1}x+c_{n}$ then $c_{i}=\sum\limits_{L\in\pounds_{i}}(-1)^{p(L)}\prod\limits_{Z\in c(L)}s(Z),$ for all $i=1,2,\cdots,n$, where $\pounds_{i}$ is the set of all linear subdigraphs $L$ of $S$ of order $i$, $p(L)$ denotes number of components of $L$ and $c(L)$ denotes the set of all cycles of $L$ and $s(Z)$ the sign of cycle $S$. If $S$ is an undirected sigraph then $S$ can be viewed as a digraph $\bar{S}$ by identifying each signed edge of $S$ with a directed cycle of length $2$ with each arc having same sign as that of the corresponding edge. For undirected sigraphs Theorem $1.1$ takes the following form [10]. Theorem 1.2. If $S$ is a sigraph with characteristic polynomial $\phi_{S}(x)=x^{n}+c_{1}x^{n-1}+\cdots+c_{n-1}x+c_{n}$ then $c_{i}=\sum\limits_{L\in\pounds_{i}}(-1)^{p(L)}2^{c(L)}\prod\limits_{Z\in c(L)}s(Z),$ for all $i=1,2,\cdots,n$, where $\pounds_{i}$ is the set of all basic figures $L$ of $S$ of order $i$, $p(L)$ denotes number of components of $L$ and $c(L)$ the number of cyclic components of $L$. The spectral criterion for cycle balance of sidigraphs (sigraphs) given by Acharya is as follows. Theorem 1.3[1, 2]. A sidigraph (sigraph) is cycle balanced (balanced) if and only if it is cospectral with the underlying unsigned digraph (graph). ## 2\. Energy of sidigraphs For spectra and energy of graphs and digraphs see [5, 6, 11, 12, 15]. Germina, Hameed and Zaslavsky [7] defined energy of a sigraph to be the sum of absolute values of sigraph eigenvalues. In this section, we extend the concept of energy to sidigraphs. Unlike sigraphs the adjacency matrix of a sidigraph need not be real symmetric, so eigenvalues can be complex numbers. Definition 2.1. Let $S$ be a sidigraph of order $n$ having eigenvalues $z_{1},z_{2},\cdots,z_{n}$. The energy of $S$ is defined as $E(S)=\sum\limits_{j=1}^{n}\|\Re z_{j}\|,$ where $\Re z_{j}$ denotes real part of complex number $z_{j}.$ If $S$ is a signed graph and $\overleftrightarrow{S}$ be its symmetric sidigraph, then clearly $A(S)=A(\overleftrightarrow{S})$ and so $E(S)=E(\overleftrightarrow{S})$. In this way, definition $2.1$ generalizes the concept of energy of undirected sigraphs. Example 2.2. Let $S$ be a sidigraph shown in Figure $1$. Dotted arrows denote negative arcs and plane ones denote positive arcs. Clearly, $S$ is non cycle balanced sidigraph. By Theorem $1.1$, the characteristic polynomial of $S$ is $\phi_{S}(x)=x^{10}+x^{7}=x^{7}(x^{3}+1)$. The eigenvalues of $S$ are $0^{7},1,\frac{1\pm\sqrt{3}\iota}{2}$, where $\iota=\sqrt{-1}$, so $E(S)=2$. SFigure 1 Example 2.3. Let $S$ be an acyclic sidigraph. Then by Theorem $1.1$, the characteristic polynomial of $S$ is $\phi(S)=x^{n}$, so that $spec(S)=\\{0^{n}\\}$ and hence $E(S)=0$. Example 2.4. Consider ${\bf S}_{n}$, the skew symmetric sidigraph on $n\geq 2$ vertices, then by Theorem $1.1$, $\phi_{{\bf S}_{n}}(x)=x^{n-2}(x^{2}+(n-1))$ so that eigenvalues are $0^{n-2},\pm\iota\sqrt{n-1}$. Thus all skew symmmetric sidigraphs are cospectral with energy equal to zero. Example 2.5. If $S$ is the sidirected cycle on $n$ vertices, then the characteristic polynomial of $S$ is $\phi(S)=x^{n}+(-1)^{[s]}$, where the symbol $[s]$ is defined as $[s]=1$ or $0$ according as $S$ is positive or negative. If $S=C_{n}$, then eigenvalues are $e^{\frac{2\iota j\pi}{n}},~{}j=0,1,\cdots,n-1$. So, $E(S)=\sum\limits_{j=0}^{n-1}\|\cos(\frac{2j\pi}{n})\|$. If $S={\bf C}_{n}$, then the eigenvalues are $e^{\frac{\iota(2j+1)\pi}{n}},~{}j=0,1,\cdots,n-1$. Therefore, $E(S)=\sum\limits_{j=0}^{n-1}\|\cos(\frac{(2j+1)\pi}{n})\|$. In particular if $S={\bf C}_{4}$, the eigenvalues are $\frac{1\pm\iota}{\sqrt{2}},\frac{-1\pm\iota}{\sqrt{2}}$ and $E(S)=2\sqrt{2}$. Example 2.6. Let $S$ be a sidigraph having $n$ vertices and unique cycle of length $r$, where $2\leq r\leq n$. Then by Theorem $1.1$, $\phi_{S}(x)=x^{n}+(-1)^{[s]}x^{r}=x^{r}(x^{n-r}+(-1)^{[s]})$, where the symbol $[s]$ is defined as $[s]=1$ or $0$ according as $S$ is cycle balanced or non cycle balanced. Clearly, energy equals to the energy of the unique cycle. Given $t$ sidigraphs $S_{1},S_{2},\cdots S_{t}$, their direct product denoted by $S_{1}\oplus S_{2}\oplus\cdots\oplus S_{t}$ is the sidigraph with $V_{S_{1}\oplus S_{2}\oplus\cdots\oplus S_{t}}=\bigcup\limits_{j=1}^{t}V(S_{j})$ and arc set $\mathscr{A}_{S_{1}\oplus S_{2}\oplus\cdots\oplus S_{t}}=\bigcup\limits_{j=1}^{t}\mathscr{A}(S_{j})$. Theorem 2.7. Let $S$ be a sidigraph on n vertices and $S_{1},S_{2},\cdots,S_{k}$ be its strong components. Then $E(S)=\sum\limits_{j=1}^{k}E(S_{j}).$ Proof. Let $Y=\\{a\in A(S):a\notin c(S)\\}$, where $c(S)$ is the set of all cycles of $S$. By Theorem $1.1$, $\phi_{S}(x)=\phi_{S-Y}(x)$, where $S-Y$ is the sidigraph obtained from $S$ by deleting the non-cyclic arcs. Clearly, $S-Y=S_{1}\oplus S_{2}\oplus\cdots\oplus S_{k}$ and adjacency matrix of sidigraph $S-Y$ is in block diagonal form with diagonal blocks as the adjacency matrices of strong components. Therefore $\phi_{S-Y}(x)=\phi_{S_{1}}(x)\phi_{S_{2}}(x)\cdots\phi_{S_{k}}(x)$ and so $E(S)=\sum\limits_{j=1}^{k}E(S_{j})$. ∎ Remark 2.8. From Theorem $1.1$, $c_{i}=\sum\limits_{L\in\pounds_{i}}(-1)^{p(L)},i=1,2,\cdots,n$. Clearly, this sum contains positive and negative ones. $+1$ arises if and only if $(a)$ Number of components of $L\in\pounds_{i}$ are odd and $s(L)<0$. We call such linear sidigraphs as type $a$ linear sidigraphs. $(b)$ Number of components of $L\in\pounds_{i}$ is even and $s(L)>0$. We call such linear sidigraphs as type $b$. $-1$ will occur if and only if $(c)$ Number of components of $L\in\pounds_{i}$ is odd and $s(L)>0$. We call such linear sidigraphs as type $c$. $(d)$ Number of components of $L\in\pounds_{i}$ is even and $s(L)<0$. We call such linear sidigraphs as type $d$. From above we observe that $c_{i}=0$ if and only if either $S$ is acyclic or for each $i$, number of linear sidigraphs of order $i$ of type $a$ or type $d$ is equal to the number of linear sidigraphs of order $i$ of type $b$ or type $c$. An immediate consequence of Remark $2.7$ is the following Lemma. Lemma 2.9. An integral $(-1,0,1)$ matrix is nilpotent if and only if its underlying sidigraph $S$ is either acyclic or in $S$, for each $i=1,2,\cdots,n$, the number of linear sidigraphs of order $i$ of type $a$ or type $d$ is equal to number of linear sidigraphs of order $i$ of type $b$ or type $c$. ∎ Unlike unsigned strong component, energy of a signed directed strong component can be zero, for example, sidigraph $S_{1}$ in Figure 2. The following result characterizes sidigraphs having energy equal to zero. Theorem 2.10. Let $S$ be a sidigraph of order $n$. Then $E(S)=0$ if and only if $S$ satisfies one of the following conditions $(I)$ $S$ is acyclic $(II)$ $S$ is skew symmetric $(III)$ for each $i=1,2,\cdots,n$, the number of linear sidigraphs of order $i$ of type $a$ or type $d$ is equal to number of linear sidigraphs of order $i$ of type $b$ or type $c$. Proof. Let $S$ be a sidigraph of order $n$. If $S$ is acyclic, then by Lemma $2.9$, $\phi_{S}(x)=x^{n}$ and so $E(S)=0.$ If $S$ is skew symmetric sidigraph, then the eigenvalues of $S$ are $0^{n-2},\pm\iota\sqrt{n-1}$, therefore $E(S)=0$. If $S$ satisfies $(III)$, then by Lemma $2.9$, $\phi_{S}(x)=x^{n}$, so that $E(S)=0.$ Conversely, if $E(S)=0$, then either eigenvalues of $S$ are $0^{n-2},\pm\iota\sqrt{n-1}$ or all equal to zero. Using the fact that a real matrix is skew symmetric if and only if all its eigenvalues are of the form $\pm\iota\alpha$, where $\alpha\in\mathbb{R}$ and by Lemma $2.9$, the result follows. ∎ ## 3\. Computation of energy of signed directed cycles We first calculate energy formulae for positive cycles. Let $C_{n}$ be a positive cycle on $n\geq 2$ vertices. The characteristic polynomial of $C_{n}$ is $\phi_{C_{n}}(x)=x^{n}-1$, so that the eigenvalues are $e^{\frac{2\pi\iota j}{n}},j=0,1,\cdots,n-1,$ $\iota=\sqrt{-1}$. Consequently energy of $C_{n}$ is $E(C_{n})=\sum\limits_{j=0}^{n-1}\|\cos\frac{2j\pi}{n}\|.$ Given a positive integer $n$, it has one of the forms $4k$, or $2k+1$, or $4k+2$, where $k\geq 0$. If $n=4k$, then $\displaystyle E(C_{n})$ $\displaystyle=\sum\limits_{j=0}^{4k-1}\|\cos\frac{2j\pi}{4k}\|=\sum\limits_{j=0}^{4k-1}\|\cos\frac{j\pi}{2k}\|=2\sum\limits_{j=0}^{2k-1}\|\cos\frac{j\pi}{2k}\|$ $\displaystyle=2+4\sum\limits_{j=1}^{k-1}\cos\frac{j\pi}{2k}=2+4\\{\frac{-1}{2}+\frac{\sin\frac{(k-\frac{1}{2})\pi}{2k}}{2\sin\frac{\pi}{4k}}\\}=2\cot\frac{\pi}{n}.$ If $n=2k+1$, then $\displaystyle E(C_{n})$ $\displaystyle=\sum\limits_{j=0}^{2k}\|\cos\frac{2j\pi}{2k+1}\|=1+2\sum\limits_{j=1}^{k}\|\cos\frac{2j\pi}{2k+1}\|=1+2\sum\limits_{j=1}^{k}\cos\frac{j\pi}{2k+1}$ $\displaystyle=1+2\\{\frac{-1}{2}+\frac{\sin\frac{(k+\frac{1}{2})\pi}{2k+1}}{2\sin\frac{\pi}{2(2k+1)}}\\}=2\csc\frac{\pi}{2n}.$ If $n=4k+2$ then $\displaystyle E(C_{n})$ $\displaystyle=\sum\limits_{j=0}^{4k+1}\|\cos\frac{2j\pi}{4k+2}\|=\sum\limits_{j=0}^{4k+1}\|\cos\frac{j\pi}{2k+1}\|=2\sum\limits_{j=0}^{2k}\|\cos\frac{j\pi}{2k+1}\|$ $\displaystyle=2+4\sum\limits_{j=1}^{k}\cos\frac{j\pi}{2k+1}=2+4\\{\frac{-1}{2}+\frac{\sin\frac{(k+\frac{1}{2})\pi}{2k+1}}{2\sin\frac{\pi}{4k+2}}\\}=2\csc\frac{\pi}{n}.$ We now calculate exact formulae for the energy of negative cycles of length $n$. Let ${\bf C}_{n}$ denote the negative cycle with $n$ vertices. Then $\phi_{{\bf C}_{n}}(x)=x^{n}+1$ and so $Spec({\bf C}_{n})=e^{\frac{(2j+1)\pi\iota}{n}},j=0,1,\cdots,n-1$, $\iota=\sqrt{-1}$. Therefore energy is given by $E({\bf C}_{n})=\sum\limits_{j=0}^{n-1}\|\cos\frac{(2j+1)\pi}{n}\|.$ If $n=4k$, then $\displaystyle E({\bf C}_{n})$ $\displaystyle=\sum\limits_{j=0}^{4k-1}\|\cos\frac{(2j+1)\pi}{4k}\|=2\sum\limits_{j=0}^{2k-1}\|\cos\frac{(2j+1)\pi}{4k}\|=4\sum\limits_{j=0}^{k-1}\cos\frac{(2j+1)\pi}{4k}$ $\displaystyle=4\\{\cos\frac{\pi}{4k}+\cos\frac{3\pi}{4k}+\cdots+\cos\frac{(2k-1)\pi}{4k}\\}=4\frac{\cos(\frac{\pi}{4k}+\frac{k-1}{2}\frac{2\pi}{4k})\sin k\frac{2\pi}{8k}}{\sin\frac{2\pi}{8k}}=2\csc\frac{\pi}{n}.$ If $n=4k+2$, then $\displaystyle E({\bf C}_{n})$ $\displaystyle=\sum\limits_{j=0}^{4k+1}\|\cos\frac{(2j+1)\pi}{4k+2}\|=4\sum\limits_{j=0}^{k-1}\cos\frac{(2j+1)\pi}{4k+2}$ $\displaystyle=4\\{\cos\frac{\pi}{4k+2}+\cos\frac{3\pi}{4k+2}+\cdots+\cos\frac{(2k-1)\pi}{4k+2}\\}=2\cot\frac{\pi}{n}.$ If $n=2k+1$, then since $-1$ is the eigenvalue of ${\bf C}_{n}$, we have $spec({\bf C}_{n})$ $=-spec(C_{n})$, and so $E({\bf C}_{n})=E(C_{n})$. Summarizing, all the above cases can be written as follows: $E(C_{n})=\left\\{\begin{array}[]{lr}2\cot\frac{\pi}{n},&\mbox{if $n=4k$},\\\ \csc\frac{\pi}{2n},{}{}{}{}&\mbox{if $n=4k+1$~{}or~{}$n=4k+3$},\\\ 2\csc\frac{\pi}{n},&\mbox{if $n=4k+2$}\\\ \end{array}\right.$ and $E(\bf{C}_{n})=\left\\{\begin{array}[]{lr}2\csc\frac{\pi}{n},&\mbox{if $n=4k$},\\\ \csc\frac{\pi}{2n},{}{}{}{}&\mbox{if $n=4k+1$~{}or $n=4k+3$},\\\ 2\cot\frac{\pi}{n},&\mbox{if $n=4k+2$}.\\\ \end{array}\right.$ Pena and Rada [15] proved that the energy of directed unsigned cycles increases monotonically with respect to order $n$. From energy formulae for positive and negative sidirected cycles, the following two results are immediate. Theorem 3.1. Among all non cycle balanced unicyclic sidigraphs on $n$ vertices, the cycle has the largest energy. Moreover, energy of negative cycles increases monotonically with respect to the order. Minimal energy is attained in ${\bf C}_{2}$, a negative cycle of order $2$. ∎ Theorem 3.2. Energy of positive and negative cycles satisfy the following: $(1)$ Energy of positive cycle of odd order equals energy of negative cycle of same order. $(2)$ Energy of negative cycle of even order is greater than energy of positive cycle of same order if and only if $n=4k$. $(3)$ Energy of negative cycle of even order is less than energy of positive cycle of same order if and only if $n=4k+2$. One of the fundamental results in theory of graph energy is Coulsons integral formula. We now state Coulsons integral formula for energy of sidigraphs. The proof is similar as in digraphs. Theorem 3.3. Let $S$ be a sidigraph with $n$ vertices having characteristic polynomial $\phi_{S}(x)$. Then $E(S)=\sum\limits_{j=1}^{n}\|\Re z_{j}\|=\frac{1}{\pi}\int\limits_{-\infty}^{\infty}(n-\frac{\iota x\phi^{\prime}_{S}(\iota x)}{\phi_{S}(\iota x)})dx,$ where $z_{1},z_{2},\cdots,z_{n}$ are the eigenvalues of sidigraph $S$ and $\int\limits_{-\infty}^{\infty}F(x)dx$ denotes principle value of the respective integral. The Coulson’s formula given above is a motivation to define energy of sidigraphs to be the sum of absolute values of real parts of eigenvalues. Example 3.4. Consider the cycle ${\bf C}_{4}$, the characteristic polynomial is $\phi_{{\bf C}_{4}}(x)=x^{4}+1$ and hence $E({\bf C}_{4})=\frac{1}{\pi}\int\limits_{-\infty}^{\infty}[4-\frac{4\iota x(\iota x)^{3}}{(\iota x)^{4}+1}]dx$ $=\frac{1}{\pi}\int\limits_{-\infty}^{\infty}\frac{4}{x^{4}+1}dx$ $=\frac{4}{\pi}\frac{\pi}{2\sin\frac{\pi}{4}}=2\sqrt{2}$, as calculated in example $2.5$. An immediate consequence of Coulson’s integral formula is the following observation, the proof of which is similar to the case in unsigned digraphs. Corollary 3.5. If $S$ is a sidigraph on $n$ vertices, then $E(S)=\frac{1}{\pi}\int\limits_{-\infty}^{\infty}\frac{1}{x^{2}}\log(x^{n}\phi_{G}(\frac{\iota}{x})dx.$ Pena and Rada [15] considered the problem of increasing property of the energy in digraphs, It is natural to consider the same problem for sidigraphs. Let ${S}_{n,h}$ denote the set of sidigraphs with $n$ vertices and every cycle having length $h$. First we shall calculate characteristic polynomial of such sidigraphs. Theorem 3.6. If $S\in S_{n,h}$, then $\phi_{S}(x)=x^{n}+\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}(-1)^{k}c^{}(S,kh)x^{n-kh},$ where $c^{}(S,kh)=$ number of positive linear sidigraphs of order $kh$ $-$ number of negative linear sidigraphs of order $kh$, for every $k=1,2,\cdots,{\lfloor{\frac{n}{h}}\rfloor}$. Proof. By Theorem $1.1$, the coefficient $c_{kh}=\sum\limits_{L\in\pounds_{kh}}(-1)^{p(L)}\prod\limits_{Z\in c(L)}s(Z)$. As each cycle of $S$ has length $h$, it follows that $c_{kh}=(-1)^{k}c^{}(S,kh)$, where $c^{}(S,kh)=$ number of positive linear sidigraphs consisting of $k$ cycles each of length $h$ $-$ number of negative linear sidigraphs consisting of $k$ cycles each of length $h$. Clearly, $c_{j}=0$ if $j$ is not a multiple of $h$, because $\pounds_{j}=\phi$ in that case. ∎ Definition 3.7. Let $S^{}_{n,h}=\\{S\in S_{n,h}:c^{}(S,kh)\geq 0,k=1,2,\cdots,\lfloor\frac{n}{h}\rfloor\\}$. Define a quasi order relation over $S^{}_{n,h}$ as follows. Let $S_{1}$ and $S_{2}$ be two elements of $S^{}_{n,h}$. Define $S_{1}\preceq S_{2}$ if for all $k=1,2,\cdots,\lfloor\frac{n}{h}\rfloor$, $c^{}(S_{1},kh)\leq c^{}(S_{2},kh)$. If $S_{1}\preceq S_{2}$ and there exists $k$ such that $c^{}(S_{1},kh)<c^{}(S_{2},kh)$ then $S_{1}\prec S_{2}$. Clearly, this is a reflexive and transitive relation over $S^{}_{n,h}$. Theorem 3.8. Let $h$ be an integer of the form $4t+2,~{}t\geq 0$. The energy increases with respect to quasi-order relation defined over $S^{}_{n,h}$, i.e., if $S_{1},~{}S_{2}\in S^{}_{n,h}$ then $S_{1}\prec S_{2}\Rightarrow E(S_{1})<E(S_{2}).$ Proof. Let $H\in S^{}_{n,h}$, by Theorem $3.6$, we have $\phi_{H}(x)=x^{n}+\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}(-1)^{k}c^{}(H,kh)x^{n-kh},$ so that $\phi_{H}(\frac{\iota}{x})=\frac{\iota^{n}}{x^{n}}+\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}(-1)^{k}c^{}(H,kh)\frac{\iota^{n-kh}}{x^{n-kh}}$ $=\frac{\iota^{n}}{x^{n}}(1+\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}(-1)^{k}c^{}(H,kh)x^{kh}\iota^{-k(4t+2)})$ $=\frac{\iota^{n}}{x^{n}}(1+\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}c^{}(H,kh)x^{kh}).$ Applying Corollary $3.5$, we have $E(S)=\frac{1}{\pi}\int\limits_{-\infty}^{\infty}\frac{1}{x^{2}}\log(x^{n}\frac{\iota^{n}}{x^{n}}(\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}c^{}(H,kh)x^{kh}))dx=$ $\frac{1}{\pi}\int\limits_{-\infty}^{\infty}\frac{1}{x^{2}}\log(\iota^{n}(1+\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}c^{}(H,kh)x^{kh}))dx$ $=\frac{1}{\pi}\int\limits_{-\infty}^{\infty}\frac{1}{x^{2}}\log(1+\sum\limits_{j=1}^{\lfloor{\frac{n}{h}}\rfloor}c^{}(H,kh)x^{kh})dx$, as $\frac{1}{\pi}p.v.\int\limits_{-\infty}^{+\infty}\log(\iota^{n})\frac{dx}{x^{2}}=0$, where $p.v.$ stands for principal value of Cauchy’s integral. It is clear from above energy expression that energy increases with respect to quasi-order relation defined over $S^{}_{n,h}$. ∎ ## 4\. NEPS in sidigraphs Recall that [13] Kronecker product of two matrices $A=(a_{ij})_{r\times s}$ and $B=(b_{ij})_{t}\times u$ denoted by $A\otimes B$ is matrix of order $rt\times su$ obtained by replacing each entry $a_{ij}$ of $A$ by a block $a_{ij}B$. Thus $A\otimes B$ consists of all $rtsu$ possible products of an entry of $A$ with an entry of $B$. The Kronecker product is a componentwise operation, i.e., $(A\otimes B)(C\otimes D)=(AC)\otimes(BD)$, provided the products $AC$ and $BD$ exist. This operation is also associative, so we can define the multiple product $A_{1}\otimes A_{2}\otimes\cdots\otimes A_{m}$. Let order of $A_{i}$ be $r_{i}\times s_{i}$. We index elements of $A_{i}$ by $a_{i};jk$, and those of multiple product by a pair of $m-$tuples, a row index $j=(j_{1},j_{2},\cdot,j_{m})$ and a column index $k=(k_{1},k_{2},\cdots,k_{m})$, where $1\leq j_{i}\leq r_{i}$ and $1\leq k_{i}\leq s_{i}$. The element $a_{jk}$ of the product matrix is $a_{jk}=a_{1;j_{1}k_{1}}a_{2;j_{2}k_{2}}\cdots a_{m;j_{m}k_{m}}.$ Lemma 4.1 [8]. Let $A_{i},i=1,2,\cdots,m$, be a square matrix of order $n_{i}$ and $\xi_{ij},~{}j=1,2,\cdots,n_{i}$ be its eigenvalues. Let $k_{1},k_{2},\cdots,k_{m}$ be non-negative integers, then the $n_{1}n_{2}\cdots n_{m}$ eigenvalues of the matrix $A^{k_{1}}_{1}\otimes\cdots\otimes A^{k_{m}}_{m}$ are $\xi_{j_{1}j_{2}\cdots j_{m}}=\xi^{k_{1}}_{1j_{1}}\cdots\xi^{k_{m}}_{1j_{m}}$ for $1\leq j_{i}\leq n_{i}.$ Let $k_{p}=(k_{p1},k_{p1},\cdots,k_{pm}),p=1,2,\cdots,q$ be vectors of non- negative integers. Then the $n_{1}n_{2}\cdots n_{m}$ eigenvalues of $\sum\limits_{p=1}^{q}A^{k_{p1}}_{1}\otimes\cdots\otimes A^{k_{pm}}_{m}$ are $\xi_{j_{1}j_{2}\cdots j_{m}}=\sum\limits_{p=1}^{q}\xi^{k_{p1}}_{1j_{1}}\cdots\xi^{k_{pm}}_{mj_{m}}.$ For NEPS in graphs see [6, 21]. The following definition extends this concept to sidigraphs. Definition 4.2. Let $\mathscr{B}$ be a set of binary $n-$tuples called basis for the product such that for every $i=1,2,\cdots,n$ there exists $\beta\in\mathscr{B}$ with $\beta_{i}=1$. The non-complete extended $p-$sum (or simply called NEPS) of sidigraphs $S_{1},S_{2},\cdots,S_{n}$ with basis $\bf\it B$ denoted by NEPS$(S_{1},S_{2},\cdots,S_{n};\mathscr{B})$ is a sidigraph with vertex set $V(S_{1})\times V(S_{2})\times\cdots\times V(S_{n})$. There is an arc from vertex $(u_{1},u_{2},\cdots,u_{n})$ to $(v_{1},v_{2},\cdots,v_{n})$ if and only if there exists $(\beta_{1},\beta_{2},\cdots,\beta_{n})\in\mathscr{B}$ such that $(u_{i},v_{i})\in A(S_{i})$ whenever $\beta_{i}=1$ and $u_{i}=v_{i}$ whenever $\beta_{i}=0$. The sign of the arc is given by $\sigma((u_{1},u_{2},\cdots,u_{n}),(v_{1},v_{2},\cdots,v_{n}))=\prod\limits_{i=1}^{n}\sigma_{i}(u_{i},v_{i})^{\beta_{i}}=\prod\limits_{i:\beta_{i}=1}\sigma_{i}(u_{i},v_{i}).$ Assume that the basis $\mathscr{B}$ has $r\geq 1$ elements, i.e., $\beta=\\{\beta_{1},\beta_{2},\cdots,\beta_{r}\\}\subseteq\\{0,1\\}^{n}\backslash\\{(0,0,\cdots,0)\\}$, we define NEPS$(S_{1},S_{2},\cdots,S_{n};\mathscr{B})=\bigcup\limits_{\beta\in\mathscr{B}}$ NEPS$(S_{1},S_{2},\cdots,S_{n};\beta).$ Example 4.3. The Kronecker product $S_{1}\otimes S_{2}\otimes\cdots\otimes S_{n}$ of sidigraphs $S_{1},S_{2},\cdots,S_{n}$ is the NEPS of these sidigraphs with basis $\beta=\\{(1,1,\cdots,1)\\}$; the cartesian product $S_{1}\times S_{2}\times\cdots\times S_{n}$ is NEPS with basis $\\{e_{i}\\},i=1,2,\cdots,n$, where $e_{i}$ is $n-$tuple with $1$ at $i$th position and $0$ otherwise. The following result shows that two different basis give disjoint arc sets. The proof is straightforward. Lemma 4.4. Let $S=$ NEPS $(S_{1},S_{2},\cdots,S_{n};\beta)$ and $S^{\prime}=$ NEPS $(S_{1},S_{2},\cdots,S_{n};\beta^{\prime})$, $\beta\neq\beta^{\prime}$. Then $\mathscr{A}(S)\cap\mathscr{A}(S^{\prime})=\phi.$ The following result gives adjacency matrix and spectra of NEPS in terms of the constituent factor sidigraphs. Theorem 4.5. If $S=$NEPS$(S_{1},S_{2},\cdots,S_{n};\mathscr{B})$, then the ajacency matrix is given by $A(S)=\sum\limits_{\beta\in\mathscr{B}}A^{\beta_{1}}_{1}\otimes\cdots\otimes A^{\beta_{n}}_{n},$ and eigenvalues are given by $z_{j_{1}j_{2}\cdots j_{n}}=\sum\limits_{\beta\in\mathscr{B}}z^{\beta_{1}}_{1j_{1}}\cdots z^{\beta_{n}}_{nj_{n}}$, where $1\leq j_{i}\leq n_{i},i=1,2,\cdots,n.$ Proof. Let $u=(u_{1j_{1}},u_{2j_{2}},\cdots,u_{nj_{n}})$ and $v=(v_{1j_{1}},v_{2j_{2}},\cdots,v_{nj_{n}})$, where $1\leq j_{i}\leq\|V(S_{i})\|$, $i=1,2,\cdots,n$ be any two vertices of $S$. Then $\displaystyle[A(S)]_{uv}$ $\displaystyle=\sum\limits_{\beta\in\mathscr{B}}(A^{\beta_{1}}_{1})_{u_{1j_{1}}v_{1k_{1}}}(A^{\beta_{2}}_{2})_{u_{2j_{2}}v_{2k_{2}}}\cdots(A^{\beta_{n}}_{n})_{u_{nj_{n}}v_{nk_{n}}}$ $\displaystyle=\sigma_{1}(u_{1j_{1}},v_{1k_{1}})^{\beta_{1}}\sigma_{2}(u_{2j_{2}},v_{2k_{2}})^{\beta_{2}}\cdots\sigma_{n}(u_{nj_{n}},v_{nk_{n}})^{\beta_{n}}$ $\displaystyle=a^{\beta_{1}}_{1;j_{1}k_{1}}a^{\beta_{2}}_{2;j_{2}k_{2}}\cdots a^{\beta_{n}}_{n;j_{n}k_{n}}=[\sum\limits_{\beta\in\mathscr{B}}A^{\beta_{1}}_{1}\otimes\cdots\otimes A^{\beta_{n}}_{n}]_{uv}.$ The second part of the result follows by Lemma $4.1$. ∎ We note two special cases of Theorem $4.5$. $(I)$ The Kronecker product $S_{1}\otimes S_{2}\otimes\cdots\otimes S_{n}$ has eigenvalues $z_{j_{1}j_{2}\cdots j_{n}}=z_{1j_{1}}z_{2j_{2}}\cdots z_{nj_{n}}$, $1\leq j_{i}\leq\|V(S_{i})\|,i=1,2,\cdots,n$. $(II)$ The cartesian product $S_{1}\times S_{2}\times\cdots\times S_{n}$ has eigenvalues $z_{j_{1}j_{2}\cdots j_{n}}=z_{1j_{1}}+z_{2j_{2}}+\cdots+z_{nj_{n}}$, $1\leq j_{i}\leq\|V(S_{i})\|,i=1,2,\cdots,n.$ In [8] the authors considered the problem of balance in NEPS of sigraphs. It is natural to consider the problem of cycle balance for sidigraphs, in view of Theorem $1.3$, the cartesian product $S_{1}\times S_{2}\times\cdots\times S_{n}$ of sidigraphs $S_{1},S_{2},\cdots,S_{n}$, is cycle balanced if and only if $S_{i}$, $i=1,2,\cdots,n$ is cycle balanced. The next result gives sufficient but not necessary condition for cycle balance of NEPS and the proof follows on same lines as that in undirected case. Theorem 4.6. NEPS$(S_{1},S_{2},\cdots,S_{n};\mathscr{B})$ is balanced if $S_{1},S_{2},\cdots,S_{n}$ are cycle balanced. Remark 4.7 $(I)$ Theorem $4.6$ does not have a general converse. A counter example is $S=$ NEPS$(-C_{3},-\overleftrightarrow{K_{2}},\\{(1,1)\\})$, where $-C_{3}$ denotes all negative directed cycle of order $3$ and $-\overleftrightarrow{K_{2}}$ is symmetric sidigraph of order $2$ with both arcs negative. $S$ is all positive, and hence cycle balanced. However $-C_{3}$ is non cycle balanced. $(II)$ In view of Theorem $1.3$, the converse of Theorem $4.6$ is always true if basis $\mathscr{B}=\\{e_{i}\\},i=1,2,\cdots,n$. Theorem 4.8. The following statements are equivalent about cartesian product $S=S_{1}\times S_{2}\times\cdots\times S_{n}$. $(I)$ $S$ is cycle balanced. $(II)$ All of $S_{1},S_{2},\cdots,S_{n}$ are cycle balanced. $(III)$ $S$ and $S^{u}$ are cospectral. Proof. Theorem $1.3$ implies equivalence of $(I)$ and $(III)$ and Theorem $4.6$ implies equivalence of $(I)$ and $(II)$. ∎ ## 5\. Upper bounds for the energy of sidigraphs Let $S$ be a sidigraph of order $n$ with adjacency matrix $A(S)=(a_{ij})$. The powers of $A(S)$ count the number of walks in signed manner. Let $w^{+}_{ij}(l)$ and $w^{-}_{ij}(l)$ respectively denote the number of positive and negative walks of length $l$ from $v_{i}$ to $v_{j}$. The following result relates integral powers of adjacency matrix with the number of positive and negative walks. Theorem 5.1. If $A$ is a adjacency matrix of a sidigraph on $n$ vertices, then $[A^{l}]_{ij}=w^{+}_{ij}(l)-w^{-}_{ij}(l)$. Proof. We prove the result by induction on $l$. For $l=1$, result is vacuously true. For $n=2$, let $n^{+}_{ij}$ denote the number of positive neighbours of distinct vertices $v_{i}$ and $v_{j}$, $n^{-}_{ij}$ the number of their common negative neighbours and $n^{\pm}_{ij}$ the number of neighbours that are positive to one vertex and negative to other. The $(i,i)$ entry of $A^{2}$ equals $w^{+}_{ii}(2)-w^{+}_{ii}(2)$. For $(i,j)$, $i\neq j$, $n^{+}_{ij}+n^{-}_{ij}=w^{+}_{ij}(2)$ and $n^{\pm}_{ij}=w^{-}_{ij}(2)$, so that $(i,j)$th entry$=w^{+}_{ij}(2)-w^{-}_{ij}(2)$. Now assume the result to be true for $l=m$. We have, $[A^{m+1}]_{ij}=[A^{m}A]_{ij}=\sum\limits_{k=1}^{n}[A^{m}]_{ik}[A]_{kj}=w^{+}_{ij}(m+1)-w^{-}_{ij}(m+1)$, by induction hypothesis. Therefore the result follows. ∎ In sidigraph $S$, let $c^{+}_{m}$ denote number of positive closed walks of length $m$ and $c^{-}_{m}$ the number of negative closed walks of length $m$. In view of the fact that sum of eigenvalues of a matrix equals to its trace, we have following Corollary. Corollary 5.2. If $z_{1},z_{2},\cdots,z_{n}$ are the eigenvalues of a sidigraph $S$, then $\sum\limits_{j=1}^{n}z^{m}_{j}=c^{+}_{m}-c^{-}_{m}$. Lemma 5.3. Let $S$ be a sidigraph having $n$ vertices and $a$ arcs and let $z_{1},z_{2},\cdots,z_{n}$ be its eigenvalues. Then $(I)$ $\sum\limits_{j=1}^{n}(\Re z_{j})^{2}-\sum\limits_{j=1}^{n}(\Im z_{j})^{2}=c^{+}_{2}-c^{-}_{2}$, $(II)$ $\sum\limits_{j=1}^{n}(\Re z_{j})^{2}+\sum\limits_{j=1}^{n}(\Im z_{j})^{2}\leq a=a^{+}+a^{-}$. Proof. By Corollary $5.2$ , we have $c^{+}_{2}-c^{-}_{2}=\sum\limits_{j=1}^{n}z_{j}^{2}=\sum\limits_{j=1}^{n}(\Re z_{j})^{2}-\sum\limits_{j=1}^{n}(\Im z_{j})^{2}+2\iota\sum\limits_{j=1}^{n}\Re z_{j}\Im z_{j}.$ Equating real and imaginary parts proves $(I)$. By Schur’s unitary triangularization, there exists a unitary matrix $U$ such that adjacency matrix $A$ of sidigraph $S$ is unitarily similar to an upper triangular matrix $T=(t_{jk})$ with $t_{jj}=z_{j}$ for each $j=1,2,\cdots,n$. Then $\sum\limits_{j,k=1}^{n}\|a_{jk}\|^{2}=\sum\limits_{j,k=1}^{n}\|t_{jk}\|^{2}$. As $A$ is $(-1,0,1)-$matrix, we have $a=\sum\limits_{j,k=1}^{n}\|\sigma(v_{j},v_{k})\|=\sum\limits_{j,k=1}^{n}\|a_{jk}\|=\sum\limits_{j,k=1}^{n}\|a_{jk}\|^{2}=\sum\limits_{j,k=1}^{n}\|t_{jk}\|^{2}\geq\sum\limits_{j=1}^{n}\|t_{jj}\|^{2}$ $~{}~{}=\sum\limits_{j=1}^{n}\|z_{j}\|^{2}=\sum\limits_{j=1}^{n}\Re z_{j}^{2}+\sum\limits_{j=1}^{n}\Im z_{j}^{2}.$ thereby proving $(II)$. ∎ Theorem 5.4. Let $S$ be a sidigraph with $n$ verices and $a=a^{+}+a^{-}$ arcs, and let $z_{1},z_{2},\cdots,z_{n}$ be its eigenvalues. Then $E(S)\leq\sqrt{\frac{1}{2}n(a+c^{+}_{2}-c^{-}_{2})}.$ Proof. Subtracting part $(I)$ of Lemma $5.3$ from $(II)$, we see $\sum\limits_{j=1}^{n}(\Im z_{j})^{2}\leq\frac{1}{2}(a-(c^{+}_{2}-c^{-}_{2}))$. Applying Cauchy-Schwartz inequality to vectors $(\|\Re z_{1}\|,\|\Re z_{2}\|,\cdots,\|\Re z_{n}\|)$ and $(1,1,\cdots,1)$, we have $E(S)=\sum\limits_{j=1}^{n}\|\Re z_{j}\|\leq\sqrt{n}\sqrt{\sum\limits_{j=1}^{n}(\Re z_{j})^{2}}=\sqrt{n}\sqrt{(c^{+}_{2}-c^{-}_{2})+\sum\limits_{j=1}^{n}(\Im z_{j})^{2}}$ $\leq\sqrt{n}\sqrt{(c^{+}_{2}-c^{-}_{2})+\frac{1}{2}(a-(c^{+}_{2}-c^{-}_{2})}=\sqrt{\frac{1}{2}n(a+c^{+}_{2}-c^{-}_{2})}.$ ∎ Remark 5.5. $(I)$ The upper bound in Theorem $5.4$ is attained by sidigraphs $S_{1}=(\frac{n}{2}\overleftrightarrow{K_{2}},+)$, $S_{2}=(\frac{n}{2}\overleftrightarrow{K_{2}},-)$, and skew-smmetric sidigraph of order $n$. Note that eigenvalues of $S_{1}$ and $S_{2}$ are $-1,+1$ each repeated $\frac{n}{2}$ times and eigenvalues of skew symmetric sidigraph of order $n$ are $0^{n-2},~{}\pm\iota\sqrt{n-1}$. $(II)$. Above result extends McClleland’s inequality for sigraphs [9] which states that $E(\Sigma)\leq\sqrt{2pq}$, holds for every sigraph with $p$ vertices and $q$ edges. Let $\overleftrightarrow{\Sigma}$ be the symmetric sidigraph of sigraph $\Sigma$, then in $\overleftrightarrow{\Sigma}$, $a=2q=c^{+}_{2}=c^{+}_{2}-c^{-}_{2}$,. By Theorem $5.4$, $E(\Sigma)=E(\overleftrightarrow{\Sigma})\leq\sqrt{\frac{1}{2}p(2q+2q)}=\sqrt{2pq}.$ The following result gives sharp upper bound of energy of sidigraphs in terms of number of arcs. The proof is same as in unsigned case. Theorem 5.6. Let $S$ be a sidigraph with $a$ arcs. Then $E(S)\leq a$ with equality if and only if $S=(\frac{a}{2}\overleftrightarrow{K_{2}},+)$ or $S=(\frac{a}{2}\overleftrightarrow{K_{2}},-)$ plus some isolated vertices. Remark 5.7. Theorem $5.6$ extends the result for sigraphs [9], which states that $E(\Sigma)\leq 2q$ for every sigraph with $q$ edges with equality if and only if $\Sigma=(\frac{q}{2}K_{2},+)$ or $\Sigma=(\frac{q}{2}K_{2},-)$ plus some isolated vertices. ## 6\. Equienergetic sidigraphs Two sidigraphs are said to be isomorphic if their underlying digraphs are isomorphic such that signs are preserved. Any two isomorphic sidigraphs are obviously cospectral. There exist nonisomorphic sidigraphs which are cospectral, e.g., consider the sidigraphs $S_{1}$ and $S_{2}$ shown in Figure $2$. Clearly, $S_{1}$ and $S_{2}$ are nonisomorphic. $Spec~{}S_{1}=\\{0^{(5)}\\}=Spec~{}S_{2}$. $S_{1}$Figure 2Figure 2$S_{2}$ Two nonisomorphic sidigraphs $S_{1}$ and $S_{2}$ of same order are said to be equienergetic if $E(S_{1})=E(S_{2})$. In [20] Rada proved the existence pairs of non-symmetric equienergetic digraphs on $n\geq 3$ vertices. Cospectral sidigraphs are obviously equienergetic, therefore problem of equienergetic sidigraphs reduces to problem of construction of noncospectral pairs of equinergetic sidigraphs such that for every pair not both sidigraphs are cycle balanced. The problem of construction of equienergetic sigraphs is an open problem [7] and for equienergetic graphs see [4, 19]. Theorem 6.1. Let $S$ be a sidigraph of order $n$ having eigenvalues $z_{1},z_{2},\cdots,z_{n}$ such that $\|\Re z_{j}\|\leq 1$ for every $j=1,2,\cdots,n$. Then $E(S\times\overleftrightarrow{K_{2}})=2n$. Proof. Let $z_{1},z_{2},\cdots,z_{t}$ be eigenvalues with nonnegative real part and $z_{t+1},\cdots,z_{n}$ be those with negative real part. Eigenvalues of cartesian product $S\times\overleftrightarrow{k_{2}}$ are $z_{1}\pm 1,z_{2}\pm 1,\cdots,z_{t}\pm 1,z_{t+1}\pm 1,\cdots,z_{n}\pm 1$. Therefore $E(S\times\overleftrightarrow{K_{2}})=\sum\limits_{j=1}^{t}(\|\Re z_{j}+1\|+\|\Re z_{j}-1\|)+\sum\limits_{j=t+1}^{n}(\|\Re z_{j}+1\|+\|\Re z_{j}-1\|).$ As $\|\Re z_{j}\|\leq 1$ for all $i=1,2,\cdots,n$, it follows that $E(S\times\overleftrightarrow{K_{2}})=\sum\limits_{j=1}^{t}(\Re z_{j}+1+1-\Re z_{j})+\sum\limits_{j=t+1}^{n}(1-\Re z_{j}+\Re z_{j}+1)=2t+2(n-t)=2n.$ ∎ Corollary 6.2. For $n\geq 2$, $E({\bf C}_{n}\times\overleftrightarrow{K_{2}})=E(C_{n}\times\overleftrightarrow{K_{2}})=2n$. Moreover, ${\bf C}_{n}\times\overleftrightarrow{K_{2}}$ and $C_{n}\times\overleftrightarrow{K_{2}}$ are noncospectral sidigraphs with $2n$ vertices. Proof. We know that eigenvalues of ${\bf C}_{n}$ are $e^{\frac{\iota(2j+1)\pi}{n}},~{}j=0,1,\cdots,n-1$ and those of $C_{n}$ are $e^{\frac{2\iota j\pi}{n}},~{}j=0,1,\cdots,n-1$. Clearly, eigenvalues of ${\bf C}_{n}$ and ${C_{n}}$ meet requirement of Theorem $6.1$, so $E({\bf C}_{n}\times\overleftrightarrow{K_{2}})=E(C_{n}\times\overleftrightarrow{K_{2}})=2n$. Moreover, $2\notin spec({\bf C}_{n}\times\overleftrightarrow{K_{2}})$ but $2\in spec(C_{n}\times\overleftrightarrow{K_{2}})$ implying that ${\bf C}_{n}\times\overleftrightarrow{K_{2}}$ and $C_{n}\times\overleftrightarrow{K_{2}}$ are non cospectral. Number of vertices in both sidigraphs is $2n$ follows by definition of cartesian product. In view of Theorem $1.3$, ${\bf C}_{n}\times\overleftrightarrow{K_{2}}$ is non cycle balanced whereas $C_{n}\times\overleftrightarrow{K_{2}}$ is cycle balanced. ∎ Example 6.3. For each odd $n$, ${\bf C}_{n}$ and $C_{n}$ is a non co-spectral pair of equienergetic sidigraphs, because $Spec({\bf C}_{n})=-Spec(C_{n})$ and $1\notin spec({\bf C}_{n})$ but $1\in spec(C_{n})$. From Corollary $6.2$ and Example $6.3$, we see that for each positive integer $n\geq 3$ there exits a pair of non co-spectral sidigraphs with one sidigraph cycle balanced and another non cycle balanced. We now construct pairs of non cospectral sidigraphs of order $2n,~{}n\geq 5$ with both constituents non cycle balanced. Let $P^{l}_{n}~{}(n\geq l+1)$ be a sidigraph obtained by identifying one pendant vertex of the path $P_{n-l+1}$ with any vertex of ${\bf C}_{l}$. Sign of non cyclic arcs is immaterial. Corollary 6.4. For each $n\geq 5$, $P^{3}_{n}\times\overleftrightarrow{K_{2}}$ and $P^{4}_{n}\times\overleftrightarrow{K_{2}}$ is a pair of noncospectral equienergetic sidigraphs of order and energy equal to $2n$. Proof. Using the fact that $\phi_{P^{l}_{n}}(x)=x^{n-l}\phi_{{\bf C}_{l}}(x)$ and Theorem $6.1$, it follows that $E(P^{3}_{n}\times\overleftrightarrow{K_{2}})=E(P^{4}_{n}\times\overleftrightarrow{K_{2}})=2n$. Now $1$ is an eigenvalue of $P^{3}_{n}\times\overleftrightarrow{K_{2}}$ with multiplicity $n-3$ but $1$ is eigenvalue of $P^{4}_{n}\times\overleftrightarrow{K_{2}}$ with multiplicity $n-4$, therefore these two sidigraphs are noncospectral. Order of both sidigraphs equals to $2n$ follows by definition of cartesian product. In view of Remark $4.7~{}(II)$, it follows that both $P^{3}_{n}\times\overleftrightarrow{K_{2}}$ and $P^{4}_{n}\times\overleftrightarrow{K_{2}}$ are non cycle balanced. ∎ Corollary 6.5. If $S$ is a sidigraph on $n$ vertices having eigenvalues $z_{1},z_{2},\cdots,z_{n}$ such that $\|\Im z_{j}\|\leq\frac{1}{\sqrt{n-1}}$ for all $j=1,2,\cdots,n$, then $E((S\otimes{\bf S}_{m})\times\overleftrightarrow{K_{2}})=2nm=$ order of sidigraph $(S\otimes{\bf S}_{m})\times\overleftrightarrow{K_{2}}$, $2\leq m\leq n$. Proof. Eigenvalues of $S$ are $z_{1},z_{2},\cdots,z_{n}$ with $\|\Im z_{j}\|\leq\frac{1}{\sqrt{n-1}}$ and eigenvalues of ${\bf S}_{m}$ are $0^{m-2},\pm\iota\sqrt{m-1}$. Then eigenvalues of Kronecker product $S\otimes{\bf S}_{m}$ meet requirement of Theorem $6.1$, therefore $E((S\otimes{\bf S}_{m})\times\overleftrightarrow{K_{2}})=2nm$. Order of sidigraph equals to $2nm$ follows from definition of Kronecker product and cartesian product. ∎ Theorem 6.6. Let $A=(a_{ij})$ be a square matrix of order $n$ having integral entries and zero trace and let $z_{1},z_{2},\cdots,z_{n}$ be its eigenvalues. Put $\alpha=\sum\limits_{j=1}^{n}\|\Re z_{j}\|$, then $\alpha$ cannot be of the form $(I)~{}(2^{t}s)^{\frac{1}{h}}$ with $h\geq 1,~{}0\leq t<h$ and $s$ odd $(II)~{}(\frac{m}{n})^{\frac{1}{r}}$ where $\frac{m}{n}$ is non-integral rational. Proof. We note that $\alpha=2\sum\limits_{\Re z_{j}\geq 0}z_{j}$. Put $z=\sum\limits_{\Re z_{j}\geq 0}z_{j}$, then $z$ being sum of algebraic integers is an algebraic integer. $(I)$ Assume $\alpha=(2^{t}s)^{\frac{1}{h}}$, so that $2z=(2^{t}s)^{\frac{1}{h}}$. Simplifying gives $z^{h}=\frac{s}{2^{l}}$, where $l=h-t\geq 1$. As $s$ is odd, therefore we see $z^{h}$ is non-integral rational algebraic integer, a contradiction. $(II)$ Proof is same as in part $(I)$. ∎ Bapat and Pati [3] characterized positive rationals which can be the energy of graph. Later Pirzada and Gutman [16] proved energy of graph cannot be square root of an odd integer. These results were generalized to digraphs by Pirzada, Mushtaq, Gutman and Rada [17]. Following result generalizes these results to sidigraphs. Theorem 6.7. Energy of a sidigraph cannot be of the form $(I)~{}(2^{t}s)^{\frac{1}{h}}$ with $h\geq 1,~{}0\leq t<h$ and $s$ odd $(II)~{}(\frac{m}{n})^{\frac{1}{r}}$ where $\frac{m}{n}$ is non-integral rational number and $r\geq 1$. Proof. Let $S$ be a sidigraph with adjacency matrix $A(S)$. Apply Theorem $6.6$ to $A(S)$ and note that $E(S)=\sum\limits_{j=1}^{n}\|\Re z_{j}\|$, the result follows. ∎ ## 7\. Open problems We conclude this paper with the following open problems. $(1)$ Characterize sidigraphs with energy equal to the number of vertices. $(2)$ Find an infinite family of noncospectral equienergetic sidigraphs on $n\geq 4$ vertices with both constituents non cycle balanced. $(3)$ Determine bases other than $\\{e_{i}\\}$ for which converse of Theorem $4.7$ holds. Acknowledgements The second author thanks University Grants Commission, New Delhi, India for providing junior research fellowship. ## References * [1] B. D. Acharya, Spectral criterion for the cycle balance in networks. J. Graph Theory 4(1980)1-11. * [2] B. D. Acharya, M. K. Gill, G. A. Patwardhan, Quasicospectral graphs and digraphs, National symposium on mathematical modelling M. R. I. Allahabad: July 19-20, 1982. * [3] R. B. Bapat, S. Pati, Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc., 1(2004), 129-132. * [4] A. S. Bonifaco, C. T. M. Vinagre, N. M. M. Abreu, Constructing pairs of equienergetic and non cospectral graphs, Applied Mathematics Letters 21(2008) 338–341. * [5] R. Brualdi, Spectra of digraphs, Linear Algebra and Appl., 432(2009), 2181-2213. * [6] D. M. Cvetkovic, M. Doob and H. Sachs, Spectra Of Graphs. Academic press, New York 1980. * [7] K. A. Germina and Shahul Hameed K, On signed paths, signed cycles and their energies. Appl. Math. Sci. 4(2010), no. 70, 3455-3466. * [8] K. A. Germina and Shahul Hameed K, and Thomas Zaslavsky, On products and line graphs of signed graphs, their eigenvalues and energy, Linear Algebra and its Applications, 435, 10 (2010) 2432-2450. * [9] K. A. Germina and Shahul Hameed K, On bounds of eigenvalues and energy of signed graphs: submitted. * [10] M. K. Gill and B. D. Acharya, A recurrence formula for computing the characteristic polynomial of a sigraph, Journal of Combinatorics, information and system sciences, 5(1980) 68-72. * [11] I. Gutman, The Energy of a graph, Ber. Math. Statist. Sekt. Forschungszenturm Graz., 103(1978), 1-22. * [12] I. Gutman, The Energy of a graph: Old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wasserman (Eds), Algebriac Combinatorics and Applications, Springer-Verlag, Berlin 2001, pp. 196-211. * [13] R. Horn and C. Johnson, Matrix Analysis; Cambridge University press, 1985. * [14] W. Lopez and J. Rada, Equienergetic digraphs, International journal of pure and applied mathematics, 36(2007) 361-372. * [15] I. Pena, J. Rada, Energy of digraphs, Linear and Multilinear Algebra, 56(5) (2008), 565-579. * [16] S. Pirzada and I. Gutman, Energy of a graph is never the square root of an odd integer, Appl. Anal. Dis. Math., 2(2008), 118-121. * [17] S. Pirzada, Mushtaq A. Bhat, I. Gutman, J. Rada, On the energy of digraphs, Bulletin of International Mathematical Virtual Institute, 3(2013), 69-76. * [18] S. Ponnusamy, Foundations of complex analysis (Narosa). * [19] H. S. Ramane, H. B. Walikar, Construction of equienergetic graphs, MATCH Commun. Math. Comput. Chem., 57(2007) 23-210. * [20] J. Rada, The McClelland inequality for the energy of digraphs, Linear Algebra and its Applications, 430(2009), 800-804. * [21] D. Stevanoic, Energy and NEPS of graphs, Linear and Multilinear Algebra, 53(2005) 67-74. * [22] T. Zaslavsky, Matrices in the theory of signed simple graphs. In: B. D. Acharya, G. O. H. Katona, and J. Nesetril, eds., Advances in Discrete Mathematics and Applications: Mysore, 2008(ICDM-2008, Mysore, India), pp. 207-229. Ramanajun Math. Soc. Lect. Notes Ser., No. 13. Ramanajun Mathematical Soc., Mysore, India, 2010.	arxiv-papers	2013-09-24T17:50:58	2024-09-04T02:49:51.455525	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Pirzada and Mushtaq A. Bhat", "submitter": "Shariefuddin Pirzada", "url": "https://arxiv.org/abs/1309.6266" }
1309.6286	# Controllable emission of a dipolar source coupled with a magneto-dielectric resonant subwavelength scatterer Brice Rolly1, Jean-Michel Geffrin1, Redha Abdeddaim1, Brian Stout1, Nicolas Bonod1∗ 1 CNRS, Aix-Marseille Université, Centrale Marseille, Institut Fresnel, UMR 7249, Campus de St Jérôme, 13397 Marseille, France ∗ Corresponding author: [email protected] We demonstrate experimentally and theoretically that a local excitation of a single scatterer of relative dielectric permittivity $\varepsilon=6$ permits to excite broad dipolar and quadrupolar electric and magnetic resonances that shape the emission pattern in an unprecedented way. By suitably positioning the feed with respect to the sphere at a $\lambda/3$ distance, this compact antenna is able to spectrally sort the electromagnetic emission either in the forward or in the backward direction, together with a high gain in directivity. Materials with $\varepsilon=6$ can be found in the whole spectrum of frequencies promising Mie antennas to become an enabling technology in numbers of applications, ranging from quantum single photon sources to telecommunications. ## Introduction Mie resonances in spherical particles of moderate refractive index were recently shown to exhibit well pronounced electric and magnetic modes of both dipole and quadrupole orders [1, 2, 3, 4, 5, 6, 7, 8]. As first predicted by Kerker in the case of particles with magnetic permeability [9], interference effects between electric and magnetic modes can strongly favour either forward or backward scattering, respectively depending on the frequency of the incident plane wave [10, 11, 12, 13, 14, 15] with applications in anti- reflecting structured surfaces for photovoltaic cells [16]. Dielectric particles reveal to be highly interesting scatterers to design highly directive, compact and lossless antennas [17, 18, 19, 20]. Recently, it was proposed to extend the so-called Kerker’s conditions to the case of near field excitations, where the interplay between electric and magnetic modes can boost the directivity of antennas [21]. It was also proposed to realize a notch inside a dielectric scatterer of refractive index $n$ around 4 in order to increase the number of multipoles excited by a near field source [26]. Such magneto-dielectric antennas offer more parameters for optimizing the emission directivity than classical metallic elements which behave principally as electric multipolar scatterers [27, 28, 29, 30, 31, 32] and they could be very interesting to direct the fluorescence emission of quantum emitters in the optical range of frequencies. Here we propose to design a versatile antenna with the use of a single spherical scatterer exhibiting a refractive index $n=2.45$. Mie resonators with refractive index typically around 3.5 have been mainly considered because they exhibit narrow and well pronounced electric and magnetic modes. A decrease of the refractive index leads to a broadening of the Mie resonances and a refractive index $n=2.45$ allows for a remarkable mixing between electric and magnetic dipolar and quadrupolar modes over a finite range of frequencies. We show that the interplay between the first two electric and magnetic modes permits to control the emission directivity by tuning the emission frequency. Also, lessening the refractive index opens the way towards the use of oxides such as rutile TiO2 to design highly directive optical antennas. ## Results We measure the emission pattern in the E-plane (i.e. $xOz$ plane, see Fig.1d) of a subwavelength scale ‘Mie’ antenna consisting of a single dielectric particle of permittivity $\varepsilon=6$ ($n$=2.45) fed in the GHz regime by a two-arm electric dipole emitter. By controlling the emitter-to-particle distance at a deep subwavelength scale, we report on the possibility to choose the emission direction by tuning the frequency from 8.7 GHz to 9.74 GHz. Experiments are carried out in an anechoic chamber (CCRM-Marseille, see Fig.1a) dedicated to amplitude and phase measurements of the electric field with a receiver antenna rotating along a circular arm 4 m in diameter, centered on the feed [13]. Experimental measurements are accompanied by a theoretical derivation of the emission pattern based on the coupling between the feed and the magnetic and electric modes of the sphere. Figure 1: (a) Photograph of the experiment in the anechoic chamber: the dipole-sphere antenna is placed on a polystyrene mast at the center of an anechoic chamber. The receiver antenna (ARA DRG 118) can rotate around the $Oy$ axis (see d) except in an exclusion zone in the angle range $[-130^{\circ},130^{\circ}]$ due to the presence of the vertical arch. (b) Close-up of the dipole-sphere antenna and of the polystyrene holders. (c) Norm of the electric field emitted by the isolated dipole antenna as a function of the frequency and angle $\phi_{\rm exp}$ in the E-plane ($xOz$ plane) in dB. At each frequency, the field is normalized by its maximum in the E-plane. (d) Sketch of the experiment with the axes and $\phi_{\rm exp}$ angle in the $(xOz)$ plane, used in Fig.1c, Fig.2 and Fig.3; the blue arrow indicates the direction of the receiving antenna. (e) Norm of the Mie coefficients $\|c_{j}^{\mathrm{e(m)}}\|$ with respect to frequency in GHz. Black line: magnetic dipole $\|c_{1}^{\mathrm{m}}\|$; red line: electric dipole $\|c_{1}^{\mathrm{e}}\|$; green line: magnetic quadrupole $\|c_{2}^{\mathrm{m}}\|$; blue line: electric quadrupole $\|c_{2}^{\mathrm{e}}\|$. The two-wire dipole is directly attached to a connector (SubMiniature version A). Each arm is 9 mm long which results in a total length smaller than the wavelength even at the highest operating frequency. The characterization of the dipole in its E-plane (Fig. 1c) shows a classical dipole-like radiation pattern, with comparable front and backward radiated amplitude (respectively $0^{\circ}$ and $180^{\circ}$), and negligible emission in the $90^{\circ}$ direction. The emitter is coupled with a single dielectric sphere, 19 mm in diameter, composed of an Eccostock HIK (Emmerson $\&$ Cumming) material of permittivity 6, presenting low losses ($\tan\delta\leq 0.02$). The distance between the dipole and the sphere can be controlled at a submillimeter scale via an expanded polystyrene holder equivalent to air at the operating frequencies (see Fig.1a-b). Measurements have been performed with emitter-to- sphere distances of 5, 10 and 20 mm and frequencies ranging from 8 to 10 GHz with a step of 20 MHz, which corresponds to a total of 300 emission pattern measurements (the emitter-to-sphere distance is defined between the emitter and the surface of the sphere, i.e. it is equal to emitter-to-sphere center distance minus the sphere radius). The amplitude of the electric field is displayed in Fig. 2 with respect to the frequency and angle $\phi_{\rm exp}$ ranging from 0 to 180∘, i.e. in a half-plane containing one of the emitter arms. At every frequency, the amplitude is normalized by the maximum measured in the [0;180∘] range, and is displayed in dB. One observes that the radiation pattern is highly sensitive to modifications of the emitter-to-particle distance on a scale much smaller than the emission wavelength. Importantly, we observe that this Mie antenna can emit either in the forward (around 8.75 GHz) or backward (around 9.5 GHz) direction efficiently at a 10 mm distance. In order to demonstrate the contribution of each mode in tuning the emission directivity, we derived a formula of the irradiance in the E-plane (derivation is detailed in the Supplementary information). Such a formula invokes the far field expression of the interference between the fields produced by (i) the dipolar source and (ii) the induced electric-magnetic dipoles and quadrupoles in the dielectric particle. The fields are derived in the spherical vector basis, $[\hat{\mathbf{e}}_{\mathrm{r}},\hat{\mathbf{e}}_{\mathrm{\theta}},\hat{\mathbf{e}}_{\mathrm{\phi}}]$ with the dipole emitter defined to lie along the $+z$ axis ($\mathbf{p}_{0}\equiv p_{0}\hat{\mathbf{z}}$) and placed at the origin of the coordinate system. The center of the spherical scatterer is placed at a distance $d$ from the emitter on the $x$-axis, $\mathbf{r}_{1}=+d\hat{\mathbf{x}}$. The total, normalized (respectively to the electric dipole emitter maximum far-field irradiance) far-field irradiance can be cast in the standard spherical coordinate system: $\displaystyle I(\theta,\phi)$ $\displaystyle=$ $\displaystyle\|\tilde{\mathbf{E}}(\mathbf{r})\|^{2}$ (1) $\displaystyle=$ $\displaystyle\bigg{\|}\sin\theta\hat{\mathbf{e}_{\theta}}$ $\displaystyle+e^{i\varphi}\gamma^{e}_{1}\tilde{\alpha}^{e}_{1}\sin\theta\hat{\mathbf{e}_{\theta}}$ $\displaystyle+e^{i\varphi}\gamma^{m}_{1}\tilde{\alpha}^{m}_{1}(\cos\phi\hat{\mathbf{e}_{\theta}}-\sin\phi\cos\theta\hat{\mathbf{e}_{\phi}})$ $\displaystyle+e^{i\varphi}\gamma^{e}_{2}\tilde{\alpha}^{e}_{2}(\cos\phi\cos 2\theta\hat{\mathbf{e}_{\theta}}-\sin\phi\cos\theta\hat{\mathbf{e}_{\phi}})$ $\displaystyle+e^{i\varphi}\gamma^{m}_{2}\tilde{\alpha}^{m}_{2}(\cos 2\phi\sin\theta\hat{\mathbf{e}_{\theta}}-\frac{\sin 2\phi\sin 2\theta}{2}\hat{\mathbf{e}_{\phi}})\bigg{\|}^{2}$ In the latter expression, each subsequent line stands for the field produced by the source and the induced electric dipole, magnetic dipole, electric quadrupole, and magnetic quadrupole respectively; $e^{i\varphi}$ is a far- field phase shift contribution; $\tilde{\alpha}_{n}^{e(m)}$ is a _dimensionless_ electric (magnetic) polarizability of order $n$, defined in terms of Mie theory and a T-matrix formalism (see Supporting Information). The first four Mie coefficients are plotted in Fig.1e with respect to the frequency and one remarks that the relative low refractive index of the resonant scatterer leads to broad resonances and that the norms of the four Mie coefficients are nearly equal near 9.75 GHz. The coupling coefficients between the emitter and the first 2 electric and magnetic modes of the sphere can be cast [33]: $\displaystyle\gamma^{e}_{1}$ $\displaystyle\equiv$ $\displaystyle e^{ikd}(k^{2}d^{2}+ikd-1)(a/d)^{3}$ $\displaystyle\gamma^{m}_{1}$ $\displaystyle\equiv$ $\displaystyle e^{ikd}(k^{2}d^{2}+ikd)(a/d)^{3}$ $\displaystyle\gamma^{e}_{2}$ $\displaystyle\equiv$ $\displaystyle-\frac{5}{3}e^{ikd}(k^{3}d^{3}+3ik^{2}d^{2}-6kd-6i)(a/d)^{4}$ $\displaystyle\gamma^{m}_{2}$ $\displaystyle\equiv$ $\displaystyle\frac{5}{3}e^{ikd}(k^{3}d^{3}+3ik^{2}d^{2}-3kd)(a/d)^{4}$ In the $(xOz)$ ($\phi=0$ or $\pi$) plane, the irradiance can be cast: $\displaystyle I^{\pm xz}(\theta)$ $\displaystyle=$ $\displaystyle\bigg{\|}\sin\theta(1+e^{-ikd\pm\sin\theta}\gamma^{e}_{1}\tilde{\alpha}^{e}_{1})$ (2) $\displaystyle-e^{-ikd\pm\sin\theta}\times\big{(}\pm\gamma^{m}_{1}\tilde{\alpha}^{m}_{1}\pm\gamma^{e}_{2}\tilde{\alpha}^{e}_{2}\cos 2\theta$ $\displaystyle+\gamma^{m}_{2}\tilde{\alpha}^{m}_{2}\sin\theta\bigg{\|}^{2}$ Figure 2: Norm of the electric field in the E-plane, in decibels, measured (top row) and simulated (bottom row) as functions of the antenna receiver angle $\phi_{\rm exp}$ in degrees (in abscissa) and emitting frequency in GHz (in ordinate). At each frequency, the field is normalized by its maximum in the E-plane. The emitter-to-sphere gaps are (a) 5 mm, (b) 10 mm and (c) 20 mm. The normalized amplitude of the electric field in the E-plane is displayed in Fig.2 as a function of the angle in the $(xOz)$ plane, and a good agreement between theory and experiments is observed for the three emitter-to-particle distances despite a small frequency shift that is likely due to an imperfect knowledge of the permittivity of the sphere (the real part is given at $\pm{5}\%$ and the imaginary part is set to 0 in the model) and to the physical length of the source, the theoretical expression assuming a point dipole. This model is general, and could also be applied to predict the scattering pattern of metallic magneto-electric antennas provided that the diagonal polarizability tensor elements are accurately determined and predominate over the extra-diagonal elements. ## Discussion Both theoretical model and experiments predict that the privileged direction of emission of the antenna can be controlled for a 10 mm separation distance by tuning the emission frequency. This feature is highlighted in Figs.3a and b where the intensities of the scattered electric field measured in the E-plane at frequencies 8.7 GHz and 9.74 GHz are plotted in polar coordinates. Measurements and theory are quantified by defining directivity gain, in isotropic decibels, $D_{\mathrm{dBi}}=10\log\big{(}4\pi I/P_{\mathrm{rad}}\big{)}$ where $I$ and $P_{\mathrm{rad}}$ are respectively the radiant intensity in the direction of interest and the total radiative power of the antenna. Theoretical values of 7.01 dBi at 8.66 GHz and 5.17 dBi at 9.58 GHz in the forward and backward directions respectively can be achieved. The sphere also effectively increases antenna gain: the presence of the sphere increases the measured electric field intensity: by a factor of 3.4 and 2.4 along the corresponding privileged directions at 8.7 GHz and 9.74 GHz respectively. Figure 3: (a,b) Norm of the electric field scattered for an emitter-to- particle gap of 10 mm in the E-plane, (red line) simulations with 20 multipoles and (blue) measurements: (a) Back-scattering observed at 9.74 GHz, (b) Forward scattering observed at 8.7 GHz. (c) Emission pattern derived with Eq. 2 when considering the induced dipoles (red line) or quadrupoles (blue line) only at 8.7 GHz. (d) Emission pattern obtained with Eq. 2 when considering electric (red line) or magnetic modes (blue line) of dipolar and quadrupolar orders, sum of electric and magnetic response of both dipole and quadrupole order (green line), electric and magnetic modes with 20 multipoles (dashed black line) at 8.7 GHz. The emission patterns plotted in Fig.3c at $f$=8.7 GHz when considering dipoles or quadrupoles only (full red and blue lines) show that neither dipole nor quadrupole excitations taken alone suffice to explain the directivity: a purely response of the sphere would predict light preferentially emitted in the backward direction, whereas a solely quadrupole response would lead to forward emission together with a low gain in directivity. A similar filtering with respect to either purely electric or magnetic modes, illustrated in Fig.3d, shows that neither purely electric nor purely magnetic modes can fit the emission pattern measured in the E-plane, while the emission pattern is accurately reproduced when both dipoles and quadrupoles are considered. Comparison with a multipolar calculation performed with a Generalized Mie theory calculation taking into account 20 multipole orders reveals the high accuracy of the quadrupolar model. These last two plots demonstrate that the emission pattern results from an efficient coupling between the electric and magnetic modes, of both dipolar and quadrupolar orders which is made possible by the broadness of the Mie resonances displayed in Fig.1e. The high sensitivity of the emission pattern with respect to the emission frequency is explained by a strong modulation of three of the relevant modes between 8.5 GHz and 10 GHz (see Fig.1e). In conclusion, a single dielectric particle of refractive index $n=2.45$ is used to efficiently tune the scattering directivity of an electric dipole emitter. The tuning of the emission frequency modulates the amplitude and phase of the first four modes of the sphere which drastically modifies the emission pattern. Dielectric particles of refractive index around 2.5 can be found over a broad spectrum making dielectric Mie antennas interesting from microwaves to nano-optics where they could acts as lossless subwavelength spectral sorters [34, 35]. ## Methods ### Measurements The measurements are carried out in an anechoic chamber (Centre Commun de Ressources en Microondes in Marseille, France) that allows electric field amplitude and phase measurements thanks to an antenna receiver placed on a circular arm of radius 2 m. For this study, the electric field is quantified by measuring the electric field component lying inside the E-plane (the electric field component normal to the E-plane being negligible). The $S_{21}$ parameter is measured to provide the norm of the electric field $20\log\|E\|$. The field is normalized at each frequency by the maximum value $E_{max}$ measured when varying the angle $\phi_{\rm exp}$ and the value of $20\log\|E/E_{max}\|$ is reported in Fig. 2. In Fig. 3, the field is also normalized by its maximum when varying the angle $\phi_{\rm exp}$ at frequencies 8.7 GHz and 9.74 GHz. The antenna receiver cannot make a full rotation around the axis due to the presence of the vertical arch. The exclusion zone is almost $100^{\circ}$. The complete $360^{\circ}$ emission pattern requires a second measurement for which the mast holder is rotated by $120^{\circ}$. In Figs.3a and b, it can be observed that the two measurements match and permit the reconstruction of the emission patterns with a high accuracy. ### Theoretical calculations The far field expressions of the electric fields radiated by the feeding electric dipole, and the electric and magnetic dipoles and quadrupoles excited in the particle are derived and summed up to obtain the final expression of the total electric field. The sources are strongly coupled in near field and the calculation of the phase differences between the coherent fields must take into account the entire field, i.e. near, intermediate and far field expressions. The expression of the emission pattern requires the calculation of the dipolar and quadrupolar polarizabilities of the scatterer, that can be easily obtained within the context of generalized Mie theory [36]. The analytical derivation is detailed in the Supporting Information. 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G. _et al._ Unidirectional Emission of a Quantum Dot Coupled to a Nanoantenna. _Science_ 329, 930–933 (2010). * [29] Esteban, R., Teperik, T. V. & Greffet, J. J. Optical Patch Antennas for Single Photon Emission Using Surface Plasmon Resonances. _Physical Review Letters_ 104, 026802 (2010). * [30] Bonod, N., Devilez, A., Rolly, B., Bidault, S. & Stout, B. Ultracompact and unidirectional metallic antennas. _Physical Review B: Condensed Matter and Materials Physics_ 82, 115429 (2010). * [31] Rolly, B., Stout, B., Bidault, S. & Bonod, N. Crucial role of the emitter–particle distance on the directivity of optical antennas. _Optics Letters_ 36, 3368–3370 (2011). * [32] Massa, E., Maier, S. A. & Giannini, V. An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas. _New Journal of Physics_ 15, 063013 (2013). * [33] Rolly, B., Stout, B. & Bonod, N. Metallic dimers: When bonding transverse modes shine light. _Physical Review B: Condensed Matter and Materials Physics_ 84, 125420 (2011). * [34] Aouani, H. _et al._ Bright Unidirectional Fluorescence Emission of Molecules in a Nanoaperture with Plasmonic Corrugations. _Nano Letters_ 11, 637–644 (2011). * [35] Shegai, T. _et al._ A bimetallic nanoantenna for directional colour routing. _Nature Communications_ 2, 481 (2011). * [36] Stout, B., Devilez, A., Rolly, B. & Bonod, N. Multipole methods for nanoantennas design: applications to Yagi-Uda configurations. _Journal of the Optical Society of America B: Optical Physics_ 28, 1213–1223 (2011). ## Acknowledgements The authors thank Sébastien Bidault for discussions and Jean-Pierre Spinelli for his involvement in the anechoic chambers. This work is supported by Agence Nationale de la Recherche via project ANR 11 BS10 002 02 TWINS. ## Author contributions N.B, R.A. and J.M.G. conceived the experiment; R. A. and J.M.G. carried out the experiment; B. R. derived the analytical expression, performed the numerical calculations with B. S. and N. B.; All the authors contributed to the discussion and analysis of the results; N. B. wrote the manuscript with contribution from all the authors. ## Additional information The authors declare no competing financial interests. Supplementary information accompanies this paper and include the detailed derivation of expression 2. Reprints and permissions information is available at www.nature.com/reprints. Correspondence and requests for materials should be addressed to N. B.	arxiv-papers	2013-09-22T10:56:20	2024-09-04T02:49:51.466704	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Brice Rolly, Redha Abdeddaim, Jean-Michel Geffrin, Brian Stout,\n Nicolas Bonod", "submitter": "Nicolas Bonod", "url": "https://arxiv.org/abs/1309.6286" }
1309.6386	# Flashing coherently rotating carbon sticks in 24Mg+24Mg collision M. H. Zhao S. Kun [email protected] O. Merlo M. R. Huang Y. Li J. S. Wang Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000, People’s Republic of China University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China Canberra, Australia Zurich University of Applied Sciences, Institute of Applied Simulation Grüental, P.O. Box CH-8820 Waedenswil, Schweiz ###### Abstract We analyze quasi-periodic oscillations in the angle-averaged ($\Delta\theta_{cm}\simeq 90^{\circ}\pm 25^{\circ}$) excitation functions for the ${}^{24}Mg+^{24}Mg$ elastic-inelastic scattering and $\alpha$-transfer channels on the energy interval $E_{cm}=44.86-47.76$ MeV. The period of the energy structures, $\simeq$0.81 MeV, is interpreted as inverse half-period ($\simeq 5\times 10^{-21}$ sec.) of coherent rotation of highly excited short- lived ($\simeq 3.6\times 10^{-21}$ sec.) chain of a length $\simeq 30$ fm. The rotational wave packet coherence survives (i) the energy relaxation (fully mixing ergodic dynamics) for the strongly overlapping states with fixed total spin and parity values, and (ii) the strong perturbation of the motion upon a change of the total spin. The present discussion rises a number of the questions. For example, is rotational coherence of large molecules necessarily destroyed in the conventionally statistical limit of structureless non- selective continuum (for fixed total spin and parity values) under the conditions of complete intramolecular energy redistribution and vibrational dephasing in the regime of strong ro-vibrational coupling? For the slow cross- symmetry phase relaxation, quantum coherent superpositions of a large number of complex configurations with, e.g., many different total angular momenta produce image of a rotation of macroscopic object with classically fixed (single) total angular momentum. Suppose that the quantum coherent superpositions involving a very large number of different good quantum numbers play a role, in a hidden form, in a formation of macroscopic world. Then why these quantum superpositions are so stable against quick aging/decay of ordered complex structures preventing or slowing down tendencies towards uniform occupation of the available phase space as prescribed by the random matrix theory? And what kind of complex macroscopic phenomena may reveal traces of partially coherent quantum superpositions involving a huge number of quantum-mechanically different integrals of motion behind of what is referred to as conservation laws in classical physics employed for the description of the macroscopic world? ###### keywords: coherent rotation , phase relaxation , ${}^{24}Mg+^{24}Mg$ collision , nuclear molecules , quantum macroscopic transition Femtochemistry provides powerful tools in studying molecular structure and dynamics in chemistry and biology [1], [2]. Nowadays, the use of ultrashort femtosecond and attosecond laser pulses is a widely used technique for the real-time monitoring of chemical reactions. One of the successes of femtochemistry is its ability for direct probe of vibrational and rotational wave packets and their coherent evolution as classical-like, essentially due to the quantum interference, system-specific trajectories of a transitional state of a given chemical reaction. Precondition for survival of the characteristic time and length scales for the system-specific wave packet dynamics is that the isolated intermediate complex is not in ergodic state. This means that phase relaxation (dephasing) or dynamical decoherence [3], responsible for a spread of the wave packets, is relatively slow, not faster than the life time of the intermediate complex. This regime of incomplete mixing is beyond applicability of random matrix theory [4] which addresses a featureless universal behavior of complex systems in the ergodic limit of complete lose of phase memory and uniform occupation of the available phase space (simple pictorial explanation of the domain of applicability of random matrix theory, for not experts in the field, is given in Sect. V of [5]). Unfortunately, the femtochemistry experimental methods are inapplicable to study reactions which are not initiated by the laser pulses. In particular, time evolution of heavy ion collisions is not accessible for the direct monitoring. Additional experimental difficulty to measure time scales of reactions involving relatively light heavy ions is their short duration. Namely, time resolution of order of zeptosecond (10-21 sec) would be needed for an accurate reconstruction of heavy ion collisional dynamics. Therefore, the only possible way to access time evolution of heavy ion reactions is provided by the detailed energy dependence of the cross sections, i.e. excitation functions. In this respect, the most favorable situation is the presence of both fast (direct) and time delayed processes [6], [7], [8], [9]. Then, under the condition of a major contribution of direct processes, the energy variations in the excitation functions are mainly given by the interference between the collision amplitudes corresponding to the direct and time delayed processes. Since the direct process amplitude is energy smooth function, this allows to obtain time dependent amplitude, ${\cal P}(t,\theta)$ of the time delayed process [6], [7], [8], [9]. Clearly, the direct process plays a role of the pump pulse switching on the clock at the initial moment of time [8], [9]. When, for that or another reason, the interference term between the amplitudes of the direct and time delayed processes does not contribute to the cross section, the energy variations in the excitations functions originate from the modulus square of the energy oscillating around zero amplitude, $F(E,\theta)$, corresponding to the time delayed processes. In this situation, an information on the time evolution, though not as complete as in the presence of direct processes, is still encoded in the incident energy dependence of the cross sections. Indeed, we represent the cross section $\sigma(E,\theta)=\|F(E,\theta)\|^{2}$ as $\displaystyle\sigma(E,\theta)\propto\int_{-\infty}^{\infty}d\tau\exp(iE\tau/\hbar)<{\cal P}(t+\tau/2,\theta)$ $\displaystyle{\cal P}(t-\tau/2,\theta)^{\ast}>_{t}$ (1) with $\displaystyle<{\cal P}(t+\tau/2,\theta){\cal P}(t-\tau/2,\theta)^{\ast}>_{t}=\int_{0}^{\infty}dt$ $\displaystyle{\cal P}(t+\tau/2,\theta){\cal P}(t-\tau/2,\theta)^{\ast}.$ (2) In the above relations, $E>0$ is the total excitation energy, $t$ is the time, $\theta$ is the emission (scattering) angle and ${\cal P}(t,\theta)\propto\int_{E_{1}}^{E_{2}}dE\exp(-iEt/\hbar)F(E,\theta)$ with ${\cal P}(t\leq 0,\theta)=0$ has a finite time resolution, $\Delta t\simeq 2\pi\hbar/(E_{2}-E_{1})$. Suppose that a pronounced quasi-periodic component, with a period of $\delta E$, is present in $\sigma(E,\theta)$. From model independent Eqs. (1) and (2), this implies quasi-periodicity or recurrences, at least one recurrence, in ${\cal P}(t,\theta)$ with a period of $\simeq 2\pi\hbar/\delta E$. If, in addition, the quasi-periodic energy components correlate for given well resolved different exit channels, this implies that $<{\cal P}(t+\tau/2,\theta){\cal P}(t-\tau/2,\theta)^{\ast}>_{t}$ is significantly enhanced at approximately the same, for these different correlated channels, moments $\tau\simeq 2M\pi\hbar/\delta E$ with $M=1,2,...$, at least with $M=1$. Quasi-periodic structures in the excitation functions of heavy ion collisions often reveal presence of non-statistical effects. For example, the oscillating energy structures in the ${}^{24}Mg+^{24}Mg$ elastic-inelastic scattering [10], [11] were interpreted in terms of the highly excited coherently rotating hyper-deformed intermediate complex [12]. However, the identification of the non-statistical quasi-periodic structures from the analysis of the cross section energy autocorrelation functions in [12] may look questionable for the following reason. In the statistical regime of random phases of the strongly overlapping resonance levels with different total spin values, the energy autocorrelation functions, $C(\varepsilon)$, have a Lorentzian shape [13]. This result is obtained for very long energy interval on which excitation functions are measured. Yet, for the finite data range analysis of the concrete experimental data sets, one encounters deviations from the theoretical prediction, in particular, in the form of fluctuations around the Lorentzians [14]. Therefore, even though the oscillations in $C(\varepsilon)$’s have approximately the same period for the different exit channels [11], [12], which is inconsistent with the statistical interpretation, additional supportive argumentation for the quasi-periodic non-chaotic behavior is highly desirable. The interpretation [12] in terms of the relatively stable coherent rotation, for the slow phase relaxation, was based on the assumption of the highly excited ($\simeq 10-15$ MeV above Yrast line) intermediate complex with strongly overlapping resonances. Therefore, it should be insightful to also test this interpretation for transfer channels in the ${}^{24}Mg+^{24}Mg$ collision. Do the energy oscillating structures, with the same quasi-period as that for the ${}^{24}Mg(^{24}Mg,^{24}Mg)^{24}Mg$ elastic-inelastic scattering, present in the transfer channels? Affirmative answer to this question would provide additional supportive indication that both ${}^{24}Mg(^{24}Mg,^{24}Mg)^{24}Mg$ elastic-inelastic scattering and the transfer reaction channels originate from the decay of the same highly excited coherently rotating hyper-deformed intermediate complex. The reasonings presented above has motivated us to analyze the angle-averaged, $\Delta\theta_{cm}\simeq 90^{\circ}\pm 25^{\circ}$, excitation functions for the ${}^{24}Mg+^{24}Mg$ elastic-inelastic scattering and $\alpha$-transfer channels measured on the energy interval $E_{cm}=44.86-47.76$ MeV [15], which is considerably smaller than that ($E_{cm}=42-56$ MeV) on which the data were taken in [10], [11]. The analysis in [15] confidently confirmed the non- statistical behavior of the energy structures in these processes. We extend the analysis [15] concentrating on the quantitative interpretation of the pronounced, transparently visible in Figs. 2 and 3 of [15], strongly correlated for different channels, quasi-periodic energy structures. Figure 1: (Color online) The normalized cross section energy autocorrelations functions for the ${}^{24}Mg(^{24}Mg,^{20}Ne)^{28}Si$ reaction and ${}^{24}Mg(^{24}Mg,^{24}Mg)^{24}Mg$ elastic-inelastic scattering. The data (dots) are obtained from the analysis of the summed deviation functions in Fig. 3 of [15]. Solid lines are the fits for the relatively pure angular resolution (though under the neglect of the interference term between the near and far side collision amplitudes) obtained with $\hbar\omega=0.41$ MeV, $\beta=0.02$ MeV and $\Gamma=0.2$ MeV. The fit for the poor angular resolution (short-dashed lines) is obtained with $\hbar\omega=0.404$ MeV, $\beta=0.0205$ MeV and $\Gamma=0.17$ MeV. The Lorentzians with $\Gamma=$0.2 MeV (long-dashed lines) are obtained for the short spin off-diagonal phase memory, $\beta>>\Gamma,\hbar\omega$, corresponding to the limit of the random matrix theory [4]. In Fig. 1 we present the normalized cross section energy autocorrelations functions $C(\varepsilon)$’s constructed from the summed deviation functions in Fig. 3 of [15]. In the upper panel of our Fig. 1, $C(\varepsilon)$ is obtained from the three lowest transitions in the reaction ${}^{24}Mg(^{24}Mg,^{20}Ne)^{28}Si$. In the lower panel of our Fig. 1, $C(\varepsilon)$ is obtained from addition of the three lowest well resolved transitions in the reaction ${}^{24}Mg(^{24}Mg,^{20}Ne)^{28}Si$ and the three lowest well resolved transitions in the ${}^{24}Mg(^{24}Mg,^{24}Mg)^{24}Mg$ elastic-inelastic scattering (see Fig. 3 in [15]). The experimental $C(\varepsilon)$’s demonstrate oscillations with the period close to that in Fig. 1 of [12]. Therefore, we fit the data with the formula [12] $\displaystyle C(\varepsilon\geq 0)/C(\varepsilon=0)\simeq{\rm Re}\\{\exp[i\pi\varepsilon/(\hbar\omega-i\beta)][1-$ $\displaystyle\exp[i\pi(\varepsilon+i\Gamma)/(\hbar\omega-i\beta)]]^{-1}\\}$ $\displaystyle/{\rm Re}\\{1/[1-\exp[-\pi\Gamma/(\hbar\omega-i\beta)]]\\}.$ (3) Here, $\Gamma$ is total decay width of the intermediate complex, $\omega$ is a real part of the angular velocity of the coherent rotation and $\beta/\hbar$ has a physical meaning of its imaginary part with $\beta$ being the spin off- diagonal phase relaxation width [16]. One observes that $C(\varepsilon)$ (3) oscillates with period $\simeq 2\hbar\omega$. The oscillations are damped for a non-vanishing $\beta$-width. The fit (solid lines in Fig. 1) is obtained with $\hbar\omega=0.41$ MeV, $\beta=0.02$ MeV and $\Gamma=0.2$ MeV. In the limit of short phase memory, $\beta>>\Gamma,\hbar\omega$, $C(\varepsilon)$ has a Lorentzian shape [13], [4] (long-dashed lines in Fig. 1 with $\Gamma=$0.2 MeV). A few comments on the derivation and physical interpretation of Eq. (3), in relation to the analyzed data, are in order. 1) The starting point for the derivation of Eq. (3) is a diagonal approximation, with respect to the levels of the intermediate complex, for $S$-matrix elements which can be justified under the condition of the fast energy redistribution-relaxation within the states with fixed total spin and parity values [5]. This diagonal approximation of the Bethe unitary $S$-matrix is valid under the Bethe random sign assumption for partial width amplitudes with given fixed total spin and parity values in a regime of strongly overlapping resonances providing the partial decay widths are small as comparing with the average level spacing of the highly excited intermediate complex, see Section 56D in [17]. It is important for the present consideration that, unlike the shell-model approach (Section IV(A2) in [4]), the formulation [17] leads to the Bethe unitary $S$-matrix which is not restricted to the nucleon and $\gamma$-channels only but is also applicable for binary collisions of composite reaction partners including heavy ions. Indeed, the physical meaningfulness of the diagonal approximation similar to that in [17] is recognized for heavy ion induced collisions even for a multiple nucleon cascade emission provided the reaction proceeds through an equilibrated intermediate system [18]. We understand that the equilibration in [18] implies (i) energy relaxation and ergodicity within the states with fixed total angular momentum and parity values on each step of the evaporation cascade, and (ii) the conventional assumption of a vanishing of the total spin and parity off-diagonal correlations on each step of the evaporation cascade. Yet, while the entrance channel orbital momentum off-diagonal interference was neglected making directions along and opposite the incident beam undistinguishable, the strong correlation between different orbital momenta of the evaporated nucleons, on each step of the evaporation cascade, assumed in [18] (Eqs. (5) and (6) in [18]) clearly contradicts to the conventional postulate on maximal randomness of the partial width amplitudes (Eq. (2.2) in [19]). We mention in passing that the orbital momenta correlations [18], if exist in reality, may result in a complete uncertainty of the standard evaluation of the spin dependence of the nuclear level densities from angular anisotropy around 90∘ in the evaporation processes [20], [21], where such orbital momenta correlations were ignored in a literal consistency with the assumption on maximal randomness of the partial width amplitudes. As a next step, we extend the argumentation [5] to justify the relations (8.1) and (8.2) in [22] between the product of the entrance ($a$) and the exit ($b$) channel partial width amplitudes for different $J$-values: $\gamma_{\nu}^{Ja}\gamma_{\nu}^{Jb}=\sum_{\mu}\gamma_{\mu}^{Ia}\gamma_{\mu}^{Ib}Q_{\nu\mu}^{JI}+R_{inc},$ (4) where $Q_{\nu\mu}^{JI}=(1/\pi)D\beta\|J-I\|/[(E_{\nu}^{J}-E_{\mu}^{I}-(J-I)\hbar\omega)^{2}+\beta^{2}(J-I)^{2}]$ (5) with $E_{\nu}^{J}$ and $E_{\mu}^{I}$ being resonance energies for the states with total spin values $J$ and $I$, respectively. The relations (4) hold true independent of whether the energy redistribution process is taken explicitly into account or not [5]. We are interested in a regime of strongly overlapping resonances, $\Gamma/D\gg 1$, where both the total decay resonance width $\Gamma$ and the average level spacing $D$ are taken $J$-independent. Under the condition of relatively slow spin off-diagonal phase relaxation, $\beta\ll\Gamma$ (but still $\beta\geq D$), $R_{inc}$ in Eq. (4) can be neglected [22] for it produces insignificant contribution into the r.h.s. of Eq. (6). We obtain $\displaystyle S_{ab}^{J}(E)=W[\|J-{\bar{J}}(E)\|/d]^{1/2}\sum_{\mu}\gamma_{\mu}^{Ia}\gamma_{\mu}^{Ib}/[E-\hbar\omega(J-I)$ $\displaystyle-E_{\mu}^{I}+(i/2)\Gamma+i\beta\|J-I\|],$ (6) where we have omitted energy smooth phase shifts which originate from the potential mean field scattering and direct interaction taking place on the relatively short time intervals of formation and disintegration of the intermediate complex. In case of elastic scattering, $(a=b)$, the products $\gamma_{\nu}^{Ja}\gamma_{\nu}^{Jb}$ and $\gamma_{\mu}^{Ia}\gamma_{\mu}^{Ib}$ in Eq. (4) and Eq. (6) should be changed to $[(\gamma_{\nu}^{Ja})^{2}-\overline{(\gamma_{\nu}^{Ja})^{2}}^{\nu}]$ and $[(\gamma_{\mu}^{Ia})^{2}-\overline{(\gamma_{\mu}^{Ia})^{2}}^{\mu}]$, respectively [5]. In Eq. (6), $W[\|J-{\bar{J}}(E)\|/d]$ is the energy averaged decay probability. It was taken in a bell shaped form with the maximum at $J={\bar{J}}(E)$ and with the width $d$ of about an effective range of total spin values coherently excited in the collision process. For the data analyzed in this Letter this range is $\Delta J\simeq 34-38$. In the derivation of Eq. (3), we took ${\bar{J}}(E)=I+(E-{\bar{E}})/(\hbar\omega)$, where ${\bar{E}}$ is the average energy corresponding to $J({\bar{E}})=I\gg 1$. Under the neglect of the interference term between the near and far side collision amplitudes due to the angle-averaging, such a choice of ${\bar{J}}(E)$ results in $C(\varepsilon)$ (3) to be approximately independent of $d$. However, for the other choices, e.g. ${\bar{J}}(E)=I=const$, the additional damping factor, $\simeq W[-\|\varepsilon\|/(d\hbar\omega)]$, would appear in the r.h.s. of Eq. (3). 2) The data fitted in Fig. 1 were obtained from the angle-averaged excitation functions, $\Delta\theta_{cm}\simeq 90^{\circ}\pm 25^{\circ}$. Yet, the formula (3) was derived for the pure angle-resolution (though under the neglect of the interference term between the near and far side collision amplitudes as well as between direct and time-delayed processes [12]). Generalization for the angle-averaged excitation functions, $\Delta\theta_{cm}\simeq 90^{\circ}\pm 25^{\circ}$, yields the additional damping factor in the r.h.s. of Eq. (3). This factor is approximately given by $\sin^{2}[\varepsilon\Delta\theta/(2^{3/2}\hbar\omega)]/[\varepsilon\Delta\theta/(2^{3/2}\hbar\omega)]^{2}$ with $\Delta\theta=50^{\circ}=0.88$. The corresponding fit of the data on the lower panel of Fig. 1 is obtained with $\hbar\omega=0.404$ MeV, $\beta=0.0205$ MeV and $\Gamma=0.17$ MeV. Clearly, the additional damping factor, due to the poor angular resolution, does not affect quasi-period of the oscillations in $C(\varepsilon)$. 3) The time dependent intensity, $\|{\cal P}(t,\theta)\|^{2}$, closely relates to the return probability for a finite time resolution, while modulus square of Fourier component of ${\cal P}(t,\theta)$, i.e. the cross section (1), is analogous to the low-resolution version the spectrum [23] providing the Bethe $S$-matrix diagonal approximation is applicable (Eq. (260) and Section 56D in [17]). In our case the finite time resolution is $\hbar/\Gamma$, and the spectrum is not resolved due to the strong overlap of the resonance levels, $\Gamma/D\gg 1$. Clearly, $C(\varepsilon)$ has a physical meaning of the autocorrelation function of the low-resolution version of the spectrum [23]. In a view of the relatively short energy interval of the measurements [15], $\Delta E_{cm}=2.9$ MeV, as comparing with $\Gamma$ the standard evaluation of errors due to the finite data range [14] would result in the statistical insignificance of the quasi-periodic structures in Fig. 1. However, the method of evaluation of the statistical uncertainties [14] is not applicable in the presence of the $S$-matrix spin off-diagonal correlations for $\beta\ll\Gamma$. The essence of the matter can be explained as follows. Consider a given realization of $\sigma^{I}(E)=\|S_{ab}^{I}(E)\|^{2}$, where $S_{ab}^{I}(E)$ is given by Eq. (6), i.e. like in [14]. Then, in a regime of strongly overlapping resonances, the standard evaluation of errors due to the finite data range [14] would be certainly applicable and quasi-periodic oscillations in the corresponding $C(\varepsilon)$, if occur at all, would not be of the sufficient statistical significance for the short energy interval, $\Delta E_{cm}=2.9$ MeV. Consider next $\sigma(E)=\sum_{n=-q}^{q}\sigma^{I}(E+2n\hbar\omega)$, where $n$ and $q$ are natural numbers and $n$ in the summation changes with the step of unity. For $q\gg 1$, $\sigma(E)$ is close to periodic function with the period of $2\hbar\omega$ independent of actual realization of $\sigma^{I}(E)$ the latter being generated like in [14]. Clearly, the correspondent $C(\varepsilon\leq r\hbar\omega)$, where $1\ll r\ll q$, will show periodic behavior even if it constructed from $\sigma(E)$ restricted to the relatively short energy interval $\Delta E\simeq(6-7)\hbar\omega$. For $q=1$ (the three terms in the sum), similar to our case, $\sigma(E)$ will not be exactly periodic but will still have the strong quasi-periodic component which will show up in the statistically significant way in the corresponding $C(\varepsilon)$ extracted even on the short energy interval $\Delta E\simeq(6-7)\hbar\omega$. The numerical simulations would represent the straightforward way to demonstrate the statistically significant registration of the quasi-periodicity for the short energy intervals. In a view of a simplicity of the simulations these can be performed during couple of days by a science or nowadays even, e.g., economy or linguistic undergraduate student. The results could be reported, e.g., during the workshop “Information and statistics in nuclear experiment and theory ISNET-3” at the ECT∗ on November 16-20, 2015. In addition to the ${}^{24}Mg+^{24}Mg$ collisions the simulations may also include the ${}^{12}C+^{24}Mg$ elastic and inelastic scattering which demonstrate strong quasi-periodic structures (Fig. 1 in [7]). This is not a joke for the issue needs clarification because when one of us (S.K.) 15 years ago discussed the experimentally observable oscillations in Fig. 1 of [7] with A. Richter (previous Director of the ECT∗) he was told that these oscillations are due to the finite data range effects and the data are consistent with the random matrix description in terms of Ericson fluctuations [4]. Then one of the questions to be answered by the student is: How many statistically independent generations, of the type of [14], of the excitation functions using the standard algorithms of the random matrix theory (the standard statistical model) [4] must be performed in order to reproduce the oscillations in Fig. 1 of [7]? Obviously the student must have an optimistic vision for the future and be brave enough to improve the world we live in. This is because the opposition is too strong, crossed all possible red lines and, therefore, has nothing to lose [5]. For we are speaking about those who demonstrated the inquisitory attitude and are killers of the new ideas which does not leave much choice at this stage. We do not mean here and put aside the Australian way proved to bring this kind of attitude up to the national operational policy which, bedazzled by the ideas of universality of the random matrix theory, is spiritually oriented on the future with no distinguishability between living human beings and corpses [5]. The work under the extended version of this Letter, where the derivation of the cross section energy autocorrelation (3) and some other relevant issues will be addressed, is under way. This will include a discussion of adequacy of the conventional interpretation of the non-statistical structures in terms of isolated resonances. For example, what is angular dependence of $C(\varepsilon)$, in particular, for the elastic ${}^{24}Mg(^{24}Mg,^{24}Mg)^{24}Mg$ scattering in the presence of dominant potential scattering contribution ($\theta\leq 70^{\circ}$), predicted within our treatment in terms of the coherent rotation of the intermediate complex with strongly overlapping resonances as compared with that resulted from the interpretation in terms of isolated resonances as identified in [24]? Stability of the coherent rotation may be appreciated by noticing that the spreading of the angular orientation during one revolution due to the finite $\beta$-width is $2\pi\beta/\hbar\omega\simeq 15^{\circ}$. Therefore, it takes about 10 complete revolutions, i.e. $\simeq 10^{-19}$ sec., for the complex to loose the coherent nature of its rotation. Energy relaxation time scale for excited nuclei is $\simeq\hbar/\Gamma_{spr}\simeq 10^{-22}$ sec., where $\Gamma_{spr}\simeq 5$ MeV is the spreading width [4]. We observe that the coherent rotation persists for about three orders of magnitude longer than it takes to complete a process of energy equilibration (vibrational relaxation by the molecular physics terminology). The strong channel correlation between individual well resolved channels in both the ${}^{24}Mg(^{24}Mg,^{24}Mg)^{24}Mg$ elastic-inelastic scattering and ${}^{24}Mg$ $(^{24}Mg,^{20}Ne)^{28}Si$ transfer reaction, as well as between the channels for these two processes, is transparently visible in Fig. 2 of [15]. Therefore, it is obvious that strongly correlated regular oscillations with the approximately channel-independent quasi-period, $\simeq 0.81$ MeV, are present in all the six channels. This rules out statistical origin of these strongly correlated quasi-periodic structures and, therefore, their interpretation in terms of Ericson fluctuations [13]. This is because Ericson fluctuations, which is a particular case of the Bethe statistical fluctuations (to be reported in the extended version of this Letter), produce uncorrelated, for different channels, irregular energy structures. As a result, the deviations from the Lorentzian must be uncorrelated for different channels instead of demonstrating regular oscillations with the approximately channel- independent quasi-period for all the six channels. The fact that the present analysis of the data on the energy interval $E_{cm}=44.86-47.76$ MeV revealed a value of the quasi-period, $\simeq 0.81$ MeV, very close to that obtained for the much longer energy interval $E_{cm}=42-56$ MeV [12], supports our interpretation. A possibility of further test of our interpretation, e.g. for the ${}^{24}Mg(^{24}Mg,^{24}Mg)^{24}Mg$ elastic scattering, is suggested by the expected dominance of direct processes (potential scattering) for $\theta\leq 70^{\circ}$ (see Figs. 6 and 7 in [24]). Then the energy variations in the excitation functions originate mainly from the interference between the collision amplitudes corresponding to the direct and time delayed processes. As a result, the characteristic quasi-periods of leading harmonics in $\sigma(E,\theta)$ become strongly $\theta$-dependent [7], [8], [9], which can be tested in experiments with pure angular resolution. We mean the numerical experiments against the previously measured relevant available data since the experimental excavation into the matter as well as the understanding of the underlying phenomena are no longer among interests of the nuclear physics community. The numerical experiments are useful to illustrate simple algorithms of transforming/processing the complex quantum information. The initial wave packet is given by a coherent superposition of partial waves with different orbital momenta describing the colliding objects in their ground states. In a process of the collision the initial quantum superpositions are transformed into the coherent superpositions of a large number of strongly overlapping resonance configurations of the intermediate complex with different total spin values. The spin off-diagonal correlations produce the coherent rotation of the intermediate complex with a well defined angular velocity. Yet, the intermediate complex represents a quantum superposition of the two objects simultaneously rotating in opposite directions. For these coherently rotating in opposite directions alternatives, each of them individually is itself result of the spin off-diagonal quantum interference, produce interference fringes [9]. For small $\beta$-width, the rotating wave packets slowly spread and interference fringes, related to the spin off-diagonal correlations, disappear [25]. In this way the complex collision experiments provide a physical visualization of executing the quantum computing algorithms for a simple description of the quantum-macroscopic transition, where the convergence to the macroscopic-like dynamics (rotation) on the final stage of the “calculations” is due to partial dynamical decoherence but still images essentially from the quantum interference. Each step of the “quantum computing”, presumably run by nature, can be easily mapped on the fundamental double slit experiment (since the many slit experiment is a combination of the double slit ones). The double slit experiment “has been designed to contain all of the mystery of quantum mechanics …” [26] and, Feynman continues, “Any other situation in quantum mechanics, it turns out, can always be explained by saying, ’You remember the case of the experiment with the two holes? It’s the same thing’.” [26]. The above simple picture of transformation of the complex quantum information is noticeably supported by the state of art numerical simulations for, e.g., H+D${}_{2}\to$HD+D molecular reaction (Fig. 1 in [27]) even though the calculations [28] rules out a picture of the isolated resonances of the intermediate complex (Fig. 1 in [28]). We mention in passing that the described above algorithms are incompatible with basic ideas of the random matrix theory as applied to classically chaotic systems [4]. Therefore it is inappropriate to rely on mentality of indistinguishability and selective blindness of the random matrix theory [5], which unfortunately found its way beyond the mere academic and education activities, as a point of reference in discussing the above mentioned system specific algorithms. Angular velocity of coherent rotation, $\omega$, can not be defined within the states with fixed $J$-value. This transparently follows from the conjecture [5] which relates $Q_{\nu\mu}^{JI}$ (5) to the spin off-diagonal correlations between squares of the individual resonance wave functions of the intermediate complex. Indeed, it is obvious that distribution of the spacial density, including the radial extension, is different for the state with fixed $J$-value from that for the coherently rotating intermediate complex. Therefore, energy of the coherent rotation as well as the associated moment of inertia and, thus, the characteristic length scale of the coherently rotating intermediate complex are also undefinable within the states with fixed $J$-value. Let us assume that the coherent rotation may be considered as the macroscopic motion. Then, an assumption of the spherical intermediate complex having rigid body moment of inertia is ruled out for, in this case, $\hbar\omega\simeq 4$ MeV, resulting in the rotational energy ($\simeq 70$ Mev) which exceeds the total excitation energy ($\simeq 60-63$ MeV). Instead, the small value of $\hbar\omega\simeq 0.4$ MeV, corresponding to the period of the rotation $\simeq 10^{-20}$ sec, indicates an anomalously strong deformation of the coherently rotating complex with $J\simeq 34-38$. We calculate moment of inertia of this coherently rotating complex, ${\cal J}_{coh}$, and find that it corresponds to a chain-state of the length $\simeq 30$ fm. This is close to the length $\simeq 24$ fm of the chain of four touching carbon nuclei, calculated with $r_{0}=1.3$ fm. We evaluate deformation energy of the chain as a sum of Coulomb energy of the two touched ${}^{24}Mg$ nuclei and double Coulomb energy of the two touched ${}^{12}C$ nuclei. For $r_{0}$=1.3 fm, this gives $\simeq$38 MeV for the deformation energy. The energy of the coherent rotation, for $J\simeq 34-38$, is $\simeq 7$ MeV. Therefore, the energy of the intrinsic excitation (heat) is estimated $\geq 15$ MeV, i.e. our “carbon sausages” are really hot. Let us evaluate average level spacing ($D$) of the intermediate complex with $J\simeq 34-38$. We use the standard statistical model formula (the Weisskopf evaluation [29], [30], [31]) for the total decay width with fixed $(J,\pi)$-values: $\Gamma=(1/2\pi)Dn{\bar{T}}$. Here, $n$ is a number of the open channels and ${\bar{T}}$ is the averaged over the channels transmission coefficient. The total number of channels, specified by the channel spins, orbital momenta and internal quantum numbers of the fragments, for the ${}^{24}Mg+^{24}Mg$ elastic and inelastic scattering, up to excitation of the (6+,4+) states of the ${}^{24}Mg$ collision fragments, is $57$. We estimate number of the $\alpha$-transfer and two-$\alpha$-transfer channels and the proton and $\alpha$-emission channels to be $\simeq 200$, where we have taken into account all possible the channel spins and orbital momenta for given internal quantum numbers of the fragments. Then our estimate is $n\simeq 250$. Since the heavy ion collisions under consideration are characterized by a closeness to the potential barrier (centrifugal plus Coulomb barriers) or even are sub-barrier, and the proton and $\alpha$-particle evaporation is mostly sub-barrier, we estimate ${\bar{T}}\simeq 0.5$. Then, for $\Gamma=0.2$ MeV, we obtain $D\simeq$10 keV. This indicates the relatively weak coupling to the continuum, $\Gamma/(nD)\simeq 0.08$, in the regime of strongly overlapping resonances, $\Gamma/D\simeq 20$. Therefore the Bethe diagonal approximation (Section 56D in [17]) obtained from the Bethe unitary $S$-matrix for binary reactions involving complex collision partners is applicable (to be presented in the extended version of this Letter). Note that the standard statistical model evaluation [17], [31], [32], for the intrinsic excitation of 15 MeV of the intermediate nucleus, yields $D\simeq$0.01-0.1 keV. This is 2-3 orders of magnitude smaller than the average level spacing evaluated above. We interpret this reduction of $D$ as an indication of a sizable admixture of clustering configurations in the wave function of the highly excited strongly deformed intermediate complex with large spin values $J\simeq 34-38$. These clustering configurations should not be considered as vibrational modes of the linear chain of four carbon nuclei each being in its ground state. Indeed it seems hardly probable that such a configuration would be stable with respect to decay. Therefore, it is perhaps more realistic to consider the hyper-deformed highly excited states as the approximately linear chain of four relatively heavy clusters in combination with some “valent” nucleons and $\alpha$-particles. Up to now the idea of rotational coherence of ergodic, with respect to the energy and phase relaxation within the states with fixed $J$-values, nuclear molecules in continuum has been employed to extract spectroscopic information ($\hbar\omega,{\cal J}_{coh}$) from dynamics for a few colliding systems, ${\sl e.g.}$, ${}^{58}Ni+^{58}Ni$ [33], ${}^{58}Ni+^{62}Ni$ [33], [34], ${}^{46}Ti+^{58}Ni$ [34], ${}^{12}C+^{24}Mg$ [6], [7], [8], [9], ${}^{24}Mg+^{24}Mg$ and ${}^{28}Si+^{28}Si$ [12], ${}^{24}Mg+^{28}Si$ [35]. Therefore, it may be of interest to apply this method for analysis of a large number of available data sets for many heavy ion colliding systems. On the other hand, the question is whether the stable coherent rotation survives violation of rotational symmetry and other external perturbations of the Hamiltonian? The conjecture [5] offers a possible specification of this question in terms of the correlation properties of eigenstates of the intermediate complex. Experimental manifestation of rotational coherent motion in complex molecules was initially met with scepticism “because of the general belief that Coriolis interactions, anharmonicity and other interactions would destroy the coherence.” (see subsection “Changing a Dogma: Development of RCS” in [2]). The present discussion leads us to ask: Is rotational coherence of complex molecules necessarily destroyed in the conventionally statistical ergodic limit of structureless (non-selective) continuum under the conditions of complete intramolecular energy redistribution (thermalization) and vibrational dephasing in the regime of strong ro-vibrational coupling? Though challenging experimentally, the subject is of interest to search for fingerprints of transformation of the quantum coherent wave packet dynamics into macroscopic motion. More specifically, in the quantum regime, the spreading of the wave packets is determined by $\beta$-width, which has essentially quantum origin of the spin off-diagonal correlations [5]. Then what is a classical analog of $\beta$-width, for classically chaotic systems, when macroscopic non-linear dynamics takes over from the coherent rotation of the spreading quantum wave packets? Addressing the questions above should include a comparison of the nature of the dephasing analyzed in [1] with that discussed in this Letter (see also [5] and references therein). ## Acknowledgments We thank Sergei Maydanyuk for useful discussions. Writing this Letter started when one of us (S.K.) was supported by Chinese Academy of Sciences Visiting Professorship for Senior International Scientists (2012-2013). ## References * [1] A. Zewail, Femtochemistry, Vols. I and II, World Scientific, Singapore, 1994. * [2] A. Zewail, www.nobelprize.org/nobel-prizes/chemistry/laureates/1999/zewail-lecture.pdf, and references therein. * [3] G. Casati, B.V. Chirikov, Phys. Rev. Lett. 75 (1995) 350. * [4] G.E. Mitchell, A. Richter, H.A. Weidenmüller, Rev. Mod. Phys. 82 (2010) 2845, and references therein. * [5] S. Kun, Y. Li, H. Zhao, M.R. Huang, e-print arXiv: quant-ph/1307.4490. * [6] S.Yu. Kun, A.V. Vagov, O.K. Vorov, Phys. Rev. C 59 (1999) R585. * [7] S.Yu. Kun, A.V. Vagov, W. Greiner, Phys. 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Strutinsky, in Proceeding of Conference on Nuclear Reactions at Low and Medium Energies, Moscow 1957, p. 522, Academy of Sciences USSR Publishing, 1958. * [21] T. Ericson, V.M. Strutinsky, Nucl. Phys. 8 (1958) 284; 9 (1958/1959) 689. * [22] S.Yu. Kun, Z. Phys. A 357 (1997) 255. * [23] E.J. Heller, Phys. Rev. Lett. 53 (1984) 1515. * [24] A.H. Wuosmaa, R.W. Zurmühle, P. Kutt, S.F. Pate, S. Saini, M.L. Halbert, D.C. Hensley, Phys. Rev. C 41 (1990) 2666. * [25] L. Benet, L.T. Chadderton, S.Yu. Kun, Wang Qi, Phys. Rev. A 75 (2007) 062110; e-print arXiv: quant-ph/0610091. * [26] R. Feynman, The Character of Physical Law, based on the Messenger Lectures given by R. Feynman at Cornell University in 1964, Lecture 6: Probability and Uncertainty - the Quantum Mechanical view of Nature, first published by BBC, 1965. * [27] P.D.D. Monks, J.N.L. Connor, S.C. Althorpe, J. Phys. Chem. A 110 (2006) 741. * [28] P.D.D. Monks, J.N.L. Connor, S.C. Althorpe, J. Phys. Chem. A 111 (2007) 10302. * [29] L.D. Landau, Phys. Zeits. Sow. 11 (1937) 556. * [30] J.M. Blatt, V.F. Weisskopf, Theoretical Nuclear Physics, Wiley, 1952. * [31] L. Landau, Ya. Smorodinsky, Lectures on Theory of Atomic Nucleus, Moscow, Technical-Theoretical Publishing, 1955. * [32] A. Bohr, B. Mottelson, Nuclear Structure, Vol. 1, Benjamin, New York, 1969. * [33] L. Vannucci, U. Abbondanno, M. Bettiolo, M. Bruno, N. Cindro, M. D’Agostino, P.M. Milazzo, R.A. Ricci, T. Ritz, W. Scheid, G. Vannini, Z. Phys. A 355 (1996) 41. * [34] S.Yu. Kun, U. Abbondanno, M. Bruno, N. Cindro, M. D’Agostino, P.M. Milazzo, R.A. Ricci, T. Ritz, B.A. Robson, W. Scheid, A.V. Vagov, G. Vannini, L. Vannucci, Z. Phys. A 359 (1997) 145. * [35] L. Benet, L.T. Chadderton, S.Yu. Kun, O.K. Vorov, Wang Qi, Phys. At. Nucl. 71 (2008) 819, e-print arXiv: 0705.0502.	arxiv-papers	2013-09-25T02:49:05	2024-09-04T02:49:51.478876	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. H. Zhao, S. Kun, O. Merlo, M. R. Huang, Y. Li, J. S. Wang", "submitter": "Meirong Huang", "url": "https://arxiv.org/abs/1309.6386" }
1309.6494	# BENCHMARKING HEADTAIL WITH ELECTRON CLOUD INSTABILITIES OBSERVED IN THE LHC H. Bartosik W. Höfle G. Iadarola Y. Papaphilippou G. Rumolo CERN Geneva Switzerland ###### Abstract After a successful scrubbing run in the beginning of 2011, the LHC can be presently operated with high intensity proton beams with 50 ns bunch spacing. However, strong electron cloud effects were observed during machine studies with the nominal beam with 25 ns bunch spacing. In particular, fast transverse instabilities were observed when attempting to inject trains of 48 bunches into the LHC for the first time. An analysis of the turn-by-turn bunch-by- bunch data from the transverse damper pick-ups during these injection studies is presented, showing a clear signature of the electron cloud effect. These experimental observations are reproduced using numerical simulations: the electron distribution before each bunch passage is generated with PyECLOUD and used as input for a set of HEADTAIL simulations. This paper describes the simulation method as well as the sensitivity of the results to the initial conditions for the electron build-up. The potential of this type of simulations and their clear limitations on the other hand are discussed. ## 1 INTRODUCTION At the early phase of the 2011 LHC run, seven days were devoted to scrubbing using the 50 ns bunch spacing beam with gradually increasing number of bunches circulating in the machine. This allowed to sufficiently condition the inner surface of the LHC beam screens and vacuum chambers for running with the same beam for physics production in routine operation throughout 2011. First attempts to inject a beam with the nominal 25 ns bunch spacing were performed at the end of June 2011 with bunch trains of 24 bunches. Pressure rise and increased heat load in the arcs were observed. The studies continued with injections of 48 bunches in August, where the beams became transversely unstable after about 1000 turns with the transverse damper switched on and after about 500 turns without transverse damper, for two injections respectively. As will be discussed in more detail below, an analysis of the turn-by-turn bunch-by-bunch data from the transverse damper pick-ups [1] points to the observation of coherent electron cloud instabilities. During these injection tests the chromaticity was set to about $Q^{\prime}\approx 2$ in both planes, as usually used on the LHC flat bottom in routine operation. Only after increasing the chromaticity to about $Q^{\prime}\approx 15$ in both planes during further studies in October, it was possible to perform nominal injections of 288 bunches with 25 ns bunch spacing from the SPS. The high chromaticity suppressed the fast instabilities observed before. However, the beam suffered from slow losses and transverse emittance blow-up along the bunch train and, as before, pressure rise and increased heat load were measured in the cold arcs. The conditioning of the LHC beam screens due to the beam based electron bombardment was demonstrated by estimating the secondary electron yield (SEY) from a comparison of the measured heat load data in the arcs with PyECLOUD simulations [2, 3]. Figure 1 shows the evolution of the obtained maximum SEY ($\delta_{\text{max}}$) as a function of beam time in the LHC together with the total intensity for both beams [4]. Since the heat load data can be read only per half cell in the arcs, the SEY can be estimated only for both beams at the same time (grey markers) unless there is only one beam present in the machine (red markers for beam 2). A clear conditioning effect from $\delta_{\text{max}}\\!=\\!2.1$ at the end of June when running with 50 ns bunch spacing to $\delta_{\text{max}}\\!=\\!1.52$ after approximately 50 h beam time with 25 ns bunch spacing is observed. The analysis and simulation studies presented in this paper will concentrate on the aforementioned injection tests with 48 bunches in August, where $\delta_{max}$ was estimated to be still around 2.1 (cf. Fig. 1). Figure 1: Evolution of $\delta_{max}$ in the LHC arcs as obtained by reproducing the measured heat load on the LHC beam screens using PyECLOUD [4]. ## 2 Analysis of turn-by-turn data Figure 2: Transverse oscillations measured with the pick-ups of the transverse feedback for 73 turns close before the beam dump, for the first case with transverse damper (left) and the second case without transverse damper (right) in the horizontal (top) and vertical plane (bottom). As mentioned before, a first attempt to inject bunch trains of 48 bunches spaced by 25 ns into the LHC with the an intensity of about $1.0\\!\times\\!10^{11}$ p/b was made for beam 2 during a machine development session on August 26, 2011. Shorter bunch trains of 12 and 24 bunches with the same bunch spacing had been successfully injected earlier in the session. It was planned to establish the injection of longer bunch trains and complete the transverse damper set-up for the nominal bunch spacing of 25 ns. A first injection of 48 bunches with the transverse damper switched on was dumped after 1000 turns due to a beam excursion interlock (referred to as dump 1). Another injection of 48 bunches without transverse damper was aborted already after around 500 turns due to a beam loss interlock (referred to as dump 2). In both cases, chromaticity was set to the values usually used during operation ($Q^{\prime}\approx 2$). Data of the damper pick-ups for about 73 turns are stored for both beams in the post mortem system at a beam abort. Each data set (channel) represents one of the allowed 3564 bunch positions per ring. Figure 2 shows the turn-by-turn oscillation of each of the 48 bunches around the closed orbit in both planes for the 2 injections just before the respective beam dump. The oscillation amplitude is very small for the first 25 bunches, especially when the damper is switched on. In this case the last bunches of the bunch train reach peak values of around 1 mm in the horizontal plane and up to 6 mm in the vertical plane, thus the instability is mainly observed in the vertical plane. Without damper, the last bunches of the train exhibit oscillations of up to 3 mm amplitude in both planes. These observations are compatible with a coupled bunch instability (slightly stronger in the horizontal plane than in the vertical plane), which can be suppressed by the damper, and a high frequency instability with a broad spectrum mainly in the vertical plane. The frequency spectrum without damper is dominated by coupled bunch modes up to 1 MHz in the horizontal plane and up to 2 MHz in the vertical plane. With damper on, instabilities in both planes are damped up to a frequency of about 15 MHz [1]. This can be explained by a “single bunch” instability in the vertical plane and a coupled bunch instability (mainly) in the horizontal plane, as has been observed in the past already in the SPS [5]. The beam observations during the injection tests in the LHC are used to benchmark numerical simulations as described in the following sections. ## 3 Simulation procedure Figure 3: Electron density per unit length along the bunch train as obtained from a PyECLOUD simulation. A snapshot of the electron distribution before each bunch entrance as indicated by the red dots is used as input for individual HEADTAIL simulations. The above described experimental observations were studied by combining PyECLOUD [2] and HEADTAIL [6] simulations. As first step the electron cloud build-up in the LHC dipoles is simulated with PyECLOUD for an r.m.s. bunch length of 12 cm and equal bunch intensities of $N=1.0\times 10^{11}$ p/b along the train111It was not possible to retrieve the actual bunch-by-bunch intensity variation from the data logging during the injection tests due to the short time of circulating beam before the dump.. Figure 3 shows the obtained electron density per unit length along the bunch train for a secondary electron yield of $\delta_{\text{max}}\\!=\\!2.1$. The simulation was initialized with a uniform distribution using the same number of primary electrons as in the simulations for determining the evolution of the SEY discussed above [3]. The importance of the number of primary electrons and their spatial distribution for the onset of the e-cloud instability will be discussed in more detail in the next section. The red markers indicate the time steps in the simulation just before the bunch passages. The electron distributions sampled at these points are used as input for a set of 48 HEADTAIL simulations, one for each bunch. Figure 4: Electron distribution before the passage of the last bunch of the train as obtained from the PyECLOUD simulation. Note that the cross section of the LHC beam screen is approximated as ellipse. Figure 4 shows for example the electron distribution before the entrance of the last bunch of the train. Note the accumulation of electrons in two stripes, as typically obtained for the build-up with 25 ns bunch spacing in a geometry as the LHC beam screen when the electrons move in a strong dipolar magnetic field. In HEADTAIL the electron cloud is represented by thin slices lumped at several accelerator sections and the electron motion is frozen in the horizontal plane in order to account for the effect of the dipole magnetic field. At each electron cloud section the electron proton interaction is computed consecutively for longitudinal bunch slices. After a complete bunch passage the electron cloud is reset to the initial distribution for the next interaction section. Figure 5 shows the evolution of the transverse emittance for a few selected bunches in the middle of the bunch train obtained with HEADTAIL for the simulation of 500 turns at the LHC injection plateau using the initial electron distribution as described above. Figure 5: Evolution of the horizontal (top) and vertical (bottom) emittance along 500 turns for a few selected bunches in the center of the bunch train. No instability or coherent excitation of the horizontal bunch motion is observed, as was expected since the electron cloud is located in dipole regions. On the other hand the electron cloud drives a single bunch instability in the vertical plane. This instability can be observed in the form of an exponential emittance growth, as in the case studied here for all bunches after bunch number 25. Figure 6 shows the bunch-by-bunch oscillations in the vertical plane during the instability (from turn 50 to turn 120) as obtained from HEADTAIL. Figure 6: Bunch-by-bunch oscillations in the vertical plane as obtained with HEADTAIL. For better comparison, only 70 turns during the instability (from turn 50 to turn 120) are shown. The simulation is in good agreement with the experimental observations during the first injection of 48 bunches in August with the transverse damper on, where the second half of the bunch train is unstable mainly in the vertical plane (cf. Fig. 2), as the damper is suppressing horizontal coupled bunch instabilities. Horizontal instabilities are not observed in the simulation, since the coupling between bunches is not taken into account but each bunch is treated by an independent HEADTAIL simulation. To include this coupling would require a combined simulation of PyECLOUD and HEADTAIL, which is envisaged to be implemented at a later stage. This will allow to reproduce better the instability observed during the injection without transverse damper. ## 4 Sensitivity studies It was shown in the previous section that the instability observed at LHC injection can be reproduced with the presented simulation procedure. In the following the sensitivity of the simulation result to the number and distribution of the primary electrons will be studied. Furthermore, it is observed that the electron cloud is formed mainly in two symmetric vertical “stripes” around the beam center (cf. Fig. 4) and only a small number of electrons is concentrated in the central part of the chamber. Thus it is interesting to assess which part of the electron cloud distribution is mainly responsible for driving the beam unstable. ### 4.1 Electron density Figure 7 shows an example for a bunch in the middle of the train, which encounters an electron density which is just above the instability threshold. The top graph shows a histogram of the horizontal electron distribution before the bunch arrival as obtained from the PyECLOUD simulation and used as input for the HEADTAIL simulation. Note again Figure 7: From top to bottom: Histogram of the horizontal electron distribution; transverse electron distribution divided into colored regions; evolution of the vertical emittance for the interaction with the electrons of the respective colored area; evolution of the vertical emittance for different horizontal cuts of the electron distribution. that most of the electrons are concentrated in the two stripes far away from the beam (the $3\,\sigma$ beam envelope covers roughly the two central bins). The graph below shows for the same case the actual positions of the macro- electrons in the transverse plane. The macro-particle distribution is divided here into five regions as indicated by the color code (corresponding to the bins in the histogram). In the following, the contribution from these five regions to the vertical instability is studied. First, each of these regions is considered independently and used as input for individual HEADTAIL simulations. The evolution of the vertical emittance for these five cases is shown for 500 turns in the third graph of Fig. 7. In the case studied here, only the central part of the electron distribution is able to drive the beam unstable. The bottom graph shows the vertical emittance evolution in case all electrons enclosed by the respective colored areas have been removed, i.e. cutting the electron distribution at the inner borders of the colored regions. The instability appears only when the electrons in the central part of the distribution are taken into account. It follows that for the typical electron cloud distributions encountered in the simulations of the LHC dipole regions, it is mainly the central electron density which determines the onset of the instability. Therefore the instability threshold can be inferred roughly from the central electron density. Only in cases where the central electron density is _very_ small compared to the density in the stripes this approximation may not hold. In the following, the central density of the electron distribution is thus used to determine the instability threshold. It should be noted that the instability thresholds found in this manner are consistent with thresholds previously found in HEADTAIL simulations assuming a uniform electron distribution before the bunch passage, which is around $1\times 10^{12}\,e^{-}/$m3 [7]. ### 4.2 Dependence on number of primary electrons One of the main uncertainties of the electron cloud build-up simulations for the LHC injection energy is the number of primary (seed) electrons. Since the synchrotron radiation at 450 GeV is inefficient to generate photo electrons, it is assumed that the primary electrons are created by rest gas ionization. Therefore the number of primary electrons depends on the pressure. However the static pressure without beam is much lower compared to the pressure levels measured after beam injection. Figure 8 shows the central density along the bunch train of 48 bunches for $\delta_{\text{max}}=2.1$ and two different cases: for a small number of seed electrons equivalent to the static pressure in the LHC cold sections (top) and for a large number of seed electrons (bottom). The dashed red line shows the central electron density at the instability threshold. As expected, it takes more bunch passages to build up the electron cloud beyond the instability threshold for a smaller number of primary electrons. The onset of the instability along the bunch train depends thus on the pressure level assumed in the build-up simulation. For completeness it should be emphasized that the saturation level of the central density is very similar in both cases (as it depends mostly on $\delta_{\text{max}}$). Figure 8: Comparison of the central electron density along the bunch train for a number of seeding electrons comparable to the static pressure in the LHC dipole sections (top) and for a larger number of seeding electrons (bottom). ### 4.3 Dependence on primary electron distribution Figure 9: Example for an electron cloud build-up simulation assuming a uniform transverse distribution of the primary electrons all across the vacuum chamber: central density along the bunch train (left), horizontal distribution for bunch 33 (middle) and for the last bunch (right). Figure 10: Example for an electron cloud build-up simulation assuming a Gaussian distribution of the primary electrons on top of a 10% uniform background all across the vacuum chamber: central density along the bunch train (left), horizontal distribution for bunch 33 (middle) and for the last bunch (right). In addition to the number of primary electrons, also their distribution is not well known in the case of the LHC at injection energy. In order to reproduce the experimentally observed vertical stripes of the electron distribution in saturation, primary electrons have to be assumed across the entire cross section of the vacuum chamber (since the electrons are bound by the vertical magnetic field lines). Therefore a uniform distribution of the primary electrons can be optionally used in PyECLOUD simulations. Figure 9 shows an example for a uniform distribution of the seed electrons together with the histogram of the horizontal electron distribution for two selected bunches (a bunch in the second half of the train where the saturation level is reached, and the last bunch). The central density is very similar in the two cases, but the stripes build-up further outside towards the end of the bunch train. Considering that the generation of the primary electrons is caused by rest gas ionization, it can be argued that the primary electrons should follow a Gaussian distribution similar to the proton beam. In this case the formation of the stripes can be achieved by adding about 10% of the total number of electrons in the form of a uniform background all across the vacuum chamber. Figure 10 shows the evolution of the central electron density along the bunch train assuming such a distribution of seed electrons together with the horizontal electron distribution for selected bunches as in Fig. 9. Here a smaller number of seed electrons was chosen in order to obtain the instability onset roughly at the same bunch along the bunch train as for the case of the uniform distribution of seed electrons. However, the evolution of the electron distribution along the bunch train is quite different for the two cases. In comparison to the case of the uniform distribution, the central density reaches much higher peak values in the saturated part of the train and it takes longer to develop the stripes in the outer part of the vacuum chamber assuming the Gaussian distribution. This can be understood intuitively, since the center of the chamber is seeded with a larger number of electrons while the outer regions of the vacuum chamber are seeded with a comparably smaller number of electrons. ## 5 Compatible parameter space In fact the number and distribution of the primary electrons are among the biggest uncertainties in the present understanding of the electron cloud build-up in the LHC dipoles at injection energy. On the other hand, the estimation of $\delta_{\text{max}}$ from the reproduction of the measured heat load in the LHC beam screens with PyECLOUD [4] is not so sensitive to the assumptions on the seed electrons if the bunch train is long and the electron saturation level is reached for many bunches. The estimation of $\delta_{\text{max}}$ for the case studied here was not done for the same time as the observations of the injections of the 48 bunches and therefore might be slightly smaller than $\delta_{\text{max}}=2.1$. In the following, the available parameter space is thus scanned for compatible solutions reproducing the observed instability at LHC injection in August 2011. Figure 11 shows compatible solutions, i.e. cases where the onset of the instability is close to bunch 25 (between bunch 23 and bunch 27), for a range of possible values for $\delta_{\text{max}}$ and for the number of primary electrons per bunch passage. The two colored regions correspond to the cases of a purely uniform distribution of the primary electrons and a Gaussian distribution with a 10% uniform background, respectively. As expected, higher pressure levels (or equivalently a larger number of primary electrons) and a larger secondary electron yield is needed to reproduce the observations in the case of a uniform electron distribution, due to the smaller central electron density compared to the case with a Gaussian electron distribution. Note that the range of the number of seed electrons per bunch passage explored in this study corresponds to room-temperature equivalent pressures between 10 nTorr and 200 nTorr (the static pressure in the LHC cold sections is around 32 nTorr room- temperature equivalent). Further studies will be needed in the future in order to better understand the mechanism of the primary electron production and their distribution at LHC injection energy. Figure 11: Range of parameters reproducing the observed instability in the LHC, as indicated by the colored areas assuming a uniform primary electron distribution (blue) and the Gaussian distribution on top of a 10% background (green). ## 6 CONCLUSIONS AND OUTLOOK Fast instabilities were observed during the first two attempts of injecting bunch trains of 48 bunches with the nominal 25 ns bunch spacing in August 2011. The onset of vertical single bunch instability can be reproduced in good agreement by electron cloud simulations using a combination of PyECLOUD for the build-up and HEADTAIL for the beam dynamics part. The simulations are based on values of $\delta_{\text{max}}$ as estimated from the measured heat load data in the LHC using PyECLOUD. The presented studies based on a combination of PyECLOUD with the HEADTAIL code can therefore be considered as consistency check of the current model of the electron-cloud effects on the LHC flat bottom. The simulations show a vertical instability, fast emittance growth and strong losses, similar to observations in the LHC. The instability onset depends strongly on the central electron density seen by the beam, which itself depends on the initial conditions assumed for the build-up, namely the number of primary electrons and their distribution. Future studies should aim at improving the understanding of the mechanism responsible for the generation of the primary electrons. This will become important also for the estimation of $\delta_{\text{max}}$ from the measured heat load close to the threshold of the electron cloud build-up, despite the fact that the beam is not expected to suffer from the instability any more. Apart from this, it would be interesting to include the coupling between the bunches due to the electron cloud in the simulations. This would require a self consistent model of the electron cloud effects like a combination of PyECLOUD and HEADTAIL in one big simulation for all bunches, which would then allow to study the observed horizontal coupled bunch instability. Finally it might be interesting to include the effect of the transverse feedback in HEADTAIL, as this would help to estimate better the required settings of chromaticity and octupoles to stabilize the beam in a strong electron-cloud regime. ## 7 ACKNOWLEDGEMENTS The authors would like to thank D. Valuch for help extracting and analyzing the turn-by-turn data from the damper pick-ups and feedback setup, G. Arduini for valuable discussion, and K. Li for support for the HEADTAIL simulations. ## References * [1] H. Bartosik and W. Höfle, “Analysis of bunch by bunch oscillations with bunch trains at injection into LHC at 25 ns bunch spacing”, CERN-ATS-Note-2012-027 MD (LHC), Geneva 2012. * [2] G. Iadarola and G. Rumolo, “PyECLOUD and Build Up Simulations at CERN”, these proceedings. * [3] G. Rumolo et al., “Electron Cloud Effects in the LHC in 2011”, in Proceedings of the LHC Beam Operation Workshop - Evian 2011, Evian 2011. * [4] G. Rumolo et al., “LHC Experience with Different Bunch Spacings in 2011 (25, 50 and 75ns)”, in Proceedings of Chamonix 2012 workshop on LHC Performance, Chamonix 2012. * [5] A. Gianluigi et al., “Beam observations with electron cloud in the CERN PS and SPS complex”, 31st Advanced ICFA Beam Dynamics Workshop on Electron-Cloud Effects, Napa 2004. * [6] G. Rumolo and F. Zimmermann, “Electron cloud simulations: beam instabilities and wakefields”, Phys. Rev. STAccel. Beams 5, 121002, 2002. * [7] K. Li and G. Rumolo, “Review of Beam Instabilities in the Presence of Electron Clouds in the LHC”, in Proceedings of IPAC11, San Sebastian 2011.	arxiv-papers	2013-09-25T13:26:37	2024-09-04T02:49:51.493017	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "H. Bartosik (CERN), W. Hofle (CERN), G. Iadarola (CERN), Y.\n Papaphilippou (CERN), G. Rumolo (CERN)", "submitter": "Scientific Information Service CERN", "url": "https://arxiv.org/abs/1309.6494" }
1309.6516	# Prediction of stable hafnium carbides: their stoichiometries, mechanical properties, and electronic structure Qingfeng Zeng [email protected] Science and Technology on Thermostructural Composite Materials Laboratory, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Junhui Peng Science and Technology on Thermostructural Composite Materials Laboratory, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Artem R. Oganov Department of Geosciences, Center for Materials by Design, and Institute for Advanced Computational Science, State University of New York, Stony Brook, NY 11794-2100, USA Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Qiang Zhu Department of Geosciences, Center for Materials by Design, and Institute for Advanced Computational Science, State University of New York, Stony Brook, NY 11794-2100, USA Congwei Xie Science and Technology on Thermostructural Composite Materials Laboratory, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Xiaodong Zhang Institute of Modern Physics, Northwest University, Xi’an, Shaanxi 710069, PR China Science and Technology on Thermostructural Composite Materials Laboratory, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Dong Dong Science and Technology on Thermostructural Composite Materials Laboratory, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Litong Zhang Science and Technology on Thermostructural Composite Materials Laboratory, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Laifei Cheng Science and Technology on Thermostructural Composite Materials Laboratory, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China ###### Abstract Hafnium carbides are studied by a systematic search for possible stable stoichiometric compounds in the Hf-C system at ambient pressure using variable-composition _ab initio_ evolutionary algorithm implemented in the USPEX code. In addition to well-known HfC, we predicted two additional compounds Hf3C2 and Hf6C5. The structure of Hf6C5 with space group $C2/m$ contains 11 atoms in the primitive cell and this prediction revives the earlier proposal by A. I. Gusev. The stable structure of Hf3C2 also has space group $C2/m$, and is more energetically favorable than the $Immm$, $P\bar{3}m1$, $P2$ and $C222_{1}$ structures put forward by A. I. Gusev. Dynamical and mechanical stability of the newly predicted structures have been verified by calculations of their phonons and elastic constants. The bulk and shear moduli of Hf3C2 are 195.8 GPa and 143.1 GPa, respectively, while for Hf6C5 they are 227.9 GPa and 187.2 GPa, respectively. Their mechanical properties are inferior to those of HfC due to the presence of structural vacancies. Chemical bonding, band structure, and Bader charge are presented and discussed. ††preprint: version 0 ## I Introduction Hafnium carbides, known as ultra-high temperature ceramics, have attracted growing attention because of their unique features. These include extremely high melting temperature and hardness, high thermal and electrical conductivity, and chemical stability, and make them promising advanced materials even in extreme thermal and chemical environments. Levine _et al._ (2002); Opeka _et al._ (2004); Savino _et al._ (2005) Hafnium carbide is known to crystallize in the NaCl-type structure (space group $Fm\bar{3}m$) and to have the composition HfC. This is a relatively well-studied material. Its elastic properties and phonon spectra have been studied experimentallyBrown _et al._ (1966); Smith and Gläser (1970), and its structural, elastic, electronic and phonon properties have been computed using first-principles methodsLi _et al._ (2011a, b). However, according to the results of theoretical calculation by A. I. Gusev Gusev and Rempel (1993), ordered stoichiometric phases Hf3C2 and Hf6C5 should exist, with possible space groups $Immm$, $P\bar{3}m1$, $P2$ or $C222_{1}$ for Hf3C2, and $C2/m$, $P3_{1}$ or $C2$ for Hf6C5. Experimental synthesis and structure determination of these subtle ordered states encounters problems due to the lack of direct methods. Gusev and Zyryanova Gusev and Zyryanova (2000) studied the order- disorder transition of Hf-C system by measuring the magnetic susceptiblity and confirmed the existence of Hf3C2 and the possible existence of Hf6C5. Here we explore the stable compounds in the Hf-C system and their crystal structures at ambient pressure using the variable-composition _ab initio_ evolutionary algorithm Oganov _et al._ (2010, 2011), and discuss their structures, elastic properties and chemical bonding. In Section 2, we describe the computational methods that were used in this work. In Section 3, we present the results - crystal structures, elastic properties and analysis of the electronic structure. Section 4 presents conclusions of this study. ## II Computational methodology The prediction of stable compounds and their crystal structures was performed using evolutionary algorithm implemented in USPEXOganov and Glass (2006); Lyakhov _et al._ (2013) code developed by Oganov’s group. This approach features global optimization with real-space representation and flexible physically motivated variation operators. For every candidate structure generated by USPEX, we use first principles structural relaxation, based on density functional theory (DFT) within the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)Perdew _et al._ (1996), as implemented in the VASP code Kresse and Furthmüller (1996). The all-electron projector-augmented wave (PAW) method Blöchl (1994), with the plane-wave kinetic energy cutoff of 900 eV and k-point meshes with reciprocal-space resolution of 2$\pi\times$ 0.07 Å-1 were used. These settings enable excellent convergences of the energy differences, stress tensors and structural parameters. The calculations of phonon dispersion, elastic properties, band structure, density of state were done using CASTEP codeSegall _et al._ (2002) and utilizing the modification of the Perdew-Burke-Ernzerhof GGA functional for solids (PBEsol) Perdew _et al._ (2008). Norm-conserving scheme Hamann _et al._ (1979) is used to generate pseudopotentials for Hf and C with the choices of electronic configuration of [Xe]6s25d2, and [He]2s22p2, respectively. In these plane-wave calculations, the cutoff energy is 750 eV, and k-point separation is 2$\pi\times$ 0.05 Å-1. Bader charge analysisBader (1990) was performed using Bader Charge Analysis codeTang _et al._ (2009); Sanville _et al._ (2007); Henkelman _et al._ (2006). ## III Results and discussions ### III.1 Crystal structure prediction and structural properties Variable-composition evolutionary algorithmOganov _et al._ (2010, 2011) used in this work is very effective in simultaneously predicting stable compositions and their structures for multi-component systems. In our searches, we allowed all possible compositions in the Hf-C system with structures containing up to 30 atoms in the unit cell. The initial generation consisted of 50 structures, and all subsequent generations had 40 structures. 50% of new structures were produced by heredity, 20% by softmutation, 10% by transmutation, 20% by random symmetric structure generator. Stable structures and their compositions were determined using the convex hull construction: a compound is thermodynamically stable if the enthalpy of its decomposition into any other compounds is positive. We explored the possible stable crystal structures for Hf-C system at ambient pressure and zero kelvin. In addition to rocksalt-type (B1) HfC (space group $Fm\bar{3}m$), we found another two compounds Hf3C2 and Hf6C5, both belonging to space group $C2/m$. The enthalpies of formation of the predicted structures are shown in Fig. 1. It can be clearly seen that HfC, Hf6C5 and Hf3C2 are thermodynamically stable compounds. Figure 1: Convex hull of Hf-C system at ambient pressure. The circle presents different structure and those structures locating on lines are thermodynamically stable. The crystal structures of Hf3C2 and Hf6C5 are shown in Fig. 2, and their crystallographic data and enthalpies of formation are listed in Table 1. For comparison, we also present computational results on HfC in Table 1. Structure of Hf3C2 has space group of $C2/m$ and 20 atoms in the conventional unit cell (Fig. 2b). Two more structures of Hf3C2 ($Immm$ and $P\bar{3}m1$) proposed by A. I. GusevGusev and Rempel (1993) were also found during our searches. However, their enthalpies are higher than that of $C2/m$, which is therefore more stable. Crystal structure of Hf6C5 is shown in Fig. 2c. Its space group is also $C2/m$ and its structure has 22 atoms in the conventional unit cell, its structural parameters are presented in Table 1, and agree with the theoretical calculation of A. I. GusevGusev and Rempel (1993). Figure 2: The crystal structures of (a) HfC with one unit cell, and (b) Hf3C2 (c) Hf6C5 with twice unit cells. The space group of HfC, Hf3C2 and Hf6C5 is $Fm\bar{3}m$,$C2/m$, $C2/m$, respectively. Table 1: Crystallographic data and enthalpies of formation of Hf3C2 and Hf6C5, the data of HfC is also presented as a comparison Compound \| Space group \| Volume \| Lattice constants \| Atom position \| Enthalpy of formation ---\|---\|---\|---\|---\|--- \| (No.) \| (Å3/f.u.) \| (Å) \| (Wyckoff position) \| (eV/atom) Hf3C2 \| $C2/m$ \| 75.91 \| $a$=5.720 \| Hf(4i) (0.737, 0.500, 0.744) \| -0.901 \| (12) \| \| $b$=9.893 \| Hf(8j) (0.248, 0.161, 0.264) \| \| \| \| $c$=5.701 \| C(2a) (0.0, 0.0, 0.0) \| \| \| \| \| C(2d) (0.0, 0.5, 0.5) \| \| \| \| \| C(4h) (0.0, 0.834, 0.5) \| Hf6C5 \| $C2/m$ \| 153.14 \| $a$=5.729 \| Hf(4i) (0.739, 0.000, 0.260) \| -0.995 \| (12) \| \| $b$=9.900 \| Hf(8j) (0.241, 0.327, 0.746) \| \| \| \| $c$=5.731 \| C(4g) (0.0, 0.333, 0.0) \| \| \| \| \| C(2d) (0.0, 0.5, 0.5) \| \| \| \| \| C(4h) (0.0, 0.832, 0.5) \| HfC \| $Fm\bar{3}m$ \| 25.34 \| $a$=4.675 \| Hf(4a) (0.0, 0.00, 0.00) \| -1.027 \| (225) \| \| $a$=4.637Li _et al._ (2011b) \| C(4b) (0.5, 0.5, 0.5) \| \| \| \| $a$=4.639Nartowski _et al._ (1999) \| \| All the stable hafnium carbides are strongly related structures and can be derived from HfC structure with the highest symmetry. HfC has a structure of cubic-packing hafnium atoms, and carbon atoms fill all octahedral voids(Fig. 2a), which is an ideal cubic rocksalt-type structure. The octahedra shown in Fig. 2b and c with green color are empty, i.e. formed by six Hf atoms but without interstitial C atoms. In Hf3C2 structure, only 2/3 of carbon octahedral voids are filled (and 1/3 are vacant), and in Hf6C5 5/6 are filled (and 1/6 are vacant). In both Hf3C2 and Hf6C5, the vacancies appear in every second octahedral layer, with 1/3 of in-layer octahedra occupied (Hf3C2) or 2/3 of in-layer octahedra occupied (Hf6C5). Ordering of the vacancies in both cases lowers the symmetry from cubic ($Fm\bar{3}m$) to monoclinic ($C2/m$). Moreover, due to the vacancies, the coordination number of Hf atoms varies in different systems: 6 in HfC, 5 in Hf6C5, and 4 in Hf3C2, while in all these structures carbon atoms invariably had the coordination number 6 (octahedral coordination). In this way, Hf6C5 and Hf3C2 can be described as defective rocksalt-type structures. It is instructive to look at molecular volumes (see Table 1). The volumes per formula unit (f.u.) of Hf3C2 (75.91 Å3/f.u.), Hf6C5 (153.14 Å3/f.u.) and HfC (25.536 Å3/f.u.) correspond to practically constant volume per Hf atom (25.3-25.5 Å3 per Hf atom). Fig. 1 shows that the most prominent stable state is HfC. And Hf3C2 and especially Hf6C5 will be stable only in a narrow range of chemical potentials in hafnium-rich conditions. This explains why HfC is well known from experiment, while Hf3C2 and especially Hf6C5 are more elusive. To verify the dynamical stability of the newly predicted Hf3C2 and Hf6C5, we computed their phonon dispersions (Fig. 3). No imaginary phonon frequencies were found throughout the Brillouin zone, suggesting dynamical stability of these phases. Figure 3: Phonon dispersion curves of (a) Hf3C2 (b) Hf6C5 at ambient pressure ### III.2 Elastic properties The elastic constants of a material describe its response to an applied stress or, conversely, the stress required to maintain a given deformation, and can be used to evaluate the elastic properties. These properties are directly related to mechanical stability. The criteria of mechanical stability of a monoclinic crystal are as followsCowley (1976): $K_{2}=\det\left\|{C_{ij}}\right\|,\;\;i,j\leq 5,\;\;K_{2}>0,\;\;C_{44}C_{66}-C_{46}^{2}>0$ (1) The calculated elastic constants of Hf3C2 and Hf6C5 at the ground state are listed in Table 2. It is obvious that these criteria are satisfied, suggesting that Hf3C2 and Hf6C5 are mechanically stable. The bulk modulus characterizes the response of a material to volume change, $B=P/(\Delta V/V)$. $P$ is the applied pressure and $\Delta V$ is the volume change. The shear modulus characterizes the response to shear deformation, $G=\tau/\gamma$. $\tau$ is the shear stress and $\gamma$ is shear strain. The bulk modulus $B$ and shear modulus $G$ can be obtained from elastic constants Hill (1952), and their values for Hf3C2 and Hf6C5 are presented in Table 2. For comparison, the values of the elastic properties of HfC are also presented in Table 2, and one can see a good agreement with the values reported in literature. The bulk and shear moduli of Hf3C2 and Hf6C5 are lower than those of HfC. The Pugh’s ratiosPugh (1954) of $G_{H}/B_{H}$ of the three compounds are larger than 0.57, indicating brittleness of these compounds. According to Eq. 2Chen _et al._ (2011), the Vickers hardness of Hf3C2, Hf6C5, and HfC is 22.28, 30.91 and 32.95 GPa, respectively - lowering of the hardness from HfC to Hf6C5 to Hf3C2 is an expected consequence of vacancies. $H_{V}=2(k^{2}G)^{0.585}-3$ (2) Table 2: Calculated elastic constants $C_{ij}$ , the bulk modulus, shear modulus and hardness (GPa) of Hf3C2, Hf6C5 and HfC at the ground state, and some literature values of HfC Compound \| Hf3C2 \| Hf6C5 \| \| HfC \| \| ---\|---\|---\|---\|---\|---\|--- \| \| \| This work \| Calc.He _et al._ (2008) \| Expt. Weber (1973) \| Expt. Brown _et al._ (1966) $C_{11}$ \| 391 \| 448 \| 589 \| 577 \| 500 \| $C_{22}$ \| 418 \| 471 \| \| \| \| $C_{33}$ \| 372 \| 470 \| \| \| \| $C_{44}$ \| 126 \| 181 \| 192 \| 171 \| 180 \| $C_{55}$ \| 153 \| 213 \| \| \| \| $C_{66}$ \| 143 \| 199 \| \| \| \| $C_{12}$ \| 198 \| 118 \| 98 \| 117 \| \| $C_{13}$ \| 102 \| 117 \| \| \| \| $C_{15}$ \| -9.7 \| -1.4 \| \| \| \| $C_{23}$ \| 91 \| 96 \| \| \| \| $C_{25}$ \| 12 \| 28 \| \| \| \| $C_{35}$ \| -0.6 \| -21 \| \| \| \| $C_{46}$ \| -7.2 \| 26 \| \| \| \| $B_{H}$ \| 195.8 \| 227.9 \| 261.7 \| 270 \| \| 242 $G_{H}$ \| 143.1 \| 187.2 \| 212.2 \| 230 \| \| 195 $k$111Pugh’s ratio: $k=G_{H}/B_{H}$ \| 0.73 \| 0.82 \| 0.81 \| \| \| $H_{V}$ \| 22.28 \| 30.91 \| 32.95 \| \| \| ### III.3 Chemical bonding The band structures and DOS of these three compounds are shown in Fig. 4. All the stable hafnium carbides are weak metals, as seen from finite but small DOS at the Fermi level - 3.28, 2.41, and 0.312 electrons/eV for Hf3C2, Hf6C5 and HfC, respectively. Taking into account the number of atoms in the primitive cells of Hf3C2, Hf6C5 and HfC, which were used in the calculations of the band structure and DOS, we see that the DOS at the Fermi level normalized per valence electron decreases as the number of vacancies decreases - from Hf3C2 (0.082 states/eV) to Hf6C5 (0.054 states/eV) and to HfC (0.039 states/eV). In all these three compounds there are pronounced pseudogaps at the Fermi level, and bonding can be characterized as mixed metallic-covalent. Indeed, orbital- projected DOS indicates strong hybridization of C-$p$ and Hf-$d$ valence states below the Fermi energy, i.e. presence of significant covalency in all three compounds. Bader charge analysis show that each Hf atom gives 1.734 electrons to each C atom in HfC. In case of Hf6C5, Hf atoms contribute 1.521$\pm 0.013$ electrons/atom, and C atoms get 1.826$\pm 0.019$ electrons/atom. In case of Hf3C2, Hf atoms contribute 1.267$\pm 0.026$ electrons/atom, and C atoms get 1.90$\pm 0.044$ electrons/atom. This, in agreement with the DOS, shows lower metallicity of HfC compared that of Hf6C5 and Hf3C2. Thus, HfC could have the highest hardness and melting point among these three compounds. Figure 4: The band structure and density of states of (a) Hf3C2 (b) Hf6C5 (c) HfC ## IV Conclusions In this paper we explored the possible stable compounds and structures in the Hf-C system at ambient pressure using variable-composition evolutionary algorithm. Besides the well-known HfC ($Fm\bar{3}m$), another two stoichiometric compounds (Hf3C2, $C2/m$ and Hf6C5, $C2/m$) are found. All three stable hafnium carbides have rocksalt-type structures: HfC with ideal cubic structure without vacancies, and defective rocksalt-type phases Hf3C2 and Hf6C5 with monoclinic symmetry ($C2/m$) due to ordering of carbon vacancies. Their elastic constants and phonon dispersions are also calculated, which verify their mechanical and dynamical stabilities. Their bulk modulus and shear modulus are lower than that of HfC due to the presence of structural vacancies. We found that all three compounds are weak metals, with increasing metallicity as the concentration of vacancies increases. We also found significant covalency in all these weakly metallic compounds, while their hardness values fall with increasing concentration of vacancies. ###### Acknowledgements. We thank the Research Fund of the State Key Laboratory of Solidification Processing of NWPU(No. 65-TP-2011), the Basic Research Foundation of NWPU (No. JCY20130114), the Foreign Talents Introduction and Academic Exchange Program (No. B08040), the National Science Foundation (Nos. EAR-1114313, DMR-1231586), DARPA (Nos. W31P4Q1310005, W31P4Q1210008), and the Government of the Russian Federation (No. 14.A12.31.0003) for financial support. The authors also acknowledge the High Performance Computing Center of NWPU for the allocation of computing time on their machines. USPEX code, with options for global optimization of the thermodynamic potential (energy, enthalpy, free energy), hardness, bandgap, dielectric constant, and other properties, is available at: http://uspex.stonybrook.edu. ## References * Levine _et al._ (2002) S. R. Levine, E. J. Opila, M. C. Halbig, J. D. Kiser, M. Singh, and J. A. Salem, J. Eur. Ceram. Soc. 22, 2757 (2002). * Opeka _et al._ (2004) M. M. Opeka, I. G. 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B 8, 5082 (1973).	arxiv-papers	2013-09-25T14:21:35	2024-09-04T02:49:51.500467	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Qingfeng Zeng, Junhui Peng, Artem R. Oganov, Qiang Zhu, Congwei Xie,\n Xiaodong Zhang, Dong Dong, Litong Zhang and Laifei Cheng", "submitter": "Qingfeng Zeng", "url": "https://arxiv.org/abs/1309.6516" }
1309.6530	# The role of fast magnetic reconnection in acceleration zones of microquasars and AGNs ###### Abstract Fast magnetic reconnection events, like those associated to solar flares, can be a very powerful mechanism operating at the jet launching region in the core of microquasars and AGNs. We have recently found that the magnetic power released by reconnection between the magnetic field lines of the coronal inner disk region and the lines anchored into the black hole is able to accelerate relativistic particles through a first-order Fermi process and produce the observed radio luminosity from both microquasars and low luminous AGNs. We also found that the observed correlation between the radio luminosity and the mass of these sources, spanning $10^{9}$ orders of magnitude in mass, is naturally explained by this process. Furthermore, recent observations of $\gamma$-ray emission with strong variability from AGNs and galactic compact sources indicate extremely small emission regions. In this work, assuming that this high energy emission is probably originated in the same acceleration zones that produce the radio emission, we have applied the scenario above of magnetic reconnection-driven acceleration in the magnetized corona around the accretion disks to investigate the origin of the high energy outcomes from an extensive number of sources including high and low luminous AGNs, microquasars, GRBs and the Crab nebula. As in the earlier analysis, we find that the radio emission of all microquasars and low luminous AGNs are well explained by our model. On the other hand, we find correlation of our model with the $\gamma$-ray emission only for microquasars and a few low luminous AGNs (e.g., M$87$), while none of the high luminous AGNs or GRBs are fitted, neither in radio (as previously found) nor in $\gamma$ emission. We attribute the lack of correlation of the $\gamma$ emission for most of the low luminous AGNs to the fact that this processed emission does not depend only on the local magnetic field activity around the source accretion disk (as the radio Synchrotron emission), but also on other environmental factors like the photon and density fields. We also find that the emission from both the high luminous AGNs and the GRBs actually anti-correlates with our model indicating that their emission is processed further out in the jet (as we claimed in [6], and in consistency with recent findings in [24]). We conclude from these results that the emission we see from the low luminous AGNs and microquasars comes from the nuclear region of their sources and therefore, can be driven by nuclear magnetic activity connected to the source/accretion disk corona, as in our model. However, in the case of the high luminous AGNs (e.g. BL Lacs) and GRBs, the nuclear emission is blocked by the surrounding density and photon fields and, therefore, we can only see the jet emission further out (which has already lost correlation with the nuclear conditions). This also explains why high luminous AGNs do not fit the fundamental plane of luminosity source versus mass correlation. ## 1 Introduction Galactic and extragalactic objects such as microquasars and AGNs often exhibit variability and quasi-periodic ejections of matter that may offer important clues about the physical processes that occur in their inner regions. A model to explain the origin of these ejections and the associated Synchrotron flare radio emission, which has characteristics that resemble those of solar flares, was developed by de Gouveia Dal Pino and Lazarian ([5]; see also [6]) where they invoked a process of fast magnetic reconnection between the magnetic field lines that arise from the accretion disk and the lines of the magnetosphere of the central source. In accretion episodes where the ratio between the effective disk pressure and magnetic pressure decreases to values smaller than the unity and the accretion rate approaches the critical Eddington rate, the magnetic reconnection may become very fast and release large amounts of magnetic energy power. Part of this energy heats the coronal and the disk gas and part accelerates particles to relativistic velocities through a first-order Fermi-like process (see also [12], [13], [8]) that results in a Synchrotron radio power-law spectrum compatible with the observations. Using this model, [5] and [6] also found that the observed correlation between the radio luminosity and the mass from microquasars to low luminous AGNs, spanning $10^{9}$ orders of magnitude in mass, is naturally explained by this process. In recent years, the very high energy emission (VHE) from AGNs have also revealed strong variability, with timescales of the order of days (e.g., M$87$), which points to extremely compact emission regions (corresponding to only a few Schwarzschild radii; e.g. [2]). However, the localization of the emitting zones for several sources is until now unclear. Magnetic reconnection events occurring close to the black holes (BH) could offer appropriate conditions for producing particle acceleration and the associated VHE $\gamma$-ray emission in these sources. Similarly, microquasars are also expected to emit high-energy $\gamma$-rays owing to their general similarities to quasars (e.g., [11]). Until now, only one source of this type, namely Cyg X-$3$, has been unambiguously detected in the GeV $\gamma$-rays, by the Agile and Fermi observatories ([21], [1]). At TeV energies, only the flux upper limits are available, in spite of intensive monitoring. There is also some evidence of sporadic GeV-TeV $\gamma$-ray emission from another source of this type, Cyg X-$1$. In the framework of high luminous AGNs and GRBs, Nemen et al. ([24]) have recently shown that the jets produced by both classes of sources exhibit nearly the same correlation between the kinetic power carried by accelerated particles and the $\gamma$-ray luminosity. This is an indication that the bulk of the high-energy emission comes from the jet region in these sources. On the other hand, the lack of correlation with low luminous AGNs may be an indication that the acceleration zones responsible for the high (and also low) energy emission in these cases are mostly in the core region, rather than further out in the jet, just like in the model proposed above. Since the $\gamma$-ray emission is probably originated in the same acceleration zones that produce the radio emission, in the present work we apply the de Gouveia Dal Pino et al. scenario above of reconnection-driven acceleration in the magnetized corona of the accretion disks ([5] and [6]) and investigate if the high energy outcomes of microquasars and low luminous AGNs can be also interpreted in the light of this mechanism. ## 2 Rate of magnetic energy released by magnetic reconnection: application to radio emission To evaluate the amount of magnetic energy that can be extracted through violent magnetic reconnection, it is adopted the standard model for the radiation-dominated accretion disk by [22] and the model by [4] to quantify the parameters of the corona. Also, it is assumed that the inner radius of the accretion disk corresponds approximately to the last stable orbit around the BH ($R_{X}=3R_{S}$, where $R_{S}$ is the Schwarzschild radius). To determine the accretion rate immediately before an event of violent magnetic reconnection, it is assumed the equilibrium between the disk gas ram pressure and the magnetic pressure of the magnetosphere anchored at the BH horizon. It is assumed further that the intensity of the BH horizon field is of the order of that of the inner disk. Under these conditions one can show that the magnetic energy power released during violent magnetic reconnection for microquasars is approximately given by [6]: $\dot{W}_{B}\simeq 10^{35}\alpha_{0.5}^{-19/16}\beta_{0.8}^{-9/16}M_{14}^{1/2}l_{100R_{X}}^{11/16}\leavevmode\nobreak\ \leavevmode\nobreak\ erg/s,$ (1) and for AGNs by [6]: $\dot{W}_{B}\simeq 10^{40}\alpha_{0.5}^{-19/16}\beta_{0.8}^{-9/16}M_{8}^{1/2}l_{10^{4}R_{X}}^{11/16}\leavevmode\nobreak\ \leavevmode\nobreak\ erg/s,$ (2) where $\alpha_{0.5}=\alpha/0.5$ is the disk viscosity and $\beta_{0.8}=\beta/0.8$ is defined as the ratio between the effective disk pressure and the magnetic pressure. The BH mass $M_{14}=M/14M_{\odot}$ and $M_{8}=M/10^{8}M_{\odot}$ are parameters suitable for the microquasars and AGNs, respectively, and the parameters $l_{100R_{X}}=l/100R_{X}$ and $l_{10^{4}R_{X}}=l/10^{4}R_{X}$ are the scale height of the magnetic reconnection zone in the corona. Figure 1 extracted from [6] depicts a synthesis of the magnetic reconnection scenario for relativistic sources including both microquasars and AGNs. The diagram shows the calculated magnetic power released in violent magnetic reconnection events as a function of the central source mass for a suitable choice of the parameter space. The symbols correspond to the observed radio luminosities of superluminal components (stars for microquasars, circles and triangles for the low luminous AGNs, i.e. LINERs and Seyfert galaxies, respectively, and squares for luminous AGNs). Figure 1: $\dot{W}_{B}$ versus the BH mass for both microquasars and AGNs. The stars represent the observed radio luminosities for microquasars. The circles, triangles and squares are observed radio luminosities of jets at parsec scales from LINERS, Seyfert galaxies, and luminous AGNs, respectively. The thick bars correspond to the calculated magnetic reconnection power and encompass the parameter space that spans $5M_{\odot}\leq M\leq 10^{10}M_{\odot}$, $0.05\leq\alpha\leq 0.5$, $0.1\leq\beta\leq 1$, and $1R_{S}\leq l\leq 1000R_{S}$ (or $0.3R_{X}\leq l\leq 333R_{X}$), with $1R_{S}\leq l\leq 10R_{S}$ in black, $10R_{S}<l\leq 30R_{S}$ in light gray, and $30R_{S}<l\leq 1000R_{S}$ in dark gray. Extracted from [6]. The diagram indicates that the magnetic power released during violent reconnection events obeys a correlation that is maintained throughout this interval, spanning $10^{9}$ orders of magnitude. This correlation implies an almost linear dependence (in a log-log diagram), which is approximately independent of the physical properties of the accretion disks of these sources. Moreover, it is compatible with the so-called “fundamental plane” obtained empirically, which correlates the radio and X-rays emission of microquasars and AGNs with the masses of their BH (see [3]). Thus, the model of [5] provides a simple physical interpretation for the existence of this empirical correlation as due to coronal magnetic activity in these sources. The diagram also reveals that the more luminous AGNs do not obey the same correlation, possibly because the density around the coronal region in these sources is so high that it “masks” the emission due to the magnetic activity. The radio emission in these cases is possibly due to regions further out at the supersonic jet, where it has already expanded enough to become optically thin and visible and where the relativistic electrons are probably accelerated in shocks (see [6]). This conclusion in de Gouveia Dal Pino et al. [6] is compatible with the recent correlation found by Nemem et al. ([24]). ## 3 The same scenario for the $\gamma$-ray emission Employing the model described above of [6], we selected a preliminary set of compact sources including high and low luminous AGNs, GRBs and galactic sources (e.g., the pulsar Crab and the microquasars Cyg-X$1$ and Cyg-X$3$) with observed $\gamma$ and radio emission and compared their luminosities with the calculated magnetic reconnection power as a function of their mass. Figure 2: Magnetic power versus the central source mass. The open blue symbols correspond to the $\gamma$-ray emission from blazars, low luminous AGNs and galactic black holes (Cyg-X$1$ and Cyg-X$3$). The blue vertical line and blue star correspond, respectively, to the $\gamma$ emission from GRBs and the Crab pulsar. The filled red symbols correspond to the radio emission of different types of sources (Seyferts, liners, radio galaxies and microquasars). Figure 2 depicts the preliminary results of this study. As in Figure 1, it compares the calculated magnetic power released by fast magnetic reconnection at the corona/accretion disk region as a function of the source mass with the observed $\gamma$ and radio luminosities. We have considered the $\gamma$ emission (open blue symbols in Figure 2) of an extensive number of high luminous AGNs and GRBs (all obtained from [24]), low luminous AGNs (Seyferts, liners and radio galaxies; see [18], [9], [10] and [15]) and microquasars (Cyg-X$1$ [14] and Cyg-X$3$ [16]). In the case of GRBs (blue vertical line in the diagram), we have assumed that all sources have a mass around a few tens of solar masses. We have also included the radio emission counterparts of several of these sources (filled red symbols; obtained from [3], and [23]) and the recently detected $\gamma$-ray emission from the Crab pulsar (blue star) which apparently comes from the surrounds of the central source (see [20]). ## 4 Discussion and Conclusions As in the previous works ([5] and [6]), we clearly see in Figure 2 that the observed radio luminosity of the low luminous AGNs and microquasars can be explained by the magnetic power released by fast reconnection in the core region of these sources. This emission is due to Synchrotron radiation from relativistic electrons which can be accelerated by a first-order Fermi process directly within the magnetic reconnection site in the coronal region around the accretion disk ([5], [12], [13], [8]). The corresponding $\gamma$-ray emission from these sources, which is produced from the interaction of the accelerated relativistic electrons and protons with the surrounding photon and density fields (through inverse Compton, and/or pp inelastic collisions and photon-meson decay) can in principle be also associated with the same emission zone in the surroundings of the core of these sources. However, direct evidence for this association, though found for the microquasars and the Crab, is not found for most of the low luminous AGNs. This is explained by the fact that the $\gamma$ emission (contrary to the radio Synchrotron emission) does not depend only on the local magnetic fields, but also on the photon and density fields in the surroundings of the source/accretion disk, as stressed above, and these factors can provoke the loss of correlation with our nuclear emission model (induced by magnetic activity around the accretion disk). The high luminous AGNs and the GRBs in Figure 2, on the other hand, clearly do not have their radio or $\gamma$ emission correlated with the magnetic reconnection power released at the core regions. This result confirms the previous findings of [6] and [24] (see also [7]) which suggested that the $\gamma$ and radio emission observed in such sources is originated further out at the relativistic jet associated to these sources (as the nuclear emission is being screened by the surrounding strong photon and density fields). The results above connecting both the radio and $\gamma$ emission from low luminous compact sources to magnetically dominated reconnection processes in their nuclear regions, though preliminary are very promising as they suggest a unifying single process of relativistic particle acceleration in the core region of low luminous AGNs and compact galactic sources which naturally interpret the Fundamental Plane [3]. 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Nemmen et al., Science 338 (2012) 1445	arxiv-papers	2013-09-25T14:46:22	2024-09-04T02:49:51.508503	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "L.H.S. Kadowaki and E.M. de Gouveia Dal Pino", "submitter": "Lu\\'is Henrique Sinki Kadowaki", "url": "https://arxiv.org/abs/1309.6530" }
1309.6534	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP-2013-176 LHCb-PAPER-2013-053 18 December 2013 Measurement of $D^{0}$–$\kern 4.14793pt\overline{\kern-4.14793ptD}{}^{0}$ mixing parameters and search for $C\\!P$ violation using $D^{0}\rightarrow K^{+}\pi^{-}$ decays The LHCb collaboration†††Authors are listed on the following pages. Measurements of charm mixing parameters from the decay-time-dependent ratio of $D^{0}\rightarrow K^{+}\pi^{-}$ to $D^{0}\rightarrow K^{-}\pi^{+}$ rates and the charge-conjugate ratio are reported. The analysis uses data, corresponding to $3\mbox{\,fb}^{-1}$ of integrated luminosity, from proton-proton collisions at 7 and 8 $\mathrm{\,Te\kern-1.00006ptV}$ center-of-mass energies recorded by the LHCb experiment. In the limit of charge-parity ($C\\!P$) symmetry, the mixing parameters are determined to be $x^{\prime 2}=(5.5\pm 4.9)\times 10^{-5}$, $y^{\prime}=(4.8\pm 1.0)\times 10^{-3}$, and $R_{D}=(3.568\pm 0.066)\times 10^{-3}$. Allowing for $C\\!P$ violation, the mixing parameters are determined separately for $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons yielding $A_{D}=(-0.7\pm 1.9)\%$, for the direct $C\\!P$-violating asymmetry, and $0.75<\|q/p\|<1.24$ at the $68.3\%$ confidence level, where $q$ and $p$ are parameters that describe the mass eigenstates of the neutral charm mesons in terms of the flavor eigenstates. This is the most precise determination of these parameters from a single experiment and shows no evidence for $C\\!P$ violation. © CERN on behalf of the LHCb collaboration, license CC-BY-3.0. LHCb collaboration R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A. Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben- Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles54, Ph. Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, P. Dornan52, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P. 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Zvyagin37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Padova, Padova, Italy 22Sezione INFN di Pisa, Pisa, Italy 23Sezione INFN di Roma Tor Vergata, Roma, Italy 24Sezione INFN di Roma La Sapienza, Roma, Italy 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland 26AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland 27National Center for Nuclear Research (NCBJ), Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 41Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands 42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 43Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 44University of Birmingham, Birmingham, United Kingdom 45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 47Department of Physics, University of Warwick, Coventry, United Kingdom 48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 50School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 52Imperial College London, London, United Kingdom 53School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 54Department of Physics, University of Oxford, Oxford, United Kingdom 55Massachusetts Institute of Technology, Cambridge, MA, United States 56University of Cincinnati, Cincinnati, OH, United States 57University of Maryland, College Park, MD, United States 58Syracuse University, Syracuse, NY, United States 59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11 61Celal Bayar University, Manisa, Turkey, associated to 37 aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oHanoi University of Science, Hanoi, Viet Nam pInstitute of Physics and Technology, Moscow, Russia qUniversità di Padova, Padova, Italy rUniversità di Pisa, Pisa, Italy sScuola Normale Superiore, Pisa, Italy Mass eigenstates of neutral charm mesons are linear combinations of flavor eigenstates $\|D_{1,2}\rangle=p\|D^{0}\rangle\pm q\|\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$, where $p$ and $q$ are complex parameters. This results in $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillation. In the limit of charge-parity ($C\\!P$) symmetry, the oscillation is characterized by the difference in mass $\Delta m\equiv m_{2}-m_{1}$ and decay width $\Delta\Gamma\equiv\Gamma_{2}-\Gamma_{1}$ between the $D$ mass eigenstates. These differences are usually expressed in terms of the dimensionless mixing parameters $x\equiv\Delta m/\Gamma$ and $y\equiv\Delta\Gamma/2\Gamma$, where $\Gamma$ is the average decay width of neutral $D$ mesons. If $C\\!P$ symmetry is violated, the oscillation rates for mesons produced as $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ can differ, further enriching the phenomenology. Both short- and long-distance components of the amplitude contribute to the time evolution of neutral $D$ mesons [1, 2, 3]. Short- distance amplitudes could include contributions from non-standard-model particles or interactions, possibly enhancing the average oscillation rate or the difference between $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson rates. The study of $C\\!P$ violation in $D^{0}$ oscillation may lead to an improved understanding of possible dynamics beyond the standard model [4, 5, 6, 7]. The first evidence for $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillation was reported in 2007 [8, 9]. By 2009, the hypothesis of no oscillation was excluded with significance in excess of 10 standard deviations [10] by combining results from different experiments [8, 9, 11, 12, 13, 14, 15, 16, 17]. In 2012, the LHCb experiment reported the first observation from a single measurement with greater than 5 standard deviation significance [18], which has been recently confirmed by the CDF experiment [19]. This Letter reports a search for $C\\!P$ violation in $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing by comparing the decay- time-dependent ratio of $D^{0}\rightarrow K^{+}\pi^{-}$ to $D^{0}\rightarrow K^{-}\pi^{+}$ rates with the corresponding ratio for the charge-conjugate processes. An improved determination of the $C\\!P$-averaged charm mixing parameters with respect to our previous measurement [18] is also reported. The analysis uses data corresponding to $1.0\mbox{\,fb}^{-1}$ of integrated luminosity from $\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions recorded by LHCb during 2011 and $2.0\mbox{\,fb}^{-1}$ from $\sqrt{s}=8\,\mathrm{\,Te\kern-1.00006ptV}$ collisions recorded during 2012. The neutral $D$ flavor at production is determined from the charge of the low- momentum pion $\pi_{\rm s}^{+}$ in the flavor-conserving strong-interaction decay $D^{+}\rightarrow D^{0}\pi_{\rm s}^{+}$. The inclusion of charge- conjugate processes is implicit unless stated otherwise. The $D^{+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi_{\rm s}^{+}$ process is denoted as right sign (RS), and $D^{+}\rightarrow D^{0}(\rightarrow K^{+}\pi^{-})\pi_{\rm s}^{+}$ is denoted as wrong sign (WS). The RS decay rate is dominated by a Cabibbo-favored amplitude. The WS rate arises from the interfering amplitudes of the doubly Cabibbo-suppressed $D^{0}\rightarrow K^{+}\pi^{-}$ decay and the Cabibbo-favored $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decay following $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillation, each of similar magnitude. In the limit of $\|x\|,\|y\|\ll 1$, and assuming negligible $C\\!P$ violation, the time-dependent ratio $R(t)$ of WS- to-RS decay rates is [1, 2, 3, 4] $R(t)\approx R_{D}+\sqrt{R_{D}}\ y^{\prime}\ \frac{t}{\tau}+\frac{x^{\prime 2}+y^{\prime 2}}{4}\left(\frac{t}{\tau}\right)^{2},$ (1) where $t$ is the decay time, $\tau$ is the average $D^{0}$ lifetime, and $R_{D}$ is the ratio of suppressed-to-favored decay rates. The parameters $x^{\prime}$ and $y^{\prime}$ depend linearly on the mixing parameters as $x^{\prime}\equiv x\cos\delta+y\sin\delta$ and $y^{\prime}\equiv y\cos\delta-x\sin\delta$, where $\delta$ is the strong-phase difference between the suppressed and favored amplitudes $\mathcal{A}(D^{0}\rightarrow K^{+}\pi^{-})/\mathcal{A}(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-})=-\sqrt{R_{D}}e^{-i\delta}$. Allowing for $C\\!P$ violation, the WS rates $R^{+}(t)$ and $R^{-}(t)$ of initially produced $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons are functions of independent sets of mixing parameters $(R_{D}^{\pm},\,x^{\prime 2\pm},\,y^{\prime\pm})$. A difference between $R^{+}_{D}$ and $R^{-}_{D}$ arises if the ratio between the magnitudes of suppressed and favored decay amplitudes is not $C\\!P$ symmetric (direct $C\\!P$ violation). Violation of $C\\!P$ symmetry either in mixing $\|q/p\|\neq 1$ or in the interference between mixing and decay amplitudes $\phi\equiv\arg\left[q\mathcal{A}(\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-})/p\mathcal{A}(D^{0}\rightarrow K^{+}\pi^{-})\right]-\delta\neq 0$ are usually referred to as indirect $C\\!P$ violation and would result in differences between $(x^{\prime 2+},\,y^{\prime+})$ and $(x^{\prime 2-},\,y^{\prime-})$. The LHCb detector [20] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. Detector components particularly relevant for this analysis are the silicon vertex detector, which provides reconstruction of displaced vertices of $b$\- and $c$-hadron decays; the tracking system, which measures charged particle momenta with relative uncertainty that varies from $0.4\%$ at $5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to $0.6\%$ at $100\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, corresponding to a typical mass resolution of approximately $8\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for a two-body charm-meson decay; and the ring-imaging Cherenkov detectors, which provide kaon-pion discrimination [21]. The magnet polarity is periodically inverted and approximately equal amounts of data are collected in each configuration to mitigate the effects of detection asymmetries. The online event-selection system (trigger) [22] consists of a first-level hardware stage based on information from the calorimeter and muon systems, followed by a software high-level trigger. Events with $D^{+}$ candidates consistent with being produced at the $pp$ collision point (primary vertex) are selected following Ref. [18]. In addition, a WS candidate is discarded if resulting from a $D^{0}$ candidate that, associated with another pion, also forms a RS candidate with $M(D^{0}\pi_{\rm s}^{+})$ within $3\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $D^{+}$ mass. This removes about 15% of the WS background with negligible signal loss. The two-body $D^{0}\pi_{\rm s}^{+}$ mass $M(D^{0}\pi_{\rm s}^{+})$ is computed using the known $D^{0}$ and $\pi^{+}$ masses [23] and their reconstructed momenta [18]. In Ref. [18], we used events selected by the hardware trigger based on hadron calorimeter transverse-energy depositions that were geometrically matched with signal final-state tracks. In the present analysis, we distinguish two trigger categories. One category consists of events that meet the above trigger requirement (triggered-on- signal, TOS). The other comprises events with candidates failing the track- calorimeter matching and events selected based on muon hardware triggers decisions ($\overline{\text{TOS}}$). The two subsamples contribute approximately equal signal yields with similar purities. However, they require separate treatment due to their differing kinematic distributions and trigger- induced biases. Figure 1: Distribution of $M(D^{0}\pi_{\rm s}^{+})$ for selected (a) right- sign $D^{0}\rightarrow K^{-}\pi^{+}$ and (b) wrong-sign $D^{0}\rightarrow K^{+}\pi^{-}$ candidates. The RS and WS signal yields are determined by fitting the $M(D^{0}\pi_{\rm s}^{+})$ distribution of $D^{0}$ candidates with reconstructed mass within $24\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known value. The time- integrated $M(D^{0}\pi_{\rm s}^{+})$ distributions are shown in Fig. 1. The smooth background is dominated by favored $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decays associated with random $\pi_{\rm s}^{+}$ candidates. The sample contains $1.15\times 10^{5}$ ($1.14\times 10^{5}$) signal WS $D^{0}$ ($\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) decays and approximately 230 times more RS decays. Yield differences between $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays are dominated by differences in charm-anticharm production rates and reconstruction efficiencies. Each sample is divided into 13 subsamples according to the candidate’s decay time, and signal yields are determined for each using shape parametrizations determined from simulation and tuned to data [18]. We assume that for a given $D^{}$ meson flavor, the signal shapes are common to WS and RS decays, while the descriptions of the background can differ. The decay- time-dependent WS-to-RS yield ratios $R^{+}$ and $R^{-}$ observed in the $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples, respectively, and their difference are shown in Fig. 2. These are corrected for the relative efficiencies for reconstructing $K^{-}\pi^{+}$ and $K^{+}\pi^{-}$ final states. Figure 2: Efficiency-corrected ratios of WS-to-RS yields for (a) $D^{+}$ decays, (b) $D^{-}$ decays, and (c) their differences as functions of decay time in units of $D^{0}$ lifetime. Projections of fits allowing for (dashed line) no $C\\!P$ violation, (dotted line) no direct $C\\!P$ violation, and (solid line) full $C\\!P$ violation are overlaid. The abscissa of the data points corresponds to the average decay time over the bin; the error bars indicate the statistical uncertainties. The mixing parameters are determined by minimizing a $\chi^{2}$ variable that includes terms for the difference between the observed and predicted ratios and for systematic deviations of parameters $\chi^{2}=\sum_{i}\left[\left(\frac{r^{+}_{i}-\epsilon_{r}^{+}\widetilde{R}_{i}^{+}}{\sigma_{i}^{+}}\right)^{2}+\left(\frac{r^{-}_{i}-\epsilon_{r}^{-}\widetilde{R}_{i}^{-}}{\sigma_{i}^{-}}\right)^{2}\right]+\chi^{2}_{\epsilon}\,+\,\chi^{2}_{B}+\,\chi^{2}_{\rm p}\,.$ (2) The measured WS-to-RS yield ratio and its statistical uncertainty in the decay-time bin $i$ are denoted by $r_{i}^{\pm}$ and $\sigma_{i}^{\pm}$, respectively. The predicted value for the WS-to-RS yield ratio $\widetilde{R}_{i}^{\pm}$ corresponds to the time integral over bin $i$ of Eq. (1) including bin-specific corrections. These account for small biases due to the decay-time evolution of the approximately $3\%$ fraction of signal candidates originating from $b$-hadron decays ($\Delta_{B}$) and of the about $0.5\%$ component of peaking background from RS decays in which both final- state particles are misidentified ($\Delta_{\rm p}$) [18]. The relative efficiency $\epsilon_{r}^{\pm}$ accounts for instrumental asymmetries in the $K\pi$ reconstruction efficiencies, mainly caused by $K^{-}$ mesons having a larger interaction cross section with matter than $K^{+}$ mesons. These asymmetries are measured in data to be in the range $0.8$–$1.2\%$ with $0.2\%$ precision and to be independent of decay time. They are derived from the efficiency ratio $\epsilon_{r}^{+}=1/\epsilon_{r}^{-}=\epsilon(K^{+}\pi^{-})/\epsilon(K^{-}\pi^{+})$, obtained from the product of $D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$ and $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}(\rightarrow\pi^{+}\pi^{-})\pi^{+}$ event yields divided by the product of the corresponding charge-conjugate decay yields. No $C\\!P$ violation is expected or experimentally observed [23] in these decays. Asymmetries due to $C\\!P$ violation in neutral kaons and their interaction cross-sections with matter are negligible. The $1\%$ asymmetry between $D^{+}$ and $D^{-}$ production rates [24] cancels in this ratio, provided that the kinematic distributions are consistent across samples. We weight the $D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$ events so that their kinematic distributions match those in the $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ sample. Similarly, these samples are weighted as functions of $K\pi$ momentum to match the RS momentum spectra. The parameters associated with $\Delta_{B}$, $\Delta_{\rm p}$, and $\epsilon_{r}$ are determined separately for TOS and $\overline{\text{TOS}}$ subsets and vary independently in the fit within their Gaussian constraints $\chi^{2}_{B}$, $\chi^{2}_{\rm p}$, and $\chi^{2}_{\epsilon}$ [18]. To avoid experimenters’ bias in the $C\\!P$ violation parameters, the measurement technique is finalized by adding arbitrary offsets to the WS-to-RS yield ratios for the $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples, designed to mimic the effect of different mixing parameters in the two samples. To rule out global systematic uncertainties not accounted for in Eq. (2), the data are first integrated over the whole decay-time spectrum and subsequently divided into statistically independent subsets according to criteria likely to reveal biases from specific instrumental effects. These include the number of primary vertices in the events, the $K$ laboratory momentum, the $\pi_{\rm s}$ impact parameter $\chi^{2}$ with respect to the primary vertex, the $D^{0}$ impact parameter $\chi^{2}$ with respect to the primary vertex, the magnetic field orientation, and the hardware trigger category. The variations of the time- integrated charge asymmetry in WS-to-RS yield ratios are consistent with statistical fluctuations. Then, we investigate decay-time-dependent biases by dividing the time-binned sample according to the magnet polarity and the number of primary vertices per event. In the TOS sample, differences of WS-to- RS yield ratios as functions of decay time for opposite magnet polarities yield $\chi^{2}$ values of 12, 17, and 14 (for 12 degrees of freedom), for events with one, two, and more than two primary vertices, respectively. The corresponding $\chi^{2}$ values in the $\overline{\text{TOS}}$ sample, 9, 11, and 8, suggest a systematically better consistency. Hence, the statistical uncertainty of each of the WS-to-RS ratios in the TOS samples is increased by a factor of $\sqrt{17/12}$, following Ref. [23]. These scaled uncertainties are used in all subsequent fits. Independent analyses of the 2011 and 2012 data yield consistent results. The ratio between RS $D^{0}$ to $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decay rates is independent of decay time with a $62\%$ $p$ value and a standard deviation of $0.16\%$, showing no evidence of correlations between particle identification or reconstruction efficiency and decay time. Table 1: Results of fits to the data for different hypotheses on the $C\\!P$ symmetry. The reported uncertainties are statistical and systematic, respectively; ndf indicates the number of degrees of freedom. See App. A for fits results including correlation coefficients. Parameter \| Value ---\|--- Direct and indirect $C\\!P$ violation $R_{D}^{+}$ [ \| $10^{-3}$] \| $3.545$ $\pm$ \| $0.082$ $\pm$ \| $0.048$ $y^{\prime+}$ [ \| $10^{-3}$] \| $5.1$ $\pm$ \| $1.2$ $\pm$ \| $0.7$ $x^{\prime 2+}$ [ \| $10^{-5}$] \| $4.9$ $\pm$ \| $6.0$ $\pm$ \| $3.6$ $R_{D}^{-}$ [ \| $10^{-3}$] \| $3.591$ $\pm$ \| $0.081$ $\pm$ \| $0.048$ $y^{\prime-}$ [ \| $10^{-3}$] \| $4.5$ $\pm$ \| $1.2$ $\pm$ \| $0.7$ $x^{\prime 2-}$ [ \| $10^{-5}$] \| $6.0$ $\pm$ \| $5.8$ $\pm$ \| $3.6$ $\chi^{2}/\text{ndf}$ \| $85.9/98$ No direct $C\\!P$ violation $R_{D}$ [ \| $10^{-3}$] \| $3.568$ $\pm$ \| $0.058$ $\pm$ \| $0.033$ $y^{\prime+}$ [ \| $10^{-3}$] \| $4.8$ $\pm$ \| $0.9$ $\pm$ \| $0.6$ $x^{\prime 2+}$ [ \| $10^{-5}$] \| $6.4$ $\pm$ \| $4.7$ $\pm$ \| $3.0$ $y^{\prime-}$ [ \| $10^{-3}$] \| $4.8$ $\pm$ \| $0.9$ $\pm$ \| $0.6$ $x^{\prime 2-}$ [ \| $10^{-5}$] \| $4.6$ $\pm$ \| $4.6$ $\pm$ \| $3.0$ $\chi^{2}/\text{ndf}$ \| $86.0/99$ No $C\\!P$ violation $R_{D}$ [ \| $10^{-3}$] \| $3.568$ $\pm$ \| $0.058$ $\pm$ \| $0.033$ $y^{\prime}$ [ \| $10^{-3}$] \| $4.8$ $\pm$ \| $0.8$ $\pm$ \| $0.5$ $x^{\prime 2}$ [ \| $10^{-5}$] \| $5.5$ $\pm$ \| $4.2$ $\pm$ \| $2.6$ $\chi^{2}/\text{ndf}$ \| $86.4/101$ Three fits are performed to the data shown in Fig. 2. The first allows direct and indirect $C\\!P$ violation; the second allows only indirect $C\\!P$ violation by constraining $R_{D}^{\pm}$ to a common value; and the third is a $C\\!P$-conserving fit that constrains all mixing parameters to be the same in the $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples. The fit results and their projections are shown in Table 1 and Fig. 2, respectively. Figure 3 shows the central values and confidence regions in the $(x^{\prime 2},\,y^{\prime})$ plane. For each fit, 104 WS-to-RS ratio data points are used, corresponding to 13 ranges of decay time, distinguishing $D^{+}$ from $D^{-}$ decays, TOS from $\overline{\text{TOS}}$ decays, and 2011 data from 2012 data. The consistency with the hypothesis of $C\\!P$ symmetry is determined from the change in $\chi^{2}$ between the fit without and with $C\\!P$ violation, taking into account the difference in number of degrees of freedom. The resulting $p$ value, for the fit with direct and indirect (indirect only) $C\\!P$ violation allowed, is $91\%$ ($81\%$), showing that the data are compatible with $C\\!P$ symmetry. Figure 3: Two-dimensional confidence regions in the $(x^{\prime 2},y^{\prime})$ plane obtained (a) without any restriction on $C\\!P$ violation, (b) assuming no direct $C\\!P$ violation, and (c) assuming $C\\!P$ conservation. The dashed (solid) curves in (a) and (b) indicate the contours of the mixing parameters associated with $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ ($D^{0}$) decays. The best-fit value for $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ ($D^{0}$) decays is shown with an open (filled) point. The solid, dashed, and dotted curves in (c) indicate the contours of $C\\!P$-averaged mixing parameters at 68.3%, 95.5%, and 99.7% confidence level (CL), respectively. The best-fit value is shown with a point. The uncertainties incorporate both statistical and systematic contributions, since all relevant systematic effects depend on the true values of the mixing parameters, and are thus incorporated into the fit $\chi^{2}$. These include the uncertainty in the fraction of charm mesons from $b$-hadron decays, and their bias on the observed decay time; the uncertainty in the fraction of peaking background; and the uncertainty in the determination of the instrumental asymmetry. The statistical uncertainty is determined in a separate fit and used to calculate the systematic component by subtraction in quadrature. Direct $C\\!P$ violation would produce a nonzero intercept at $t=0$ in the efficiency-corrected difference of WS-to-RS yield ratios between $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons shown in Fig. 2 (c). It is parametrized by the asymmetry measured in the first fit $A_{D}\equiv{(R_{D}^{+}-R_{D}^{-})/(R_{D}^{+}+R_{D}^{-})}=(-0.7\pm 1.9)\%$. Indirect $C\\!P$ violation results in a time dependence of the efficiency- corrected difference of yield ratios. The slope observed in Fig. 2 (c) is about 5% of the individual slopes of Figs. 2 (a) and (b) and is consistent with zero. From the results of the fit allowing for direct and indirect $C\\!P$ violation, a likelihood for $\|q/p\|$ is constructed using the relations $x^{\prime\pm}=\|q/p\|^{\pm 1}(x^{\prime}\cos\phi\pm y^{\prime}\sin\phi)$ and $y^{\prime\pm}=\|q/p\|^{\pm 1}(y^{\prime}\cos\phi\mp x^{\prime}\sin\phi)$. Confidence intervals are derived with a likelihood-ratio ordering and assuming that the correlations are independent of the true values of the mixing parameters. The magnitude of $q/p$ is determined to be $0.75<\|q/p\|<1.24$ and $0.67<\|q/p\|<1.52$ at the $68.3\%$ and $95.5\%$ confidence levels, respectively. Significantly more stringent bounds on $\|q/p\|$ and additional information on $\phi$ are available by combining the present results with other measurements [10], in particular when also using theoretical constraints, such as the relationship $\tan\phi=x(1-\|q/p\|^{2})/y(1+\|q/p\|^{2})$ [25, 26], which applies in the limit that direct $C\\!P$ violation is negligible. In summary, $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillation is studied using $D^{+}\rightarrow D^{0}(\rightarrow K^{+}\pi^{-})\pi^{+}$ decays reconstructed in the full sample of $pp$ collisions, corresponding to $3\mbox{\,fb}^{-1}$ of integrated luminosity collected by the LHCb experiment in 2011 and 2012. Assuming $C\\!P$ conservation, the mixing parameters are measured to be $x^{\prime 2}=(5.5\pm 4.9)\times 10^{-5}$, $y^{\prime}=(4.8\pm 1.0)\times 10^{-3}$, and $R_{D}=(3.568\pm 0.066)\times 10^{-3}$. The observed parameters are consistent with, $2.5$ times more precise than, and supersede the results based on a subset of the present data [18]. Studying $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays separately shows no evidence for $C\\!P$ violation and provides the most stringent bounds on the parameters $A_{D}$ and $\|q/p\|$ from a single experiment. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. ## References [1] S. Bianco, F. Fabbri, D. Benson, and I. Bigi, A Cicerone for the physics of charm, Riv. 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Direct and indirect $C\\!P$ violation --- Results \| Correlations Parameter \| Fit value \| $R_{D}^{+}$ \| $y^{\prime+}$ \| $x^{\prime 2+}$ \| $R_{D}^{-}$ \| $y^{\prime-}$ \| $x^{\prime 2-}$ $R_{D}^{+}$ [ \| $10^{-3}$] \| $3.545\pm 0.095$ \| $1.000$ \| $-0.942$ \| $\phantom{-}0.862$ \| $-0.016$ \| $-0.007$ \| $\phantom{-}0.006$ $y^{\prime+}$ [ \| $10^{-3}$] \| $5.1\pm 1.4$ \| \| $\phantom{-}1.000$ \| $-0.968$ \| $-0.007$ \| $\phantom{-}0.007$ \| $-0.007$ $x^{\prime 2+}$ [ \| $10^{-5}$] \| $4.9\pm 7.0$ \| \| \| $\phantom{-}1.000$ \| $\phantom{-}0.005$ \| $-0.007$ \| $\phantom{-}0.008$ $R_{D}^{-}$ [ \| $10^{-3}$] \| $3.591\pm 0.094$ \| \| \| \| $\phantom{-}1.000$ \| $-0.941$ \| $\phantom{-}0.858$ $y^{\prime-}$ [ \| $10^{-3}$] \| $4.5\pm 1.4$ \| \| \| \| \| $\phantom{-}1.000$ \| $-0.966$ $x^{\prime 2-}$ [ \| $10^{-5}$] \| $6.0\pm 6.8$ \| \| \| \| \| \| $\phantom{-}1.000$ No direct $C\\!P$ violation --- Results \| Correlations Parameter \| Fit value \| $R_{D}$ \| $y^{\prime+}$ \| $x^{\prime 2+}$ \| $y^{\prime-}$ \| $x^{\prime 2-}$ $R_{D}$ [ \| $10^{-3}$] \| $3.568\pm 0.066$ \| $1.000$ \| $-0.894$ \| $\phantom{-}0.770$ \| $-0.895$ \| $\phantom{-}0.772$ $y^{\prime+}$ [ \| $10^{-3}$] \| $4.8\pm 1.1$ \| \| $\phantom{-}1.000$ \| $-0.949$ \| $\phantom{-}0.765$ \| $-0.662$ $x^{\prime 2+}$ [ \| $10^{-5}$] \| $6.4\pm 5.5$ \| \| \| $\phantom{-}1.000$ \| $-0.662$ \| $\phantom{-}0.574$ $y^{\prime-}$ [ \| $10^{-3}$] \| $4.8\pm 1.1$ \| \| \| \| $\phantom{-}1.000$ \| $-0.950$ $x^{\prime 2-}$ [ \| $10^{-5}$] \| $4.6\pm 5.5$ \| \| \| \| \| $\phantom{-}1.000$ No $C\\!P$ violation --- Results \| Correlations Parameter \| Fit value \| $R_{D}$ \| $y^{\prime}$ \| $x^{\prime 2}$ $R_{D}$ [ \| $10^{-3}$] \| $3.568\pm 0.066$ \| $1.000$ \| $-0.953$ \| $\phantom{-}0.869$ $y^{\prime}$ [ \| $10^{-3}$] \| $4.8\pm 1.0$ \| \| $\phantom{-}1.000$ \| $-0.967$ $x^{\prime 2}$ [ \| $10^{-5}$] \| $5.5\pm 4.9$ \| \| \| $\phantom{-}1.000$	arxiv-papers	2013-09-25T14:54:08	2024-09-04T02:49:51.515341	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C. Baesso, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th.\n Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous,\n I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy,\n R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A.\n Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci,\n A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E.\n Bowen, C. Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M.\n Britsch, T. Britton, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F.\n Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek,\n P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J.\n Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M.\n Coombes, S. Coquereau, G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, M.\n Cruz Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, P. David, P.N.Y. David,\n A. Davis, I. De Bonis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda,\n L. De Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del\n Buono, N. D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H.\n Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, P. Dornan, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, A. Falabella, C. F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M.\n Frank, C. Frei, M. Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L.\n Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, P. Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, P. Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E.\n Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew,\n S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A.\n Hicheur, E. Hicks, D. Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt,\n T. Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P.\n Ilten, R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M.\n John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W.\n Kanso, M. Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, O.\n Kochebina, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac,\n J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S.\n Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R.\n Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N.\n Lopez-March, H. Lu, D. Lucchesi, J. Luisier, H. Luo, O. Lupton, F. Machefert,\n I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J.\n Maratas, U. Marconi, P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A.\n Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins\n Tostes, A. Martynov, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E.\n Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B.\n Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J.\n Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J.\n Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, S.\n Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pearce, A. Pellegrino,\n G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, G. Pessina,\n K. Petridis, A. Petrolini, A. Phan, E. Picatoste Olloqui, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F. Polci, G. Polok, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B.\n Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz\n Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V.\n Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, M.\n Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J.\n Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, W. Sutcliffe, S. Swientek, V. Syropoulos, M.\n Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V.\n Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. 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1309.6700	# Spectral conditions for the existence of specified paths and cycles in graphs ††thanks: Supported by the NNSF of China (No.11101057), China Postdoctoral Science Foundation (No.20110491443) and Zhejiang Provincial Natural Science Foundation of China(LY12A01016). Mingqing Zhaia , Huiqiu Linb , Shicai Gongc aSchool of Mathematical Science, Chuzhou University, Anhui, Chuzhou, 239012, China bDepartment of Mathematics, East China University of Science and Technology, Shanghai, 200237, China cSchool of Science, Zhejiang A & F University, Zhejiang, Linan, 311300, China Corresponding author. E-mail addresses: [email protected] (M.Zhai); [email protected] (H.Lin); [email protected](S.Gong). ###### Abstract Let $G$ be a graph with $n$ vertices and $\lambda_{n}(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal graphs. This result gives spectral conditions for the existence of specified paths and cycles in graphs. AMS Classification: 05C50; 05C38 Keywords: Path; Cycle; Spectral radius; Least eigenvalue; Adjacency matrix ## 1 Introduction Let $G=(V,E)$ be the graph with vertex set $V$ and edge set $E$. Let $\|G\|$ and $e(G)=\|E(G)\|$ be the order and the size of $G$, respectively. A graph $G$ is said $H$-free, if $G$ does not contain a copy of $H$. Denote by $P_{t}$ and $C_{t}$ the path and cycle of order $t$, respectively. Let $N^{d}(u)=\\{v\|v\in V(G),d_{G}(v,u)=d\\}$, where $V(G)$ is the vertex set of $G$ and $d_{G}(v,u)$ is the distance between $u$ and $v$. Particularly, denote $N(u)=N^{1}(u)$ and $d(u)=\|N(u)\|$. For two disjoint vertex sets $V_{1}$ and $V_{2}$, let $e(V_{1},V_{2})$ be the number of edges with one endpoint in $V_{1}$ and another in $V_{2}$. Let $A(G)$ be the adjacency matrix of a graph $G$. The largest modulus of an eigenvalue of $A(G)$ is called the spectral radius of $G$ and denoted by $\rho(G)$. It is known that for a connected graph $G$, there exists a positive unit eigenvector corresponding to $\rho(G)$, which is called the Perron vector of $G$. A central problem of extremal graph theory is as follows: for a given graph $H$, what is the maximum number of edges of an $H$-free graph of order $n$? In the past decades, much attention has been paid to the spectral version of above question, that is, what is the maximum spectral radius of an $H$-free graph of order $n$? It is known that if $H$ is a star $K_{1,r+1}$, then the extremal graph is an $r$-regular graph (see [8]). In recent years, Fiedler and Nikiforov [6] solved the question for $H\cong C_{n}$ or $H\cong P_{n}$. Nikiforov solved the case $H\cong C_{2k+1}$ for large enough $n$ (see [9]) and the case that $H$ is a complete graph $K_{r+1}$ or a complete bipartite graph $K_{2,r+1}$ (see [10]). In [11], Nikiforov showed if $G$ is a $C_{4}$-free graph with $n$ vertices, then $\rho^{2}(G)-\rho(G)-(n-1)\leq 0$ with equality if and only if $G$ is the friendship graph (now $n$ is odd). Moreover, he conjectured if $G$ is a $C_{4}$-free graph with even vertices, then $\rho^{3}(G)-\rho^{2}(G)-(n-1)\rho(G)+1\leq 0$ with equality if and only if $G$ is a star of order $n$ with $\frac{n}{2}-1$ disjoint additional edges. Zhai and Wang [18] showed this conjecture is true. In [12], Nikiforov obtained the extremal graph for $H\cong P_{2k+2}$ or $H\cong P_{2k+3}$ and gave a conjecture on $H\cong C_{2k+2}$ (see [12], Conjecture 15.). This paper solves these two questions in the class of bipartite graphs (see Theorems 2.1 and 2.2). Let $\lambda_{n}(G)$ be the least eigenvalue of a graph $G$ of order $n$. It is known that $\lambda_{n}(G)=-\rho(G)$ for a bipartite graph $G$ (see [4]). Recently, researchers have begun to pay attention to the least eigenvalues of graphs with a given value of some well-known integer graph invariant: for instance: order and size [1, 2, 5, 13], unicyclic graphs with a given number of pendant vertices [7], matching number and independence number [14], number of cut vertices [15], connectivity, chromatic number [16], domination number [17]. This paper also gives a spectral extremal characterization on the least eigenvalue of graphs (see Theorems 1.3 and 1.4). ###### Lemma 1.1 Let $G=<X,Y>$ be a bipartite graph, where $\|X\|\geq k$, $\|Y\|\geq k-1$. If $G$ does not contain a copy of $P_{2k+1}$ with both endpoints in $X$, then $e(G)\leq(k-1)\|X\|+k\|Y\|-k(k-1).$ $None$ Equality holds if and only if $G$ is isomorphic to a complete bipartite graph $K_{\|X\|,\|Y\|}$, where $\|X\|=k$ or $\|Y\|=k-1$. ###### Lemma 1.2 Let $G$ be a $P_{2k+3}$-free bipartite graph. If $G$ does not contain a connected component isomorphic to $K_{k+1,k+1}$, then for an arbitrary vertex $u\in V(G)$, $e(N(u),N^{2}(u))\leq(k-1)\|N(u)\|+k\|N^{2}(u)\|-k(k-1),$ $None$ and equality holds if and only if the subgraph induced by $N(u)\cup N^{2}(u)$ is isomorphic to one of the following: (i) $K_{\|N(u)\|,\|N^{2}(u)\|}$, where $\|N(u)\|=k$ or $\|N^{2}(u)\|=k-1$; (ii) a graph obtained from $K_{k,\|N^{2}(u)\|}$ and an additional vertex $v\in N(u)$ by joining it with $k-1$ vertices of $N^{2}(u)$. ###### Theorem 1.3 Let $G$ be a $C_{t}$-free graph of order $n$. (i) If $t$ is odd or $n<t$, then $\lambda_{n}(G)\geq-\sqrt{\lfloor\frac{n}{2}\rfloor\lceil\frac{n}{2}\rceil}$, and equality holds if and only if $G\cong K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$. (ii) If $t=2k+2$ and $n\geq t$, then $\lambda_{n}(G)\geq-\sqrt{k(n-k)}$, and equality holds if and only if $G\cong K_{k,n-k}$. ###### Theorem 1.4 Let $G$ be a $P_{t}$-free graph of order $n$. (i) If $n<t$, then $\lambda_{n}(G)\geq-\sqrt{\lfloor\frac{n}{2}\rfloor\lceil\frac{n}{2}\rceil}$. Equality holds if and only if $G\cong K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$. (ii) If $n\geq t$, then $\lambda_{n}(G)\geq-\sqrt{k(n-k)}$, where $k=\lfloor\frac{t-2}{2}\rfloor$. Equality holds if and only if $G\cong K_{k,n-k}$ or $G\cong K_{2,2}\cup K_{1}$ for $(n,t)=(5,5)$. In the remaining part of this paper, we give the proofs of Lemmas 1.1 and 1.2, and Theorems 1.3 and 1.4. ## 2 Proofs Proof of Lemma 1.1. We use induction on $\|Y\|$. If $\|Y\|=k-1$, then $e(G)\leq e(K_{\|X\|,k-1})=(k-1)\|X\|.$ So, (1) follows and the equality holds if and only if $G\cong K_{\|X\|,k-1}$. Now we suppose that the inequality holds for $\|Y\|\leq M$, where $M\geq k$. Case 1. There exists a vertex $u\in Y$ such that $d_{G}(u)\leq k-1$, then by the hypothesis, $e(G-u)\leq(k-1)\|X\|+k(\|Y\|-1)-k(k-1).$ Hence we have $e(G)=e(G-u)+d_{G}(u)<(k-1)\|X\|+k\|Y\|-k(k-1).$ Case 2. For each vertex $u\in Y$, $d_{G}(u)\geq k$. We use induction on $\|X\|$. If $\|X\|=k$, then $e(G)\leq e(K_{k,\|Y\|})=k\|Y\|,$ so (1) follows and the equality holds if and only if $G\cong K_{k,\|Y\|}$. Suppose that the inequality holds for $\|X\|\leq M$ where $M\geq k+1$. Subcase 2.1. There exists a vertex $v\in X$, such that $d_{G}(v)\leq k-2$, then by the hypothesis, we have $e(G-v)\leq(k-1)(\|X\|-1)+k\|Y\|-k(k-1).$ Hence $e(G)=e(G-u)+d_{G}(u)<(k-1)\|X\|+k\|Y\|-k(k-1).$ Subcase 2.2. For each vertex $v\in X$, $d_{G}(v)\geq k-1$. We may assume that $G$ is connected. Otherwise, we use the induction hypothesis on each component of $G$ and the inequality follows immediately. Now we suppose that $P$ is a longest path in $G$ with two endpoints $w_{1},w_{2}$. Subcase 2.2.1. Either $w_{1}$ or $w_{2}$ in $Y$, say $w_{1}\in Y$. Then $\|P\cap Y\|\geq\|P\cap X\|$. Note that $G$ is $P_{2k+2}$-free, then $\|P\cap X\|\leq k$. Since $d_{G}(w_{1})\geq k$, thus we have $\|P\cap X\|=k$ and $N_{G}(w_{1})=P\cap X$. One can find now each vertex in $P\cap Y$ is an endpoint of a path of the same length with $P$. Similarly, we have $N_{G}(w)=P\cap X$ for any $w\in P\cap Y$. Then $X=P\cap X$ (otherwise we can find a path with length longer than $\|P\|$ which is forbidden). Thus we have $d_{G}(u)=k$ for each $u\in Y$ since $d_{G}(u)\geq k$ and $\|X\|=\|P\cap X\|=k$. Therefore $G\cong K_{k,\|Y\|}$ and then the inequality in (1) is an equality. Subcase 2.2.2. Both $w_{1}$ and $w_{2}$ are in $X$. Then $\|P\cap X\|=\|P\cap Y\|+1$. Note that $G$ does not contain a copy of $P_{2k+1}$ with both endpoints in $X$, then $\|P\cap Y\|\leq k-1$. Since $P$ is a longest path in $G$, thus $N_{G}(w_{1})\subset P\cap Y$. Hence we have $N_{G}(w_{1})=P\cap Y$ and $d_{G}(w_{1})=\|P\cap Y\|=k-1$ since $d_{G}(w_{1})\geq k-1$. Further, each vertex $u\in X$ is an endpoint of a path of the same length with $P$. So we have $d_{G}(u)=k-1$ for each vertex $u\in X$. Similarly, we have $Y=P\cap Y$. Therefore, $G\cong K_{\|X\|,k-1}$ and then the inequality in (1) is an equality. Thus we complete the proof. $\Box$ Proof of Lemma 1.2. Denote by $H$ the subgraph of $G$ induced by $N(u)\cup N^{2}(u)$. By Lemma 1.1, if $H$ does not contain a copy of $P_{2k+1}$ with both endpoints in $N(u)$, then Inequality (2) holds, and equality holds if and only if $G\cong K_{\|N(u)\|,\|N^{2}(u)\|}$ with $\|N(u)\|=k$ or $\|N^{2}(u)\|=k-1$. Now suppose that $H$ contains a path $P$ of order $2k+1$ with both endpoints in $N(u)$. Since $G$ is $P_{2k+3}$-free, $N(u)\backslash V(P)=\emptyset$ which implies that $\|N(u)\|=k+1$ and $\|N^{2}(u)\|\geq k$. We distinguish the following two cases. Case 1. There is a vertex $v\in N(u)$ such that $d_{H}(v)\leq k-1$. Now, $e(H-v)\leq e(K_{k,\|N^{2}(u)\|})=k\|N^{2}(u)\|=(k-1)\|N(u)\|+k\|N^{2}(u)\|-(k+1)(k-1),$ and equality holds if and only if $H-v\cong K_{k,\|N^{2}(u)\|}$. Therefore, $e(H)=e(H-v)+d_{H}(v)\leq(k-1)\|N(u)\|+k\|N^{2}(u)\|-k(k-1)$ If equality holds, then $d_{H}(v)=k-1$. Case 2. For each vertex $v\in N(u)$, $d_{H}(v)\geq k$. Now let $w\in N(u)$ be an endpoint of $P$. Note that $P$ is a longest path in $H$. Then $d_{H}(w)\geq k$ and $N_{H}(w)\subseteq V(P)\cap N^{2}(u)$. However, $\|V(P)\cap N^{2}(u)\|=k$. Hence $N_{H}(w)=V(P)\cap N^{2}(u)$. Further, we can find that each vertex $x\in N(u)$ must be an endpoint of a path of order $2k+1$ in $H$. Correspondingly, $N_{H}(x)=V(P)\cap N^{2}(u)$. This implies that $V(P)$ induces a copy of $K_{k+1,k}$. Since $G$ is $P_{2k+3}$-free, $N^{2}(u)\backslash V(P)=\emptyset$. Thus $H\cong K_{k+1,k}$. Since $G$ does not contain a connected component isomorphic to $K_{k+1,k+1}$, $N^{3}(u)\neq\emptyset$. We then have a path of order $2k+3$ in $G$, a contradiction. This completes the proof. $\Box$ ###### Theorem 2.1 Let $G$ be a $C_{t}$-free bipartite graph of order $n$. (i) If $t$ is odd or $n<t$, then $\rho(G)\leq\sqrt{\lfloor\frac{n}{2}\rfloor\lceil\frac{n}{2}\rceil}$. Equality holds if and only if $G\cong K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$. (ii) If $t=2k+2$ and $n\geq t$, then $\rho(G)\leq\sqrt{k(n-k)}$. Equality holds if and only if $G\cong K_{k,n-k}$. Proof. (i) Recall that $K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$ has maximal spectral radius among all bipartite graphs of order $n$. If $t$ is odd or $n<t$, then $\rho(G)\leq\sqrt{\lfloor\frac{n}{2}\rfloor\lceil\frac{n}{2}\rceil}$ since $G$ is bipartite. Equality holds if and only if $G\cong K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$. (ii) Let $G$ be a bipartite graph without $C_{2k+2}$ and $B=(b_{ij})_{n\times n}=A(G)^{2}-k(n-k)I,$ where $I$ is the identity matrix of order $n$. Since $G$ is $C_{2k+2}$-free, $G$ does not contain a copy of $P_{2k+1}$ with both endpoints in $N(u)$ for any $u\in V(G)$. Let $b(u)$ be the sum of the $u$-th row of $B$. Note that $(A(G)^{2})_{u}$ is the number of walks of length 2 which are started from $u$. Then by Lemma 1.1, for any $u\in V(G)$, $\displaystyle b(u)$ $\displaystyle=$ $\displaystyle\|N(u)\|+e(N(u),N^{2}(u))-k(n-k)$ $\displaystyle\leq$ $\displaystyle\|N(u)\|+(k-1)\|N(u)\|+k\|N^{2}(u)\|-k(k-1)-k(n-k)$ $\displaystyle=$ $\displaystyle k(\|N(u)\|+\|N^{2}(u)\|)-k(n-1)$ $\displaystyle\leq$ $\displaystyle k(n-1)-k(n-1)$ $\displaystyle=$ $\displaystyle 0.$ If the equality holds, then by the second inequality, $N^{3}(u)=\emptyset$. Further, by the first inequality and Lemma 1.1, $G-u\cong K_{\|N(u)\|,\|N^{2}(u)\|}$ with $\|N(u)\|=k$ or $\|N^{2}(u)\|=k-1$. This implies that $G\cong K_{k,n-k}.$ Let $X$ be an eigenvector of $G$ corresponding to $\rho(G)$ with $\sum_{i=1}^{n}x_{i}=1$. Then $\rho^{2}(G)-k(n-k)=\sum_{i=1}^{n}[\rho^{2}(G)-k(n-k)]x_{i}=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}b_{ij}x_{j}=\sum\limits_{u\in V(G)}b(u)x_{u}\leq 0.$ Thus we have $\rho(G)\leq\sqrt{k(n-k)}$. If the equality holds, then $b(u)=0$ for any $u\in V(G)$. Thus $G\cong K_{k,n-k}$. On the contrary, if $G\cong K_{k,n-k}$, then $\rho(G)=\sqrt{k(n-k)}.$ This completes the proof. $\Box$ ###### Theorem 2.2 Let $G$ be a $P_{t}$-free bipartite graph of order $n$. (i) If $n<t$, then $\rho(G)\leq\sqrt{\lfloor\frac{n}{2}\rfloor\lceil\frac{n}{2}\rceil}$. Equality holds if and only if $G\cong K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$. (ii) If $n\geq t$, then $\rho(G)\leq\sqrt{k(n-k)}$, where $k=\lfloor\frac{t-2}{2}\rfloor$. Equality holds if and only if $G\cong K_{k,n-k}$ or $G\cong K_{2,2}\cup K_{1}$ for $n=t=5$. Proof. Since $K_{\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil}$ has maximal spectral radius among all bipartite graphs of order $n$, (i) holds clearly. Next consider (ii). If $t=2k+2$, that is, $G$ is $P_{2k+2}$-free, then $G$ is also $C_{2k+2}$-free. Note that $K_{k,n-k}$ does not contain a copy of $P_{2k+2}$. By (ii) of Theorem 2.1, the result holds. If $t=2k+3$, then $n\geq 2k+3$. Note that $\rho(K_{k,n-k})=\sqrt{k(n-k)}\geq\sqrt{k(k+3)}\geq(k+1)=\rho(K_{k+1,k+1}).$ Above inequalities become equalities if and only if $n=2k+3=5$, that is, $n=t=5$. So we may assume that $G$ does not contain a connected component isomorphic to $K_{k+1,k+1}$. Then by Lemma 1.2, $e(N(u),N^{2}(u))\leq(k-1)\|N(u)\|+k\|N^{2}(u)\|-k(k-1)$ for any $u\in V(G)$. Let $X$ be an eigenvector of $G$ corresponding to $\rho(G)$ with $\sum_{i=1}^{n}x_{i}=1$ and $B=(b_{ij})_{n\times n}=A^{2}(G)-k(n-k)I.$ Similar to the proof of Theorem 2.1, we have $\rho^{2}(G)-k(n-k)=\sum_{i=1}^{n}[\rho^{2}(G)-k(n-k)]x_{i}=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}b_{ij}x_{j}=\sum\limits_{u\in V(G)}b(u)x_{u}\leq 0.$ Hence, $\rho(G)\leq\sqrt{k(n-k)}$. If the equality holds, then $b(u)=0$ for any $u\in V(G)$. This also implies that $N^{3}(u)=\emptyset$ and (2) becomes an equality for any $u\in V(G)$. Note that if the case (ii) of Lemma 1.2 occurs, then $d_{G}(u)=k+1$ and $d_{G}(v)=k$. Thus either $G-u\cong K_{k,\|N^{2}(u)\|}$ or $G-u\cong K_{\|N(u)\|,k-1}$ (otherwise, there will exist a vertex $v\in V(G)$ such that $b(v)<0$). So $G\cong K_{k,n-k}$. we complete the proof. $\Box$ The following lemma is very useful in characterizing the extremal graph with minimal least eigenvalue. ###### Lemma 2.3 ([3]) If $G$ is a simple graph with $n$ vertices, then there exists a spanning bipartite subgraph $H$ of $G$ such that $\lambda_{n}(G)\geq\lambda_{n}(H)$ with equality if and only if $G\cong H$. Using Lemma 2.3 and Theorems 2.1 and 2.2, we immediately get Theorems 1.3 and 1.4. Acknowledgment Thanks to Prof. Guanghui Xu who organized a workshop in Zhejiang A & F University during Aug. 04-10. ## References * [1] F.K. Bell, D. Cvetković, P. Rowlinson, S.K. Simić, Graph for which the least eigenvalues is minimal, I, Linear Algebra and its Applications, 429 (2008) 234-241. * [2] F.K. Bell, D. Cvetković, P. Rowlinson, S.K. Simić, Graph for which the least eigenvalues is minimal, II, Linear Algebra and its Applications, 429 (2008) 2168-2179. * [3] D.S. Cao, Y. Hong, The distribution of eigenvalues of graphs, Linear Algebra and its Applications, 216 (1995) 211-224. * [4] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs, third ed., Barth, Heidelberg, 1995. * [5] Y.Z. Fan, Y. Wang, Y.B. Gao, Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread, Linear Algebra and its Applications, 429 (2-3) (2008) 577-588. * [6] M. Fiedler, V. Nikiforov, Spectral radius and Hamiltonicity of graphs, Linear Algebra and its Applications, 432 (2010) 2170-2173. * [7] R.F. Liu, M.Q. Zhai, J.L. Shu, The least eigenvalues of unicyclic graphs with $n$ vertices and $k$ pendant vertices, Linear Algebra and its Applications, 431 (5-7) (2009) 657-665. * [8] H. Minc, Nonnegative matrices, John Wiley and Sons, New York, 1988. * [9] V. Nikiforov, A spectral condition for odd cycles in graphs, Linear Algebra and its Applications, 428 (2008) 1492-1498. * [10] V. Nikiforov, Bounds on graph eigenvalues II, Linear Algebra and its Applications, 427 (2007) 183-189. * [11] V. Nikiforov, The maximum spectral radius of $C_{4}$-free graphs of given order and size, Linear Algebra and its Applications, 430 (2009) 2898-2905. * [12] V. Nikiforov, The spectral radius of graphs without paths and cycles of specified length, Linear Algebra and its Applications, 432 (2010) 2243-2256. * [13] M. Petrović, B. Borovićanin, T. Aleksić, Bicyclic graphs for which the least eigenvalue is minimum, Linear Algebra and its Applications, 430 (4) (2009) 1328-1335. * [14] Y.Y. Tan, Y.Z. Fan, The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph, Linear Algebra and its Applications, 433 (4) (2010) 790-795. * [15] Y. Wang, Y.Z. Fan, The least eigenvalue of a graph with cut vertices, Linear Algebra and its Applications, 433 (1) (2010) 19-27. * [16] M.L. Ye, Y.Z. Fan, D. Liang, The least eigenvalue of graphs with given connectivity, Linear Algebra and its Applications, 430 (4) (2009) 1375-1379. * [17] B.X. Zhu, The least eigenvalue of a graph with a given domination number, Linear Algebra and its Applications, 437 (2012) 2713-2718. * [18] M.Q. Zhai, B. Wang, Proof of a conjecture on the spectral radius of $C_{4}$-free graphs, Linear Algebra and its Applications, 437 (2012) 1641-1647.	arxiv-papers	2013-09-26T01:14:21	2024-09-04T02:49:51.535567	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Mingqing Zhai, Huiqiu Lin and Shicai Gong", "submitter": "HuiQiu Lin", "url": "https://arxiv.org/abs/1309.6700" }
1309.6711	# A Spin-Light Polarimeter for Multi-GeV Longitudinally Polarized Electron Beams Prajwal Mohanmurthy, and Dipangkar Dutta P. Mohanmurthy and D. Dutta are with the Department of Physics and Astronomy at Mississippi State University, Mississippi State, MS 39762.Manuscript received July 11, 2013 ###### Abstract The physics program at the upgraded Jefferson Lab (JLab) and the physics program envisioned for the proposed electron-ion collider (EIC) include large efforts to search for interactions beyond the Standard Model (SM) using parity violation in electroweak interactions. These experiments require precision electron polarimetry with an uncertainty of $<$ 0.5 %. The spin dependent Synchrotron radiation (SR), called ”spin-light,” can be used to monitor the electron beam polarization. In this article we develop a conceptual design for a “spin-light” polarimeter that can be used at a high intensity, multi-GeV electron accelerator. We have also built a Geant4 based simulation for a prototype device and report some of the results from these simulations. ###### Index Terms: Polarized electrons, synchrotron radiation, spin light, differential ionization chambers. ## I Introduction The determination of the longitudinal polarization of the electron beam is one of the dominant systematic uncertainties in any parity violating electron scattering (PVES) experiment. In order to achieve the desired high precision, the polarization of the electron beam must be monitored continuously with an uncertainty of $<$0.5%. These ambitious goals can be achieved if multiple independent and high precision polarimeters are used simultaneously. In addition to being precise, the polarimeters must be non-invasive and must achieve the desired statistical precision in the shortest time possible. Compton and Møller polarimeters are typically the polarimeters of choice for these experiments and are essential to achieve the desired precision. However, a complimentary polarimetry technique based on the spin dependence of synchrotron radiation, referred to as “spin-light,” can be used as a relative polarimeter. A spin-light polarimeter could provide additional means for improving the systematic uncertainties and when calibrated against a Compton/Møller polarimeter it could provide a stable continuous monitoring of the beam polarization. We develop the conceptual design for a continuous polarimeter based on “spin-light”. The proposed spin-light polarimeter can achieve statistical precision of $<$ 1% in measurement cycles of less than 10 minutes for 4 - 20 GeV electron beams with a beam currents of $\sim$ 100 $\mu$A. TABLE I: A comparison of the Compton, Møller and Spin-light polarimeters. Compton \| Spin-Light \| Møller ---\|---\|--- non-invasive, \| non-invasive, \| invasive continuous \| continuous \| analyzing power \| analyzing power \| analyzing power energy dependent \| energy dependent \| energy independent high currents \| moderately \| low currents \| high currents \| target is 100% \| no target \| target is $<$ 10% polarized \| needed \| polarized (requires stable laser) \| \| electron & photon \| beam left & right \| no independent detection are \| detectors provide \| measurements two independent \| two independent \| possible measurements \| measurements \| high precision \| high precision \| high precision absolute polarimeter \| relative polarimeter \| absolute polarimeter Best reported [3] \| expected \| Best reported [1] instrumental \| instrumental \| instrumental uncertainty: 0.4% \| uncertainty: 0.6% \| uncertainty: 0.47% Best reported [3] \| estimated \| Best achieved [2] absolute \| absolute \| absolute uncertainty: 0.5% \| uncertainty: $\sim$5% \| uncertainty: 0.85% Møller and Compton polarimeters have a proven track record of very high precision, the JLab Hall-C Møller polarimeter has an instrumental uncertainty of 0.47% [1] and absolute uncertainty of 0.85% [2], while the Compton polarimeter used in the SLD experiment achieved an instrumental uncertainty of 0.4% [3] and an absolute uncertainty of 0.5% [3], hence they are essential for any PVES program. However, a spin-light polarimeter would have a few operational and instrumental advantages over conventional polarimeters, such that when used in parallel with Compton/Møller polarimeters they might help reduce the systematic uncertainties and achieve the very high precision essential for the future PVES program. For example, Møller polarimeters use a polarized Fe target, and the polarization of the target is difficult to determine and may depend on the beam intensity. Moreover, Møller polarimeters operate at low current, and are invasive to the primary experiment. Compton polarimeters require a stable laser (the photon target) and are very sensitive to backgrounds. The proposed spin-light polarimeter is a target free device, hence it should be easier to operate over long periods, with its stability governed just by the stability of the electron beam. Moreover, this novel polarimeter would facilitate cross-checks and systematic studies when used with other conventional polarimeters. On the other hand, one of the disadvantages is that the proposed device can achieve comparable instrumental uncertainties only as a relative polarimeter, whereas the absolute polarization is what is required in the PVES experiments. Nevertheless a precise and stable relative polarimeter can be a very useful device. The spin- light polarimeter could be used in conjunction with a Compton polarimeter, such that the difficult to operate Compton polarimeter is used for calibration and the easier to operate and stable spin-light polarimeter is used to continuous monitor the beam polarization. Moreover, only the Møller and the spin-light polarimeters allow measurement of the transverse component of a longitudinally polarized electron beam. The key features of conventional polarimeters and a spin-light polarimeter are summarized in Table I. In this article we discuss the theory behind the spin-light polarimeter and present a complete conceptual design for such a polarimeter. In order to study some of the systematic uncertainties of the novel spin-light polarimeter we have built a Geant4 [4] based simulation of the polarimeter. The results from these simulations are also presented here. This work was inspired by a 1993 proposal by Karabekov and Rossmanith [5]. ## II Synchrotron Radiation At the typical energies and magnetic fields of present day terrestrial accelerators, the intensity and angular distribution of SR from an electron moving along a curved path, under the influence of a magnetic field, is described with high precision by classical electrodynamics [6]. The total radiation rate for highly relativistic electrons, $P^{clas}$ is $\propto E^{4}$ as given by the Larmor formula: $P^{clas}=\frac{2}{3}\frac{e^{2}\gamma^{4}c}{R^{2}}$, where $e$ is the electron charge, $v_{e}$ is the velocity of the electron, $c$ is the velocity of light in vacuum, $\beta=\frac{v_{e}}{c}$, $\gamma=\frac{1}{\sqrt{1-\beta^{2}}}=\frac{E}{m_{e}c^{2}}$ is the Lorentz boost, $m_{e}$ is the rest mass of the electron, and, $R$ is the radius of curvature of the electron orbit. The angular distribution of the radiated power is given by $\frac{dP}{d\Omega}=\frac{e^{2}\gamma^{4}c}{4\pi R^{2}}\frac{(1-\beta\cos\theta)^{2}-(1-\beta^{2})\sin^{2}\theta\cos^{2}\phi}{(1-\beta\cos\theta)^{5}},$ (1) where the angles $(\theta,\phi)$ are measured with respect to the direction of the electron’s motion. For highly relativistic electrons ($\gamma>>1$), the radiation is Lorentz boosted in the forward direction with an opening angle $\theta\approx 1/\gamma$. The classical theory also tells us that SR is strongly linearly polarized [7], with $P_{\sigma}=\frac{7}{8}P$ and $P_{\pi}=\frac{1}{8}P$, where $\sigma$-component is the one where the electric field lies in the plane of the electron orbit while $\pi$-component is the one where the electric field lies in the plane perpendicular to the orbital plane. Experimentally, these properties of SR have been demonstrated with high precision for a wide range of frequencies [8] \- [11]. Classical theory also tells us that the cone of SR passes over a fixed angle $\theta$ in retarded 111accounts for the finite delay between the photon emission and detection time $\Delta t^{{}^{\prime}}\approx\frac{\Delta\theta}{\omega_{0}}=\frac{1}{\gamma\omega_{0}}$, where $\omega_{0}=v_{e}/R\approx c/R$ is the Larmor frequency for the electron. For a distant observer the corresponding time interval is $\Delta t=\Delta t^{{}^{\prime}}(1-\beta)\approx 1/(2\gamma^{3}\omega_{0})$. Hence the spectral width of the radiation is $\Delta\omega\approx 1/\Delta t=2\gamma^{3}\omega_{0}$. This implies that the radiation is strong at high harmonics of the Larmor frequency $(\omega_{0})$ and can be considered as continuous. However from quantum mechanics we know that the radiation at frequency $\omega$ consists of photons of energy $\hbar\omega$. Thus there must exist a sufficiently strong acceleration such that a single photon will carry away all of the electron’s energy ($\gamma m_{e}c^{2}=\hbar\omega_{c}$). The critical magnetic field strong enough to provide this acceleration is found to be $B_{crit}=\frac{m^{2}c^{3}}{e\hbar}$ = 4.41$\times$109 Tesla. The frequency of the photon radiated under the influence of $B_{crit}$ is called the critical frequency ($\omega_{c}$) and is given by$\omega_{c}=\frac{3}{2}\gamma^{3}\omega_{0}=\frac{3}{2}\gamma^{3}c/R$. From this we get the critical energy $E_{crit}=m_{e}c^{2}\sqrt{\frac{m_{e}cR}{\hbar}}\approx$ 106 GeV. The extremely large values of the critical field and critical energy help explain why the classical theory is successful at the energies and fields accessible at present day accelerators. However, it turns out that several quantum effects appear at considerably lower electron energies. ### II-A Quantum Theory of Synchrotron Radiation The exact expression for SR intensity including quantum corrections was calculated by Sokolov, Ternov and Klepikov, based on the solution to the Dirac equation in the framework of quantum electrodynamics [12]. They showed that at energies above a few 100 MeV, there would be fluctuations in the radius of the electron orbit leading to radial oscillations [13] of the electron trajectory and quantum widening of the trajectory similar to Brownian motion. These oscillations and widening of the trajectory are essential in determining the dynamics of electrons in an accelerator, specially storage rings. Sokolov and Ternov also developed the mathematics required to describe the spin of relativistic electrons moving in an external electromagnetic field [14, 15], which allowed them to calculate the electron spin related properties of SR. The power radiated by electrons undergoing transitions $n\rightarrow n^{{}^{\prime}}$ (related to the radius of the electron orbit), $s\rightarrow s^{{}^{\prime}}$ (quadratic fluctuation of the radius) and $j\rightarrow j^{{}^{\prime}}$ (spin orientations with respect to the magnetic field), integrated over angles and summed over polarization states, is given by [14, 15]: $P=P^{clas}\times\frac{9\sqrt{3}}{16\pi}\sum_{s^{{}^{\prime}}}\int_{0}^{\infty}\frac{ydy}{(1+\xi y)^{4}}I^{2}_{ss^{{}^{\prime}}}(x)F(y),$ (2) where, $\displaystyle F(y)$ $\displaystyle=$ $\displaystyle\frac{1+jj^{{}^{\prime}}}{2}[2(1+\xi y)\int_{y}^{\infty}K_{5/3}(x)dx$ (3) $\displaystyle+$ $\displaystyle\frac{1}{2}\xi^{2}y^{2}K_{2/3}(y)-j(2+\xi y)\xi yK_{1/3}(y)]$ $\displaystyle+$ $\displaystyle\frac{1-jj^{{}^{\prime}}}{2}\xi^{2}y^{2}\left[K_{2/3}(y)+lK_{1/3}(y)\right]$ where $P^{clas}$ is the classical expression for SR power radiated, $y=\frac{\omega}{\omega_{c}}$, $x=\frac{3}{4}\frac{\xi\gamma^{3}y^{2}}{(1+\xi y)^{2}}$, $I_{ss^{{}^{\prime}}}$ are the Laguerre functions, $K_{n}(x)$ are modified Bessel functions, and $\xi=\frac{3}{2}\frac{B}{B_{crit}}\gamma$ is the critical parameter. Because, $B_{crit}$ = 4.41$\times$ 109 Tesla we have $\xi<<1$ for magnetic fields used at all man-made accelerators and the above expression can be expanded in terms of the critical parameter $\xi$ to get [16]; $\displaystyle P$ $\displaystyle=$ $\displaystyle P^{clas}[(1-\frac{55\sqrt{3}}{24}\xi+\frac{64}{3}\xi^{2})$ (4) $\displaystyle-$ $\displaystyle(\frac{1+jj^{{}^{\prime}}}{2})(j\xi+\frac{5}{9}\xi^{2}\frac{245\sqrt{3}}{48}j\xi^{2})$ $\displaystyle+$ $\displaystyle(\frac{1-jj^{{}^{\prime}}}{2})(\frac{4}{3}\xi^{2}+\frac{315\sqrt{3}}{432}j\xi^{2})+...]$ These quantum corrections to the classical expression for charge radiation ($P^{clas}$) involve contributions from the electron recoil effects of radiation, interference of the charge radiation and radiation due to the intrinsic magnetic moment of the electron, magnetic moment radiation due to Larmor precession, magnetic moment radiation due to Thomas precession, interference between the Larmor and Thomas radiation and radiation due to the anomalous magnetic moment of an electron [16]. The lowest order spin-dependent correction is of order $\xi$ and the lowest order spin-flip correction term is of order $\xi^{2}$. The difference (of order $\xi$) between the expression for power radiated by polarized and unpolarized (spin averaged) electron beams has the form, $\displaystyle P^{pol}-P^{unpol}$ $\displaystyle=$ $\displaystyle-j\xi P^{clas}\int_{0}^{\infty}\frac{9\sqrt{3}}{8\pi}y^{2}K_{1/3}(y)dy.$ (5) The above expression is directly related to the spin polarization of an electron beam $j$, hence the difference $P^{pol}-P^{unpol}=P^{spin}$ can be called “spin-light” [14, 15]. This offers a new possibility for visual or direct observation of the polarization characteristics of an electron beam by determining the SR power at a fixed range of spectral frequency. The spin dependence of the SR was verified at the VEPP-4 storage ring in Novosibirsk [17], using a 3 pole wiggler magnet (called a magnetic snake). The intensity of the SR produced by the wiggler for transversely polarized electrons was monitored while the beam was periodically depolarized using a RF field. The measured variation in SR intensity with polarization matched exactly with the expectation from the Sokolov-Ternov theory. This spin dependent part of the SR has been successfully used at the VEPP-4 to monitor the transverse polarization of the electron beam. The spin-flip term has a unique effect on circular accelerators. The radiation probability with spin flip is given by $w^{\uparrow\downarrow}=\frac{1}{\tau}\left(1+j\frac{8\sqrt{3}}{15}\right)$, where $j=$ 1 is for spin along the magnetic field and $j=$ -1 is for spin opposite to the magnetic field [14]. Using this relation, Sokolov and Ternov had predicted that, over time the beam in a circular accelerator would eventually become polarized opposite to the direction of the magnetic field. In other words the beam of a circular accelerator becomes transversely polarized over time via self polarization. This phenomena known as the Sokolov-Ternov self polarization was first observed at the French storage ring [18] at Orsay and is now routinely used to polarize beams at circular electron accelerators such as DESY. ## III Spin Light In the discussion above the spin orientation $j$ is relative to the magnetic field that produces the SR, with $j=$ 1 along the magnetic field. Polarized electron beams have longitudinal ($p_{z}$) and transverse ($p_{\perp}$) components relative to the beam direction and the transverse polarization have vertical and horizontal components. For a vertically oriented magnetic field, the total SR power from transversely polarized electrons, ignoring spin flip terms and other terms of order $\xi^{2}$, is given by [14]: $\displaystyle P_{\gamma}(tran)$ $\displaystyle=$ $\displaystyle\frac{9n_{e}}{16\pi^{3}}\frac{ce^{2}}{R^{2}}\gamma^{5}\int_{0}^{\infty}\frac{y^{2}dy}{(1+\xi y)^{4}}\oint d\Omega(1+\alpha^{2})^{2}$ (6) $\displaystyle\times$ $\displaystyle[K^{2}_{2/3}(z)+\frac{\alpha^{2}}{1+\alpha^{2}}K^{2}_{1/3}(z)$ $\displaystyle-$ $\displaystyle p_{\perp}\xi y\frac{1}{\sqrt{1+\alpha^{2}}}K_{1/3}(z)K_{2/3}(z)],$ where $n_{e}$ is the number of electrons, $z=\frac{\omega}{2\omega_{c}}(1+\alpha^{2})^{3/2}$, and $\alpha=\gamma\psi$, where $\psi$ is the vertical angle in the frame of the moving electron. The rest of the symbols are as defined in previous instances. The polarization dependent term in the above expression is an even function of the vertical angle therefore when integrated over all angles it makes the total SR power spin dependent. Thus by measuring this spin dependence in the total SR power radiated one can measure the transverse polarization of the electron beam. On the other hand, the total SR power from longitudinally polarized electrons, ignoring spin flip terms and other terms of order $\xi^{2}$, is given by [14]: $\displaystyle P_{\gamma}(long)$ $\displaystyle=$ $\displaystyle\frac{9n_{e}}{16\pi^{3}}\frac{ce^{2}}{R^{2}}\gamma^{5}\int_{0}^{\infty}\frac{y^{2}dy}{(1+\xi y)^{4}}\oint d\Omega(1+\alpha^{2})^{2}$ (7) $\displaystyle\times$ $\displaystyle[K^{2}_{2/3}(z)+\frac{\alpha^{2}}{1+\alpha^{2}}K^{2}_{1/3}(z)$ $\displaystyle+$ $\displaystyle p_{z}\xi y\frac{\alpha}{\sqrt{1+\alpha^{2}}}K_{1/3}(z)K_{2/3}(z)],$ The spin dependent term in the above expression is an odd function of the vertical angle therefore when integrated over all angles it goes to zero and the total SR power for longitudinally polarized electrons is spin independent. However, the power radiated into the space above ($0<\psi<\pi/2$) and below ($-\pi/2<\psi<0$) the orbital plane of the electron are different and the difference between them is spin dependent. Therefore, by measuring this spatial asymmetry one can monitor the longitudinal polarization of the electron beam. For $\xi y<<1$, if we divide the above expression by the energy of the radiated photon, $E_{\gamma}=\frac{3}{2}\frac{\hbar c}{R}\gamma^{3}y$ we get the total number of photons radiated into a finite horizontal angle $\Delta\theta$ as, Figure 1: (a) The total SR power radiated, per MeV (red), and the spin dependent difference in power radiated, per MeV (blue), above and below the orbital plane, for 11 GeV, longitudinally polarized electrons in a 4 T magnetic field and 100 $\mu$A current, and 10% detector efficiency. (b) The total number of SR photons per MeV and the number of “spin-light” photons per MeV, and (c) the asymmetry $\frac{\Delta N}{N}$ as a function of photon energy. (d) The time for 1% statistical uncertainty, in seconds. $\displaystyle N_{\gamma}$ $\displaystyle=$ $\displaystyle\frac{3}{4\pi^{2}}\frac{1}{137}\frac{I_{e}}{e}\gamma\Delta\theta\int_{y_{1}}^{y_{2}}ydy\int_{-\alpha}^{\alpha}(1+\alpha^{2})^{3/2}$ (8) $\displaystyle\times$ $\displaystyle\left[K^{2}_{2/3}(z)+\frac{\alpha^{2}}{1+\alpha^{2}}K^{2}_{1/3}(z)\right]d\alpha,$ where $I_{e}$ is the beam current. The difference in the photon flux radiated in the the space above and below the electron orbit is given by, $\displaystyle\Delta N_{\gamma}(p_{z})$ $\displaystyle=$ $\displaystyle\frac{3}{\pi^{2}}\frac{1}{137}\frac{I_{e}}{e}p_{z}\xi\gamma\Delta\theta\int_{y_{1}}^{y_{2}}y^{2}dy\int_{0}^{\alpha}\alpha(1+\alpha^{2})^{3/2}$ (9) $\displaystyle\times$ $\displaystyle K_{1/3}(z)K_{2/3}(z)d\alpha$ To examine the size and characteristics of the spin dependence we have numerically integrated the above two expressions for longitudinally polarized electron with 100% polarization, in a 4 Tesla magnetic field, with $I_{e}$ = 100 $\mu$A, and $E_{e}$ = 11 GeV. We have integrated over a horizontal angular acceptance of $\Delta\theta$ = 10 mrad, and a vertical acceptance of $\alpha=\pm$1\. The characteristic spectra of SR and spin-light obtained from these numerical intragrations are shown in Fig. 1. The total power radiated $P_{\gamma}$(long) and the spin dependent difference of power radiated above and below the orbital plane of the electron $\Delta P$(long) are shown as a function of photon energy in Fig 1 (a). The number of SR photons $N_{\gamma}$(long) and the number of spin-light photons $\Delta N_{\gamma}$(long), as function of photon energy are shown in Fig 1 (b). The asymmetry defined as $A=\frac{\Delta N_{\gamma}(long)}{N_{\gamma}(long)}$ as a function of photon energy is shown in Fig. 1 (c). Figure 2: (a) The spin dependent asymmetry as a function of ratio of photon energy to the critical photon energy, for electron beam energy, Ebeam = 4 – 20 GeV. (b) The spin dependent asymmetry for magnetic field, Bwigg = 2 – 5 Tesla. Figure 3: (top) The total SR power radiated as a function of the electron beam energy, for a 100 $\mu$A beam current and Bwigg = 4 Tesla. (bottom) The vertical size of the SR beam spot as a function of the electron beam energy, at a distance of 10 m from the wiggler magnet. Fig. 1 (c) indicates that one should measure the hard tail of the SR spectrum ($E_{\gamma}>$ 500 keV) and avoid the soft part of the spectrum where the asymmetry is low and changes rapidly with energy. Although the asymmetry is small $\sim$ 10-4 the photon flux is high, even at the hard tail of the spectrum, allowing a rapid determination of the asymmetry, with 1% statistical uncertainty within a few tens of seconds ($\frac{\delta A}{A}=\frac{1}{A\sqrt{2N}}$) as shown in Fig 1 (d). The energy dependence of the asymmetry for $E_{e}$ = 4 – 20 GeV and the magnetic field dependence of the asymmetry for $B_{wigg}$ = 2 – 5 T are shown in Fig 2(a) and Fig 2(b) respectively. These figures demonstrate that a spin-light based polarimetry is a very promising technique at intermediate energies and can be used to monitor the polarization of 4 – 20 GeV electrons in very rapid measurement cycles, with high statistical precision. Although the size of the asymmetry increases with increasing electron beam energy it should be noted that the total power of SR increases as the fourth power of the electron beam energy and increases linearly with beam current. Moreover, the vertical size of the SR spot decreases with increasing electron beam energy as shown in Fig. 3 (bottom). These factors impose practical limitations on the highest electron beam energies and the highest currents at which a spin-light polarimeter would be feasible. We estimate that it is best suited for the 4 - 20 GeV energy range for currents less than $\sim$ 10 mA. ## IV A Conceptual Design The two basic components of a spin-light based polarimeter are the source of SR and the X-ray detector which can measure the spatial asymmetry. ### IV-A The SR Source - Wiggler A three pole wiggler magnet with a magnetic field that has uniform magnitude but reversed direction at each pole and a short-long-short pole arrangement is well suited as a source of SR. The three poles must be symmetric about the center such that the line integral of the magnetic field in the direction of the motion of the electron, $z$, must be zero (i.e. $\int B(z)dz$ = 0), ensuring that it does not affect the electron beam transport and its spin direction (beyond the wiggler). The field being of opposite polarity at the 3 poles, flips the sign of the spin dependent spatial asymmetry from any two adjacent poles and hence when measured simultaneously it can help reduce systematic uncertainties arising from the vertical motion of the beam. The intensity and the asymmetry both increase with increasing field strength, while the pole length decreases with increasing field strength. Therefore a field strength of 4 T is a judicious choice for the wiggler field. A 10 mrad bend can be achieved with a pole length of 10 cm. Thus the total magnet length is 40 cm, and the spacing between the poles is optimized for ease of extraction and detection of the SR beam. A separation of 1 m between the poles allows for collimators to be placed that can separate the SR beams spots from the different poles. The small pole length ensures that the effect of spin- flip inducing SR and the fluctuation of the SR power are negligible ($<$ 0.1%). Wiggler magnets are regularly used at light sources around the world such as the Advanced Photon Source (APS) at Argonne National Lab and Spring8 in Japan. Some of these magnets are well suited for a spin-light polarimeter [22]. Figure 4: A position sensitive ionization chamber developed at the APS and SPring-8 with a resolution of 5$\mu$m when operated at photon flux of 5.0$\times$1012 8 keV photons/sec. [23] Note that the image shown here would constitute just one half of a differential ionization chamber required for spin-light polarimetry. #### IV-A1 Effect of the wiggler on the electron beam A non-invasive polarimeter is highly desirable and hence we must study the effect of fluctuations related to the quantum nature of SR produced by the wiggler. The effect of SR on the electron beam were carefully studied for the recirculating arcs [25], and the same methods can be used to calculate the influence of the wiggler on critical beam parameters. As described in Ref. [24] and [25], the distribution of energies lost by individual electrons in bending through some angle $\theta$ is given by a convolution of the distribution of the number of photons emitted and the distribution of energies of those photons. The number of photons emitted by a particular electron per radian bend will be distributed according to Poisson statistics about a mean value given by [24], $n=\frac{5}{2\sqrt{3}}\frac{\gamma}{137}=20.62E,$ where $n$ is the mean number of photons per radian bend, and $E$ is the beam energy in GeV. The average energy of the photons emitted is $E_{c}=\hbar\omega_{c}=\frac{3}{2}\frac{\hbar c\gamma^{3}}{R}$ [24]. Therefore the mean energy fluctuation is given by $\Delta E=\sqrt{n}E_{c}$. It is interesting to note that the energy fluctuation depends only on the electron beam energy and the bend radius of the wiggler. A beam of 11 GeV electrons in a 4T wiggler with a 10 m bend radius and a bend angle of 10 mrad, gives $n\sim$ 2 and $E_{c}$ = 199 keV. Therefore $\Delta E/E\sim$ 2.5$\times$10-5, which is comparable to the fluctuations due to the recirculating arcs of the JLab accelerator [25]. The SR power spectrum usually peaks at angles of $\pm 1/\gamma$ with respect to the electron direction. However, if an electron emits on the average two photons in a magnet, the angular distribution of the momentum kick received by each electron is peaked in the direction of the electron’s motion. The magnitude of the transverse kicks generated by the emission of a photon with energy $E_{c}$ in the direction $\theta_{\gamma}=1/\gamma$ with respect to the electron direction is given by [24], $\Delta\theta_{e}=\frac{E_{\gamma}\sin{\theta_{\gamma}}}{E_{e}}=11.3\times 10^{-9}\frac{E_{e}(GeV)}{R(m)}$. The r.m.s. kick from the emission of $n$ photons is given by $\sqrt{n}\Delta\theta_{e}$. Thus for a 11 GeV beam bend by 10 mrad the r.m.s. kick is $\sim$ 1.5 $\times$ 10-8 rad, which is negligible. Thus the wiggler magnet would have negligible influence on the electron beam and a spin-light polarimeter can be used for non-invasive monitoring of the beam polarization. #### IV-A2 Influence of the wiggler bend direction The wiggler bend direction was chosen to be beam-left for the conceptual design (see for example Figs. 8 & 9). Since the bend direction is transverse to the asymmetry direction one does not expect any systematic influence due to the choice of the bend direction. However, the design includes two symmetric pairs of ionization chambers placed on either side of the beam, and since the bend is small (10 mrad), it should be possible to build symmetric pairs of collimators and slits on both sides of the beam. With such a setup the independence of the spin-light asymmetry with respect to the wiggler bend direction can be directly verified during calibration and commissioning of the device. For the stability of operation, changes in bend direction during regular operation is not desirable. ### IV-B The X-ray Detector - Ionization Chamber The detector used to measure the spatial asymmetry must be sensitive to X-rays in the range of about 500 keV to 2.5 MeV and must be able to pick out a small asymmetry from a large spin independent background, it must be radiation hard, have low noise and be able to withstand high rates of $\sim$ 1012 photons/sec. Ionization chambers (IC) are well known for their high rate capability when operated as an integrating detector (i.e. in current mode), low electronic noise and radiation hardness. Argon/Xenon is an attractive candidate for use as an ionization medium, its high atomic number (18/54) and density (when compressed) gives it a high stopping power for hard X-rays and low energy gamma [26]. Over the last two decades, room temperature, high pressure ($>$ 50 atm, 0.55 g/cc) xenon (HPXe) ionization chambers have been developed with high detection efficiency in the 50 keV - 2.0 MeV range [19, 20, 21]. Figure 5: (left) A schematic of the split anode plate. (right) The dual differential ionization chamber for spin-light polarimetry. Another recent development, is the split collector ionization chamber that have turned the IC into a position sensitive device. Position sensitive ionization chambers are designed to have the collector plate split into two sections in a zig-zag/backgammon pattern such that each half operates as an independent ionization chamber. A prototype of such a chamber has been shown in Fig. 4. These chambers were developed at the APS at ANL and at the SPring-8 light source in Japan. They are used to measure the vertical position of X-ray beams and have been shown to have a resolution of 5 $\mu$m [23]. These chambers also have very low dark currents in the $\sim$pA range and have been operated at photon flux of 5.0$\times$1012 photons/sec. They work by measuring the difference in counts between the two halves of the chamber, i.e. they are differential ionization chambers (DIC). A position sensitive DIC operated in current mode can be used to measure the spatial asymmetry of the SR generated by longitudinally polarized electrons. A dual, 1 atm. Ar/Xe differential ionization chamber would be ideal for a relative polarimeter. A schematic for such an IC is shown in Fig. 5. The chamber would consist of Ti or stainless steel windows thick enough to cut down the low energy X-rays ($<$ 50 keV). A pair of split central anode plates (separated by a thin insulator) would be placed between the cathodes. The anode plates would be split in a backgammon pattern. The current measured on each half of the anode plates is amplified with a differential current amplifier. Figure 6: A schematic of the collimated beams of synchrotron radiation from 2 adjacent poles of the wiggler magnet incident on the differential ionization chambers. The figure on the left is a beam’s view of the electrodes of the DIC. The collimated radiation is shown as oval blobs, while the figure on the right is an isometric view of the electrodes, without showing the incident radiation. A magnified view of the synchrotron radiation from two adjacent poles of the wiggler magnet incident on a dual DIC is shown in Fig. 6. The left panel of the figure shows the beam’s view of the dual DIC. The collimated radiation is shown as oval blobs with the up-down asymmetry represented by the gray shading of the blob (the collimation scheme needed to achieve this is discussed in the next section). The right panel shows an isometric view of the electrodes in the DIC, without showing the incident radiation. The collimated SR beam from two adjacent poles will be incident on opposite sides of the anodes in the dual DIC. The spin-light spatial asymmetry (above and below the orbital plane) will have opposite sign in each half of the DIC because the magnetic field direction of the adjacent poles of the wiggler are opposite. On the other hand any spatial asymmetry due to vertical motion of the beam will have the same sign in the two halves of the dual DIC and hence should cancel to first order. Thus the dual DIC is essential to ensure that the spin-light polarimeter is insensitive to vertical beam motion. #### IV-B1 The Signal from the DICs If we denote $N^{L(R)}_{SR}$ as the number of SR photons on the left(right) of the anode plates, $N^{L(R)}_{spin}$ as the number of spin-light photons, and $\Delta N^{L(R)}_{z}$ as the difference in number of photons introduce by the vertical beam motion, then the contribution to the measured current from the top left part of the dual DIC will be (see Fig. 6); $I_{1}^{L}\propto N^{L}_{SR}+N^{L}_{spin}+\Delta N^{L}_{z},$ similarly the current contribution from the top right of the dual DIC is; $I_{1}^{R}\propto N^{R}_{SR}-N^{R}_{spin}+\Delta N^{R}_{z}.$ From the simulation studies we estimate that after collimation the size of these currents will be on the order of $\sim$ 10 nA. Note the change in the sign of the contribution from spin light photons because they are generated from adjacent poles of the wiggler while the contribution from vertical beam motion has the same sign. For the bottom left and right parts of the dual DIC we get; $I_{2}^{L}\propto N^{L}_{SR}-N^{L}_{spin}-\Delta N^{L}_{z},$ and $I_{2}^{R}\propto N^{R}_{SR}+N^{R}_{spin}-\Delta N^{R}_{z}.$ Thus, the signal from the top and bottom halves of the DIC, $S1$, and $S2$ as shown in Fig. 6, can be written as, $\displaystyle S1$ $\displaystyle\propto$ $\displaystyle(N^{L}_{SR}+N^{L}_{spin}+\Delta N^{L}_{z})-(N^{R}_{SR}-N^{R}_{spin}+\Delta N^{R}_{z})$ (10) $\displaystyle=$ $\displaystyle 2N_{spin},$ and $\displaystyle S2$ $\displaystyle\propto$ $\displaystyle(N^{L}_{SR}-N^{L}_{spin}-\Delta N^{L}_{z})-(N^{R}_{SR}+N^{R}_{spin}-\Delta N^{R}_{z})$ (11) $\displaystyle=$ $\displaystyle-2N_{spin}.$ Hence $S1-S2\propto 4N_{spin}\propto 4P_{e}$, and the vertical motion related asymmetry cancels to first order as does the corrections due to transverse polarization. However, it should be noted that the signals $S1+S2$ is proportional to the transverse polarization of the electron beam. This possibility of measuring both the longitudinal and transverse asymmetries in the same setup, provides further capability for reducing systematic uncertainties and makes the spin-light polarimeter an extremely versatile tool. Figure 7: (left) The number of photons absorbed in a 1-atm, 50 cm long Xe chamber. (right) The detector response weighted asymmetry. The number of photons absorbed in the ionization chamber can be calculated by multiplying the SR spectrum with the absorption function $A(\lambda)=1-e^{-\mu(\lambda)\cdot t}$, where $t$ is the length of the chamber, $\lambda$ is the photon wavelength and $\mu$ is the absorption coefficient which is obtained from NIST database [27]. The number of photons absorbed in a 50 cm long chamber with 1 atm Xe, is shown in Fig. 7(left). Also shown in the absorption weighted (or detector response weighted) asymmetry (right). ### IV-C Collimation The spacing between the wiggler poles was chosen to be 1 m to allow adequate room for the placement of collimators that would separate the SR beams from each pole. The flight path from the wiggler to the detector is selected to be 10 m, which implies that the SR spot size due to each of the wiggler poles will fan out over a horizontal length of $\sim$ 10 cm. The vertical width of the SR spot is only $\sim$ 1 mm. With appropriate placement of collimators on the wiggler pole entrance and exit faces, it is possible to separate the SR beam spot from the four different wiggler poles. The magnet system will wiggle the beam by 10 mrad in the horizontal plane such that each pole of the wiggler magnet produces a fan of synchrotron radiation in the horizontal plane as shown in the top view of the magnets (Fig 8). Figure 8: A schematic of the fan of synchrotron radiation produced as the electron beam traverses through each of the 4 poles of the wiggler magnet. The top view of the magnets has been shown. The two colors are used to indicate that the poles 1 and 4 have opposite polarity compared to poles 2 and 3 and therefore the sign of the asymmetry for the SR fans of the two colors are opposite. For longitudinally polarized electrons this fan of synchrotron radiation will have an up-down asymmetry in the vertical direction (due to spin-light). A series of collimators placed at the front and back faces of each pole of the wiggler magnet and at the center of the central pole of the wiggler will be used to select small angular ranges from the entire fan of synchrotron radiation as shown in the top view of the magnets in Fig 9. This collimation scheme enables separation of the synchrotron radiation from each pole of the wiggler magnet. Such a separation is necessary because the up-down asymmetry of the synchrotron radiation has opposite sign for each pole of the wiggler. Each of the collimated beams of synchrotron radiation will be separated by a few cm when they are projected into two symmetric dual DICs located at a distance of 10 m from the wiggler magnet. Figure 9: A schematic for the slits and collimators used to select small angular range of the fan of synchrotron radiation. The synchrotron beams from poles 1 and 2 will be incident on the dual DIC placed on the beam left and the beams from poles 3 and 4 are incident on the dual DIC placed on the beam right. ### IV-D The Complete Polarimeter Figure 10: A schematic for a differential spin-light polarimeter (not to scale). A 3D view of the complete spin-light polarimeter is shown in Fig 10. Each pole of the wiggler magnet is separated by a distance of 1 m and the two dual differential ionization chambers are placed 10 m from the last pole of wiggler magnet. The vertical backgammon split on the central anodes makes the DIC position sensitive in the vertical direction and hence the signal from the DIC is sensitive to the up-down asymmetry for each of the collimated beams of synchrotron radiation. The second dual DIC is necessary to provide an independent measurement of the up-down asymmetry and help reduced systematic uncertainties. The main parameters of this conceptual design are tabulated in Table II. TABLE II: Parameters of the SR polarimeter at 11 GeV \| Beam & Magnet ---\|--- Ee, current \| (GeV), ($\mu$A), \| 11.0, 100 Bwiggler \| (T) \| 4.0 Pole (total) length \| (m) \| 0.1 (0.4) Separation \| (m) \| 1.0 between poles \| \| bend angle \| (mrad) \| 10 vert. opening \| (mrad) \| 0.05 angle \| \| Flight path \| (m) \| 10 to det. \| \| \| SR and detector Nγ/s \| (Hz) \| 5.8$\times$ 1013 $\Delta$NSpin/s \| (Hz) \| 1.8$\times$ 109 Detector \| (1 atm Xe, cm) \| 50 medium \| \| Nabs/s \| (Hz) \| 3.1$\times$ 1012 $\Delta$Nfluctuation/s \| (Hz) \| 7.6$\times$ 106 vert. beam \| (mm) \| 1.0 spot after 10m \| \| flight \| \| ### IV-E Systematic Instrumental Uncertainties Some of the major sources of systematic uncertainties for a spin-light polarimeter include the background asymmetries from processes such as Bremsstrahlung and false asymmetry due to vertical beam motion, differences in chamber efficiency and magnetic field non-uniformity between adjacent poles of the wiggler. The measured experimental asymmetry from a spin light polarimeter can be written as $A_{expt}=A_{raw}(1+B/S)-A_{B}B/S+A_{F}$, where $S$ and $B$ are the signal and background, $A_{B}$ is the background asymmetry and $A_{F}$ is the false asymmetry due to factors such as vertical beam motion, differences in the chamber efficiency and differences in the field strength between adjacent poles. The main advantage of operating the ionization chambers as differential detectors is that the false asymmetries will cancel to first order. In addition the visible portion of the synchrotron light can be used to align the detectors and help control systematic uncertainties. The 3-pole design ensures that the vertical beam motion related false asymmetry also cancels to first order. However, the size of the background must be small compared to the signal. In order to address this issue, a full Geant4 [4] simulation of a proto-type spin-light polarimeter was built. The simulation was also used to study the effect of the asymmetry associated with the background. In addition to the built-in synchrotron radiation physics available in Geant4, we have implemented a spin-light generator using a parametrized model of spin-light at the nominal running conditions (11 GeV beam, 4T magnetic field and 10 mrad bend angle). Using this generator we were able to reproduce the expected photon energy spectrum and the expected asymmetry as shown in Fig. 11. Figure 11: The simulated photon spectrum and asymmetry compared to the calculated spectra for the nominal running conditions. The exact position and width of the slits/collimators on the front and back faces of the wiggler magnets were optimized iteratively to obtain the best separation between the SR photons originating from the different poles of the magnets. Currently, we are using slits only to illuminate the beam right split ionization chamber. A 3 mm sheet of lead shielding is also applied in front of the ion chamber. The ionization generated from the photons incident on the ion-chamber is integrated to obtain the signal from the chamber and this signal was used to study several of the systematic uncertainties, such as the effect of background radiation, the position/alignment of the collimators and the alignment of the ion chambers, on the asymmetry signal. The results of these studies have been tabulated in Table III. The simulated data show that the background from non-SR radiation is $\sim$ 1.6% and most of the background is at energies below 0.5 MeV. If the background related dilution can be determined to $\sim$ 1% the systematic uncertainty due of background radiation will be $\sim$ 0.5%. Moreover, in the proposed setup the background can be determined by measuring the difference in the signal from the chambers with the wiggler magnets turned on and off during calibration and commissioning of the device and at some relatively long interval during regular operations (these background measurements would be invasive in nature). The same Geant4 simulation was also used to determine the contribution to the systematic uncertainty due to the uncertainty in the exact position and width of the slits and collimators that are placed on the front and back faces of the poles of the wiggler magnet. These collimators are used to separate the SR photons originating from the various poles. Thus, any uncertainty in the position and width of the collimators can give rise to a false asymmetry due to mixing of spin-light and SR photons from different poles. The simulation demonstrates that a position/width uncertainty of 100 $\mu$m would lead to a uncertainty of $<$ 0.2%. We have also simulated the effects of the finite beam size on the up-down asymmetry. The simulations were performed for a beam with a Gaussian distribution of $\sigma$= 100 $\mu$m. The effects of the fringe fields at the tapered edges of the wiggler poles has also been studied using a Poisson Fish [28] model of the magnetic field. Although the absolute photon flux was reduced due the fringe fields the effect on the asymmetry was found to be minimal. We have studied the effects of beam halos on the spin-light asymmetry. The restrictive collimation scheme discussed in section IV-C ensures that the contribution from beam halo is very limited. For a ratio of peak to halo of $\sim~{}10^{8}$ the dilution to the spin-light asymmetry is several orders of magnitude smaller than the spin-light asymmetry. Thus, for a tightly controlled beam, halos should not pose a serious challenge. Most of the systematic uncertainties listed in Table III are independent of the electron beam energy in the 4 - 20 GeV range, however, because of the decrease in the SR spot size with increasing electron beam energy the uncertainty due the slit width scales inversely with beam energy. The proposed device is best used as a relative polarimeter, however, it can be used as an absolute device if the lower and upper bounds of the energy sensitivity of the DIC is determined accurately. The absolute value of the spin light asymmetry depends on the absolute value of the energy window over which the DIC signals are integrated. It is especially sensitive to the lower bound because of the rapid change in the SR intensity and the spin-light asymmetry as a function of decreasing photon energy. For an absolute measurement the lower bound of the integration window and the sensitivity to the vertical motion of the beam would be the two dominant sources of systematic uncertainty. Excellent energy resolutions have been demonstrated for HPXe ionization chambers [19, 20, 21]. With such high resolution ionization chambers one should be able to determine the response function and the lower bound of energy sensitivity of the chamber to better than 2%. Using a lower bound of 0.5 MeV, a variation of $\pm$ 10 keV in the lower bound results in a 2.5% change in the calculated asymmetry. An absolute measurement would be sensitive to vertical beam motion, a $\pm$ 0.1 mm variation in the vertical beam position can result in a 4% change in the calculated asymmetry. Thus, a spin-light polarimeter would only be capable of $\sim$ 5% absolute polarization measurement. A table of estimated systematic uncertainties is shown in Table III. We estimate the systematic instrumental uncertainties of a relative polarimeter to be $<$ 1%. TABLE III: Systematic instrumental uncertainties for a relative polarimeter at $E_{beam}=11$ GeV. Source \| Uncertainty \| $\frac{\delta A}{A}$ ---\|---\|--- Dark current \| $\sim$ pA \| $<$ 0.01% Intensity fluctuations \| $\Delta N\times$ 10-3 \| $<$0.1 % Beam energy \| 1.0$\times$10-3 \| $<$ 0.05 % Density of chamber gas, \| relative difference \| $<$0.01% Slit width \| 100 $\mu$m \| $<$0.2 % Background related \| known to 0.5% \| 0.5 % dilutions \| for B/S $\sim$ 0.02 \| Other dilutions \| cancel to first order \| $<$ 0.1% Halo contributions \| $10^{-8}$ \| $<$ 0.1 % Total \| \| 0.6 % ## V Conclusion Spin light based polarimetry was demonstrated over 30 years ago, but has been ignored since then. A spin-light polarimeter has several advantages over conventional polarimeters and when used in conjunction with a Compton polarimeter it could help provide a new benchmark for precision polarimetry. The 11 GeV beam at JLab or the electron beam at a future EIC would be well suited for spin light polarimetry and such a polarimeter would help achieve the $<$ 0.5 % polarimetry desired by experiments approved for the 12 GeV era and proposed for the EIC. A 3 pole wiggler with a field strength of 4 T and a pole length of 10 cm would be adequate for such a polarimeter. A dual position sensitive ionization chambers with split anode plates is ideally suited as the X-ray detector for such a polarimeter. The differential detector design would help reduce systematic uncertainties. ## Acknowledgment This work was supported in part by the U.S. Department of Energy under contract # DE-FG02-07ER41528, by the EIC Detector R&D grant from Brookhaven National Lab. One of us (P.M.) would also like the thank the Jefferson Science Associates for a JSA Fellowship. ## References * [1] M. Hauger et al., Nucl. Inst. Meth. A462, 382 (2001). * [2] To be reported for the polarization measured during the QWeak experiment at JLab, Hall-C. * [3] P. C. Rowson, D. Su, and S. Willocq, Ann. Rev. Nucl. Part. Sci. 51, 345 (2001); M. Woods, SLAC-PUB-7319 (1996). * [4] J. Allison et al., Nucl. Inst. and Meth. A506, 250 (2003); J. Allison et al., IEEE Trans. in Nucl. Science 53, 270 (2006). * [5] I. P. Karabekov, R. Rossmanith, Proc. of the 1993 PAC, Washington, v. 1, p. 457 (1993); I. P. Karabekov and S. I. Karabekian, Proceedings of 5th European Particle Accelerator Conference (EPAC 96), Sitges, Spain, 10-14 Jun 1996, pp 1743-1745 (1996); A. V. Airapetian, R. O. Avakian, I. P. Karabekov, E. L. Saldin, and M. V. Yurkov, Proc. of the SPIN-96, Amsterdam, The Netherlands, Vol.1, p762 (1996). * [6] D. D. Ivanenko, I. Pomeranchuk, Ya Zh. Eksp. Teor. Fiz. 16, 370 (1946); J. Schwinger, Phys. Rev. 75, 1912 (1947). * [7] G. A. Schott, Ann. Phys. 24, 635 (1907); A. A. Sokolov and I. M. Ternov, Zh. Eksp. Theor. Fiz. 31, 373 (1956), Sov. Phys. JETP 4, 396 (1957). * [8] F. R. Elder, R. V. Langmuir and H. C. Pollock, Phys. Rev. 74, 52 (1948). * [9] M. Yu Ado and P. A. Cherenkov, Sov. Phys. Dokl. 1, 517 (1957). * [10] F. A. Korolev, E. .N. Akimov, E. N. Markov, and O. F. Kulikov Sov. Phys. Dokl. 1, 568 (1957). * [11] P. Joos, Phys. Rev. Lett. 4, 558 (1960). * [12] A. A. Sokolov, N. P. Klepikov and I. M. Ternov, JETF 23, 632 (1952). * [13] A. A. Sokolov, and I. .M. Ternov, JETF 25, 698 (1953). * [14] A. A. Sokolov and I. .M. Ternov, Synchrotron Radiation, Pergamon Press, New York (1968); A. A. Sokolov and I. .M. Ternov, Radiation from Relativistic Electrons, A.I.P. Translation Series, New York (1986). * [15] I. M. Ternov, Physics - Uspekhi 38, 409 (1995). * [16] V. A. Bordovitsyn, Ph. D. Thesis, Moscow (1983); I. M. Ternov and V. A. Bordovitsyn, Vestn. Mosk. Univ. Ser. Fiz. Astr. 24, 69 (1983); V. A. Bordovitsyn and V. V. Telushkin, Nucl. Inst. and Meth. B266, 3708 (2008). * [17] S. A. Belomesthnykh et al., Nucl. Inst. and Meth. 227, 173 (1984). * [18] J. Le Duff, P. C. Marin, J. L. Manson, and M. Sommev, Orsay - Rapport Technique, 4-73 (1973). * [19] V. V. Dmitrenko et al., Sov. Phys.-tech. Phys. 28, 1440 (1983); A. E. Bolotnikov et al., Sov. Phys.-Tech. Phys. 33, 449 (1988); * [20] C. Levin et al., Nucl. Inst. and Meth. A332, 206 (1993). * [21] G. Tepper and J. Losee, Nucl. Inst. and Meth. A356, 339 (1995). * [22] E. Nakamura et al., J. of Elec. Spec. and Rel. Phen. 80, 421 (1996); D. E. Baynham, P. T. M. Clee, and D. J. Thompson, Nucl. Instr. and Meth., 152, 31 (1978) * [23] K. Sato, J. of Synchrotron Rad., 8, 378 (2001); T. Gog, D. M. Casa and I. Kuzmenko, CMC-CAT technical report. * [24] M. Sands, SLAC Technical note, SLAC-121 (1970). * [25] B. Norum, CEBAF Technical note, TN-0019 (1985). * [26] A. E. Bolotnikov and B. Ramsey, Nucl. Inst. and Meth. A396, 360 (1997). * [27] http://www.nist.gov/physlab/data/xraycoef/index.cfm * [28] http://library.lanl.gov/cgi-bin/getfile?00415886.pdf Prajwal Mohanmurthy Prajwal Mohanmurthy obtained his Bachelor of Science degree from Mississippi State University in 2012. He was a graduate research fellow at the High Performance Computing Collaboratory at Mississippi State University and currently is a graduate research fellow in the Laboratory for Nuclear Sciences at Massachusetts Institute of Technology. His research interests are centered around test of standard model and fundamental symmetries in search of physics beyond the standard model. His recent research involvements have been geared towards a search for Axionic Dark Matter and the precision measurement of the mass of neutrinos. He also actively collaborates to develop beam instrumentation for up and coming facilities and future accelerators. --- Dipangkar Dutta Dr. Dipangkar Dutta is an Associate Professor of Physics at the Mississippi State University Department of Physics and Astronomy. He obtained his Bachelor of Technology degree from Indian Institute of Technology, Bombay in 1992 and his doctoral degree in Physics from Northwestern University in 1999. He was a post-doctoral and senior post- doctoral fellow in the Laboratory for Nuclear Sciences at Massachusetts Institute of Technology. His research is focused primarily on precision measurement of fundamental properties of nucleons. He is also interested in precision tests of fundamental symmetries and the Standard Model. ---	arxiv-papers	2013-09-26T02:22:48	2024-09-04T02:49:51.542174	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Prajwal Mohanmurthy, Dipangkar Dutta", "submitter": "Prajwal Mohanmurthy", "url": "https://arxiv.org/abs/1309.6711" }
1309.6773	# PyECLOUD and build-up simulations at CERN G. Iadarola(1),(2) and G. Rumolo(1) (1) CERN Geneva Switzerland (2) Università di Napoli “Federico II Naples Italy ###### Abstract PyECLOUD is a newly developed code for the simulation of the electron cloud (EC) build-up in particle accelerators. Almost entirely written in Python, it is mostly based on the physical models already used in the ECLOUD code but, thanks to the implementation of new optimized algorithms, it exhibits a significantly improved performance in accuracy, speed, reliability and flexibility. Such new features of PyECLOUD have been already broadly exploited to study EC observations in the Large Hadron Collider (LHC) and its injector chain as well as for the extrapolation to high luminosity upgrade scenarios. ## 1 INTRODUCTION The analysis of the electron cloud (EC) observations in the Large Hadron Collider (LHC) and its injectors have raised new challenges for the EC build- up simulations. On one hand, for a correct understanding of machine observations it is often necessary to deal with beams with thousands of bunches and with non idealities like non uniform bunch populations and bunch lengths along the train. On the other hand, the demand for extensive parametric scans gives quite stringent requirements in terms of speed and reliability. CERN has a long experience in the EC build-up simulation, mostly carried out with the ECLOUD code, developed and maintained at CERN since 1997 [1, 2, 3, 4]. Unfortunately, due to its not modular structure and to the programming language (FORTRAN 77), this code would need a deep reorganization and a serious upgrade to enable it to fulfill the aforementioned requirements. Therefore we have decided to write a fully reorganized code, in a more modern and powerful language, considering that the initial effort would be compensated by a significantly increased efficiency in future developments and debugging. The new code has been called PyECLOUD, since it is almost entirely written in Python and is largely based on the physical models of the ECLOUD code. Nevertheless, several features and implementations have been modified, in some cases completely redesigned, with respect to ECLOUD, with substantial improvements in terms of reliability, accuracy, speed and usage flexibility. ## 2 PyECLOUD Like ECLOUD, PyECLOUD is a 2D macroparticle (MP) code, where the electrons are grouped in MPs in order to achieve a reasonable computational burden. The dynamics of the MP system is simulated following the flow diagram sketched in Fig. 1. Figure 1: Flowchart representing PyECLOUD main loop. Figure 2: Top: evolution of the number of electrons in the beam pipe for an LHC type beam with 25ns bunch spacing in the SPS (2 trains of 72 bunches); middle: evolution of the reference MP size; bottom: evolution of the number of MPs, the regeneration threshold is highlighted in red. At each time step, seed electrons, due to residual gas ionization and/or to synchrotron radiation induced photoemission from the chamber walls, are generated with a number consistent with the passing beam slice and with positions and momenta determined by theoretical or empirical models. Then the electric field acting on each MP is evaluated: the field of the beam is precomputed on a suitable rectangular grid, loaded from file and obtained at each MP location by a linear (4 points) interpolation; the space charge contribution of the electron system itself is calculated by a classical Particle in Cell (PIC) algorithm, where the finite difference method is employed to solve the Poisson equation with perfectly conducting boundary conditions on the beam chamber. Once the total electric field at each MP location is known, MP positions and momenta are updated by integrating the dynamics equation; at this stage the presence of an externally applied dipolar magnetic field can also be taken into account. At each time step, a certain number of MPs can hit the wall. In these cases a proper model of the secondary emission process is applied to generate charge, energy and angle of the emitted electrons. According to the size of the emitted charge, a rescaling of the impinging MP can be performed or new MPs can be emitted. ### 2.1 MP size management One of the peculiarities of the EC build-up process is the fact that, due to the multipacting effect, the number of electrons can spread several orders of magnitude during the passage of the bunch train (see Fig. 2 \- top). As a consequence, it is impossible to choose a MP size which is suitable for the entire simulation, allowing both a satisfactory description of the phenomena and a computationally affordable number of MPs. The MP size management in PyECLOUD has been significantly improved with respect to ECLOUD and will be briefly described in this subsection. MP sizes are not enforced throughout the simulation process but are determined step by step by “decisions” taken during the execution. For this purpose a target MP size $N_{ref}$, dynamically adapted during the simulation, is employed to control the number of electrons per MP. In particular: * • The size of MPs generated by seed mechanisms is exactly $N_{ref}$; * • When a MP hits the wall, it is simply rescaled according to the Secondary Electron Yield (SEY) if the emitted charge is below $1.5N_{ref}$, otherwise ”true” secondary MPs are generated so that the resulting MP size is as close as possible to $N_{ref}$; * • Once per bunch passage, a cleaning procedure is performed, which deletes the MPs with charge lower than $10^{-4}N_{ref}$. $N_{ref}$ is changed whenever the total number of MPs becomes larger than a certain threshold defined in the input file(typical value $\sim 10^{5}$), which means that the computational burden has become too high. When this happens, a regeneration of the set of MPs is applied, by the following procedure (see Fig. 2): * • Each MP is assigned to a cell of a uniform grid in the 5-D phase space $(x,y,v_{x},v_{y},v_{z})$ obtaining the phase space distribution of the electron gas distributed on the mesh points; * • The new $N_{ref}$ is chosen in order to get a target number of MPs (typically 5-10 times smaller than the regeneration threshold), which still allows for an accurate simulation but with a more reasonable computational effort; * • A new set of MPs, having the new reference size, is generated according to the computed distribution. The preservation of the entire phase space is very important in EC build-up simulation since the dynamics imparted by passing bunches generates very distinctive velocity distributions at the different time steps and the conservation of few specific moments would not guarantee a sufficient accuracy. Several numerical test have shown that the errors on the total charge and the total energy which are introduced by this procedure, are about 1% at the first time step after the regeneration and they become even smaller at the first bunch passage after the regeneration. Figure 3: Electron cloud build-up simulations for different time steps in ECLOUD (top) and PyECLOUD (bottom). Simulated case: SPS MBB bending magnet, 26GeV, two trains of 72 bunches with 225ns gap, 25ns bunch spacing, 1.1$\cdot 10^{11}$ protons per bunch. \| Processing time ---\|--- Time step [ps] \| ECLOUD \| PyECLOUD 200 \| 29 min \| 12 min 100 \| 1 h 27 min \| 13 min 50 \| 1 h 45 min \| 24 min 25 \| 3 h 7 min \| 40 min 12 \| 4 h 15 min \| 1 h 6 min Table 1: Computation time required by ECLOUD and PyECLOUD for the simualtions in Fig. 3. ### 2.2 Performances The passage from ECLOUD to PyECLOUD had a significant impact on the performances both in terms of accuracy and of computational efficiency. Fig. 3 shows a comparison between the two codes in terms of convergence properties with respect to the time step which is chosen for the simulation. While in ECLOUD it is quite difficult to achieve a good convergence, PyECLOUD gives a good estimate of the total number of electrons in the chamber already for quite large time steps ($\sim 0.1$ns) while a satisfactory convergence is obtained for a time step of the order of $25$ps. For the same test cases the simulation time required by the two codes is reported in Tab. 1 showing that the improvements introduced in PyECLOUD had also a positive impact on the code efficiency. Furthermore, the new code has been designed in order to offer an increased usage flexibility, allowing to deal with irregular beam structures e.g. non uniform bunch intensity and/or bunch length along the bunch train, irregular bunch spacings and bunch profiles. Thanks to these new features, PyECLOUD has been already largely exploited at CERN for several EC simulation studies for the LHC and its injector chain [5, 6, 7, 8]. In particular, as described in detail in [9], PyECLOUD simulations have been used to reconstruct the evolution of the SEY of the chambers in the LHC arcs, from the measurement of the heat load deposited on the beam screen of the cryogenic magnets. The new code also allows us to estimate the bunch by bunch energy loss due to the interaction of the beam with the EC and to export the electron distribution seen by each bunch. The first feature allowed us to benchmark the results against bunch by bunch stable phase measurements [10] while the second was used, together with HEADTAIL simulations, to analyze the instabilities observed in the LHC with 25ns bunch spacing [11]. Simulation studies have also addressed the EC formation in the common vacuum chambers of the LHC. An example of this kind of application will be described in the following section in order to show the capability of PyECLOUD to deal with beams made of thousands of bunches with irregular spacings. ## 3 EC build up in LHC common vacuum chambers Common vacuum chambers having 800mm diameter are installed on both sides of the ALICE experiments in the Long Straight Section 2 (LSS2) of the LHC. During 2011 operation with 50ns bunch spacing an important pressure rise was noticed in these chambers with a significant impact on the background observed by the ALICE experiment. The analysis of the pressure data has shown that a severe pressure increase is observed only when the two rings of the LHC are completely filled. Fig. 4 shows the pressure evolution during a proton-proton physics fill (with 50ns spacing) in which the injection from the SPS of the last two trains of 144 bunches was delayed by about one hour. It can be noticed that the pressure rise appears already at the injection energy (450GeV), but only after the last two injections have taken place. Figure 4: Pressure evolution in the 800mm chambers near the ALICE experiment of the LHC during a proton physics fill. The total beam intensity is the two rings is also included. PyECLOUD simulations have been run in order to investigate if the EC formation in these chambers could explain this pecular behavior. The presence of both counter-rotating beams in the chamber had to be taken into account since it determines different “hybrid” bunch spacings at the different sections of the $\sim$30m long vacuum chambers when both beams are circulating in the LHC (see Fig. 5). In particular the two beam configurations in Fig. 6 have been simulated, which correspond to the beam patterns in the two rings at the moments indicated by (a) and (b) in Fig. 4. The results at a certain section of the considered vacuum pipes are shown in Fig. 7. It can be noticed that, in the configuration of Fig. 6a, both beams present a gap of about one quarter of the length of the ring. Probably due to the quite large radius, the EC can develop only when both beams are passing in the chamber, while a decay of the number of electrons is observed when only one beam is passing and the gap is long enough to allow a complete reset of the EC between subsequent turns. On the other hand, after the injection of the last two trains from the SPS, the layout of the beam in the two rings looks like the one in Fig. 6b where no large gap is present in any of the two beams. As a consequence a complete decay of the EC between subsequent turns is not possible anymore. In fact, a memory effect is observed between turns with a strong enhancement of the EC activity and, as consequence, of the electron stimulated gas desorption leading to the observed pressure rise. Figure 5: Hybrid bunch spacings which are observed at different sections of the 800mm common chambers. Figure 6: Filling pattern of the two beams in the LHC before (a) and after (b) the injections of the last two trains. Beam 1 (blue) is clockwise rotating, beam 2 (red) counterclockwise. The position the straight section 2 (ALICE) is highlighted Figure 7: PyECLOUD simulation of the electron cloud build-up in the 800mm common chambers for the filling patterns in Fig. 6. ## 4 CONCLUSIONS PyECLOUD is a new EC build-up simulation code, which has been developed at CERN for a reliable and efficient analysis of the EC observations in LHC as well as for EC studies related to the high luminosity upgrade of the LHC and its injectors. Based on the physical models already implemented in ECLOUD, the new code features several improvements in terms of algorithms and implementation (e.g. a new MP size management method) with a significant impact on speed and convergence properties. Moreover the code is explicitly designed to deal with irregular beam structures (e.g. different bunch intensities and bunch lengths along the train, arbitrary spacings and profiles) in order to allow for an accurate analysis of EC observations in CERN accelerators. Several EC studies for the LHC and its injectors have been already carried out with the new code giving very encouraging indications on the reliability of the models and numerical solutions. ## 5 ACKNOWLEDGMENTS The authors would like to express their gratefulness to G. Arduini, H. Bartosik, C. Bhat, V. Baglin, R. De Maria, O. Dominguez, M. Driss Mensi, J. Esteban-Muller, K. Li, H. Maury Cuna, G. Miano, E. Méétral, H. Neupert, G. Papotti, E. Shaposhnikova, M. Taborelli, L. Tavian, C. Y. Vallgren, and F. Zimmermann for the support they provided in the code development, the simulation work and the machine data analysis as well as for valuable comments and discussions. ## References * [1] F. Zimmermann, “A Simulation Study of Electron-Cloud Instability and Beam-Induced Multipacting in the LHC”, CERN LHC Project Report 95 (February 1997) * [2] F. Zimmermann, “Electron Cloud Simulations for SPS and LHC”, Proc. Chamonix X, 17-21 January 2000, CERN-SL-2000-007 DI (2000), p. 136 * [3] G. Rumolo and F. Zimmermann, “Practical User Guide for ECloud”, CERN-SL-Note-2002-016. * [4] G. Rumolo et al.,“Simulation of the electron-cloud build up and its consequences on heat load, beam stability, and diagnostics”, Phys. Rev. ST Accel. Beams 4 (2001), 1 * [5] G. Iadarola and G. Rumolo, “Electron Cloud in the CERN accelerators (PS, SPS, LHC)”, elsewhere in these proceedings * [6] H. Maury Cuna et al., “Simulation of electron-cloud heat load for the cold arcs of the Large Hadron Collider”, Proceedings of the 3rd International Particle Accelerator Conference 2012, 20 - 25 May 2012, New Orleans, USA * [7] F. Caspers et al., “Comparison between electron cloud build-up measurements and simulations at the CERN PS”, Proceedings of the 3rd International Particle Accelerator Conference 2012, 20 - 25 May 2012, New Orleans, USA * [8] C. Bhat et al., “E-Cloud dependence on the Bunch Profile - An Experiment in the PS and an Extension to the LHC”, elsewhere in these proceedings * [9] G. Iadarola and G. Rumolo, “Electron cloud simulations with PyECLOUD”, Proceedings of the 11th International Computational Accelerator Physics Conference (ICAP), 19-24 August 2012, Rostock-Warnemünde, Germany * [10] J. Esteban Muller et al., “Synchronous Phase Shift at LHC”, elsewhere in these proceedings * [11] H. Bartosik et al., “Benchmarking HEADTAIL with e-cloud observations with LHC 25ns beam”, elsewhere in these proceedings	arxiv-papers	2013-09-26T09:41:16	2024-09-04T02:49:51.552996	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "G. Iadarola (CERN and U. Naples), G. Rumolo (CERN)", "submitter": "Scientific Information Service CERN", "url": "https://arxiv.org/abs/1309.6773" }
1309.6795	# ELECTRON CLOUD IN THE CERN ACCELERATORS (PS, SPS, LHC) G. Iadarola(1+2) and G. Rumolo(1) (1) CERN Geneva Switzerland (2) Università di Napoli “Federico II Naples Italy ###### Abstract Several indicators have pointed to the presence of an Electron Cloud (EC) in some of the CERN accelerators, when operating with closely spaced bunched beams. In particular, spurious signals on the pick ups used for beam detection, pressure rise and beam instabilities were observed at the Proton Synchrotron (PS) during the last stage of preparation of the beams for the Large Hadron Collider (LHC), as well as at the Super Proton Synchrotron (SPS). Since the LHC has started operation in 2009, typical electron cloud phenomena have appeared also in this machine, when running with trains of closely packed bunches (i.e. with spacings below 150ns). Beside the above mentioned indicators, other typical signatures were seen in this machine (due to its operation mode and/or more refined detection possibilities), like heat load in the cold dipoles, bunch dependent emittance growth and degraded lifetime in store and bunch-by-bunch stable phase shift to compensate for the energy loss due to the electron cloud. An overview of the electron cloud status in the different CERN machines (PS, SPS, LHC) will be presented in this paper, with a special emphasis on the dangers for future operation with more intense beams and the necessary countermeasures to mitigate or suppress the effect. ## 1 Introduction In the CERN PS, the electron cloud was first observed in 2001 during the last part of the cycle for the production of the the so-called LHC-type beams, i.e. the beams of the type needed for the LHC filling. The production scheme of these beams in the PS is based on two or three steps of bunch splitting in order to obtain at the exit of the PS bunch trains with 50ns or 25ns spacing, respectively. In either case, the final stage of bunch splitting takes place at the top energy (26 GeV/c) and is followed by adiabatic bunch shortening and fast bunch rotation shortly before extraction [1]. These two processes are meant to shorten the bunches from their 15 ns length after the last splitting to 12 and then 4 ns, respectively, and make them suitable to be injected into the SPS. Therefore, these beams only circulate in the PS for few tens of msec with a structure prone to electron cloud formation (beam parameters are summarized in Table 1). During this short time before extraction, an electron cloud was initially revealed in 2001 by the presence of a baseline drift in the signal from the pick up as well as beam transverse instabilities [2]. The transverse instabilities made a new appearance with 25ns beams in 2006, when the bunches were accidentally shortened to 10ns or below (instead of the nominal 12ns) during the phase of adiabatic shortening prior to the fast rotation. This again suggested that the short bunches could initiate the electron cloud build up earlier in the cycle and produce enough electron cloud for a sufficiently long time as to render the beam visibly unstable. In March 2007, an experiment for dedicated electron cloud measurements was set up at the PS to be able to directly measure the electron signal by using a shielded biased pick up [3] and confirm its presence in the machine in the last phase of the LHC beams production. The experimental setup was designed, fabricated, and mounted in straight section (SS) 98 during the accelerator shutdown 2006/2007. All the details of the setup can be found in Ref. [3]. These studies confirmed that the electron cloud develops during the last 40 to 50 ms before ejection, i.e. when the bunches are shortened by the RF gymnastics. Besides, they also showed that the electron cloud can be suppressed by putting a sufficiently large voltage of either polarity onto a clearing electrode, even if the clearing efficiency depends on the magnetic field present in the region of the measurement in a non-trivial way. Table 1: Relevant beam parameters in the PS during the flat top RF gymnastics for the two bunch spacings of 50 and 25ns. \| 50ns \| 25ns ---\|---\|--- Beam energy (GeV/c) \| 26 \| Bunch intensity --- ($\times 10^{11}$ ppb) 0.82-1.95 \| 0.83-1.33 Bunch length (ns) \| 15 $\rightarrow$ 12 $\rightarrow$ 4 Number of bunches \| 36 \| 72 Transv. norm. emittances ($\mu$m) \| 1-2 \| 2-3 In 2011, new systematic measurements of electron cloud have been performed at the CERN-PS with the goal of extracting the following information: * • Dependence of the electron cloud build up evolution on some controllable beam parameters (bunch spacing, bunch intensity, bunch length). * • A new collection of time resolved experimental data of electron cloud build up in some desired sets of beam conditions. These sets of data can serve two purposes. First, comparing them with build up simulations will allow us to validate (or improve) the simulation model on which our tools are based. Second, by matching the simulations to the experimental data in all the different beam conditions, we can pin down the surface properties of the PS vacuum chamber (secondary electron yield, $\delta_{max}$, and reflectivity of the electrons at zero energy, $R_{0}$) and extrapolate then how much electron cloud we can expect in the PS with the higher intensity beams foreseen in the frame of the LHC Injector Upgrade (LIU) project, and whether that can be detrimental to the beam. Table 2: Relevant beam parameters of the SPS 50 and 25ns beams. \| 50ns \| 25ns ---\|---\|--- Beam energy (GeV/c) \| 26 $\rightarrow$ 450 \| Bunch intensity --- ($\times 10^{11}$ ppb) 0.3-1.7 \| 0.3-1.4 Bunch length (ns) \| 4 $\rightarrow$ 2.8 $\rightarrow$ 1.6 Number of bunches \| 144 \| 288 Transv. norm. emittances ($\mu$m) \| 1-2 \| 2-3 The SPS has been suffering from electron cloud formation since it first began to take and accelerate 25ns beams produced in the PS with the scheme explained above. Observations of pressure rise, beam instability, emittance growth were first made in the early 2000 and all these effects strongly limited the capability of this accelerator of handling LHC-type beams [4]. While the coherent instabilities could be suppressed by the use of the transverse damper (against the horizontal coupled bunch oscillations) and running with sufficiently high chromaticity (against the strong single bunch effect in the vertical plane), emittance growth and positive tune shift along the bunch train could still be measured, pointing to the continuing presence of a strong electron cloud inside the beam chamber. All this led to the decision to have in 2002 the first dedicated scrubbing run, in which the SPS was operated exclusively with 25ns beams for one full week. The goal was to use the bombardment from the electron cloud itself to clean the beam chamber inner surface, and therefore lower its Secondary Electron Yield (SEY) and reduce, in turn, the amount of electron cloud build up. The strategy proved successful [5] and the week of scrubbing run was then repeated at the beginning of the 2003, 2004, 2006 and 2007 runs to provide the necessary machine cleaning. During these years, dedicated experiments were conducted in the SPS to study in detail the electron cloud formation in cold regions (COLDEX) or in NEG coated chambers [6], or to benchmark simulation codes with machine observations [7]. From 2006 on, electron cloud studies in the SPS acquired new momentum in the framework of the SPS upgrade studies [8] and the experimental activity over the following years was mainly focused to find the scaling law of the electron cloud instability with beam energy [9] and to validate the efficiency of amorphous carbon (a-C) coating of the beam chamber [10]. All the electron cloud machine development activity of the last couple of years at the SPS has been devoted to defining the status of the 25ns beams in this machine and use the direct electron cloud measurements in chambers equipped with strip monitors to understand beam induced scrubbing in different chamber geometries and with different materials. A comprehensive report of all observations in terms of beam behavior, pressure rise and dedicated ekectron cloud measurements will be soon published [11]. Recently, the nominal 25ns and 50ns LHC beams in the SPS seem to be undegraded and does not suffer any longer from the strong electron cloud effect that was present during the first years of SPS operation with this type of beams. The achievable parameters are summarized in Table 2. The three values of bunch length quoted in this table correspond to injection into 2 MV buckets, after shortening at flat bottom by increase of the RF voltage to 3 MV, and at flat top after controlled longitudinal emittance blow up during the accelerating ramp. Several studies conducted in the past predicted that also the LHC would suffer from heat load, pressure rise and beam instabilities due to electron cloud, when operating with trains made of closely spaced proton bunches (e.g. [12]). Since mid 2010 LHC entered this mode of operation. In the first phase, beams with 150 ns bunch spacing were injected, accelerated and brought to collision. During this period of operation, the only possible signature of electron cloud build up was a pressure rise observed in the common vacuum chamber, close to the Interaction Regions. Subsequently, at the end of October 2010, an attempt was made to switch to 50 ns spacing operation. After an initial physics fill with 108 nominal bunches (filling scheme with 1 pilot bunch and $9\times 12$ bunches), some important dynamic pressure rises were observed at injection when filling with trains of 24 bunches. In fact, the first attempt of injection in batches of 24 even led to the closure of the vacuum valves in point 7 after the injection of 108 nominal bunches per beam, as the interlock level of $4\times 10^{-7}$ mbar was reached on two vacuum gauges. After that, since it became clear that further improvements in the LHC performance were hampered by the electron cloud, emphasis was put on machine studies to characterize the electron cloud build-up in the LHC, its effects and possible cures. It was also decided that a comparative study with the behaviour of 75 ns beams was necessary to define a path for the 2011 run. Toward the end of the 2010 proton run, a Machine Development (MD) session was devoted to the set up of the LHC with 50 ns bunch trains. During this MD, three effective days of beam time were used for the setting-up proper as well as for studies and machine scrubbing. The study of the 75 ns beam took place in another dedicated MD period, while the LHC had already switched to ion operation. About 2.5 days were devoted to the setting-up of the injection and capture of the 75 ns beam and, later on, to comparative studies with the 50 ns beam. This MD gave a clear indication that, probably also benefiting from the previous MD’s scrubbing with 50 ns beams, the electron cloud effects with 75 ns appeared significantly less pronounced than with 50 ns beams, such that this bunch spacing could be regarded as a relatively safe option [13]. Figure 1: MD sessions labeled (a), (c), (d) and (e): injected beams. The LHC operation was therefore resumed in 2011 directly with 75 ns beams. After the scrubbing run in 2010 it was expected that up to 200-300 bunches could be injected and accelerated without major problems. This was confirmed during the start-up with beam. After about one month of operation, the LHC could successfully collide trains of 200 bunches distributed in batches of 24 bunches each. At the beginning of April, 10 days were devoted to scrubbing of the LHC with 50 ns beams. The goal was to prepare the machine to switch to 50 ns beams and thus extend the luminosity reach for the 2011 run. During the scrubbing run, up to 1020 bunches per beam were injected into the LHC in batches of 36 and stored at injection energy. The strategy consisted of constantly topping the total beam intensity in the LHC with the injection of more trains, such that the vacuum activity, and therefore the electron cloud, could be kept at a constant level and efficiently reduce the Secondary Electron Yield (SEY) of the walls to a value below the threshold for build up. The success of the scrubbing run was proved by the subsequent smooth LHC physics operation with 50 ns spaced beams. Between mid April and end June the number of bunches collided in the LHC was increased up to its maximum value of 1380 per beam, while the intensity per bunch and the transverse emittances remained constant at their nominal values (i.e., 1.15$\times 10^{11}$ ppb and 2.5 $\mu$m). The switch to 50 ns beams with lower transverse emittances (1.5 $\mu$m) and the adiabatic increae of the bunch current to $1.5\times 10^{11}$ ppb did not cause any significant recrudescence of the electron cloud effects, probably also owing to the MD sessions with 25ns beams that took place in the second half of 2011, which created enough margin in the machine cnnditioning to ensure electron cloud free 50ns operation. Beams with 25ns spacing were injected into the LHC only during five MD sessions of the 2011 run, which are listed and briefly described here: 1. (a) 29 June, 2011: first injections of 25ns beams into the LHC. The filling scheme consisted of nine batches of 24 bunches separated by increasing gaps (2.28, 5.13 and 29.93 $\mu$s). Pressure rise around the machine as well as heat loads in the arcs were observed. All the last bunches of each batch suffered losses and emittance growth [14]; 2. (b) 26 August, 2011: first injections of a 48-bunch train into the LHC with 25ns spacing. Two attempts were made to inject a 48-bunch train from the SPS, which led to beam dump triggered by large beam excursion and beam loss interlocks, respectively. During the first injection test, the transverse damper was on and it is believed that the beam suffered a coherent electron cloud instability in both planes (more critical in vertical) soon after injection. During the second test, the transverse damper was switched off and the beam was affected by a coupled bunch instability [16]. This MD session had then to be interrupted because of a cryo failure caused by a thunderstorm; 3. (c) 7 October, 2011: injection tests and first ramp. In the first part of the MD, trains with 48-72-144-216-288 bunches from the SPS were injected into the LHC. Given the experience during the previous MD, the chromaticity $Q^{\prime}$ was set to around 15-20 units in both the horizontal and vertical planes in order to keep the beams stable against the electron cloud effect. In the second part, only 60 bunches per beam were injected in trains of $12+2\times 24$, were accelerated to 3.5 TeV and collided during approximately 5h; 4. (d) 14 October, 2011: first long stores of 25ns beams at injection energy in the LHC. During this session up to 1020 bunches per beam were injected in batches of 72. The chromaticity was kept high in both planes ($Q^{\prime}_{x,y}\approx 15$) in order to preserve the beam stability. First, a dedicated fill for pressure measurements was made, with batches injected at gradually reduced distances from 4 to 2 $\mu$s (in steps of 1$\mu$s). Subsequently, the batch spacing was kept constant for each of the next three fills and it was set to 6.3, 3.6 and 1 $\mu$s (rounded values). Strong emittance growth and slow losses affecting the last bunches of each train were observed throughout this MD session; 5. (e) 24–25 October, 2011: record number of bunches in the LHC. Four long fills took place (average store time was approximately 4h), with 25ns beams injected into both rings in batches of 72 separated by 1$\mu$s. In the third and fourth fills, 2100 bunches were injected for beam 1, while the number of bunches could not exceed 1020 for beam 2, due to a vacuum interlock on one of the injection kickers (MKI). Although the situation seemed to improve over the MD, slow losses and emittance growth kept affecting both beams. Before starting the fourth fill, the horizontal chromaticity $Q^{\prime}_{x}$ was lowered from 15 to 3 units and the horizontal damper gain was slightly increased. Probably due to that, some horizontal instabilities could be observed from the signal of the damper pick up during the fourth fill, but the overall performance did not appear degraded from the previous fill. The MD ended with a 30’ fill with only beam 1, during which batches of 72 bunches were injected into the LHC at different spacings in order to provide the stable pressure measurements needed for the modeling of the electron cloud build up in the straight sections (see next Section). Figure 1 shows the detailed story, in terms of injected beams 1 & 2, of the sessions (a), (c), (d) and (e). Experimental data from these MDs will be used in the next section to extrapolate the evolution of $\delta_{\mathrm{max}}$ on the beam screen in the arcs and in proximity of the vacuum gauges. For sake of compactness, we have chosen to concatenate these three sessions and represent them as a function of a continuous time coordinate (interpretable as hours with 25ns beam), which will be systematically used throughout this paper when referring to the studies with the 25ns beams. ## 2 Studies in the different machines ### 2.1 PS measurements In 2011, the MD program in the PS for electron cloud studies took place in November and extended over several sessions to cover different sets of beam parameters. In particular, electron cloud build up data were recorded for 25ns and 50ns beams. The bunch intensities were scanned in the ranges indicated in Table 1. The trigger for the data acquisition was set at extraction, when in normal conditions each bunch of the beam has been already fully rotated (4ns bunch length). However, specifically for these measurements, the bunch length at this time for a fixed bunch intensity was also set to 6.5ns or 15ns by simply adjusting or fully removing, respectively, the final step of the fast bunch rotation. This allowed studying the dependence of the electron cloud build up not only on the bunch intensity but also on the bunch length. The threshold for electron cloud formation with 50ns beams was found to lie at about $10^{11}$ ppb and the measured signal increases monotonically with the bunch intensity. This is not entirely surprising, since the measurements were taken with zero magnetic field while the non-monotonic behaviour of the electron cloud build up with the bunch intensity is more frequent in dipole regions. The shielded pick up is installed inside a C-magnet, which was kept off during the MD sessions because the orbit perturbation it introduces would have required a specific correction. Scans with 25ns beams were also made and the threshold for electron cloud formation was found to be below $8\times 10^{10}$ ppb, with a behavior of the electron cloud signal increasing with the bunch intensity. We have tried to fit the PS data with those from electron cloud build up simulations [17]. First of all, the output of the code that should be compared with the measured signal is the electron flux to the wall. In a first approximation, we do not consider the holes in the vacuum chamber, which are expected to cause only a minor perturbation in a field-free region. In general, the simulated electron flux to the wall vanishes during the bunch passage, because initially all the electrons are drawn to the center of the vacuum chamber by the passing bunch (e.g. during the first $\sim$2ns of a 4ns long bunch) and they are gradually released only during the falling edge of the bunch, when they may reach the walls again. The fact that the measured signal does not exhibit this feature makes plausible a low pass filtering of the signal (inherent to the measurement technique or due to electronics and/or cables) with a corner frequency in the range of some hundreds of MHz. Figure 2 shows measured and simulated signal, where the simulated signal, obtained with $\delta_{\mathrm{max}}=1.6$ and $R_{0}=0.5$, was low pass filtered with a corner frequency of 200 MHz. The impressive resemblance between the two suggests that our electron cloud model correctly describes the phenomenon and the rationale applied for the data analysis is promising. Figure 2: E-cloud build up simulation (top) and measurement (bottom) for a 25ns beam with $1.33\times 10^{11}$ ppb and 4ns long. It is therefore clear that the electron cloud is present in the CERN PS with both 50ns and 25ns beams when they reach the final beam structure, shorty before being ejected. However, since it only makes a short appearance in the last few ms of the production cycle of these beams, with the present beam parameters, there is not enough time to render beam unstable or let incoherent effects develop. On the negative side, very low electron doses are deposited on the chamber walls, making it basically impossible to rely on efficient machine scrubbing if the electron cloud should ever become a limiting factor. The question to be addressed is whether this effect may become a bottleneck for the LHC Injector Upgrade (LIU) beams, envisaging bunch intensities of up to $3\times 10^{11}$ ppb within lower transverse emittances. A full simulation study including both the build up and instability part is needed to assess the margins. Figure 3: Threshold SEY for electron cloud formation in the two types of chambers in SPS drift spaces, as a function of the bunch intensity. ### 2.2 SPS studies One of the key points to be addressed to understand the electron cloud in the SPS is to determine the values of SEY thresholds for its formation in the different beam chambers and try to deduce what parts are critical for both present and future LHC beams. In the SPS there are six main different types of vacuum chamber: two types are used in the main dipoles, two in the quadrupoles and two in the drift spaces, depending on the beta functions in the nominal optics. We have studied the electron cloud build up in both dipole chambers and driift space chambers. Since the magnetic fields from quadrupoles have not been implemented yet in the PyECLOUD code, these chambers, which however cover only less than 10% of the total circumference, have not yet been simulated. The drift chambers are of A or B type, both circular and with a radius of 78 or 65 mm, respectively. Figure 3 shows the SEY threshold as function of bunch intensity at both 26 and 450 GeV/c. The following interesting features can be observed: * • The SEY thresholds are mostly decreasing with bunch current, but tend to change slope for 50ns beams with bunch populations above $2\times 10^{11}$ ppb. * • There exist regions in which 50ns can create a worse electron cloud than 25ns. * • The SEY thresholds become very low (close to 1.05) for 25ns beams in Drift B and with bunch currents above $2\times 10^{11}$ ppb. The vacuum chambers in dipoles also come in two different sorts with almost rectangular shape: the MBB-type, characterized by a height of about 56.5mm and 132mm width; and the MBA-type, flatter than the MBBs and thus more suited to regions with lower vertical betatron functions, characterized by a height of 43mm and 156mm width. In Fig. 4 the SEY threshold is displayed as a function of the bunch intensity for both 25 and 50ns beams as well as at injection and top energy. Also in this case the dependencies are not trivial and exhibit the following features: * • The SEY thresholds do mostly increase with bunch current. When they do not, the behavior tends to be flat, indicsting then little dependence of the SEY threshold on the bunch intensity in these intensity ranges. * • The SEY thresholds of the 50ns beam lie above 2.0 in the MBA chambers. * • The SEY thresholds can become in general very low (around 1.2) for 25ns beams in MBB chambers. Figure 4: Threshold SEY for electron cloud formation in the two types of chambers in SPS dipoles, as a function of the bunch intensity. Considering all the results of the above study, it is evident that the most dangerous chambers in the SPS, in terms of favoring electron cloud build up for the present and future LHC beam intensities, are the drift B and the MBB pipes, which exhibit the lowest SEY thresholds at almost all intensity ranges. In particular, it is specially worrisome that both these chambers exhibit SEY thresholds below 1.3, which seems the saturation value for scrubbing of StSt in laboratory measurements [18]. Besides, StSt samples exposed to the SPS beam and then extracted from the machine have never shown SEY values below 1.5. Presently, it still remains unclear whether we still have electron cloud in some of the SPS regions, because the observed pressure rise is several order of magnitude lower than the one observed in previous years and the nominal 25ns beam is not really affected anymore by significant electron cloud effects [11]. While more studies are ongoing to try to characterize the present status of the SPS and draw conclusions on future strategies against electron cloud, it is however clear that the critical regions that might need coating (if scrubbing is insufficient or too long) would amount to about 40% of the whole machine (Drift B + MBB). Experimentally, we can say that, thanks to the regular scrubbing runs the SPS had from 2003 to 2008 with 25ns beams at every start up (plus several MD sessions with this type of beams every year), the performance with 25ns beams has been constantly improving over the years and in 2011, nominal 25ns beams with transverse emittances below 3$\mu$m were first produced and extracted. This leads us to believe that presently the electron cloud has weakened or disappeared in most parts of the SPS and might be still only surviving in the MBBs for operation with nominal intensity 25ns beams. In these conditions, it seems to be efficiently kept under control and does not give rise to detrimental effects on the beam. An Increase in bunch intensity may awaken the electron cloud in the Drifts (and MBAs, because the stripes move to unscrubbed regions) with the consequent effect on beam stability and emittance evolution. It is clear that a scrubbing run will be necessary after the Long Shutdown 2013-2014, but its length and efficiency are difficult to estimate. The experience after LS1 will therefore give an indication whether we really need to coat the most critical parts of the SPS, or we can afford to rely on scrubbing also for future operation. Figure 5: Top picture: Heat load measured during four fills from the MD session (d) and (e), in the same time coordinate as in Fig.1. The five vertical bars represent the measurement points used to compare heat load with electron cloud simulations. Bottom picture: Estimated evolution of $\delta_{\mathrm{max}}$ on the inner surface of the beam screen in the dipole chambers ### 2.3 LHC observations The heat load data from the cryogenic system give the total power dissipated (in W/half-cell) on the beam screens of both beams 1 and 2. Using the measured heat load it is possible to estimate the SEY of the arc chamber walls. The exact procedure is explained in Ref. [15] and is based on the comparison of the heat load data with PyECLOUD simulations, run with realistic bunch-by- bunch intensities and lengths (data from the fast BCT and the BQM). Heat load observations in the arcs were made with 50ns before the scrubbing tun and then with 25ns beams. From the heat load data with 50ns beams before and after the scrubbing run, we could estimate the SEY on the chamber wall of the arcs to have reached a value between 2.1 and 2.2, sufficient to suppress electron cloud build up with 50ns beams. Measurements in some reference cells from the first LHC MD with 25ns beams (MD session (a), 29 June, 2011) can be found in Ref. [14]. Figure 5 shows the heat load data, sector by sector, collected during the MD sessions (d) and (e). We can notice that the additional heat load peaked to values of nearly 50 W/half-cell (i.e. approximately an average of 0.5 W/m/beam) during the last fill with 2100 bunches for beam 1 and 1020 bunches for beam 2. A decay of the measured heat load between injections, and in any case after the last injection, is also clearly visible in the examined cases, due to the weakening of the electron cloud activity from scrubbing and also from intensity loss (e.g., compare with the BCT signal in Fig. 1, acquired at the same time). Using the bunch-by-bunch intensity and length data at the times marked with vertical bars in the top plot of Fig. 5 plus the data from the injection tests on the 29 June, PyECLOUD simulations were run scanning $\delta_{\mathrm{max}}$, so that the curves of the simulated heat loads as a function of $\delta_{\mathrm{max}}$ could be produced for all these measurement points. The electron reflectivity at zero energy was fixed to the value of 0.7. The $\delta_{\mathrm{max}}$ corresponding to each heat load measurement was then found matching the simulation to the measured value and the results are in the curve displayed in the bottom part of Fig. 5. Figure 6: Snapshots of the horizontal and vertical emittance measurements for beam 1 during the last fill of 14 October and the last two fills of 24–25 October MDs. While the 50ns beam proved to be stabilized in the LHC by the electron cloud mitigation achieved with the scrubbing run, the 25ns beam has exhibited clear signs of transverse instability and emittance growth throughout all the dedicated MD sessions. Despite a clearly improving trend from one fill to the next one, these signs have not completely disappeared. During the first tests on 29 June, when only batches of 24 bunches were injected from the SPS, the beam could be kept inside the machine because the level of electron cloud reached along each batch was enough to cause significant emittance growth, but no coherent instability and fast beam loss [14]. When, on the following MD session, batches of 48 bunches were for the first time transferred from the SPS to the LHC, the beam was twice dumped after few hundreds of turns, due to the excitation of a transverse instability leading to unacceptable beam losses. During the successive MD sessions, this problem was circumvented by injecting the beam into the LHC with high chromaticity settings. Values of $Q^{\prime}_{x,y}$ around 15 were chosen, as they had been found to be sufficiently stabilizing in HEADTAIL simulations [19]. Using these settings, the beam could be kept inside the LHC, albeit with degraded transverse emittances (see bunch-by-bunch emittance plots from the MDs of 14, 24 and 25 October, Fig. 6). Since the BSRT needs about 2 sec to measure the emittances of each bunch, each of the snapshots in the figure does not represent an instantaneous photograph of the beam at a certain time, but results from a sweep over the bunches that can last as much as several minutes. Although the batch spacing was decreased from 2 $\mu$s during the measurement of 14 October to the 1 $\mu$s of the last MD session, the vertical emittance blow up exhibits signs of improvement. No significant further change is observed then in the vertical plane between the measurements taken in the last two fills (consistently with a slight scrubbing effect between them). The situation looks more complicated in the horizontal plane. Here a deterioration can be noticed from the 14/10 measurement to the 24/10 one. If this is related solely to the decreased batch spacing, which has enhanced the electron cloud along the full train owing to the stronger memory effect between batches, we could not explain why we observed an improvement in the vertical plane, instead. It is interesting that the situation appears improved for the 25/10 measurement, when the LHC was run with lowered horizontal chromaticity settings. This fact may suggest that by lowering chromaticity we have moved from a regime of strong incoherent emittance growth driven by electron cloud and high chromaticity to a new one, in which the beam suffers a fast instability, but later evolves with a better lifetime [20, 21]. In any case, as a general consideration, a clear weakening of the electron cloud effect from 14 to 25 October is witnessed by the improved quality of the first two–three batches. The first two seem to be hardly affected by emittance growth in both transverse planes by the time of the last 25ns fill. ## 3 Conclusions and outlook In conclusion, we have reached quite a deep knowledge of the electron cloud in the different CERN accelerators, and presently it does not seem to be a limiting factor with the present operation parameters: * • In the PS the electron cloud only appears in the last milliseconds of the LHC beam production cycle and does not stay long enough as to affect the beam * • The SPS currently benefits from several years of scrubbing with nominal 25ns beams. Therefore, it seems that now the electron cloud has been either suppressed in the whole machine or it still survives in some more sensitive parts, but at a level not harmful to the beams (50ns, nominal 25ns) * • In the LHC, the electron cloud does not have important adverse effects on operational 50ns beams, however it still affects the 25ns beams and additional scrubbing is needed to further lower it and permit operation with this type of beams. However, some questions are still open, like the performance of SPS and LHC with 25ns beams after LS 2013-2014 and whether the electron cloud can become a serious bottleneck for the beams required by the LIU project. ## References * [1] R. Garoby, “Status of the Nominal Proton Beam for LHC in the PS”, CERN/PS 99-13 (RF) * [2] R. Cappi, et al., Phys. Rev. ST Accel. Beams 5, 094401 (2002). * [3] E. Mahner, T. Kroyer and F. Caspers, Phys. Rev. ST Accel. Beams 11, 094401 (2008) * [4] G. Arduini, K. Cornelis, W. Hoefle, G. Rumolo, and F. Zimmermann, in Proceedings of PAC 2001 (18-23 June 2001, Chicago, USA) and CERN-SL-2001-0050 * [5] J.M. Jimenez et al., LHC-Project-Report-632 (2003) * [6] V. Baglin, A. Rossi, et al. in Proceedings of ECLOUD04 (19-23 April 2004, Napa California, USA) * [7] D. Schulte, G. Arduini, V. Baglin, J.M. Jimenez, F. Ruggiero, and F. Zimmermann , in Proceedings of PAC2005 (16-20 May 2005, Knoxville Tennessee, USA) and LHC Project Report 847 * [8] G. Rumolo, E. Métral and E. Shaposhnikova, in Proceedings of LHC LUMI 2006 (16-20 October, Valencia, Spain) * [9] G. Rumolo, G. Arduini, E. Métral, E. Shaposhnikova, E. Benedetto, R. Calaga, G. Papotti and B. Salvant, Phys. Rev. Letters 100 (2008) 144801 * [10] C. Yin Vallgren et al., Phys. Rev. ST Accel. Beams 14, 071001 * [11] H. Bartosik, G. Iadarola et al., to be published * [12] F. Zimmermann, in Proceedings of Chamonix X & XI, CERN-SL-2000-001 DI (2000) and. CERN-SL-2001-003 DI (2001) * [13] G. Arduini et al., CERN-ATS-Note-2011-046 MD (2011) * [14] B. Goddard et al., CERN-ATS-Note-2011-050 MD (2011) * [15] G. Rumolo et al., in Proceedings of the LHC Beam Operation Workshop - Evian 2011 (12-14 December, 2011, Evian, France) * [16] H. Bartosik and W. Höfle, CERN-ATS-Note-2012-027 MD (2012) * [17] G. Iadarola and G. Rumolo, “Improved electron cloud build up modeling with PyECLOUD”, elsewhere in these proceedings * [18] C. Yin Vallgren, Ph.D. thesis, CERN-THESIS-2011-063 (2011) * [19] K. Li and G. Rumolo, MOPS069 in proceedings of IPAC’11 (San Sebastian, Spain) * [20] E. Benedetto, G. Franchetti, and F. Zimmermann, Phys. Rev. Lett. 97 (2006) 034801 * [21] K. Ohmi and F. Zimmermann, Phys. Rev. Lett. 85 (2000) 3821-3824	arxiv-papers	2013-09-26T11:03:55	2024-09-04T02:49:51.561276	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "G. Iadarola (CERN and U. Naples), G. Rumolo (CERN)", "submitter": "Scientific Information Service CERN", "url": "https://arxiv.org/abs/1309.6795" }
1309.6914	# Chemical concrete machine Marius Buliga Institute of Mathematics, Romanian Academy P.O. BOX 1-764, RO 014700 Bucureşti, Romania [email protected] (This version: 26.09.2013) ###### Abstract The chemical concrete machine is a graph rewriting system which uses only local moves (rewrites), seen as chemical reactions involving molecules which are graphs made up by 4 trivalent nodes. It is Turing complete, therefore it might be used as a model of computation in algorithmic chemistry. ## 1 Introduction Moe-Behrens explains [10] that there is a trend to use the architecture of a silicon computer for building a biological computer. Whatever happens in a living cell which resembles to computation (in a more vague sense than Turing computation) is not necessarily implemented as in a silicon computer, though. Instead, synthetic biology tries to import concepts from silicon computers into the bio realm, mainly because these concepts are already familiar. Hence the accent on bits and boolean logic, although not exclusive (other studies concern for example membrane computing, or neural networks, or continuously evolving dynamical systems). More important, all these studies seem to concentrate on imperative programming. Functional programming is the other main programming paradigm, a very elegant one, receiving more and more attention in the competition with the more well known imperative programming. Functional programming has its roots in the lambda calculus, one of the two pillars of computation, along with the Turing machine (of equivalent power, but with different philosophies behind). In 1992 Berry and Boudol [2] introduce the chemical abstract machine, which models asynchronous concurrent computations (in particular a concurrent lambda calculus) by using a chemical metaphor. The chemical abstract machine has states which are seen as chemical solutions and the evolution from a state to another is ruled by chemical reactions, heating and cooling. Chemical solutions are multisets of molecules, i.e. list of numbers of molecules of each type. Between 1994-1996, Fontana and Buss [6] [7] [8] introduce the idea that lambda calculus is a kind of natural formalization of the bare bones of chemistry. Individual molecules are seen as lambda terms, reactions between molecules are seen as the application operation in lambda calculus, and the abstraction operation from lambda calculus ”captures the role of functional groups embedded in the reactants”, [7] p. 11. Fontana and Buss use lambda calculus with eta reduction, i.e. the terms (or molecules) are identified with functions, in the sense that the molecule $A$ is identified with the function $B\mapsto AB$. The work of Fontana and Buss started the ”algorithmic chemistry” research field. In a sense, this can be seen as a part of an older research subject, the very difficult one, of course, concerning the organization and functioning of living systems. They cite in [7] the seminal paper by Varela, Maturana and Uribe [12]. The chemical concrete machine is a graphic formalism which uses a chemical metaphor. It is a modification of graphic lambda calculus, introduced in [3]. (See also the graphic lambda calculus web tutorial. Moreover, there is already a web tutorial for the chemical concrete machine.) The graphic lambda calculus has been constructed for reasons independent from any of the subjects evoked until now. It is a graph rewriting system which has a sector which is equivalent with untyped lambda calculus, but it has other interesting sectors as well, like the one which is equivalent with knot diagrams, or the one which contains finite differential calculus in spaces with dilations [4], i.e. emergent algebras [5]. It is therefore Turing complete, but in order to achieve this, it uses a GLOBAL FAN-OUT move. Moves (i.e. graph rewrites) can be characterized as local, when they involve a fixed number of nodes and arrows in the graphs, or global, otherwise. By modifying the graphic lambda calculus, we obtain the chemical concrete machine, which is Turing complete by using only local moves. Another difference from graphic lambda calculus is that we reformulate the moves as chemical reactions between molecules (represented by trivalent graphs made by 4 types of trivalent nodes). Thus a computation in the chemical concrete machine is simply a chemical reaction network involving molecules and chemical reactions from the formalism. There are also some differences between the algorithmic chemistry of Fontana and Buss and the chemical concrete machine. The most important one is that that the application and the abstraction operations from lambda calculus are, in the chemical concrete machine, trivalent nodes (say, like atoms which compose the molecules), and moves (corresponding to reductions in lambda calculus) are assimilated with enzymes, thus reductions are certain chemical reactions. Also, the chemical concrete machine can be used for implementing lambda calculus without extensionality, mainly because the eta reduction appears as a global move in graphic lambda calculus, which does not seem possible to be replaced by local moves, as it has been done with the GLOBAL FAN-OUT. This leads to the last point: it is intriguing to search if ”extreme” functional programming (i.e. functional programming without extensionality) is easier to be implemented in biological computing than imperative programming. Already, lambda calculus looks like a kind of chemistry. In the chemical concrete machine the somehow esoteric application and abstraction are made concrete. Moreover, because the chemical concrete machine uses the graphic lambda calculus formalism, it follows that simple spatial manipulations, like grouping molecules together, releasing them in some order or all of them simultaneously, are more straightforward to achieve in graphic lambda calculus than in the usual lambda calculus. Another possibility would be to try to implement the chemical concrete machine in silico, which will be discussed in another article. ### Acknowledgments. I benefited from various discussions with the following (in no particular order): Stephen P. King, Eugenio Battaglia, Gerd Moe-Behrens, Louis Kauffman, ine (nickname in comments made at the chorasimilarity blog), Lucius Meredith, Mike Stay. ## 2 Chemical concrete machine The chemical concrete machine is a graph rewriting system, derived from graphic lambda calculus [3], which can be seen as a model of chemical, or why not biological, computing. The distinction from other such models is that it works with very specific graphs, called ”molecules”, which interact in very specific ways. This is the source of the word ”concrete” in the denomination, as opposed to the chemical abstract machine of Berry and Boudol [2]. I hope it can be implemented in reality (that is in a real chemical reaction network), but nevertheless it can be seen as a proof of principle for the fact that the chemistry of a very small number of molecules can manifest Turing completeness. In this respect, the chemical concrete machine appears as applied algorithmic chemistry in a virtual world with chemistry rules which are made up by a mathematician. Molecules. In the following I shall use the name ”molecule” in the following sense. A molecule is seen as made by other smaller molecules (for example atoms are molecules) which are connected by chemical bonds, or by other molecules called ”arrows”. Therefore, a molecule is seen as a graph with nodes which are smaller molecules and arrows which connect those nodes. A collection of molecules is a molecule. The building block of molecules are the following elementary, or essential ones. On the first row you see the list of essential molecules, named respectively 1. - ”application”, corresponding to the application gate in graphic lambda calculus, or to the application operation in lambda calculus 2. - ”fan-out” , corresponding to the FAN-OUT gate in graphic lambda calculus 3. - ”fan-in”, which will replace the $\varepsilon$ gate from graphic lambda calculus 4. - ”abstraction”, corresponding to the $\lambda$ gate in graphic lambda calculus, or to the lambda abstraction from lambda calculus. On the second row we see a loop, an arrow (which are considered molecules, as written before), and a molecule called ”terminal”, which corresponds in graphic lambda calculus to the termination gate. We shall work also with a list of ”other molecules”, which can react (or not) one with another, this is left to the choice of the user of the chemical concrete machine, which can add more reactions in the formalism. There is also a collection $GARB$ of garbage. ###### Definition 2.1 The set $MOLECULES$ of molecules is formed by all locally planar graphs which can be formed by using the essential molecules (and the ”other molecules” with unspecified valences) as nodes, and arrows obtained by connecting the essential molecules such that they respect the arrows orientation. The graphs can have free arrows (with one of the ends of an arrow not connected to any other molecule), also they can have several connected components (i.e. a finite union of molecules is a molecule). Loops and arrows are molecules. The rules of building molecules are exactly the ones for building graphs in the set $GRAPH$ described in section 2 [3], or go directly to the graphic lambda calculus tutorial, with the understanding that for the chemical concrete machine we use colored nodes instead of the gates from graphic lambda calculus, as described in the following figure. The only new thing is that we admit also a list $A,B,C,...$ of unspecified nodes with unspecified valences, which model ”other molecules”. Here are some examples of molecules. We imagine that all molecules float inside a container. There may be several copies of the same molecule in the container. Enzymes and moves. Besides molecules, I shall also use ”enzymes”. A chemical reaction will always involve a molecule and an enzyme. Enzymes are not molecules. Instead, enzymes are names of moves (graph rewrites). What is to be noticed is that any enzyme comes in two flavors: ”+” and ”-”. The reason for this is the following. Any chemical reaction which involves a molecule and an enzyme will correspond to a move in the list of moves of the chemical concrete machine formalism. The list of moves of the chemical concrete machine, which will be described soon, contains the local moves of graphic lambda calculus which involves the fan-out, application and abstraction gates, supplemented with a list of moves related to the fan-in gate, which replace the emergent algebra moves from the graphic lambda calculus. All the moves of the graphic lambda calculus are bi-directional. Therefore, any such move can be seen as a pair of unidirectional moves. By convention, the moves from left to right are denoted by names of enzymes with ”+” and the moves from right to left are denoted by names of enzymes with ”-”. We cannot simply write chemical reactions as molecules + enzyme = molecules + GARB because these molecules may be complicated beasts (graphs) and because, as we shall see, the enzymes prefer to react with certain patterns (subgraphs) of molecules. For specifying how the reaction takes place we need: 1. - molecules 2. - an enzyme 3. - and a ”reaction site”, which is a small part of the initial collection of molecules. The reaction site have to be present in the molecule, otherwise the reaction cannot happen. In order to explain how to translate from the language of chemical reactions and enzymes to the language of moves of the chemical concrete machine, let’s contemplate the following figure. I consider two examples of reactions (which will turn out to correspond to graphic beta moves in the realm of graphic lambda calculus). In the first row we see a reaction between a molecule and the enzyme $\beta^{+}$, which results into two other molecules and some GARB. There is a small region in the initial molecule, marked by a dashed red closed curve, which represents a reaction site for the $\beta^{+}$ molecule. In the second row is written the same reaction, but in a simpler form. The red ”+” sign is eliminated, the two molecules which are obtained are juxtaposes, as if they are floating in the 3D container, and the GARB is ignored. Moreover, the enzyme $\beta^{+}$ points towards the reaction site. The rows 3 and 4 describe another reaction. At closer inspection, it’s a reaction which can be interpreted as the inverse of the first one. Let’s examine directly the 4th row (which is obtained from the 3rd row by the same procedure as the 2nd row was obtained from the 1st). The reaction site of the enzyme $\beta^{-}$ is a pair of arrows from two different molecules, The resulting molecule is the same as the initial molecule from the previous reaction. ###### Definition 2.2 Here is the list of moves, with the names taken from graphic lambda calculus (where it is the case), by using the dictionary for translation (links to the web tutorials are given). 1. - graphic beta move and FAN-IN move : 1. - local FAN-OUT moves : 1. - DIST moves (”dist” comes from the word ”distributivity”): 1. - LOCAL PRUNING moves: 1. - Elimination of loops (i.e. moving loops from and to $GARB$) While all of these moves define the chemical concrete machine, some of them, seen in isolation, appear in other parts. I don’t know if there is any, more profound, reason for this, but nevertheless here are some examples of such places: 1. - in [1], the merge move (2.28) section 2.6 looks like the pair of graphic beta move - fan-in move; the blow up move (13.1) section 13.1 looks even more like the graphic beta move, 2. - in [11], the unzip move from section 3 looks almost alike the graphic beta move, 3. - in [9], after definition 3.1.3 of a bialgebra in a monoidal category, a DIST move and a LOC PRUNING move appear under the form of graphical identities. Of course, CO-COMM and CO-ASSOC moves can be seen as related to co-commutative and co-associative comonoids. ## 3 Using the chemical concrete machine (I) Let’s see what the chemical concrete machine can do. ### Lists and locks. Suppose you have a family of molecules which you want to free in the medium in a given order. This corresponds to having a list of molecules, which is ”read” sequentially. I shall model this with the help of the zipper from graphic lambda calculus. Suppose that the molecules we want to manipulate have the form $A\rightarrow A^{\prime}$, with $A$ and $A^{\prime}$ from the family of ”other molecules” and $\rightarrow$ an arrow. Here are three zippers. The first zipper, called a $\beta$ zipper, behaves in the following way. In the presence of $\beta^{+}$ enzymes, there is only one reaction site available, namely the one involving the red and green nodes in the neighbourhood of the $D,D^{\prime}$. So there is only one reaction possible with a $\beta^{+}$ enzyme, which has a a result the molecule $D\rightarrow D^{\prime}$ and a new, shorter $\beta$ zipper. This new zipper has only one reaction site, this time involving nodes in the neighbourhood of $C,C^{\prime}$, so the reaction with the enzyme $\beta^{+}$ gives $C\rightarrow C^{\prime}$ and a new, shorter zipper. The reaction continues like this, freeing in order the molecules $B\rightarrow B^{\prime}$, then $A\rightarrow A^{\prime}$ and $E\rightarrow E^{\prime}$. The second zipper is called a FAN-IN zipper (or a $\phi$ zipper). It behaves the same as the previous one, but this time in the presence of the FAN-IN enzyme $\phi^{+}$. In the case of a mixed zipper, he first molecule $D\rightarrow D^{\prime}$ is released only in the presence of a $\phi^{+}$ enzyme, then we are left with a $\beta$ zipper. This can be used to lock zippers. Look for example at the following molecule: called a locked $\beta$ zipper. In the presence of only $\beta^{+}$ enzymes, nothing happens, because there is no reaction site compatible with the $\beta^{+}$ enzyme. . If we add into the reactor also $\phi^{+}$ enzymes, then the zipper unlocks, by releasing a loop (that’s seen as garbage) and a $\beta$ zipper which starts to react with $\beta^{+}$ enzymes. The same idea can be used for keeping a molecule inactive unless both $\phi^{+}$ and $\beta^{+}$ enzymes are present in the reactor. Say that w have a molecule $A\rightarrow A^{\prime}$ which is made inactive under the form presented in the following figure The molecule is locked, but it has two reaction sites, one sensible to $\beta^{+}$, the other sensible to $\phi^{+}$. Both enzymes are needed for unlocking the molecule, but there is no preferred order of reaction with the enzymes (in particular these reactions can happen in parallel). ### Sets. Suppose now that we don’t want to release the molecules in a given order. We need to prepare a molecule which has several reaction sites available, so that multiple reactions can happen in parallel, as in the last example. Mathematically, that could be seen as a representation of the set of molecules we want to free, instead of the list of them. This is easy, as described in the next figure: On the first row we see what is called a $\beta$ set. It has 4 possible reaction sites with the enzyme $\beta^{+}$, therefore, in the presence of this enzyme, the molecules $A\rightarrow A^{\prime}$, … , $E\rightarrow E^{\prime}$ are released at the same moment. A FAN-IN, or $\phi$ set, behaves the same as the previous one, but this time in the presence of the FAN-IN $\phi^{+}$ enzyme. Finally a mixed set releases different molecules, depending on the presence of $\phi^{+}$ or $\beta^{+}$ enzymes. ### Pairs. As another example, here is a more involved molecule, which produces different pairs of molecules, according to the presence of $\phi^{+}$ or $\beta^{+}$ enzymes. In the following figure we see how we model a pair of molecules, then two possible reactions a represented. The idea is that we can decide, by controlling the amount of $\beta^{+}$ or $\phi^{+}$, to couple $A$ with $D$ and $C$ with $D$, or to couple $A$ with $B$ and $C$ with $D$. 4\. Multipliers and comultipliers. Multipliers and co-multipliers are molecules which self-multiply. More precisely, in the next figure we see the definition of those: Here $A$ and $A^{\prime}$ are molecules from the formalism of the chemical concrete machine and $1$ and $2$ are labels. The blue arrow means any finite sequence of chemical reactions (moves) from the formalism. In certain configurations, zippers are multipliers. In the following figure we see what happens in the presence of DIST enzymes: The reaction continues: Now, the zipper multiplied into two zippers, but they are still connected. We need more information about $A,B,C,D$ and $A^{\prime},B^{\prime},C^{\prime},D^{\prime}$. Remark that: In conclusion: if $A,B,C,D$ are multipliers and $A^{\prime},B^{\prime},C^{\prime},D^{\prime}$ are co-multipliers, then the zipper is a multiplier. This will be used in the next section. ## 4 Using the chemical concrete machine (II). Turing universality The B,C,K,W system ”is a variant of combinatory logic that takes as primitive the combinators $B,C,K$, and $W$. This system was discovered by Haskell Curry in his doctoral thesis”. I shall explain first which are the correspondents of the B, C, K, W, combinators in the formalism of the chemical concrete machine. (Via the red- green vs black-white change of notation, they can be deduced from their expressions in graphic lambda calculus, which are obtained by using the algorithm described in section 3 [3]. ) ###### Definition 4.1 The class of combinator molecules (i.e. those which correspond to combinators written in the B,C,K,W system) is defined by the following procedure: 1. - the B, C, K, W molecules are combinator molecules, 2. - if A, D are combinator molecules then the molecule AD, defined as in the next figure, is a combinator molecule, 3. - if A is a combinator molecule and B is another molecule obtained from A by the application of the moves of the chemical concrete machine, then B is a combinator molecule. The purpose of this section is to prove the following. ###### Theorem 4.2 The chemical concrete machine contains combinatory logic under the form of the B,C,K,W system. ### Proof. By Theorem 3.1 and Proposition 3.2 [3], using also the dictionary between the notations in graphic lambda calculus and those of the chemical concrete machine, it follows that the combinatory algebra (in particular in the form of the B,C,K,W system) can be expressed in the chemical concrete machine if: 1. - we replace the GLOBAL PRUNING move from graphic lambda calculus with the management of the GARB (garbage) class of molecules (alternatively, we might simply neglect GLOBAL PRUNING and we obtain something a bit more expressive than combinatory algebra), 2. - we add to the chemical machine list of moves the following GLOBAL FAN-OUT move: Here, the meaning of the red dashed circle is that the GLOBAL FAN-OUT move can be applied only if the molecule A has no arrows outside the red dashed circle, with the exception of the one connecting it with the fan-out molecule. In Proposition 3.2 [3] was used the S,K,I combinators system, instead of the B, C, K, W, but the Theorem 3.1 [3] shows that already untyped lambda calculus can be expressed with the chemical concrete machine, provided that the GLOBAL FAN-OUT move is added. The choice of the B,C,K, W system instead of the S,K,I system is only for exposition purposes. The goal is to show that for any combinator molecule, the GLOBAL FAN-OUT move can be replaces by a finite chain of local moves of the chemical concrete machine. As a consequence, we don’t need the GLOBAL FAN-OUT move in order to have combinatory logic in the chemical concrete machine. By looking at the last part of the previous section, we want to prove that any combinator molecule is a multiplier. This is done in the following two steps. Step 1. If A, D are combinator molecules which are multipliers then AD is a multiplier. The proof is given in the next figure, by using DIST moves: Step 2. Is only left to prove that B,C, K, W are multipliers. By the conventions of the chemical concrete machine, I mention here the enzymes which are involved in the reactions, instead of writing the moves, like in the graphic lambda calculus. The proofs for B and C are very much alike, therefore I put here only the proof that B is a multiplier: The proof that K is a multiplier is the following: As a side remark, notice how, in both cases, the reactions seem feasible, in the sense that they can be accomplished by a linear process, because at any step there is only one kind of reaction site available. For the W combinator (molecule), things get a bit more complex. There is a reaction (or move) which needs explanations. I called it DISENTANGLE (CO-ASSOC) reaction. It is this: It can clearly be done by a succession of CO-ASSOC moves (reactions). This ends the proof of the theorem. $\square$ From the point of view of the feasibility in the real world (provided a real implementation of the chemical concrete machine will appear), it seems hard to control the exact order of applications of CO-ASSOC moves which gives the DISENTANGLE move as an effect. So, probably, we shall need a ”disentangle enzyme” dedicated to this. As an alternative, remark that for proving that W is a multiplier we need an application of the DISENTANGLE composite move, described in the next figure: For practical (or theoretical as well) purposes, it is enough to take this as a move. In other words, what would get rid of needing a controlled sequence of CO-ASSOC reactions for multiplying the molecule W is this: assume that the molecule which is connected to the ”y” essential molecule (i.e. to the input of a FAN-OUT gate) is a ”propagator”. Propagators are defined in the next figure: Propagators are different from multipliers, because they are molecules with one selected input and one selected output which ”propagate along a FAN-OUT gate”, while multipliers are multiplied by a FAN-OUT gate. Propagators can serve in fact as labels, or names, which propagate along a tree of FAN-OUT gates. Let’s see, as an application, how the IFTHENELSE construct appear as a combinator molecule in the chemical concrete machine. In lambda calculus, there are terms called TRUE, FALSE and IFTHENELSE, which are the of the booleans true, false and if-then-else. The associated graphs in the chemical concrete machine are: Take two molecules A, B, with one exit each. Then IFTHENELSE TRUE A B should become A. In the chemical concrete machine, with only $\beta^{+}$ enzymes, the following chain of reactions happens: Along this chain of reactions, there is no other choice than the ones from the figure, because at every step there is only one reaction site available to the enzyme $\beta^{+}$. The result is, unsurprisingly, compatible with the lambda calculus version, with the exception that A and B are not supposed to be (graphs corresponding to) lambda terms. They can be anything, for example, from the family of ”other molecules”. In lambda calculus IFTHENELSE FALSE A B should become (by reductions) B. In the chemical concrete machine the following chain of reactions happens: With a little bit of imagination, if we look closer to what TRUE, FALSE and IFTHENELSE are doing, we see that it is possible to adapt the IFTHENELSE to a molecule which releases, under the detection of one molecule (like TRUE), the ”medicine” A, and under the detection of another molecule (like FALSE) the ”medicine” B. ## References * [1] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, J. Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 * [2] G. Berry, G. Boudol, The Chemical Abstract Machine, Theoretical Computer Science, 96(1):217–248, 1992 * [3] , M. Buliga, Graphic lambda calculus, to appear in Complex Systems, arXiv:1305.5786 * [4] M. Buliga, Dilatation structures I. Fundamentals, J. Gen. Lie Theory Appl., 1 (2007), 2, 65-95. arXiv:math/0608536 * [5] M. Buliga, Emergent algebras, arXiv:0907.1520 * [6] W.Fontana and L.W.Buss, The Barrier of Objects: From Dynamical Systems to Bounded Organizations, in: Boundaries and Barriers, J.Casti and A.Karlqvist (eds.), pp.56–116, Addison-Wesley, 1996 (pdf) * [7] W.Fontana and L.W.Buss, ’The Arrival of the Fittest’: Toward a Theory of Biological Organization, Bull. Math. Biol., 56, 1-64 (1994) (pdf) * [8] W.Fontana and L.W.Buss , What would be conserved if ‘the tape were played twice’?, Proc. Natl. Acad. Sci. USA, 91, 757–761 (1994) (pdf) * [9] A. Kissinger, Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing, arXiv:1203.0202 * [10] G. Moe-Behrens, The biological microprocessor, or how to build a computer with biological parts, Computational and Structural Biotechnology Journal, 7, 8 (Apr 2013) e201304003 * [11] D.P. Thurston, The algebra of knotted trivalent graphs and Turaev’s shadow world, Geom. Topol. Monogr. 4 (2002) 337-362, arXiv:math/0311458 * [12] Varela, F., H. R. Maturana and R. Uribe. 1974. Autopoiesis: the organization of living systems, its characterization and a model, BioSystems 5, 187-196.	arxiv-papers	2013-09-26T14:24:23	2024-09-04T02:49:51.572031	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Marius Buliga", "submitter": "Marius Buliga", "url": "https://arxiv.org/abs/1309.6914" }
1309.6923	CERN–2013–00512 August 2013Fifty years of the CERN Proton SynchrotronVolume IIEditors:S. GilardoniD. ManglunkiGENEVA2013 ISBN \| 978–92–9083–391–8 ---\|--- ISSN \| 0007–8328 DOI \| 10.5170/CERN–2013–005 Copyright © CERN, 2013 Creative Commons Attribution 3.0 Knowledge transfer is an integral part of CERN’s mission. CERN publishes this report Open Access under the Creative Commons Attribution 3.0 license (http://creativecommons.org/licenses/by/3.0/) in order to permit its wide dissemination and use. This monograph should be cited as: Fifty years of the CERN Proton Synchrotron, volume II edited by S. Gilardoni and D. Manglunki, CERN-2013-005 (CERN, Geneva, 2013), DOI: 10.5170/CERN–2013–005 Dedication The editors would like to express their gratitude to Dieter Möhl, who passed away during the preparatory phase of this volume. This report is dedicated to him and to all the colleagues who, like him, contributed in the past with their cleverness, ingenuity, dedication and passion to the design and development of the CERN accelerators. Abstract This report sums up in two volumes the first 50 years of operation of the CERN Proton Synchrotron. After an introduction on the genesis of the machine, and a description of its magnet and powering systems, the first volume focuses on some of the many innovations in accelerator physics and instrumentation that it has pioneered, such as transition crossing, RF gymnastics, extractions, phase space tomography, or transverse emittance measurement by wire scanners. The second volume describes the other machines in the PS complex: the proton linear accelerators, the PS Booster, the LEP pre-injector, the heavy-ion linac and accumulator, and the antiproton rings. Preface It was on 24 November 1959 that the proton beam in the CERN Proton Synchrotron was accelerated to a kinetic energy of 24 GeV. Thus the first strong-focusing proton synchrotron ever built has been faithfully serving the international physics community for 50 years. It has been the subject of a virtually continuous upgrade boosting its intensity per pulse from $10^{10}$ protons by more than three orders of magnitude to $3\times 10^{13}$ protons. Various injectors have been added and it has been modified such that, in addition to protons, light and heavy ions, positrons and electrons, as well as antiprotons could be accelerated or even decelerated often within the same supercycle. This would not have been possible had the initial design not been solid and sound allowing for maintainability, flexibility, and versatility and whose intrinsic potential was brought to fruition by the efforts and the ingenuity of generations of accelerator physicists, engineers, operators, and technicians. This report has been written to mark the fiftieth anniversary of the first operation of this unique accelerator. Volume I outlines the euphoric spirit in the European physics community in which such a bold design could be suggested, and gives an overview of the evolution of this unique accelerator described in a wealth of publications. This volume provides also a description in more depth of the outstanding achievements and highlights in its development. Volume II provides an overview of the injectors of the PS and of the accelerator system used for antiproton accumulation and storage, which has been closely associated with the PS. Contributors Jean-Paul Burnet, Christian Carli, Michel Chanel, Roland Garoby, Simone Gilardoni, Massimo Giovannozzi, Steven Hancock, Helmut Haseroth, Kurt Hübner, Detlef Küchler, Julian Lewis, Alessandra Lombardi, Django Manglunki, Michel Martini, Stephan Maury, Elias Métral, Dieter Möhl, Günther Plass, Louis Rinolfi, Richard Scrivens, Rende Steerenberg, Charles Steinbach, Maurizio Vretenar, Thomas Zickler Contents Preface id1 List of contributors id1 Linac 1 LABEL:S01-Linac1 The Proton Synchrotron Booster (PSB) LABEL:S02-PSB Linac 2 LABEL:S03-Linac2 The Antiproton Accumulator, Collector and Decelerator Rings LABEL:S05-AA-AC-AD The Low-Energy Antiproton and Ion Rings LEAR and LEIR LABEL:S06-LEAR-LEIR The LEP Pre-injector (LPI) LABEL:S04-LPI Linac 3 LABEL:S07-Linac3 See pages 1 of 01-Linac1.pdf See pages 2- of 01-Linac1.pdf See pages 1 of 02-PSB.pdf See pages 2- of 02-PSB.pdf See pages 1 of 03-Linac2.pdf See pages 2- of 03-Linac2.pdf See pages 1 of 05-AA-AC-AD.pdf See pages 2- of 05-AA-AC- AD.pdf See pages 1 of 06-LEAR-LEIR.pdf See pages 2- of 06-LEAR-LEIR.pdf See pages 1 of 04-LPI.pdf See pages 2- of 04-LPI.pdf See pages 1 of 07-Linac3.pdf See pages 2- of 07-Linac3.pdf	arxiv-papers	2013-09-26T14:44:55	2024-09-04T02:49:51.579618	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Simone Gilardoni (ed.) (CERN), Django Manglunki (ed.) (CERN),\n Jean-Paul Burnet, Christian Carli, Michel Chanel, Roland Garoby, Massimo\n Giovannozzi, Steven Hancock, Helmut Haseroth, Kurt H\\\"ubner, Detlef\n K\\\"uchler, Julian Lewis, Alessandra Lombardi, Michel Martini, Stephan Maury,\n Elias M\\'etral, Dieter M\\\"ohl, G\\\"unther Plass, Louis Rinolfi, Richard\n Scrivens, Rende Steerenberg, Charles Steinbach, Maurizio Vretenar, Thomas\n Zickler", "submitter": "Scientific Information Service CERN", "url": "https://arxiv.org/abs/1309.6923" }
1309.7083	# Characterization of EASIROC as Front-End for the readout of the SiPM at the focal plane of the Cherenkov telescope ASTRI D. Impiombato [email protected] S. Giarrusso [email protected] T. Mineo [email protected] M. Belluso S. Billotta G. Bonanno O. Catalano A. Grillo G. La Rosa D. Marano G. Sottile INAF, Istituto di Astrofisica Spaziale e Fisica cosmica di Palermo, via U. La Malfa 153, I-90146 Palermo, Italy INAF, Osservatorio Astrofisico di Catania, via S. Sofia 78, I-95123 Catania, Italy ###### Abstract The Extended Analogue Silicon Photo-multiplier Integrated Read Out Chip, EASIROC, is a chip proposed as front-end of the camera at the focal plane of the imaging Cherenkov ASTRI SST-2M telescope prototype. This paper presents the results of the measurements performed to characterize EASIROC in order to evaluate its compliance with the ASTRI SST-2M focal plane requirements. In particular, we investigated the trigger time walk and the jitter effects as a function of the pulse amplitude. The EASIROC output signal is found to vary linearly as a function of the input pulse amplitude with very low level of electronic noise and cross-talk ($<$1%). Our results show that it is suitable as front-end chip for the camera prototype, although, specific modifications are necessary to adopt the device in the final version of the telescope. ###### keywords: Front-end, ASIC for SiPM, Front-end Electronics for Detector, Readout Analogue Electronics Circuits, Electronic Detector Readout Concepts, Trigger Concepts and Systems, Cherenkov Telescope ††journal: Nuclear Physics A ## 1 Introduction The use of Cherenkov telescopes in gamma-ray astronomy opened a new frontier in the study of the emission from very energetic sources such as supernovae, neutron stars and supermassive black-holes. Since the discovery of TeV emission from the Crab nebula with Whipple [1] in 1989, the number of known sources emitting in this range of energy has rapidly increased to hundreds in a relatively small number of years. The Cherenkov Telescope Array (CTA) is a new generation of ground based gamma-ray telescopes aiming at observing the sky from a few tens of GeV to beyond 100 TeV. It is an array comprising telescopes with three different sizes each designed to cover a specific range of energy [2]. The low energy band (from a few tens of GeV up to a hundred GeV) will be covered by 24 m mirror diameter telescopes that observe a Field of View (FoV) of 4∘-5∘. The range 100 GeV-1 TeV will be observed by 10-12 m mirror diameter telescopes having a FoV of 6∘-8∘. The highest energy region will be observed by small size telescopes (4-6 m mirror diameter) having a FoV of about 10 degrees. ASTRI (Astrofisica con Specchi a Tecnologia Replicante Italiana; [3]), is a ’Flagship Project’ of the Italian Ministry of Education, University and Research led by the Italian Institute of Astrophysics (INAF), whose main object is to build a prototype for the small size CTA telescopes. The prototype, hereafter named ASTRI SST-2M telescope, will adopt innovative solutions to explore the 1-100 TeV range of the electromagnetic spectrum: a wide field dual mirror Schwarzschild-Couder optical system and Silicon Photo- Multipliers (SiPM) as sensors of the camera at the focal plane. The telescope design foresees a focal ratio F# = 0.5, an equivalent focal length of 2150mm, an average effective area of about 6.5m2, and, considering the size of the focal plane surface, it covers a Field of View (FoV) of $\sim$9.6∘ in diameter. The detection of Very High Energy (VHE) gamma photons is possible by observing the Cherenkov light produced by the relativistic secondary particles in the air shower generated by the interaction with the Earth atmosphere. This light, emitted in the ultraviolet band within a cone of about 1.3∘, reaches the ground in a ” pool ” having a radius of $\sim$120 m. It is very faint and lasts only a few ns with a duration that depends mainly on the distance between the shower core and the telescope axis [4]. SiPMs are suitable for the detection of the Cherenkov flashes, as already demonstrated by the FACT project [5], because they are very fast and sensitive to the light in the range 300-900 nm. Their drawbacks with respect to the traditional PMTs (PhotoMultiplier Tubes) are: a very high dark count, after pulses, optical cross-talk between elementary diodes inside a pixel and a gain strongly dependent on chip temperature. Nevertheless, for a single pixel the rate of the dark counts is lower than the rate of the Night Sky Background (NSB), so that the high instrumental background does not degrade the telescope sensitivity. Moreover, the effects of optical cross-talk and after pulses, that may affect the sensitivity, are typically lower than 20%[6] and the gain can be kept stable with an adequate strategy of temperature control. In this paper, we present a set of measurements aimed at characterizing the performance of EASIROC, a device devoted to read the SiPM output, and compare the results with ASTRI SST-2M focal plane requirements. ## 2 The ASTRI Camera The camera at the focal plane of the ASTRI SST-2M is based on monolithic SiPMs Hamamatsu S11828-3344m111http://www.hamamatsu.com/sp/hpe/HamamatsuNews/HEN111.pdf with 4x4 squared pixels, 3$\times$3mm large, made up of 3600 elementary diodes of 50$\mu$m pitch giving a filling factor of 62%. The physical pixels are 2$\times$2 grouped in logical pixels (6.2$\times$6.2mm) shown with a yellow square in Fig. 1. Therefore, considering that, at the focal plane, 1∘ corresponds to 37.5mm, the logical pixel has a size of 0.17∘ matching the optics angular resolution. The detector units are organized in an array of 37 Photon Detection Modules (PDM) with 8$\times$8 logical pixels each (see Fig. 1), for modularity and a fast read out of the focal plane after any trigger. The trigger threshold is chosen to have a maximum rate of $\sim$300 Hz from the whole focal plane that ensures a dead time $<$3%. The energy working range of the the ASTRI SST-2M telescope prototype is 1-100 TeV and, considering the optics area, the requirements for the maximum number of photoelectrons detected in one pixel is 1000 with a goal of 2000. The very short duration of gamma events requires a dedicated FEE (Front-End Electronics) capable, not only to catch the very fast pulses of Cherenkov light, but also to provide auto-trigger capability. The device proposed for the ASTRI SST-2M telescope prototype is the Extended Analogue Silicon Photo- Multiplier Integrated Read Out Chip (EASIROC; [7]) equipped with 32-channels each with the capability of measuring charge from 1 to 2000 photoelectrons assuming a SiPM gain of 106. Two EASIROC devices are then devoted to read a single PDM. Figure 1: A PDM of ASTRI SST-2M focal plane composed of 4$\times$4 SiPMs. The yellow square shows the logical pixels obtained grouping 2$\times$2 physical pixels ## 3 The EASIROC chip EASIROC, produced by Omega Micro222http://omega.in2p3.fr, is a 32 channel fully analogue front-end Application-Specific Integrated Circuit (ASIC) dedicated to the readout of SiPM detectors. The architecture of its analog core and the characteristics are shown in Fig. 2 and Table 1, respectively. In each channel, two separate electronics chains allow High- and Low-gain (HG/LG) of the signal in order to measure charge from 160 fC up to 320 pC covering a range up to 2000 photoelectrons for the adopted $10^{6}$ SiPM gain. The high voltage of each logical pixel can be individually tuned by varying the bias voltage in a 8-bit DAC from 0 to 4.5V in order to equalize the gains. Each of the two chains is composed of an adjustable preamplifier followed by a tunable shaper and a track and hold circuit. A third chain is implemented to generate a trigger using a fast shaper followed by a discriminator, whose threshold is set by a 10-bit DAC common to all the 32 channels (see Fig. 2). The signal shaping time is set to 50 ns according to our needs, while the trigger shaping time is fixed to 15 ns. The power consumption is lower than 5 mW/channel and unused features can be disabled to further save power. All EASIROC main parameters can be programmed by downloading a configuration table through a slow control serial line. The processing of the analog signal takes place in the front-end channels of the device, while the read-out is handled at the internal back-end of the ASIC. The outputs of all the channels can be readout from multiplexers that, running in parallel, can sequentially switch the sampled signal of the two chains (LG and HG). An evaluation board has been designed and realized by Omega Micro to test the functional characteristics and performance of this ASIC. It allows an easy access to the EASIROC output and provides many test points to the Field Programmable Gate Array (Altera Corporation - Cyclone FPGA Model EP1C6Q240C6N) dedicated lines (see Fig. 3). It is equipped with two external Analog-to- Digital Converters (ADC) to allow ASIC digital data acquisition. The trigger of each channel and the OR output of the 32 trigger channels ($OR32$) are available in parallel. The 32 triggers are sent to the FPGA that, according to the set threshold, generates a common hold signal ($HOLD-B$) for the output read-out. A LabVIEW software of the National Instruments, developed by the LAL (Laboratoire de l’Acc$\acute{e}$l$\acute{e}$rateur Lin$\acute{e}$aire) Tests group333 http://www.lal.in2p3.fr/, has been provided together with the evaluation board to command the EASIROC chip and to receive the output via Universal Serial Bus (USB) connection. It allows one to send the ASIC configuration and to receive the output bits via a USB cable connected to the test board. In our measurements, we used a new version of the chip, EASIROC-A, kindly provided by Omega Micro. Table 1: Main characteristics of the EASIROC chip. Technology: \| Austria-Micro-Systems (AMS) SiGe 0.35 $\mu$m ---\|--- Dimensions : \| 16.6 mm2 (4.15$\times$4.01mm) Power Supply : \| 4.5V/0V Consumption: \| 4.84mW per channel Package : \| Naked (PEBS) TQFP160 Figure 2: Architecture of the front-end EASIROC (Omega Micro courtesy) Figure 3: Architecture of the PCB (Printed Circuit Board) of the EASIROC (Omega Micro courtesy) ## 4 Laboratory set-up A pulse function arbitrary generator is used to create the input charge injected in EASIROC. The signal is similar, as much as possible, to the SiPM one characterized by very fast rise time (a few hundreds of ps) followed by an exponential decay a few tens of ns). The amplitude of this signal for an input charge of 0.16 pC, equivalent to 1 pe for a SiPM gain of 106, is 300$\pm$4$\mu$V. According to the goal on the ASTRI SST-2M maximum rate per pixel, the gain of the LG chain preamplifier is set to 3 that produces a monotonic response up to 2000 pe. Moreover, we decided to fix the gain of the HG chain preamplifier to 150 allowing an almost linear working range up to $\sim$50 pe (see Sect. 6.3). The shaping time of the signal was fixed to 50 ns, considering that the higher energy events, mainly coming from large shower core distance, are expected to last up to $\sim$30 ns. In our sets of measurements, we first characterised the trigger behaviour and the delays introduced by the jitter and by the time walk. We then sampled the shape of the output signal for both HG and LG chains and investigated its linearity with respect to the injected charge. Finally, we evaluated the level of the cross-talk. All the measurements were performed using only one of the 32 channels (Ch 31). However, to check the validity of the results, some measurements were repeated for other channels. ## 5 Trigger The trigger chain, derived from the HG preamplifier, is composed of a dedicated 15 ns fast shaper followed by a discriminator which provides the trigger signal. The threshold of the discriminator is common to all the 32 channels and it is set by a 10-bit DAC whose output range goes from 1.06V up to 2.38V, in steps of 1.28mV. To characterise the DAC-controlled discriminator, we tested the threshold voltage at the 1024 DAC values using a Keithley 2000 Digital Multimeter and we found that Voltages follow a straight line with an average discrepancy of $\sim$0.5$\%$. The trigger output was evaluated varying the discriminator threshold for fixed injected charge in the range 0-8 pC (0-50 pe) using a reference clock (10kHz) synchronous to the signal. As an example, we show in Fig. 4 the efficiency curves for 0.33 pe, 1 pe, 2 pe and 3 pe. The last dotted line at $\sim$930 DAC corresponds to the trigger efficiency with no injection charge, obtained keeping the same reference clock. In this case, triggers are due to the electronics noise. From simulations, we evaluated that the requirement of 300Hz on the total trigger rate at the focal plane is ensured if at least four contiguous pixels are triggered, with the threshold set at 3 pe. The trigger efficiency as function of the injected charge is shown in Fig. 5 for a threshold set to 3 pe. Figure 4: Trigger efficiency for fixed injected charges as function of the discriminator threshold. The dotted line at 930 DAC counts corresponds to the trigger efficiency with no injection charge Figure 5: Trigger efficiency vs injected charge setting the discriminator threshold to 3 pe. ### 5.1 Time Walk and Jitter The time walk and jitter of the trigger were investigated by measuring, at the oscilloscope, the delay between the trigger at the chip and the strobe of the generator. The range of injected charge is 0.16–3.2 pC (1-20 pe) and the discriminator threshold level is sequentially set to 0 pe (the pedestal), 1 pe, 2 pe and 3 pe. Since these measurements could be influenced by the jitter in the signal, we first investigated its level and found that it is negligible being of the order of a few tens of ps. The time jitter, measured as the RMS of the delay distribution, are presented in Fig. 6 as a function of injected charge. As expected, it is higher at pedestal but, in the sampled charge range, its value is always lower than 1.23 ns and decreases with increasing threshold. For a discriminator threshold of 3 pe and injected charges higher than 0.64 pC (4 pe) it is lower than 0.3 ns. We show in Fig. 7 the values of the time walk as function of injected charges. We measured them as the mean of the delay distribution and shifted them so that the time walk corresponding to the largest injected charge is zero in each case. The time walk is always below 5.5 ns and it is reduced to 1 ns for signals higher than 1.92 pC (12 pe) for a discriminator threshold of 3 pe. Figure 6: Jitter of the trigger vs injected charge at several discriminator thresholds. Figure 7: Time walk of the trigger vs injected charge at several discriminator thresholds. The zero level was set to the time walk corresponding to the largest injected charge. ## 6 Output signal The expected range of the Cherenkov signal (1-2000 pe) requires the characterization of both HG and LG chain. Our measurements investigated the level of the electronic noise for different working conditions of the chip, the shape of the output signal and the linearity as function of the injected charge. ### 6.1 Electronic Noise The level of the noise affecting the signal was evaluated as function of the preamplifier gain. We defined the noise as the $RMS$ of the analogue output signal with no charge injected. To convert the voltages into Equivalent Noise Charge (ENC), we used the following formula: $ENC\,=\,K\times V_{RMS}\hskip 14.22636pt{\rm with}\hskip 14.22636ptK=\frac{Q_{Npe}}{V_{Npe}}$ (1) where $V_{RMS}$ is the rms spread in output voltage due to the noise, ${Q_{Npe}}$ and ${V_{Npe}}$ are the charge and the output voltage corresponding to a signal $Npe$, respectively. We found that $K=0.045$ [pe mV-1] for gain = 150 in the HG and $K=1.15$ [pe mV-1] for gain = 3 in the LG chain. The feedback capacitance (see Fig. 2) was varied in its working range 0.1-1.5 pF corresponding to the gain range 150-10 in the HG chain and to 15–1 in the LG one. The average noise together with its variability range are shown in Fig. 8, for both gains. In the HG chain, the noise is 2.1 mV at gain =150, which converts to 0.1 pe using the Equation 1. The noise in the LG chain is constant for all the investigated gains and its average value is 0.91$\pm$0.05 mV, equivalent to $\sim$1 pe. Figure 8: Average electronic noise in mV measured at the oscilloscope vs the preamplifier gain. The vertical dashed lines indicate the gains at which the EASIROC will be operated. The shaded areas represent the noise variability ranges. The left panel relates to the HG and the right panel to the LG. ### 6.2 Slow-shaper sampling The chip saves the amplitude of the preamplified and shaped signal at a given time, the hold time, directly set by the user using a track and hold cell (see Fig. 2). Two external ADCs, one for the HG and one for the LG, digitize the signal level measured at the hold time (see Fig. 2). We sampled the shaped signal for the two electronics chains at several hold times with the main task of evaluating the peaking time. Each point was derived averaging over 1000 samplings and several different injected charges were investigated fixing the shaping time to 50 ns. The pedestal-subtracted signal was normalized to give a peak amplitude of one, and the time axis was adjusted so that this peak was centred at 0. The curves as a function of time in ns are shown in 9 and 10 for the HG and LG chains, respectively. We note that, 10 ns after the peaking time, the amplitude of the signal produced by an injected charge of 5.6 pC is reduced by $\sim$25 ADC counts with respect to the peak (corresponding to $\sim$1.4% in the normalized curve of Fig. 9) for the HG chain, and by $\sim$1 ADC counts (corresponding to $\sim$1% in the curve of Fig. 10) for the LG chain. In order to compute the peaking time, the peak of the signal was fitted with a Gaussian whose best fit values are presented in Fig. 11 for the two gain chains. We note that, the peaking time of the LG is quite constant: it varies by 0.6 ns with respect to its average (47.1 ns) over the entire investigated range (56-338 pe). The case is different in the 6-50 pe range of the HG chain where the peaking time varies by about 6% with respect to the average of 74.0 ns that is moreover 24 ns higher than the designed shaping time (50 ns). Figure 9: Amplitude normalized to the ADC value at the peaking time versus the time in ns shifted at the peaking time for the HG chain. Figure 10: Amplitude normalized to the ADC value at the peaking time versus the time in ns shifted at the peaking time for the LG chain. Figure 11: Peaking time vs the injected charge. Red points are relate to the LG chain, blue points to the HG one. ### 6.3 Charge linearity The linearity of the system was checked by analyzing the slow shaper amplitude at the time 50 ns for the LG chain and 80 ns for the HG one as function of the injected charge. We assumed as error of each point the sigma of the distributions obtained performing 1000 tests. These uncertainties are always lower than 1%. Furthermore, the Integral and Differential Nonlinearity specified for the device AD9220 is much smaller than this uncertainty444http://www.alldatasheet.com/datasheet- pdf/pdf/237229/AD/AD9220.html. The ADC values as function of injected charges are shown in Fig. 12 and Fig. 13 for the LG and the HG. We note that the ADC values curve has a monotonic behaviour up to 8 pC (50 pe) for the HG and up to 320 pC (2000 pe) for the LG. These two working ranges are then suitable for the ASTRI SST-2M requirements (see Sect.4). Moreover, we established the range where the correlation between the ADC values and the charge can be modelled with a straight line. Starting from a few points we increased the fitting range up to the maximum charge that still gives an acceptable $\chi^{2}$. We find that the HG chain is linear in the range 0.16-6.4 pC and the LG chain in the range 0.16-64 pC, where $\chi^{2}$ =6.5 for 6 degrees of freedom(dof) and $\chi^{2}$ =13 (7 dof) were obtained for HG and LG, respectively. The measured points and the fitting lines in the validity ranges are shown in Fig. 12 and Fig. 13. Figure 12: ADC values vs injected charges for the HG chain. The continuous line is the linear best fit up to the injected charge of 6.4 pC (see text). Figure 13: ADC values vs injected charges for the LG chain. The continuous line is the linear best fit up to the injected charge of 64 pC (see text). ## 7 Cross-Talk Our last step of measurements were devoted to investigate the cross-talk as function of the injected charge. This was performed measuring the output signal in the two $Ch\leavevmode\nobreak\ 31$ neighboring channels ($Ch\leavevmode\nobreak\ 0$ and $Ch\leavevmode\nobreak\ 30$) for input charges ranging from 1 to 320 pC (6-2000 pe). We assumed that a cross-talk signal is detected if its intensity is 2.5 $\sigma$ above the noise. We find that no significant cross-talk was observed for the LG in the whole investigated range and for the HG below 5 pC (31 pe). The ratios, in percentage, between the outputs in $Ch\leavevmode\nobreak\ 0$ and $Ch\leavevmode\nobreak\ 30$ and the output in $Ch\leavevmode\nobreak\ 31$ for the HG chain for input charges above 5 pC are presented in Fig. 14. We note that the effect of the cross-talk is always lower than 1% in the ASTRI SST-2M HG working range ($<$50 pe) and its level is $<$3% in the range where HG saturates (bottom panel of Fig. 14). These values must be compared with the cross-talk between logical pixels, which is not known at the moment. Dedicated measurements are foreseen in the SiPM characterization. Figure 14: The ratios, in percentage, between the outputs in $Ch\leavevmode\nobreak\ 0$ and $Ch\leavevmode\nobreak\ 30$ and the output in $Ch\leavevmode\nobreak\ 31$ vs injected charge for the HG chain. The top panel corresponds to the ASTRI working range ($<$50 pe); the bottom panel shows the cross talk in the range where the HG chain saturates. ## 8 Summary and Conclusion We performed a set of measurements to investigate the performance of EASIROC as the ASIC front-end for the camera at the focal plane of the ASTRI SST-2M telescope prototype. The measurements were mainly aimed at understanding the triggering capability and the shape of the output signals. Table 2: Experimental results of EASIROC characterization. Jitter(discriminator thresholds 3 pe) \| 0.3 ns ---\|--- Time walk(discriminator thresholds 3 pe) \| $<$5.5ns Noise HG \| $\sim$0.1 pe Noise LG \| $\sim$1 pe Cross Talk HG \| $<$1$\%$ Cross Talk LG \| No The main experimental results are presented in the Table 2. They are summarized in the following points and compared with the ASTRI SST-2M requirements: * 1. Trigger: EASIROC is able to trigger events with a number of counts $>$0.3 pe, well below the value foreseen for the ASTRI SST-2M prototype. In fact, the requirement on the maximum trigger rate (300 Hz) can be obtained if at least four contiguous pixels each have a signal greater than 3 pe. The trigger time walk and jitter are always below 5.5 ns and 0.3 ns, respectively, for a trigger threshold set to 3 pe. They introduce an uncertainty in the charge measurements of a few percent in both chains, with the shaping function of EASIROC configured to give a signal with a broad peak. * 2. Signal: The output signal has a stable peaking time in the LG chain and its value corresponds to the set shaping time. Some problems are detected in the HG chain. The peaking time is on average shifted by $\sim$20 ns with respect to the shaping time with a significant drift as function of the injected charge. Since the hold signal $HOLD-B$ is common to both the chains in the present version of the chip, a reduction in the amplitude of the HG signal is measured (see Fig.9). This effect can be reduced, without degrading the performance of the LG chain, by choosing a suitable sampling time. However, a new version of the chip should correct this shortcoming either by anticipating the HG peaking time or by providing two different hold signals. We detected a monotonic behaviour up to 8 pC for HG chain and up to 320 pC for the LG one. The levels of the electronic noise and the cross-talk between channels introduced by EASIROC are negligible. In conclusion, our measurements showed that EASIROC is suitable for the front- end of the camera at the focal plane of the ASTRI SST-2M telescope prototype. Minor ad-hoc modifications in its re-design, as for example a different $HOLD$ signal for each electronics chain, are however necessary to tailor this device to be fully compliant with the telescope requirements. ## Acknowledgements The work presented in this paper was partially supported by the ASTRI, ”Flagship Project” financed by the Italian Ministry of Education, University, and Research (MIUR) and lead by the Italian National Institute of Astrophysics (INAF). We are deeply grateful to S. Callier, C. De La Taille, and L. Raux of the Omega Micro at Orsay and to M.C. Maccarone, A. Rubini and G. Tosti of the ASTRI collaboration for useful discussions and suggestions. ## References * [1] T.C. Weekes, et al., ”Observation of TeV gamma rays from the Crab nebula using the atmospheric Cerenkov imaging technique”, ApJ, 342, p.379-395, 1989. * [2] M. Actis et al., The CTA Consortium, ”Design concepts for the Cherenkov Telescope Array CTA: an advanced facility for ground-based high-energy gamma-ray astronomy”, Experimental Astronomy, Vol.32, Issue 3, p. 193-316, 2011. * [3] R. Canestrari, O. Catalano, P. Conconi, G. Pareschi, B. Sacco, S. Vercellone, on behalf of the ASTRI collaboration, ”The Italian ASTRI program: an end-to-end dual-mirror telescope prototype for Cherenkov light imaging above few TeV”, 32nd International Cosmic Ray Conference, Beijing, 2011. * [4] M. He$\ss$, K. Bernlöhr, A. Daum, et al., ”The time structure of Cherenkov images generated by TeV $\gamma$-rays and by cosmic rays”, Astroparticle Physics 11, 363-377, 1999. * Anderhub et al. [2009] H. Anderhub, M. Backes, A. Biland, et al., ”A novel camera type for very high energy gamma-ray astronomy based on Geiger-mode avalanche photodiodes”, Journal of Instrumentation, 4, 10010, 2009. * [6] P. Eckert, H.C. Schultz-Coulon, W. Shen, R. Stamen, A. Tadday, ”Characterisation Studies of Silicon Photomultipliers”, Nuclear Instruments and Methods in Physics Research Section A, Vol. 620, Issues 2 3, p. 217-226, 2010. * [7] S. Callier, C. de La Taille, G. Martin-Chassarda, L. Raux, ”EASIROC, an easy & versatile ReadOut device for SiPM”, TIPP 2011 Technology and Instrumentation in Particle Physics 2011. * [8] D. Impiombato, S. Giarrusso, et al., ”Characterization of the front-end EASIROC for read-out of SiPM in the ASTRI camera”, Procs. SciNeGHE 2012 Workshop, 20-22 June 2012, Lecce, Italy, Nuclear Physics B: Proceedings Supplements, in press.	arxiv-papers	2013-09-26T22:34:11	2024-09-04T02:49:51.588973	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "D. Impiombato, S. Giarrusso, T. Mineo, M. Belluso, S. Billotta, G.\n Bonanno, O. Catalano, A. Grillo, G. La Rosa, D. Marano, G. Sottile", "submitter": "Domenico Impiombato", "url": "https://arxiv.org/abs/1309.7083" }
1309.7148		arxiv-papers	2013-09-27T08:30:12	2024-09-04T02:49:51.597744	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Guo-Long Tan, Li-Min Liu, Weibing Wu", "submitter": "Guolong Tan", "url": "https://arxiv.org/abs/1309.7148" }
1309.7352	# Evolution of star clusters in a cosmological tidal field Steven Rieder1,2, Tomoaki Ishiyama3, Paul Langelaan1, Junichiro Makino4, Stephen L. W. McMillan5 and Simon Portegies Zwart1 1 Sterrewacht Leiden, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands 2 Section System and Network Engineering, University of Amsterdam, Amsterdam, The Netherlands 3 Center for Computational Sciences, University of Tsukuba, Japan 4 Interactive Research Center of Science, Graduate School of Science and Engineering Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan 5 Department of Physics, Drexel University, Philadelphia, PA 19104, USA [email protected] (Accepted …. Received …; in original form …) ###### Abstract We present a method to couple $N$-body star cluster simulations to a cosmological tidal field, using the Astrophysical Multipurpose Software Environment. We apply this method to star clusters embedded in the CosmoGrid dark matter-only $\Lambda$CDM simulation. Our star clusters are born at $z=10$ (corresponding to an age of the Universe of about 500 Myr) by selecting a dark matter particle and initializing a star cluster with 32,000 stars on its location. We then follow the dynamical evolution of the star cluster within the cosmological environment. We compare the evolution of star clusters in two Milky-Way size haloes with a different accretion history. The mass loss of the star clusters is continuous irrespective of the tidal history of the host halo, but major merger events tend to increase the rate of mass loss. From the selected two dark matter haloes, the halo that experienced the larger number of mergers tends to drive a smaller mass loss rate from the embedded star clusters, even though the final masses of both haloes are similar. We identify two families of star clusters: native clusters, which become part of the main halo before its final major merger event, and the immigrant clusters, which are accreted upon or after this event; native clusters tend to evaporate more quickly than immigrant clusters. Accounting for the evolution of the dark matter halo causes immigrant star clusters to retain more mass than when the z=0 tidal field is taken as a static potential. The reason for this is the weaker tidal field experienced by immigrant star clusters before merging with the larger dark matter halo. ###### keywords: galaxies: star clusters, galaxies: evolution, cosmology: dark matter, methods: N-body simulations ††pagerange: Evolution of star clusters in a cosmological tidal field–LABEL:lastpage ## 1 Introduction Globular clusters are dense self gravitating systems of a few $10^{4}$ to $\sim 10^{6}$ stars (Brodie & Strader, 2006). With an age of about 12.6 Gyr (Krauss & Chaboyer, 2003) they are among the oldest objects in the universe and tend to populate the haloes of galaxies. Their age is consistent with being born between $z=12$ to 7, which is consistent with the results of $\Lambda$CDM simulations (Kravtsov & Gnedin, 2005). From the time the clusters were born on galaxies grow by about a factor 100 in mass via mergers to their current mass, size and morphology (White & Rees, 1978; Peacock, 1999). The environment in which the globular clusters evolved since their birth has consequently changed quite dramatically over their lifetimes. These changes may have a profound effect on the evolution of star clusters. Most modern star-cluster simulations take some sort of background potential of the host galaxy into account. This started already in the early 1990s with Chernoff & Weinberg (1990), and soon afterwards became a lively industry. Many simulations have been performed with a fixed tidal limit (Vesperini & Heggie, 1997; Portegies Zwart et al., 1998; Giersz, 2001) whereas other include some sort of tidal potential with a more fluent description of the tidal field (Gnedin & Ostriker, 1997; Takahashi & Portegies Zwart, 2000; Lamers, Baumgardt & Gieles, 2010). In most of these simulations the cluster orbit was circular and did not change with time. In a few cases the orbit was allowed to be eccentric, but still did not change with time (Baumgardt & Makino, 2003; Giersz & Heggie, 2011, 2009). The next refinement was the relaxing of the orbital parameters, allowing the cluster orbit to change (Tanikawa & Fukushige, 2005, 2010; Praagman, Hurley & Power, 2010; Renaud, Gieles & Boily, 2011; Berentzen & Athanassoula, 2012; Renaud & Gieles, 2013). The evolution of star clusters in a live galactic potential combined with parametrized cluster evolution was studied by Kruijssen et al. (2011), and Matsui et al. (2012) performed simulations in which they resolve the formation of star clusters in a single galaxy merger event. However, the mass evolution of the parent galaxy is generally ignored in all these simulations, except for the few cases with a galaxy merger (Saitoh et al., 2009; Renaud, Gieles & Boily, 2011; Kruijssen et al., 2012; Renaud & Gieles, 2013). In this paper we study the evolution of star clusters in a cosmological environment. The background potential against which the star clusters are evolved, are taken from the CosmoGrid dark-matter only $\Lambda$CDM simulation (Portegies Zwart et al., 2010). We selected two Milky Way like haloes in which the star clusters are simulated. The coupling between the cosmological simulation and the star cluster is realized via the Astrophysical Multipurpose Software Environment (AMUSE) (Portegies Zwart et al., 2009, 2011, 2013; Pelupessy et al., 2013). In these simulations we initialize a total of 30 star cluster at $z=10$, and evolve them together with the cosmological simulation up to $z=0$. ## 2 The experimental setup We simulate star clusters in a $\Lambda$CDM environment. We do this in two steps, first by calculating the $\Lambda$CDM environment and then using the tidal field from this environment as an external tidal field for the star cluster simulations. We investigate the results of two distinct regions in the $\Lambda$CDM environment. ### 2.1 The CosmoGrid $N$-body simulation The cosmological simulation employed in this article originates from the CosmoGrid calculation (Portegies Zwart et al., 2010; Ishiyama et al., 2013), which is a dark matter-only $\Lambda$CDM simulation of $2048^{3}$ particles in a $(21~{}h^{-1}{\rm Mpc})^{3}$ co-moving cosmological volume. We performed these simulations using the GreeM (Ishiyama, Fukushige & Makino, 2009; Ishiyama, Nitadori & Makino, 2012) and Sushi (Groen et al., 2011) codes. GreeM is a massively parallel TreePM code based on the implementation of Yoshikawa & Fukushige (2005). The SUSHI code is an extension of the GreeM code, which can run on a planet wide grid of supercomputers. Within both codes, the equations of motion are integrated in co-moving coordinates using the leap-frog scheme with a shared, adaptive timestep. In this simulation each particle has a mass of $1.28\times 10^{5}$ ${\rm M}_{\odot}$. In total, we have 556 snapshots, separated by $\mbox{dt}\simeq 35\mbox{Myr}$ (for $t<7.5\mbox{Gyr}$) and $\mbox{dt}\simeq 17.5\mbox{Myr}$ (for $t\geq 7.5\mbox{Gyr}$). The CosmoGrid simulation lasted from $z=65$ to $z=0$. We employed the following cosmological parameters: $\Omega_{0}=0.3,\lambda_{0}=0.7,h=0.7,\sigma_{8}=0.8,n=1.0$. For more details on the simulation see Portegies Zwart et al. (2010); Ishiyama et al. (2013). ### 2.2 Halo catalogue We use the halo finder Rockstar (Behroozi, Wechsler & Wu, 2013) to identify haloes in each snapshot. Rockstar is based on adaptive hierarchical refinement of friends-of-friends groups in six dimensions and allows for the robust tracking of subhaloes. We use the gravitationally consistent merger tree code from Behroozi et al. (2013) to construct the merger history for all haloes identified by Rockstar. We use AMUSE to find the radial density profiles for our haloes. From the $z=0$ haloes, we then select two haloes, based on their relative isolation and a mass comparable to that of the Milky-Way Galaxy. The two haloes are quite similar in many respects, but have a different merger history and the number of subhaloes at $z=0$ is different. In Fig. 1, we present the merger history of the two haloes, both schematically and visually. Halo A completes a major merger at around $t=6.5$ Gyr, the halo it merges with can be seen in the third figure from the top. At the end of the simulation (see Fig. 2 a), it is in the process of merging with another similar-sized halo. Other than that, there is no significant interaction between $t=6.5$ Gyr and $t=13.7$ Gyr. By the end of the simulation, halo A is the largest object within a radius of $6.3~{}h^{-1}{\rm Mpc}$ (see Tab. 1). Halo B exists in a denser part of the volume, and as a result more structure is visible in the outskirts of its environment (see Fig. 2b). By $z=0$, it is the largest halo within only $0.97~{}h^{-1}{\rm Mpc}$. During its history, it underwent many small merger events, and one long-lasting major merger event that completed around $t=11$ Gyr. In the mass evolution of both haloes (Fig. 3), the larger merger events are clearly visible. Since the virial mass includes the mass from subhaloes, the mergers are visible here at the start of interaction, rather than at the end as in Fig. 1. The halo density profiles at $z=0$ (see Fig. 4) are consistent with the haloes described in Ishiyama et al. (2013), with concentration parameters $c_{\rm vmax}$ of $3.53$ and $3.76$ for haloes A and B respectively. The lower concentration of halo A may be explained by its ongoing major merger event, which causes the halo to have two cores (see Fig. 2a). Table 1: Properties of the selected haloes in the final CosmoGrid snapshot. Halo \| $M_{\rm vir}$ \| $R_{\rm vir}$ \| $V_{\rm max}$ \| $c_{\rm vmax}$ \| $N_{\rm sub}$ \| $D_{\rm n}$ \| b/a \| c/a ---\|---\|---\|---\|---\|---\|---\|---\|--- \| $10^{11}$ $h^{-1}{\rm M}_{\odot}$ \| $h^{-1}{\rm kpc}$ \| km/s \| \| \| $h^{-1}{\rm Mpc}$ \| \| A \| $6.33$ \| $173.8$ \| 140.2 \| 3.53 \| 61 \| 6.30 \| 0.81 \| 0.63 B \| $4.78$ \| $159.4$ \| 133.1 \| 3.76 \| 29 \| 0.97 \| 0.73 \| 0.68 $M_{\rm vir}$ and $R_{\rm vir}$ are the virial mass and radius (Bryan & Norman, 1998), $V_{\rm max}$ is the maximum of the rotation curve, $c_{\rm vmax}$ is the concentration parameter, $N_{\rm sub}$ is the number of subhaloes with a mass larger than $10^{8}$ $h^{-1}{\rm M}_{\odot}$, $D_{\rm n}$ is the distance of the nearest more massive halo. a, b and c are the principal axes of the halo. Figure 1: The merger history of the two selected dark-matter haloes A (left) and B (right). The size of each circle is proportional to the virial mass of the halo. Only haloes and subhaloes with a peak mass larger than $5\times 10^{8}$$h^{-1}{\rm M}_{\odot}$that are accreted at $z=0$ are plotted. The bottom two halo images are identical to those in Fig. 2. (a) (b) Figure 2: Projected image of the two selected haloes A (top panel) and B (bottom panel), in the final snapshot ($z=0$). The linear dimension of the image is $400~{}h^{-1}{\rm kpc}$. The intensity represents the column density (scaled to minimum/maximum values) and the colour is scaled to the velocity dispersion. Each galaxy halo contains about 5 million dark-matter particles. Figure 3: Evolution of the virial mass of the two selected haloes A and B. Halo A experiences major merger events around 5 Gyr and 11 Gyr, while halo B a long-lasting major merger event from around 8 Gyr on. Figure 4: Radial density profiles of the two selected haloes A and B and their environment out to $400~{}h^{-1}{\rm kpc}$ at $z=0$. The clusters are selected at galactocentric radii of $3$, $6$ and $12~{}h^{-1}{\rm kpc}$. ### 2.3 The clusters We select a total of 30 dark-matter particles from the $z=0$ snapshot. For both haloes we select 15 particles at random, equally divided over three bins at galactocentric radius $3\pm 0.05~{}h^{-1}{\rm kpc}$, $6\pm 0.05~{}h^{-1}{\rm kpc}$ and $12\pm 0.05~{}h^{-1}{\rm kpc}$. These selected particles are considered the globular clusters for which we will calculate the evolution. Since we do not apply further restrictions in the selection criteria, the clusters may (and likely will) have their peri- and apocentres well outside these bins. In Fig. 5, we show a projection of these particles in the central region of their host halo at $z=0$. (a) (b) Figure 5: Projected image of the central portion of the two selected haloes A (top panel) and B (bottom panel) with their ‘globular clusters’ at $z=0$. The image size is $24~{}h^{-1}{\rm kpc}$. Red squares indicate clusters at $3~{}h^{-1}{\rm kpc}$ from the halo centre, green circles those at $6~{}h^{-1}{\rm kpc}$ and blue triangles those at $12~{}h^{-1}{\rm kpc}$. We locate the selected dark-matter particles at $z=10$, which corresponds to an age of the universe of $\sim 0.5$ Gyr. In Fig. 6 we present the $z=10$ image of the two selected haloes with their selected dark-matter particles that will represent globular clusters, and the distributions from which these particles are drawn. In both haloes, the particles that end up in the more central parts of the halo are already the most concentrated in density centres at $z=10$. The particles of halo A are largely concentrated in two regions, in which the haloes that merge around $t=6.5$ Gyr form. In contrast, the particles of halo B are more spread out, reflecting the more violent history of this halo. (a) (b) Figure 6: Projected image of halo A (top) and B (bottom) at $z=9.65$. The image size is $2~{}h^{-1}{\rm Mpc}$. Red squares indicate clusters that at $z=0$ were selected at $3~{}h^{-1}{\rm kpc}$ from the halo centre, green circles those at $6~{}h^{-1}{\rm kpc}$ and blue triangles those at $12~{}h^{-1}{\rm kpc}$. The panels on the right display (from top to bottom) the distributions of particles that end up in spherical shells at $3$, $6$ and $12~{}h^{-1}{\rm kpc}$ from the halo centres at $z=0$. At the $z=10$ snapshot we initialize a ‘globular cluster’. Our clusters have 32 000 stars distributed in a Plummer (1911) sphere with a virial radius of 3 pc. The clusters are assumed to be born in virial equilibrium. All stars in the clusters have the same mass and we did not include stellar evolution. The star clusters are simulated using the AMUSE framework (Portegies Zwart et al., 2009, 2011, 2013; Pelupessy et al., 2013). Our simulation code solves for the equations of motion using Bonsai (Bédorf, Gaburov & Portegies Zwart, 2012) and ph4 (McMillan et al., 2012). Bonsai is a Barnes & Hut (1986) tree code that runs on GPUs. It supports both shared timesteps and block timesteps, the latter allowing individual blocks to have different timesteps for increased accuracy in dense regions without slowing the simulation down too much. For this code, we adopted an opening angle (which controls the accuracy, smaller angles being more accurate) of 0.6, we set the smallest timestep to be used to 1/65536 $N$-body time units (Heggie & Mathieu, 1986) and we used a softening length of 0.00125 $N$-body length units (0.00375 parsec). Ph4 is a direct $N$-body integrator with block time steps and GPU acceleration. In order to directly compare the results to those obtained with Bonsai, we apply the same softening length as before to the runs with ph4. In order to validate the use of the tree code, we compare a cluster simulated with Bonsai to a reference simulation using ph4. In Fig. 7, we present the mass and the Lagrangian radii of this simulated cluster as a function of time, for both Bonsai and ph4. The difference in mass evolution between ph4 and Bonsai remains quite small until about 5 Gyr. After this moment, both clusters go into core collapse and the two codes start to deviate more. Until about 8.5 Gyr, the ph4 cluster displays much higher mass loss than the Bonsai cluster as it expands following core collapse. After 8.5 Gyr, both codes again show similar behaviour. The Lagrangian radii of the clusters are nearly equal until core collapse occurs at about 5 Gyr. After this, the core collapse is initially deeper in ph4, while after 8.5 Gyr Bonsai reaches the same depth. From these results, we infer that our Bonsai simulations are not as well suited for determining the internal structure and evolution of the star clusters as ph4 would be, and underestimates mass loss due to core collapse. However, the effect of the tidal field on the mass loss rate is similar in ph4 and Bonsai. Since we investigate only the mass evolution of the clusters due to the tidal field in this article, we conclude that Bonsai gives an adequate indication of the effect of tidal fields on the cluster mass loss, and that it can be used to study the survivability of star clusters. Figure 7: Evolution of the bound mass (top panel) and the 90%, 50% and 1% Lagrangian radii (top to bottom, bottom panel) of star cluster A12-2. The cluster was simulated with Bonsai (red, solid curves) and ph4 (green, dashed curve). ### 2.4 The tidal field In each snapshot of the CosmoGrid simulation we calculate the tidal tensor at the location of the selected dark-matter particle, which represents a star cluster. The contribution to the tidal tensor $\mathbf{T}_{\rm t}$ from a particle with relative position $\mathbf{r}$ is given by the second derivative of the gravitational potential $\phi$: $T_{t}^{ij}(\mathbf{r^{\prime}})=-\frac{\partial^{2}\phi}{\partial{r^{\prime i}\partial r^{\prime j}}}$ (1) where $\mathbf{r}^{\prime}=\mathbf{r}+\epsilon$. For the CosmoGrid simulation, the value for the softening length $\epsilon$ was 175 parsec. The strength of the tidal field scales as $\frac{\partial^{2}\phi}{\partial^{2}\mathbf{r}}\sim\frac{1}{r^{3}}.$ Any particle at a distance of about $\epsilon$ will have as much effect on the tidal tensor as the largest halo in the CosmoGrid simulation (containing $\sim 3\times 10^{8}$ particles) would have at a distance of $\sim 117$ kpc. We therefore include the contribution from all particles within a radius of 125 kpc from our clusters to determine $\mathbf{T}_{\rm t}$. The strength of the tidal field is calculated from the eigenvalues $\lambda_{i}$ and eigenvectors $\mathbf{\nu}_{i}$ of this tensor $\mathbf{T}_{\rm t}$. The eigenvalues give the magnitude of tidal field, whereas the eigenvectors give the direction along which the system is stretched. This method for calculating the tidal tensor is similar to the one employed in Renaud, Gieles & Boily (2011). Since we calculate the tidal tensor $\mathbf{T}_{\rm t}$ from snapshots of the CosmoGrid simulation and the number of snapshots is limited, we do not have a continuous tidal field. In order to prevent large, sudden changes, we linearly interpolate the tidal tensor between snapshots to create a continuous tidal tensor. In Sec. 2.7, we validate this interpolation method. ### 2.5 Combining the clusters and the tidal field We use the tidal tensor to calculate the external potential acting on each of the stars in the simulated clusters. We integrate the internal potential of the clusters with this external potential using a Bridge-like scheme (Fujii et al., 2007), which is implemented in AMUSE (Portegies Zwart et al., 2013). This scheme can be used to combine interacting systems that are calculated in different instances and/or using different codes, i.e. multiple interacting star clusters, globular clusters in a galactic environment, a galactic disk in a halo potential or embedded star clusters (Pelupessy & Portegies Zwart, 2011). In this scheme, the cluster experiences the gravity from the external field through periodic velocity kicks. It alternates between these velocity kicks and a drift due to self gravity evolution of the system. During one time step, the system first experiences a kick of the velocities over a time step $dt/2$, then a drift over a time step $dt$, and finally another kick over $dt/2$. In our setup, the external potential is derived from the CosmoGrid simulation and therefore necessarily fixed, while the clusters receive velocity kicks from the external tidal field. ### 2.6 Escaping and bound stars In the simulations, we calculate a tidal radius from the cluster mass and the strength of the tidal field (given by the largest eigenvalue of the tidal tensor $\lambda_{\rm max}$). This tidal radius is equal to $R_{\rm tidal}=\left(\frac{GM}{\lambda_{\rm max}}\right)^{1/3}.$ (2) Particles at a distance from the cluster centre larger than $R_{\rm tidal}$ will experience a larger force from the external tidal field than from the cluster’s own internal mass. It is then considered an ‘escaping particle’, and not included in the cluster’s bound mass (defined as the total mass inside $R_{\rm tidal}$). If the particle returns to a position within the tidal radius, this is reversed. If it moves to a distance $>10R_{\rm tidal}$, the particle is removed from the simulation. At times when the cluster is located in the centre of a local subconcentration of dark matter or near a halocentre, the value of $\lambda_{\rm max}$ may become negative for a short period. In such cases, the tidal radius is not defined. At such times, the bound mass of the cluster is not evaluated and no mass loss is experienced by the cluster. Stars that may have escaped from the cluster during this period will however still be removed once the tidal radius is again defined. ### 2.7 Validation We validate the simulation environment by comparing our results with those obtained using a different method and to a star cluster in isolation. As a reference model, we simulate a star cluster in an orbit with $R_{\rm apo}=12~{}h^{-1}{\rm kpc}$ and $e=0.71$ around a point-mass of $10^{10}$ ${\rm M}_{\odot}$, by including this point-mass in the simulation. The simulated cluster contains 8 000 equal-mass stars of 1 ${\rm M}_{\odot}$within a Plummer (1911) sphere with a virial radius of 3 pc. We compare this model to a simulation where we first calculated the tidal field that would be experienced by such a cluster and used this as an external field for the simulation using the Bridge scheme. For this test, we run simulations with ph4, as the large difference in particle masses would make the reference simulation unsuitable for a single-precision tree code like Bonsai. The mass evolution of these test simulations are presented in Fig. 8. The difference in the mass evolution of the cluster in the simulations with and without Bridge is quite small, indicating the validity of this method for this set of parameters. Also, we validate the effect of the discretisation at which the tidal tensor is evaluated. Since the time-resolution of the cosmological simulation is limited to about 35 Myr at high $z$ and 17.5 Myr at low $z$, anomalies will be visible for clusters with an orbital period of this order. Ideally, one would like to increase the number of snapshots for the cosmological simulation in order to obtain a higher time-resolution for the tidal tensor. In Fig. 9 we present the mass evolution of a cluster in a static halo potential (Paczynski, 1990) with a core mass of $10^{9}$ ${\rm M}_{\odot}$ and a core radius of $1$ kpc, now using tidal tensors sampled with a time resolution ranging from 1 to 35 Myr. The orbital parameters of the cluster in this potential are $R_{\rm apo}=15$ kpc, $e=0.85$, and the cluster orbits the potential in $670$ Myr. The mass loss rates for the clusters using a tidal tensor sampled with a time interval of 1 Myr and 9 Myr have converged in this orbit, while the clusters using tensors sampled with a 17.5 or 35 Myr time interval show a slightly reduced mass loss rate and therefore longer lifetimes. For clusters with closer orbits, this effect will be stronger. In a cosmological setting, any change in the potential that takes place on a timescale similar to or smaller than the sampling rate cannot be taken into account properly, and may also lead to errors in the mass loss rate. Figure 8: Mass evolution of a 8 000 star cluster in isolation (solid curve), and in orbit around a point mass of $10^{10}$ ${\rm M}_{\odot}$(dashed and dotted curves). The dashed curve gives the mass-evolution of the cluster when the tidal field is incorporated in the gravitational $N$-body simulation using Bridge. The tidal field in the latter case was resolved at the resolution of the $N$-body integrator. The dotted curve gives the mass-evolution when the point mass is directly included in the simulation. Figure 9: Mass evolution of a 8 000 star cluster with tidal field calculated from a halo potential using the tidal tensor and evaluated using Bridge at discrete time intervals. The cluster orbits in the halo potential with $R_{\rm apo}=15\,{\rm kpc}$ and $e=0.85$. In order to study the effect of the time resolution of the tidal field, we evaluate the tidal tensor with time intervals of 1, 9, 18 and 35 Myr. ## 3 Results ### 3.1 The evolution of the selected haloes We perform simulations of star clusters in two selected haloes, which we call halo A and halo B. Halo A experiences two major merger events, the last of which is not completed by $z=0$ (see Figs.1 and 3), but otherwise its evolution is rather gradual from $z=65$ to $z=0$. By the end of the simulation (at $z=0$) its mass is $6.3\times 10^{11}\,\mbox{$h^{-1}{\rm M}_{\odot}$}$. Halo B has a more violent history; it experiences two major mergers, one between 4 Gyr and 5 Gyr and one with comparable mass between 8 Gyr and 10 Gyr (see also Figs. 1 and 3). By the end of the simulation at $z=0$ it has a mass of $4.8\times 10^{11}$ $h^{-1}{\rm M}_{\odot}$ (see Tab. 1). Even though the total mass of halo B is slightly smaller than that of halo A, both haloes contain a mass of $7\times 10^{9}$ $h^{-1}{\rm M}_{\odot}$ within 3 kpc from the halo centre (see Fig. 4). In each halo we have selected 15 dark-matter particles which are initialized at $z=10$ as star clusters, and evolved with the cosmological simulation as a background potential. ### 3.2 The evolution of the star clusters We perform 30 simulations of star clusters with a tidal field; 15 are initialized in halo A and 15 in halo B. An additional cluster is simulated in isolation, to identify the mass loss component caused by relaxation. For this cluster, a radius of 200 pc is used to determine if stars are bound. All clusters are born at $z=10$ (corresponding to an age of the Universe of about 500 Myr) with a total mass of $\sim 32\,000$ ${\rm M}_{\odot}$ and an initial virial radius of 3 parsec for each cluster. All stars have the same mass and we did not include stellar evolution. In Fig. 10 and Tab. 2 we present the mass evolution and final masses of these simulations. The mass evolution of all simulated clusters is rather gradual irrespective of the sudden events in the growth of the host haloes. The small and rather sudden changes in mass are caused by the pericentre passages of the clusters in its orbit around the dark matter host. The clusters with a smaller orbital separation at z=0 tend to lose mass at a higher rate. The averaged final mass of the star clusters in halo A is smaller than the final mass of those that evolved in halo B for each of the radial bins (see Tab. 2). This is consistent with halo B originating from a larger number of less massive haloes, causing the tidal forces experienced by clusters in this halo over time to be smaller. For both haloes the clusters selected around $3~{}h^{-1}{\rm kpc}$ show the strongest mass loss; this is noticeable from the first few Gyr on. This behaviour is as expected from the distribution of particles at $z=10$ (see Fig. 6), where we see that the particles that end up in the central parts are already more concentrated at high redshift. The rate of mass loss for these clusters proceeds more gradual compared to the clusters in wider orbits. Integrated over time clusters lose mass at a rather constant rate. Figure 10: Bound mass in the simulated clusters for haloes A (top figure) and B (bottom figure). The red, solid; green, dashed and blue, dotted lines indicate clusters selected at $3$, $6$ and $12~{}h^{-1}{\rm kpc}$, respectively. The black line indicates the mass of a cluster without an external tidal field. Table 2: Results for star cluster simulations using Bonsai, with various tidal tensors. Each cluster contains 32 000 equal mass stars of about 1${\rm M}_{\odot}$, and has an initial radius of 3 parsec. We use a softening length of $40/N$. Cluster type is Immigrant (I) or Native (N). The clusters indicated with a bold font are displayed in Figs.12 and 14. Cluster ID \| Halo \| Type \| Distance at z=0 \| Mass at z=0 ---\|---\|---\|---\|--- Isolated \| \| \| \| 28174 $\mathbf{{\rm\bf M}_{\odot}}$ A3-1 \| A \| I \| $3~{}h^{-1}{\rm kpc}$ \| 10157 ${\rm M}_{\odot}$ A3-2 \| A \| I \| $3~{}h^{-1}{\rm kpc}$ \| 15171 ${\rm M}_{\odot}$ A3-3 \| A \| N \| $3~{}h^{-1}{\rm kpc}$ \| 10969 ${\rm M}_{\odot}$ A3-4 \| A \| N \| $3~{}h^{-1}{\rm kpc}$ \| 11085 ${\rm M}_{\odot}$ A3-5 \| A \| N \| $3~{}h^{-1}{\rm kpc}$ \| 10169 ${\rm M}_{\odot}$ Average \| \| \| \| 11510 $\mathbf{{\rm\bf M}_{\odot}}$ A6-1 \| A \| I \| $6~{}h^{-1}{\rm kpc}$ \| 17906 ${\rm M}_{\odot}$ A6-2 \| A \| N \| $6~{}h^{-1}{\rm kpc}$ \| 24130 ${\rm M}_{\odot}$ A6-3 \| A \| N \| $6~{}h^{-1}{\rm kpc}$ \| 14964 ${\rm M}_{\odot}$ A6-4 \| A \| N \| $6~{}h^{-1}{\rm kpc}$ \| 17978 ${\rm M}_{\odot}$ A6-5 \| A \| N \| $6~{}h^{-1}{\rm kpc}$ \| 18430 ${\rm M}_{\odot}$ Average \| \| \| \| 18682 $\mathbf{{\rm\bf M}_{\odot}}$ A12-1 \| A \| I \| $12~{}h^{-1}{\rm kpc}$ \| 21677 ${\rm M}_{\odot}$ A12-2 \| A \| I \| $12~{}h^{-1}{\rm kpc}$ \| 16487 ${\rm M}_{\odot}$ A12-3 \| A \| N \| $12~{}h^{-1}{\rm kpc}$ \| 17574 ${\rm M}_{\odot}$ A12-4 \| A \| N \| $12~{}h^{-1}{\rm kpc}$ \| 15873 ${\rm M}_{\odot}$ A12-5 \| A \| N \| $12~{}h^{-1}{\rm kpc}$ \| 16904 ${\rm M}_{\odot}$ Average \| \| \| \| 17703 $\mathbf{{\rm\bf M}_{\odot}}$ B3-1 \| B \| I \| $3~{}h^{-1}{\rm kpc}$ \| 19422 ${\rm M}_{\odot}$ B3-2 \| B \| N \| $3~{}h^{-1}{\rm kpc}$ \| 18974 ${\rm M}_{\odot}$ B3-3 \| B \| N \| $3~{}h^{-1}{\rm kpc}$ \| 16366 ${\rm M}_{\odot}$ B3-4 \| B \| N \| $3~{}h^{-1}{\rm kpc}$ \| 15633 ${\rm M}_{\odot}$ B3-5 \| B \| N \| $3~{}h^{-1}{\rm kpc}$ \| 20839 ${\rm M}_{\odot}$ Average \| \| \| \| 18247 $\mathbf{{\rm\bf M}_{\odot}}$ B6-1 \| B \| I \| $6~{}h^{-1}{\rm kpc}$ \| 24774 ${\rm M}_{\odot}$ B6-2 \| B \| N \| $6~{}h^{-1}{\rm kpc}$ \| 18446 ${\rm M}_{\odot}$ B6-3 \| B \| I \| $6~{}h^{-1}{\rm kpc}$ \| 15988 ${\rm M}_{\odot}$ B6-4 \| B \| N \| $6~{}h^{-1}{\rm kpc}$ \| 21128 ${\rm M}_{\odot}$ B6-5 \| B \| N \| $6~{}h^{-1}{\rm kpc}$ \| 21398 ${\rm M}_{\odot}$ Average \| \| \| \| 20347 $\mathbf{{\rm\bf M}_{\odot}}$ B12-1 \| B \| I \| $12~{}h^{-1}{\rm kpc}$ \| 19986 ${\rm M}_{\odot}$ B12-2 \| B \| I \| $12~{}h^{-1}{\rm kpc}$ \| 23479 ${\rm M}_{\odot}$ B12-3 \| B \| N \| $12~{}h^{-1}{\rm kpc}$ \| 25819 ${\rm M}_{\odot}$ B12-4 \| B \| N \| $12~{}h^{-1}{\rm kpc}$ \| 26374 ${\rm M}_{\odot}$ B12-5 \| B \| I \| $12~{}h^{-1}{\rm kpc}$ \| 22137 ${\rm M}_{\odot}$ Average \| \| \| \| 23559 $\mathbf{{\rm\bf M}_{\odot}}$ In Tab. 2 we distinguish between two types of clusters: those that were part of the main halo before the final completed major merger event (at $t=6$ Gyr and $t=8$ Gyr for haloes A and B, respectively), and those that are accreted upon or after this event. The former we identify with ‘native’ clusters, and the others as ‘immigrants’. The difference between immigrant clusters and native clusters is apparent in the Figs. 11. (a) (b) (c) (d) (e) (f) Figure 11: Co-moving distance of the clusters to the main halo centre for clusters in halo A (left) and halo B (right). Top, middle and bottom figures show clusters selected at $3$, $6$ and $12~{}h^{-1}{\rm kpc}$, respectively. Merger events in both haloes are visible as particles fall towards the halo centre. The orbital periods of the clusters are clearly visible. In Fig. 11 we present the orbital evolution of the selected dark-matter particles (i.e. the clusters) from halo A (left) and halo B (right). In the following paragraphs, we discuss the evolution of two clusters from each halo; one immigrant and one native cluster in more detail. #### 3.2.1 The clusters in halo A In Fig. 12, we show the mass evolution, mass loss rate over intervals of 10 Myrs and tidal field strength for an immigrant cluster (nr. A6-1, left) and a native cluster (nr. A6-4, right). This halo experiences a major merger which starts at about 6 Gyr, at that time the immigrant cluster is also captured by the main halo. The merger lasts until about 8 Gyr (see also Fig. 1). When the merger is finished, the tidal field strength $\lambda_{\rm max}$ shows more frequent peaks, indicating it has a shorter orbital period than before the merger. However, there is little change in the mass-loss rate. The native cluster of halo A experiences the same merger but was already member of the major halo. Its orbit becomes somewhat less eccentric after the merger, while its apocentric distance and orbital period decrease. The mass- loss rate from this cluster is mostly unaffected by the merger. (a) (b) Figure 12: Evolution of two star clusters in orbit around halo A. To the left is a typical immigrant cluster identified with dark-matter particle nr A6-1 and to the right we present a native cluster, particle nr A6-4 (see also Tab.2). From top to bottom the panels show the bound mass, the mass-loss rate, the strength of the tidal field ($\lambda_{\rm max}$) and the co-moving distance of the cluster to the centre of the dark matter halo. The decrease seen in the co-moving distance, apart from the merger event, is caused by the expanding cosmic volume. In Fig. 13, we show the result of two star clusters (A12-1 and A12-4), both using the tidal tensor calculated from the evolving CosmoGrid halo and using a tidal tensor calculated from the static $z=0$ CosmoGrid halo. For the static halo case, we sampled the tensor using the orbital trajectory of the cluster around its parent halo. For the native cluster, the resulting mass evolution differs only marginally, while the immigrant cluster suffers considerably higher mass loss in the static halo case, especially around the time the halo merger takes place, when its orbit is erratic. The static halo is a good approximation for the native cluster, while it falls short for the immigrant cluster. Figure 13: Evolution of two star clusters, A12-1 (immigrant) and A12-4 (native). Both are simulated in an evolving halo as well as a static halo. #### 3.2.2 The clusters in halo B In Fig. 14 we present the details of the evolution of an immigrant cluster (nr. B6-3, left) and a native cluster (nr. B6-2, right) of halo B. The immigrant cluster is captured during the major merger event that starts around $t=8$ Gyr. Different from the clusters in halo A, the orbital period of the cluster around the dark matter halo is hardly visible in the cluster’s mass evolution (see Fig. 14, left, top panel). Just before the merger (around $t=7$ Gyr) the mass-loss rate is slightly smaller than before or after the merger. This is caused by the distortion of the infalling halo of which this cluster is a member at that time. When the merger is completed the cluster mass-loss rate has resumed to be as high as before the merger (see Fig. 14, left, second panel). The native cluster becomes part of the main halo during its first major merger, at around $t=5$ Gyr. After this merger, the tidal forces experienced by the cluster are stronger than before, visible in Fig. 14 (right, third panel) as a sudden increase of $\lambda_{\rm max}$ after $t=6$ Gyr. Its mass- loss rate is also increased, as can be seen from Fig. 14 (right, second panel) and the difference in slope of $M_{\rm bound}(t)$ at $t=4$ Gyr and $t=8$ Gyr. The second major merger event leads to a temporary reduction in mass-loss for the cluster (at around $t=10$ Gyr). After this second halo merger the orbital period is evidently visible in the mass evolution of the cluster. (a) (b) Figure 14: Evolution of two star clusters in orbit around halo B. To the left is a typical immigrant cluster identified with dark-matter particle nr B6-3 and to the right we present a native cluster, particle nr B6-2 (see also Tab.2). From top to bottom the panels show the bound mass, the mass-loss rate, the strength of the tidal field ($\lambda_{\rm max}$) and the co-moving distance of the cluster to the centre of the dark matter halo. Negative values of $\lambda_{\rm max}$ correspond to times where the cluster is located inside a local (sub)concentration of matter. The spikes in the bottom panels for the immigrant and the native clusters is associated with a confusion in identifying the main parent in the halo finder. ## 4 Discussion and conclusions We presented a method to simulate star clusters within a pre-calculated tidal field, using the AMUSE environment. As a proof of concept, we apply this method to calculate the mass loss rates for star clusters in two live $\Lambda$CDM haloes from the CosmoGrid simulation. Our method compares well to self-consistent simulations. We find that the mass loss rate strongly depends on the cluster’s orbital parameters around the halo centre, as well as the central mass of the halo. Also, tidal disruption due to the $\Lambda$CDM environment is weaker in haloes that experienced many mergers. Finally, we find that in a Milky Way-like halo, the contribution of the $\Lambda$CDM environment to $\Delta M/M_{\rm init}$ can be up to 0.6 for clusters with an initial mass of 32 000${\rm M}_{\odot}$. In this article, several effects have not been taken into account, most notably we used a dark matter-only simulation. In this section, we discuss the relevance of these effects and how we intend to address these shortcomings. Since the CosmoGrid simulation is a dark matter-only simulation, we do not account for the effect of baryons. Because the formation of the large-scale environment is dominated by dark matter, the lack of baryons has little influence on the formation of the haloes. Star clusters however likely contain little or no dark matter (Baumgardt et al., 2010; Conroy, Loeb & Spergel, 2011). Because our simulation lacks baryons, there is no indication where star clusters would have formed or ended up had they been formed in our cosmological environment. We therefore have to resort to our method of identifying tracer particles for the star clusters in the final snapshot. We selected particles based on their distance to the halo centre. However, it is very likely that globular clusters would have formed in more specific locations, and possibly followed paths quite different from the ones in our simulation. In a simulation that includes baryonic matter and star formation, it would be possible to detect locations and masses of star clusters as they form. The difference between old and young clusters herein is large: old clusters (like the globular clusters in the Milky Way) formed before there was a Galactic disk, and remain relatively free of its influence. Young clusters however form in the galactic disk, and the tides experienced by these clusters are dominated by encounters with giant molecular clouds and spiral arms (Lamers & Gieles, 2006; Gieles, Athanassoula & Portegies Zwart, 2007), the effect of which is about four times larger than the tidal field (e.g. Lamers & Gieles, 2006, Figure 1). For this reason, we focus on old stellar clusters and initialize our simulated star clusters at an early epoch, before the galactic environment would have formed. In order to simulate young clusters, a galaxy simulation including baryons would be required. However, when the old clusters formed, the GMC density in the star-forming environment was likely very high, causing early disruption of low-mass globular clusters (Elmegreen, 2010). This effect is not included in our simulations. In order to investigate a large number of clusters with a reasonable amount of stars, we used the Barnes & Hut (1986) tree code Bonsai for most of our star cluster simulations. We compared the results for two distinct star clusters to similar simulations with the direct $N$-body code ph4. The results for both runs are similar for both codes, the direct code showing enhanced mass loss around the time of core collapse. However, the tree code fails to accurately describe the inner structure of the star cluster, and requires the distance between stars to be softened. Another limit of our environment is its resolution. The spatial resolution of CosmoGrid is given by the softening length employed (175 parsec). Forces that occur on a scale similar to or smaller than this softening length are not accurately taken into account. Likewise, the mass resolution of dark matter particles in CosmoGrid is $1.28\times 10^{5}$ ${\rm M}_{\odot}$, about 4 times larger than the initial mass of our simulated clusters. The effects of tidal forces caused by a small dark matter object passing at close range to our clusters (such as a subhalo) are therefore limited, creating a possible bias against the effect of such structures. However, this resolution effect would be more important for baryons than it is for dark matter. The benefit of using a large-scale cosmological simulation however, is that the formation of a dark-matter halo is followed. A simulation that only models a collision between galaxies would not take the earlier history and distribution of star clusters into account. In order to have the benefits of both a cosmological environment and high resolution, one could use re- simulation, where galaxies are simulated at high resolution within a lower- resolution environment. Another limit imposed by our use of a pre-calculated simulation is its limited number of snapshots. We lack continuous information about the tidal tensor. In order to prevent sudden changes in the tidal field, we interpolate the tidal tensor between snapshots. However, it remains impossible to accurately track sudden changes in the tidal field on timescales shorter than our time resolution, such as those occurring during halo mergers. Since the orbital periods of our clusters around the halo centre are larger than this time resolution, we do not expect this to have a large influence on the evolution of the tidal field. However, short-lasting passages of nearby objects may not be taken into account accurately, and a method in which the tidal field is sampled at more intervals remains preferable. In a follow-up article (Rieder et al., in prep.), we will apply the method described in this article to the evolution of star clusters in the disk of a simulated Milky Way-type galaxy. In this follow-up, we will address several of the limitations discussed above, especially the lack of baryons and as a result the orbits and origins of the clusters. ## Acknowledgements It is our pleasure to thank the anonymous referee for very helpful suggestions and comments that greatly helped to improve the article. Also, we are grateful to Arjen van Elteren, Derek Groen, Inti Pelupessy, Jeroen Bédorf, Mark Gieles and Nathan de Vries for support, interesting discussions and useful suggestions. This work was supported by NWO (grants IsFast [#643.000.803], VICI [#639.073.803], LGM [#612.071.503] and AMUSE [#614.061.608]), NCF (grants [#SH-095-08] and [#SH-187-10]), NOVA and the LKBF in the Netherlands, and by NSF grant AST-0708299 in the U.S. T.I. is financially supported by MEXT HPCI STRATEGIC PROGRAM and MEXT/JSPS KAKENHI Grant Number 24740115. We thank the DEISA Consortium (EU FP6 project RI-031513 and FP7 project RI-222919) for support within the DEISA Extreme Computing Initiative (GBBP project). 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1309.7446	# estimates of the gaps between consecutive eigenvalues of Laplacian Daguang Chen∗, Tao Zheng, Hongcang Yang∗∗ ###### Abstract. By the calculation of the gap of the consecutive eigenvalues of $\mathbb{S}^{n}$ with standard metric, using the Weyl’s asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ We conjecture that this order is also right for general Dirichlet problem of the Laplace operator, which is optimal if this conjecture holds, obviously. In this paper, using new method, we solve this conjecture in the Euclidean space case intrinsically. We think our method is valid for the case of general Riemannian manifolds and give some examples directly. ###### Key words and phrases: Laplacian, consecutive eigenvalues, test function, Riemannian manifold, Hyperbolic space ###### 2010 Mathematics Subject Classification: Primary 35P15, 58C40; Secondary 58J50. The work of the first named author was partially supported by NSFC grant No. 11101234. The work of the third named author was partially supported by NSFC and SF of CAS. ## 1\. introduction Let $\Omega$ be a bounded domain in an $n$-dimensional complete Riemannian manifold $M$ with boundary (possible empty). Then the Dirichlet eigenvalue problem of Laplacian on $\Omega$ is given by $\left\\{\begin{aligned} \Delta u=&-\lambda u,\;\mbox{in}\;\Omega,\\\ u=&0,\;\;\;\;\;\;\;\;\mbox{on}\;\partial\Omega,\end{aligned}\right.$ (1.1) where $\Delta$ is Laplaican on $M$. It is well known that the spectrum of (1.1) has the real and purely discrete eigenvalues $0<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\cdots\nearrow\infty,$ (1.2) where each $\lambda_{i}$ has finite multiplicity which is repeated according to its multiplicity. The corresponding orthonormal basis of real eigenfunctions will be denoted $\\{u_{j}\\}_{j=1}^{\infty}$. We go forward under the assumption that $L^{2}(\Omega)$ represents the real Hilbert space of real-valued $L^{2}$ functions on $\Omega$. We put $\lambda_{0}=0$ if $\partial\Omega=\emptyset$. An important aspect of estimating higher eigenvalues is to obtain as precise as possible the estimate of gaps of consecutive eigenvalues of (1.1). In this regard, we will review some important results on the estimates of eigenvalue problem (1.1). For the upper bound of the gap of consecutive eigenvalues of (1.1), when $\Omega$ is a bounded domain in an $2$-dimensional Euclidean space $\mathbb{R}^{2},$ in 1956, Payne, Pólya and Weinberger (cf.[29] and [30]) proved $\lambda_{k+1}-\lambda_{k}\leq\frac{2}{k}\sum_{i=1}^{k}\lambda_{i}.$ (1.3) C. J. Thompson [34], in 1969, extended (1.3) to $n$-dimensional case and obtained $\lambda_{k+1}-\lambda_{k}\leq\frac{4}{nk}\sum_{i=1}^{k}\lambda_{i}.$ (1.4) Hile and Protter [25] improved (1.4) to $\sum_{i=1}^{k}\frac{\lambda_{i}}{\lambda_{k+1}-\lambda_{i}}\geq\frac{nk}{4}.$ (1.5) Yang (cf. [37] and more recently [18]) has obtained a sharp inequality $\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})\left(\lambda_{k+1}-\left(1+\frac{4}{n}\right)\lambda_{i}\right)\leq 0.$ (1.6) From (1.6), one can infer $\lambda_{k+1}\leq\frac{1}{k}\left(1+\frac{4}{n}\right)\sum_{i=1}^{k}\lambda_{i}.$ (1.7) The inequalities (1.6) and (1.7) are called Yang’s first inequality and second inequality, respectively (cf. [8, 9, 7, 23]). Also we note that Ashbaugh and Benguria gave an optimal estimate for $k=1$ (cf. [2, 3, 4]). From the Chebyshev’s inequality, it is easy to prove the following relations $(\ref{CY41})\Longrightarrow(\ref{CY42})\Longrightarrow(\ref{hp1})\Longrightarrow(\ref{ppwtype}).$ From (1.6), Cheng and Yang [16] obtained $\lambda_{k+1}-\lambda_{k}\leq 2\left[\left(\frac{2}{n}\frac{1}{k}\sum\limits_{i=1}^{k}\lambda_{i}\right)^{2}-\left(1+\frac{4}{n}\right)\frac{1}{k}\sum\limits_{i=1}^{k}\left(\lambda_{i}-\frac{1}{k}\sum\limits_{j=1}^{k}\lambda_{j}\right)^{2}\right]^{\frac{1}{2}}.$ (1.8) Cheng and Yang [18], using their recursive formula, obtained $\lambda_{k+1}\leq C_{0}(n)k^{\frac{2}{n}}\lambda_{1},$ (1.9) where $C_{0}(n)\leq 1+\frac{4}{n}$ is a constant (see Cheng and Yang’s paper [18]). From the Weyl’s asymptotic formula (cf. [35]), we know that the upper bound (1.9) of Cheng and Yang is best possible in the meaning of the order on $k$. For a complete Riemannian manifold $M$, from the Nash’s theorem [26], there exists an isometric immersion $\psi\,:\;M\longrightarrow\mathbb{R}^{N},$ where $\mathbb{R}^{N}$ is Euclidean space. The mean curvature of the immersion $\psi$ is denoted by $H$ and $\|H\|$ denotes its norm. Define $\Phi=\\{\psi\;\|\psi\;\mbox{is an isometric immersion from $M$ into Euclidean space}\\}.$ When $\Omega$ is a bounded domain of a complete Riemannian manifold $M$, isometrically immersed into a Euclidean space $\mathbb{R}^{N}$, Cheng and the first author [14] (cf. [20, 24])obtained $\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq\frac{4}{n}\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})\left(\lambda_{i}+\frac{n^{2}}{4}H_{0}^{2}\right),$ (1.10) where $H_{0}^{2}=\inf_{\psi\in\Phi}\sup_{\Omega}\|H\|^{2}.$ (1.11) Using the recursive formula in Cheng and Yang [18], Cheng and the first author in [14] also deduced $\lambda_{k+1}+\frac{n^{2}}{4}H_{0}^{2}\leq C_{0}(n)k^{\frac{2}{n}}\left(\lambda_{1}+\frac{n^{2}}{4}H_{0}^{2}\right),$ (1.12) where $H_{0}^{2},C_{0}(n)$ are given by (1.11) and (1.9) respectively. From (1.10), we can get the gaps of the consecutive eigenvalues of Laplacian $\lambda_{k+1}-\lambda_{k}\leq 2\left(\left(\frac{2}{n}\frac{1}{k}\sum\limits_{i=1}^{k}\lambda_{i}+\frac{n}{2}H_{0}^{2}\right)^{2}-\left(1+\frac{4}{n}\right)\frac{1}{k}\sum\limits_{i=1}^{k}\left(\lambda_{i}-\frac{1}{k}\sum\limits_{j=1}^{k}\lambda_{j}\right)^{2}\right)^{\frac{1}{2}}.$ (1.13) ###### Remark 1.1. When $\Omega$ is an $n$-dimensional compact homogeneous Riemannian manifold, a compact minimal submanifold without boundary and a connected bounded domain in the standard unit sphere $\mathbb{S}^{N}(1)$, and a connected bounded domain and a compact complex hypersurface without boundary of the complex projective space $\mathbb{CP}^{n}(4)$ with holomorphic sectional curvature 4, many mathematicians have sutudied the universal inequalities for eigenvalues and the difference of the consecutive eigenvalues (cf. [16, 17, 19, 21, 22, 23, 28, 38, 27, 33]). ###### Remark 1.2. Another problem is the lower bound of the gap of the first two eigenvalues. In general, there exists the famous fundamental gap conjecture for the Dirichlet eigenvalue problem of the Schrödinger operator(cf.[11, 1, 39, 32, 40]and the references therein). The fundamental gap conjecture was solved by B. Andrews and J. Clutterbuck in [10] . From (1.8) and (1.13), it is not difficult to see that both Yang’s estimate for the gap of consecutive eigenvalues of (1.1) implicited in [37] and the estimate from [14] are on the order of $k^{\frac{3}{2n}}$. However, by the calculation of the gap of the consecutive eigenvalues of $\mathbb{S}^{n}$ with standard metric, using the Weyl’s asymptotic formula, we know the order of the upper bound of this gap is $k^{\frac{1}{n}}.$ Hence we conjecture that ###### Conjecture 1.1. Let $\Omega$ be a bounded domain in an $n$-dimensional complete Riemannian manifold $M$.For the Dirichlet problem (1.1), the upper bound for the gap of consecutive eigenvalues of Laplacian should be $\lambda_{k+1}-\lambda_{k}\leq C_{n,\Omega}k^{\frac{1}{n}},$ (1.14) where $C_{n,\Omega}$ is a constant dependent on $\Omega$ itself and the dimension $n$. ###### Remark 1.3. The famous Panye-Pólya-Weinberger conjecture(cf.[29, 30, 34, 5, 6])claims that, when $M=\mathbb{R}^{n}$, for Dirichlet eigenvalue problem (1.1), one should have $\dfrac{\lambda_{k+1}}{\lambda_{k}}\leq\left.\dfrac{\lambda_{2}}{\lambda_{1}}\right\|_{\mathbb{B}^{n}}=\left(\dfrac{j_{n/2,1}}{j_{n/2-1,1}}\right)^{2},$ (1.15) where $\mathbb{B}^{n}$ is the $n$-dimensional unit ball in $\mathbb{R}^{n},$ and $j_{p,k}$ is the $k^{th}$ positive zero of the Bessel function $J_{p}(t)$. From the Weyl’s asymptotic formula and (1.15), we know that the order of the upper bounder of the consective eigenvalues of eigenvalue problem (1.1) is just $k^{\frac{2}{n}}$. This is why we make this conjecture. In the following, the constant $C_{n,\Omega}$ are allowed to be different in different cases. When $\Omega$ is a bouded domain in $\mathbb{R}^{n}$, for the Dirichlet eigenvalue problem (1.1), we give the affirmative answer to the conjecture (1.14). ###### Theorem 1.1. Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain in Euclidean space $\mathbb{R}^{n}$ and $\lambda_{k}$ be the $k^{th}$ eigenvalue of the Dirichlet eigenvalue problem (1.1). Then we have $\lambda_{k+1}-\lambda_{k}\leq C_{n,\Omega}k^{\frac{1}{n}},$ (1.16) where $C_{n,\Omega}=4\lambda_{1}\sqrt{\frac{C_{0}(n)}{n}}$, $C_{0}(n)$ is given by $(\ref{diffyong})$. We think our method of proving Theorem 1.1 will be valid for the case of $n$-dimensional complete Riemannian manifold. Here, we give some examples directly. ###### Corollary 1.2. Let $\Omega\subset\mathbb{H}^{n}(-1)$ be a bounded domain in hyperbolic space $\mathbb{H}^{n}(-1)$, and $\lambda_{k}$ be the $k^{th}$ eigenvalue of the Dirichlet eigenvalue problem (1.1). Then we have $\lambda_{k+1}-\lambda_{k}\leq C_{n,\Omega}k^{\frac{1}{n}},$ (1.17) where $C_{n,\Omega}$ depends on $\Omega$ and the dimension $n$, given by $C_{n,\Omega}=4\left[C_{0}(n)\left(\lambda_{1}-\frac{(n-1)^{2}}{4}\right)\Big{(}\lambda_{1}+\frac{n^{2}}{4}H_{0}^{2}\Big{)}\right]^{\frac{1}{2}},$ (1.18) $C_{0}(n)$ and $H_{0}^{2}$ are the same as the ones in (1.12). In fact, by the comparison theorem for the distance function in Riemannian manifold, we have ###### Corollary 1.3. Let $\Omega\subset M$ be a bounded domain of an $n$-dimensional ($n\geq 3$) simply connected complete noncompact Riemannian manifold $M$ with sectional curvature $Sec$ satisfying $-a^{2}\leq Sec\leq-b^{2},$ where $a\geq b\geq 0$ are constants. Let $\lambda_{k}$ be the $k^{th}$ eigenvalue of the eigenvalue problem (1.1). Then we have $\lambda_{k+1}-\lambda_{k}\leq C_{n,\Omega}k^{\frac{1}{n}}$ (1.19) where $C_{n,\Omega}$ depends on $\Omega$ and the dimension $n$, given by $C_{n,\Omega}=4\left[C_{0}(n)\left(\lambda_{1}-\frac{(n-1)^{2}}{4}b^{2}+\frac{a^{2}-b^{2}}{4}\right)\left(\lambda_{1}+\frac{n^{2}}{4}H_{0}^{2}\right)\right]^{\frac{1}{2}},$ (1.20) $C_{0}(n)$ and $H_{0}^{2}$ are the same as the ones in (1.12). ###### Remark 1.4. Under the same assumption of Corollary 1.3, Lu and the first two authors [15] obtained $\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})^{2}\leq 4\sum_{i=1}^{k}(\lambda_{k+1}-\lambda_{i})\left(\lambda_{i}-\frac{(n-1)^{2}}{4}b^{2}+\frac{n-1}{2}(a^{2}-b^{2})\right).$ Therefore, one can get $\displaystyle\lambda_{k+1}-\lambda_{k}\leq 2$ $\displaystyle\left[\left(\frac{2}{k}\sum_{i=1}^{k}\lambda_{i}-\frac{(n-1)^{2}}{4}b^{2}+\frac{n-1}{2}(a^{2}-b^{2})\right)^{2}\right.$ $\displaystyle\left.\quad-\frac{5}{k}\sum_{i=1}^{k}\left(\lambda_{i}-\frac{1}{k}\sum_{i=1}^{k}\lambda_{j}\right)^{2}\right]^{\frac{1}{2}}.$ ## 2\. proofs of main results In this section, we will give the proof of Theorem 1.1. In order to prove our main results, we need the following key lemma ###### Lemma 2.1. For the Dirichlet eigenvalue problem (1.1), let $u_{k}$ be the orthonormal eigenfunction corresponding to the $k^{th}$ eigenvalue $\lambda_{k}$, i.e. $\left\\{\begin{aligned} \Delta u_{k}=&-\lambda_{k}u_{k},\;\text{in}\;\Omega,\\\ u_{k}=&0,\;\;\;\;\;\;\;\;\;\;\;\text{on}\;\partial\Omega,\\\ \int_{\Omega}u_{i}u_{j}&=\delta_{ij}.\end{aligned}\right.$ Then for any complex value function $g\in C^{3}(\Omega)\cap C^{2}(\overline{\Omega})$ and $k,\,i\in\mathbb{Z}^{+},\,k>i\geq 1,$ we have $\displaystyle\Big{(}(\lambda_{k+1}-\lambda_{i})+(\lambda_{k+2}-\lambda_{i})\Big{)}\int_{\Omega}\|\nabla g\|^{2}u_{i}^{2}\leq$ $\displaystyle\int_{\Omega}\Big{\|}2\nabla g\cdot\nabla u_{i}+u_{i}\Delta g\Big{\|}^{2}$ (2.1) $\displaystyle+(\lambda_{k+1}-\lambda_{i})(\lambda_{k+2}-\lambda_{i})\int_{\Omega}\|gu_{i}\|^{2}.$ ###### Proof. For $i<k$, define $\left\\{\begin{aligned} a_{ij}=&\int_{\Omega}gu_{i}u_{j},\\\ b_{ij}=&\int_{\Omega}\left(\nabla g\cdot\nabla u_{i}+\frac{1}{2}u_{i}\Delta g\right)u_{j},\\\ \varphi_{i}=&gu_{i}-\sum_{j=1}^{k}a_{ij}u_{j},\end{aligned}\right.$ where $\nabla$ denotes the gradient operator. Obviously, $a_{ij}=a_{ji},\;\;\int_{\Omega}\varphi_{i}u_{j}=0,\;\;\mbox{for}\;\;j=1,2,\cdots,k.$ (2.2) Then, from the Stokes’ theorem, we get $\lambda_{j}a_{ij}=\int_{\Omega}gu_{i}(-\Delta u_{j})=-\int_{\Omega}(u_{i}\Delta g+g\Delta u_{i}+2\nabla g\cdot\nabla u_{i})u_{j},$ i.e. $2b_{ij}=(\lambda_{i}-\lambda_{j})a_{ij}.$ (2.3) From the Stokes’ theorem, we have $-2\int_{\Omega}gu_{i}\nabla\overline{g}\cdot\nabla u_{i}=-\int_{\Omega}g\nabla\overline{g}\cdot\nabla u_{i}^{2}=\int_{\Omega}(\nabla g\cdot\nabla\overline{g}+g\Delta\overline{g})u_{i}^{2}.$ By the definition of $a_{ij}$, $b_{ij}$ and (2.3), we obtain $\int_{\Omega}\|\nabla g\|^{2}u_{i}^{2}=-2\int_{\Omega}gu_{i}\left(\nabla\overline{g}\cdot\nabla u_{i}+\frac{1}{2}u_{i}\Delta\overline{g}\right)=-2\sum_{j=1}^{\infty}a_{ij}\overline{b_{ij}}=\sum_{j=1}^{\infty}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}.$ Similarly, from the Stokes’ theorem, (2.2) and (2.3), we have $\displaystyle\int_{\Omega}\|\nabla\varphi_{i}\|^{2}=$ $\displaystyle\int_{\Omega}\overline{\varphi_{i}}\left(-\Delta\varphi_{i}\right)$ (2.4) $\displaystyle=$ $\displaystyle\int_{\Omega}\overline{\varphi_{i}}\left(-\Delta(gu_{i})+\sum\limits_{j=1}^{k}a_{ij}\Delta u_{j}\right)$ $\displaystyle=$ $\displaystyle\int_{\Omega}\overline{\varphi_{i}}\left(-2\nabla g\cdot\nabla u_{i}-\Delta gu_{i}-g\Delta u_{i}\right)$ $\displaystyle=$ $\displaystyle-\int_{\Omega}\overline{\varphi_{i}}\left(2\nabla g\cdot\nabla u_{i}+\Delta gu_{i}-\lambda_{i}gu_{i}\right)$ $\displaystyle=$ $\displaystyle-2\sum\limits_{j=k+1}^{\infty}\overline{a_{ij}}b_{ij}+\lambda_{i}\sum\limits_{j=k+1}^{\infty}\|a_{ij}\|^{2}$ $\displaystyle=$ $\displaystyle\sum\limits_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}+\lambda_{i}\sum\limits_{j=k+1}^{\infty}\|a_{ij}\|^{2}.$ From the Rayleigh-Ritz inequality (cf. [13]) and (2.4), we have $\lambda_{k+1}\leq\frac{\int_{\Omega}\|\nabla\varphi_{i}\|^{2}}{\int_{\Omega}\|\varphi_{i}\|^{2}}=\frac{\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}}{\sum_{j=k+1}^{\infty}\|a_{ij}\|^{2}}+\lambda_{i},$ i.e. $(\lambda_{k+1}-\lambda_{i})\sum_{j=k+1}^{\infty}\|a_{ij}\|^{2}\leq\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}.$ (2.5) From the Cauchy-Schwarz inequality, we have $\left(\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}\right)^{2}\leq\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})^{2}\|a_{ij}\|^{2}\sum_{j=k+1}^{\infty}\|a_{ij}\|^{2},$ i.e. $\displaystyle\left(\int_{\Omega}\|\nabla g\|^{2}u_{i}^{2}-\sum_{j=1}^{k}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}\right)^{2}$ (2.6) $\displaystyle\leq$ $\displaystyle\left(\int_{\Omega}\|gu_{i}\|^{2}-\sum_{j=1}^{k}\|a_{ij}\|^{2}\right)\left(\int_{\Omega}\left\|2\nabla g\cdot\nabla u_{i}+u_{i}\Delta g\right\|^{2}-\sum_{j=1}^{k}(\lambda_{j}-\lambda_{i})^{2}\|a_{ij}\|^{2}\right).$ Define $\left\\{\begin{aligned} \widetilde{B}(i)=&\int_{\Omega}\|gu_{i}\|^{2}-\sum_{j=1}^{k}\|a_{ij}\|^{2}=\sum_{j=k+1}^{\infty}\|a_{ij}\|^{2}>0,\\\ \widetilde{A}(i)=&\int_{\Omega}\left\|2\nabla g\cdot\nabla u_{i}+u_{i}\Delta g\right\|^{2}-\sum_{j=1}^{k}(\lambda_{j}-\lambda_{i})^{2}\|a_{ij}\|^{2}=\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})^{2}\|a_{ij}\|^{2}\geq 0,\\\ \widetilde{C}(i)=&\int_{\Omega}\|\nabla g\|^{2}u_{i}^{2}-\sum_{j=1}^{k}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}=\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})\|a_{ij}\|^{2}.\end{aligned}\right.$ Next, we will deduce the maxima of $\widetilde{C}(i)$ by using the Lagrange method of multipliers (cf.[12]). For any sequence $\\{\phi_{ij}\\}_{j=k+1}^{\infty}$ satisfying $\sum_{j=k+1}^{\infty}\|\phi_{ij}\|^{2}\leq\infty$ we define the function, $\displaystyle\Phi(\|\phi_{ij}\|,\mu,\lambda)=$ $\displaystyle\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})\|\phi_{ij}\|^{2}+\mu\left(\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})^{2}\|\phi_{ij}\|^{2}-\widetilde{A}(i)\right)$ $\displaystyle+\lambda\left(\sum_{j=k+1}^{\infty}\|\phi_{ij}\|^{2}-\widetilde{B}(i)\right).$ where $\mu,\lambda$ are two real parameters. Assume $\\{a_{ij}\\}_{j=k+1}^{\infty}$ is the extreme point of $\varphi$. Then for any $\\{\psi_{ij}\\}_{j=k+1}^{\infty}$ satisfying $\sum\limits_{j=k+1}^{\infty}\|\psi_{ij}\|^{2}<\infty$, from $\left.\frac{\mathrm{d}}{\mathrm{d}t}\right\|_{t=0}\Phi(\|a_{ij}\|+t\|\psi_{ij}\|)=0$ we have $2\sum\limits_{j=k+1}^{\infty}\|a_{ij}\|\|\psi_{ij}\|\left((\lambda_{j}-\lambda_{i})+\mu(\lambda_{j}-\lambda_{i})^{2}+\lambda\right)=0.$ (2.7) Taking $\psi_{ij}=\left\\{\begin{aligned} 1,&\;\;j=p,\\\ 0,&\;\;\mbox{otherwise},\end{aligned}\right.$ in (2.7), we have $\|a_{ip}\|\left((\lambda_{p}-\lambda_{i})+\mu(\lambda_{p}-\lambda_{i})^{2}+\lambda\right)=0,\;p=k+1,\cdots.$ (2.8) From $\left\\{\begin{aligned} &\frac{\partial\Phi}{\partial\mu}=0,\\\ &\frac{\partial\Phi}{\partial\lambda}=0,\end{aligned}\right.$ we have the two constraint conditions $\left\\{\begin{aligned} &\sum_{j=k+1}^{\infty}(\lambda_{j}-\lambda_{i})^{2}\|a_{ij}\|^{2}=\widetilde{A}(i),\\\ &\sum_{j=k+1}^{\infty}\|a_{ij}\|^{2}=\widetilde{B}(i).\end{aligned}\right.$ (2.9) Since there are two Lagrange multipliers and $\widetilde{B}(i)>0,$ from (2.8) and (2.9), there exist $r>l>k$ such that $\|a_{ir}\|\cdot\|a_{il}\|\neq 0,\;\lambda_{r}>\lambda_{l},$ and $\|a_{ij}\|=0,\;j\neq r,l.$ Hence, we have $\left\\{\begin{aligned} &m_{r}(\lambda_{r}-\lambda_{i})^{2}\|a_{ir}\|^{2}+m_{l}(\lambda_{l}-\lambda_{i})^{2}\|a_{il}\|^{2}=\widetilde{A}(i),\\\ &m_{r}\|a_{ir}\|^{2}+m_{l}\|a_{il}\|^{2}=\widetilde{B}(i),\end{aligned}\right.$ (2.10) where $m_{r},m_{l}$ are the multiplicity of the eigenvalues $\lambda_{r}\,\mbox{and}\,\lambda_{l},$ respectively. From (2.10), we have $\widetilde{C}(i)=\frac{\widetilde{A}(i)+(\lambda_{r}-\lambda_{i})(\lambda_{l}-\lambda_{i})\widetilde{B}(i)}{(\lambda_{l}-\lambda_{i})+(\lambda_{r}-\lambda_{i})}.$ (2.11) From (2.6), we have $\widetilde{C}(i)\leq\sqrt{\widetilde{A}(i)\widetilde{B}(i)}.$ (2.12) By the definition of $\widetilde{A}(i)$ and $\widetilde{B}(i),$ we have $(\lambda_{k+1}-\lambda_{i})\leq\sqrt{\widetilde{A}(i)/\widetilde{B}(i)}.$ (2.13) From the range of the function $\frac{\widetilde{A}(i)+(\lambda_{r}-\lambda_{i})(\lambda_{l}-\lambda_{i})\widetilde{B}(i)}{(\lambda_{l}-\lambda_{i})+(\lambda_{r}-\lambda_{i})},$ (2.12), we have $r=k+2.$ From (2.13), we have $l=k+1.$ Therefore, we obtain $\widetilde{C}(i)\leq\frac{\widetilde{A}(i)+(\lambda_{k+2}-\lambda_{i})(\lambda_{k+1}-\lambda_{i})\widetilde{B}(i)}{(\lambda_{k+2}-\lambda_{i})+(\lambda_{k+1}-\lambda_{i})}.$ (2.14) From (2.14), and the definition of $\widetilde{A}(i),\;\widetilde{B}(i)$ and $\widetilde{C}(i),$ we have $\displaystyle\left((\lambda_{k+2}-\lambda_{i})+(\lambda_{k+1}-\lambda_{i})\right)\int_{\Omega}\|\nabla g\|^{2}u_{i}^{2}$ (2.15) $\displaystyle\leq$ $\displaystyle\int_{\Omega}\left\|2\nabla g\cdot\nabla u_{i}+u_{i}\Delta g\right\|^{2}+(\lambda_{k+1}-\lambda_{i})(\lambda_{k+2}-\lambda_{i})\int_{\Omega}\|gu_{i}\|^{2}$ $\displaystyle-\sum_{j=1}^{k}(\lambda_{k+1}-\lambda_{j})(\lambda_{k+2}-\lambda_{j})\|a_{ij}\|^{2}$ $\displaystyle\leq$ $\displaystyle\int_{\Omega}\left\|2\nabla g\cdot\nabla u_{i}+u_{i}\Delta g\right\|^{2}+(\lambda_{k+1}-\lambda_{i})(\lambda_{k+2}-\lambda_{i})\int_{\Omega}\|gu_{i}\|^{2}$ which finishes the proof of Lemma 2.1. ∎ Based on Lemma 2.1, we have ###### Corollary 2.2. Under the assumption of Lemma 2.1, for any real value function $f\in C^{3}(\Omega)\cap C^{2}(\overline{\Omega}),$ we have $\displaystyle\left((\lambda_{k+2}-\lambda_{i})+(\lambda_{k+1}-\lambda_{i})\right)\int_{\Omega}\|\nabla f\|^{2}u_{i}^{2}$ (2.16) $\displaystyle\leq$ $\displaystyle 2\sqrt{\left((\lambda_{k+2}-\lambda_{i})(\lambda_{k+1}-\lambda_{i})\right)\int_{\Omega}\|\nabla f\|^{4}u^{2}}+\int_{\Omega}\left(2\nabla f\cdot\nabla u_{i}+u_{i}\Delta f\right)^{2}.$ ###### Proof. Taking $g=\exp(\sqrt{-1}\alpha f),\,\alpha\in\mathbb{R}\backslash\\{0\\}$ in (2.1), we have $\displaystyle\alpha^{2}\left((\lambda_{k+1}-\lambda_{i})+(\lambda_{k+2}-\lambda_{i})\right)\int_{\Omega}\|\nabla f\|^{2}u_{i}^{2}$ (2.17) $\displaystyle\leq$ $\displaystyle\alpha^{4}\int_{\Omega}\|\nabla f\|^{4}u_{i}^{2}+\alpha^{2}\int_{\Omega}\left\|2\nabla f\cdot\nabla u_{i}+u_{i}\Delta f\right\|^{2}+(\lambda_{k+1}-\lambda_{i})(\lambda_{k+2}-\lambda_{i}).$ From (2.17), we have $\displaystyle\left((\lambda_{k+1}-\lambda_{i})+(\lambda_{k+2}-\lambda_{i})\right)\int_{\Omega}\|\nabla f\|^{2}u_{i}^{2}$ (2.18) $\displaystyle\leq$ $\displaystyle\alpha^{2}\int_{\Omega}\|\nabla f\|^{4}u_{i}^{2}+\frac{1}{\alpha^{2}}(\lambda_{k+1}-\lambda_{i})(\lambda_{k+2}-\lambda_{i})+\int_{\Omega}\left\|2\nabla f\cdot\nabla u_{i}+u_{i}\Delta f\right\|^{2}.$ Using the Cauchy-Schwarz inequality in (2.18), we have (2.16). ∎ ###### Corollary 2.3. Under the assumption of Lemma 2.1, for any real value function $f\in C^{3}(\Omega)\cap C^{2}(\overline{\Omega})$ satisfying $\|\nabla f\|^{2}=1$, we have $(\lambda_{k+2}-\lambda_{k+1})^{2}\leq 16\left(\int_{\Omega}(\nabla f\cdot\nabla u_{i})^{2}-\frac{1}{4}\int_{\Omega}(\Delta f)^{2}u_{i}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta f)\cdot\nabla f)u_{i}^{2}\right)\lambda_{k+2}.$ (2.19) Furthermore, we have $\lambda_{k+2}-\lambda_{k+1}\leq 4\left(\lambda_{i}-\frac{1}{4}\int_{\Omega}(\Delta f)^{2}u_{i}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta f)\cdot\nabla f)u_{i}^{2}\right)^{\frac{1}{2}}\sqrt{\lambda_{k+2}}.$ (2.20) ###### Proof. From Corollary 2.2 and $\|\nabla f\|^{2}=1$, we have $\left((\lambda_{k+2}-\lambda_{i})+(\lambda_{k+1}-\lambda_{i})\right)-2\sqrt{(\lambda_{k+2}-\lambda_{i})(\lambda_{k+1}-\lambda_{i})}\leq\int_{\Omega}\left(2\nabla f\cdot\nabla u_{i}+u_{i}\Delta f\right)^{2},$ i.e. $\left(\sqrt{\lambda_{k+2}-\lambda_{i}}-\sqrt{\lambda_{k+1}-\lambda_{i}}\right)^{2}\leq\int_{\Omega}\left(2\nabla f\cdot\nabla u_{i}+u_{i}\Delta f\right)^{2}.$ By integration by parts, we have $\int_{\Omega}\left(2\nabla f\cdot\nabla u_{i}+u_{i}\Delta f\right)^{2}=4\int_{\Omega}(\nabla f\cdot\nabla u_{i})^{2}-\int_{\Omega}(\Delta f)^{2}u_{i}^{2}-2\int_{\Omega}(\nabla(\Delta f)\cdot\nabla f)u_{i}^{2}.$ Hence, we have $\left(\sqrt{\lambda_{k+2}-\lambda_{i}}-\sqrt{\lambda_{k+1}-\lambda_{i}}\right)^{2}\leq 4\int_{\Omega}(\nabla f\cdot\nabla u_{i})^{2}-\int_{\Omega}(\Delta f)^{2}u_{i}^{2}-2\int_{\Omega}(\nabla(\Delta f)\cdot\nabla f)u_{i}^{2}$ (2.21) Multiplying (2.21) by $\left(\sqrt{\lambda_{k+2}-\lambda_{i}}+\sqrt{\lambda_{k+1}-\lambda_{i}}\right)^{2}$ on both sides, we can get $\displaystyle(\lambda_{k+2}-\lambda_{k+1})^{2}\leq$ $\displaystyle 4\left(\int_{\Omega}(\nabla f\cdot\nabla u_{i})^{2}-\frac{1}{4}\int_{\Omega}(\Delta f)^{2}u_{i}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta f)\cdot\nabla f)u_{i}^{2}\right)$ $\displaystyle\times\left(\sqrt{\lambda_{k+2}-\lambda_{i}}+\sqrt{\lambda_{k+1}-\lambda_{i}}\right)^{2}$ $\displaystyle\leq$ $\displaystyle 16\left(\int_{\Omega}(\nabla f\cdot\nabla u_{i})^{2}-\frac{1}{4}\int_{\Omega}(\Delta f)^{2}u_{i}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta f)\cdot\nabla f)u_{i}^{2}\right)\lambda_{k+2},$ which is the inequality (2.19). From the Cauchy-Schwarz inequality and integration by parts, we obtain $(\lambda_{k+2}-\lambda_{k+1})^{2}\leq 16\left(\lambda_{i}-\frac{1}{4}\int_{\Omega}(\Delta f)^{2}u_{i}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta f)\cdot\nabla f)u_{i}^{2}\right)\lambda_{k+2}.$ Finally we have (2.20). ∎ ###### Proof of Theorem 1.1 . Let $x_{1},\;x_{2},\cdots,x_{n}$ be the standard coordinate functions in $\mathbb{R}^{n}$. Since $\|\nabla x_{l}\|=1,l=1,\cdots,n$, we can use Lemma 2.3. Taking $f=x_{l},\,l=1,\cdots,n,\,\mbox{and}\,\,i=1$ in (2.19) and then taking sum over $l$ from $1$ to $n$, we have $\displaystyle n(\lambda_{k+2}-\lambda_{k+1})^{2}\leq$ $\displaystyle 16\lambda_{k+2}\int_{\Omega}\sum_{l=1}^{n}\left(\frac{\partial u_{1}}{\partial x_{l}}\right)^{2}$ (2.22) $\displaystyle=$ $\displaystyle 16\lambda_{1}\lambda_{k+2}.$ From Theorem 3.1 in [18] (see also (1.9)), from (2.22), we deduce $\displaystyle\lambda_{k+2}-\lambda_{k+1}\leq$ $\displaystyle 4\sqrt{\frac{\lambda_{1}}{n}}\sqrt{\lambda_{k+2}}$ $\displaystyle\leq$ $\displaystyle 4\lambda_{1}\sqrt{\frac{C_{0}(n)}{n}}(k+1)^{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle C_{n,\Omega}(k+1)^{\frac{1}{2}},$ where $C_{n,\Omega}=4\lambda_{1}\sqrt{\frac{C_{0}(n)}{n}}$, $C_{0}(n)$ is given by $(\ref{diffyong})$. Since $k$ is arbitrary, this completes the proof of Theorem 1.1. ∎ Although Corollary 1.2 can be deduced directly by taking $a=b=1$ in Corollary 1.3, its proof is interesting independently. Here we give its proof for the upper half-plane model of hyperbolic space. ###### Proof of Corollary 1.2. For convenience, we will use the upper half-plane model of the hyperbolic space, that is, $\mathbb{H}^{n}(-1)=\\{(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}\|x_{n}>0\\}$ with the standard metric $\mathrm{d}s^{2}=\frac{(\mathrm{d}x_{1})^{2}+\cdots+(\mathrm{d}x_{n})^{2}}{(x_{n})^{2}}.$ Taking $r=\log x_{n},$ we have $\mathrm{d}s^{2}=(\mathrm{d}r)^{2}+\mathrm{e}^{-2r}\sum_{i=1}^{n-1}(\mathrm{d}x_{i})^{2}.$ Since $\|\nabla r\|=1,\;\Delta r=-(n-1)$, taking $f=r\,\,\mbox{and}\,i=1$ in (2.20), we have $\displaystyle\lambda_{k+2}-\lambda_{k+1}\leq$ $\displaystyle 4\left(\lambda_{1}-\frac{1}{4}\int_{\Omega}(\Delta r)^{2}u_{i}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta r)\cdot\nabla r)u_{1}^{2}\right)^{\frac{1}{2}}\sqrt{\lambda_{k+2}}$ (2.23) $\displaystyle=$ $\displaystyle 4\left(\lambda_{1}-\frac{(n-1)^{2}}{4}\right)^{\frac{1}{2}}\sqrt{\lambda_{k+2}}.$ By the result in [14] (see also (1.12)), from (2.23), we have $\displaystyle\lambda_{k+2}-\lambda_{k+1}\leq$ $\displaystyle 4\left(\lambda_{1}-\frac{(n-1)^{2}}{4}\right)^{\frac{1}{2}}\sqrt{C_{0}(n)\left(\lambda_{1}+\frac{n^{2}}{4}H_{0}^{2}\right)}(k+1)^{\frac{1}{n}}$ $\displaystyle=$ $\displaystyle C_{n,\Omega}(k+1)^{\frac{1}{n}},$ where $C_{n,\Omega}$ is defined by (1.18). Since this inequality holds for any $k$, we can deduce (1.17). ∎ ## 3\. Proof of Corollary 1.3 ###### Proof of Corollary 1.3. Assume that $\Omega$ is a bounded domain in an $n$-dimensional complete noncompact Riemannian manifold $(M,\,g)$ with sectional curvature $Sec$ satisfying $-a^{2}\leq Sec\leq-b^{2}$, where $0\leq b\leq a$ are constants. For $p\notin\overline{\Omega}$ fixed, define the distance function by $\rho(x)=\mbox{distance}(x,\,p)$. From Proposition 2.2 in P.15 of [31], and $\|\nabla\rho\|=1,$ we have $\nabla\rho\cdot\nabla(\Delta\rho)=-\|\text{Hess}\ \rho\|^{2}-\text{Ric}(\nabla\rho,\nabla\rho).$ (3.1) Let $0\leq h\leq h_{1},\cdots,h_{n-1}\leq H$ be the eigenvalues of the $\mathrm{Hess}\rho$. Then we have $\displaystyle 2\|\mbox{Hess}\rho\|^{2}-(\Delta\rho)^{2}$ (3.2) $\displaystyle=$ $\displaystyle 2\sum_{i=1}^{n-1}h_{i}^{2}-\left(\sum_{i=1}^{n-1}h_{i}\right)^{2}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n-1}h_{i}^{2}-\sum_{i\neq j}h_{i}h_{j}$ $\displaystyle\leq$ $\displaystyle h_{n-1}^{2}+h_{1}h_{2}+\cdots+h_{n-2}h_{n-1}-\sum_{i\neq j}h_{i}h_{j}$ $\displaystyle=$ $\displaystyle h_{n-1}^{2}-h_{1}h_{2}-\cdots- h_{n-2}h_{n-1}-\sum_{i\neq j\atop i,j\leq n-2}h_{i}h_{j}$ $\displaystyle\leq$ $\displaystyle H^{2}-(n-2)^{2}h^{2}.$ From the Hessian comparison theorem (cf. [36]), under the conditions in Corollary 1.3, we have $a\frac{\cosh a\rho}{\sinh a\rho}\geq h_{n-1}\geq\cdots\geq h_{1}\geq b\frac{\cosh b\rho}{\sinh b\rho}.$ (3.3) Since $n\geq 3$ and $\frac{a^{2}}{\sinh^{2}a\rho}$ is a decreasing function of $a$, from (3.2) and (3.3), under the conditions in Corollary 1.3, we have $\displaystyle 2\|\mbox{Hess}\rho\|^{2}+2\mbox{Ric}(\nabla\rho,\,\nabla\rho)-(\Delta\rho)^{2}$ (3.4) $\displaystyle\leq$ $\displaystyle a^{2}\frac{\cosh^{2}a\rho}{\sinh^{2}a\rho}-(n-2)^{2}b^{2}\frac{\cosh^{2}b\rho}{\sinh^{2}b\rho}-2(n-1)b^{2}$ $\displaystyle=$ $\displaystyle a^{2}+\frac{a^{2}}{\sinh^{2}a\rho}-(n-2)^{2}b^{2}-(n-2)^{2}\frac{b^{2}}{\sinh^{2}b\rho}-2(n-1)b^{2}$ $\displaystyle\leq$ $\displaystyle-(n-1)^{2}b^{2}+(a^{2}-b^{2})+\frac{b^{2}}{\sinh^{2}b\rho}-(n-2)^{2}\frac{b^{2}}{\sinh^{2}b\rho}$ $\displaystyle\leq$ $\displaystyle-(n-1)^{2}b^{2}+(a^{2}-b^{2}).$ Taking $f=\rho\,\,\mbox{and}\,i=1$ in (2.20), we have $\lambda_{k+2}-\lambda_{k+1}\leq 4\left(\lambda_{1}-\frac{1}{4}\int_{\Omega}(\Delta\rho)^{2}u_{1}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta\rho)\cdot\nabla\rho)u_{1}^{2}\right)^{\frac{1}{2}}\sqrt{\lambda_{k+2}}.$ (3.5) From (3.1) and (3.4), we obtain $\displaystyle\lambda_{1}-\frac{1}{4}\int_{\Omega}(\Delta\rho)^{2}u_{1}^{2}-\frac{1}{2}\int_{\Omega}(\nabla(\Delta\rho)\cdot\nabla\rho)u_{1}^{2}$ (3.6) $\displaystyle=$ $\displaystyle\lambda_{1}+\frac{1}{4}\int_{\Omega}\left(2\|\mathrm{Hess}\rho\|^{2}+2\mathrm{Ric}(\nabla\rho,\,\nabla\rho)-(\Delta\rho)^{2}\right)u_{1}^{2}$ $\displaystyle\leq$ $\displaystyle\lambda_{1}-\frac{(n-1)^{2}}{4}b^{2}+\frac{a^{2}-b^{2}}{4}.$ By the result in [14] (see also (1.12)), from (3.5) and (3.6), we have $\displaystyle\lambda_{k+2}-\lambda_{k+1}\leq$ $\displaystyle 4\left(\lambda_{1}-\frac{(n-1)^{2}}{4}b^{2}+\frac{a^{2}-b^{2}}{4}\right)^{\frac{1}{2}}\sqrt{C_{0}(n)\left(\lambda_{1}+\frac{n^{2}}{4}H_{0}^{2}\right)}(k+1)^{\frac{1}{n}}$ $\displaystyle\leq$ $\displaystyle C_{n,\Omega}(k+1)^{\frac{1}{n}},$ where $C_{n,\Omega}$ is defined by (1.20). 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E., Payne, G. Pólya and H. F. Weinberger, _On the ratio of consecutive eigenvalues_ , J. Math. and Phys., 35 (1956), 289-298. * [31] R. Schoen and S. T. Yau, Lectures on Differntial Geometry, Boston: International Press, 1994. * [32] I. M. Singer, B. Wong, S. T. Yau and Stephen S. T. Yau, _An estimate of the gap of the first two eigenvalues in the Schrödinger operator_ , Ann. Scuola. Norm. Sup. Pisa, Series IV,v. XII, n 2(1985),3 19-333. * [33] H. Sun, Q.-M. Cheng and H. C. Yang, _Lower order eigenvalues of Dirichlet Laplacian_ , Manuscripta math., 125 (2008), 139-156. * [34] C. J. Thompson, On the ratio of consecutive eigenvalues in $n$-dimensions, Stud. Appl. Math., 48 (1969), 281-283. * [35] H. Weyl, _Der Asymptotische Verteilungsgesetz der Eigenwerte Linearer partieller Differential-gleichungen_ , Math. Ann. 71 (1912), 441-469. * [36] H. Wu, L.C. Shen and Y. L. Yu, Introduction to Riemannian Geometry (in Chinese), Peking University Press, 1989. * [37] H. C. Yang, _An estimate of the difference between consecutive eigenvalues_ , preprint IC/91/60 of ICTP, Trieste, 1991. * [38] P. C. Yang and S. T. Yau, _Eigenvalues of the Laplacian of compact Riemannian surfaces and minimal submanifolds_ , Ann. Scuola Norm. Sup. Pisa CI. Sci., 7 (1980), 55-63. * [39] S. T. Yau, _Nonlinear analysis in geometry_ , Monographies de L Enseignement Math ematique, vol. 33, L Enseignement Math ematique, Geneva, 1986. S erie des Conf erences de l Union Math ematique Internationale, 8. * [40] Q. Yu and J. Zhong, _Lower bounds of the gap between the first and second eigenvalues of the Schödinger operator_ , Trans. Amer. Math. Soc., 294 (1986), no. 1, 341-349. Daguang Chen Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China E-mail: [email protected] Tao Zheng Department of Mathematics, Beijing Institute of Technology, Beijing 100081, P. R. China E-mail: [email protected] Hongcang Yang Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100080, P. R. China E-mail:[email protected]	arxiv-papers	2013-09-28T10:19:17	2024-09-04T02:49:51.625981	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Daguang Chen, Tao Zheng and Hongcang Yang", "submitter": "Tao Zheng", "url": "https://arxiv.org/abs/1309.7446" }
1309.7465	# Fractional quantum Hall states in charge-imbalanced bilayer systems N. Thiebaut1 N. Regnault2,3 and M.O. Goerbig1 1 Laboratoire de Physique des Solides, CNRS UMR 8502, Univ. Paris-Sud, F-91405 Orsay cedex, France 2 Laboratoire Pierre Aigrain, ENS and CNRS, 24 rue Lhomond, 75005 Paris, France 3 Department of Physics, Princeton University, Princeton, NJ 08544 [email protected] ###### Abstract We study the fractional quantum Hall effect in a bilayer with charge- distribution imbalance induced, for instance, by a bias gate voltage. The bilayer can either be intrinsic or it can be formed spontaneously in wide quantum wells, due to the Coulomb repulsion between electrons. We focus on fractional quantum Hall effect in asymmetric bilayer systems at filling factor $\nu=4/11$ and show that an asymmetric Halperin-like trial wavefunction gives a valid description of the ground state of the system. ## 1 Introduction In a two-dimensional electron gas subjected to a high magnetic field repulsively interacting electrons may collectively form a gapped state insensitive to small perturbations. Such states are called fractional quantum Hall states (FQHS) and show up for partial (fractional) fillings of the highest partially occupied Landau level, i.e. for rational values of the filling factor $\nu$, which is the ratio of the number of electrons $N$ and the number of magnetic flux quanta $N_{\phi}$ threading the system. Since the discovery of the fractional quantum Hall effect (FQHE) in 1982 [1] new experimental setups nourished our understanding of this multifaceted phenomenon. The bilayer system, which consists of two parallel 2D electron systems in a semiconductor heterostructure exhibits FQHE at even-denominator filling factors (such as $\nu=1/2$ [2]) is one of them. Though it is more common to consider symmetric layers, for which charges are equally spread between the two layers ($\nu_{\text{tot}}=2/n$), we are interested in a bilayer system with charge imbalance, which is obtained by the application of an external gate voltage (fig. 1). The strength of correlations between electrons that belong to different layers depends on their distance $d$ and the energy difference between the two layers $\Delta$ is proportional to the external gate voltage $V_{G}$. Beside its own richness, the FQHE in a bilayer system provides a model for its counterpart in a wide quantum well [3, 4, 5], where even-denominator states have also been observed [3, 4, 6]. In the latter, density functional theory calculations indicate that the Coulomb repulsion favors an electron concentration at the borders of the quantum well, which may thus be viewed as a spontaneously formed bilayer system [7]. Charge-imbalanced bilayers can thus depict wide quantum wells in which electrons split in two layers and populate one of them preferentially, either spontaneously or under the influence of external gate electrodes. Here, we investigate the conditions for the existence of FQHS (at $\nu=4/11$) in an imbalanced bilayer system by means of exact diagonalization. We introduce asymmetric Halperin states and discuss their relevance for the description of the ground state of the system. Figure 1: Schematic view of the effect of a gate voltage on a bilayer system. (Left) Setup. (Right) Effect of the gate voltage on the confinement potential. We consider the limit where the width $w$ of each quantum well is small as compared to the layer separation $d$, $w\ll d$. ## 2 Model We restrict the electron dynamics to the lowest Landau level, which amounts to considering the limit in which adjacent Landau levels are separated by a large cyclotron gap, as compared to the characteristic Coulomb energy scale, $e^{2}/4\pi l_{B}$. FQHS are often fully polarized due to the Coulomb repulsion between electrons in a flat Landau energy level, and we consider such states, that is we neglect the electron spin from now on. The effective Hamiltonian of the bilayer system reads $\hat{H}=\frac{1}{2}\sum_{\\{\sigma_{i}\\}}\sum_{\\{m_{i}\\}}V_{\\{m_{i}\\}}^{\\{\sigma_{i}\\}}c_{m_{1},\sigma_{1}}^{\dagger}c_{m_{2},\sigma_{2}}^{\dagger}c_{m_{4},\sigma_{4}}c_{m_{3},\sigma_{3}}+\Delta\sum_{m}(c_{m,\downarrow}^{\dagger}c_{m,{\downarrow}}-c_{m,\uparrow}^{\dagger}c_{m,{\uparrow}})\quad,$ (1) where $\sigma=\uparrow,\downarrow$ is the layer index and $m_{i}$ the angular momentum which labels the single-electron states in the symmetric gauge111Here we assume that the confinement potential is sufficiently narrow (compared to the layer separation, $w\ll d$) to avoid electrons from populating the first excited state of the confinement potential. This permits to neglect the wavefunction extent in the confinement direction, which assumption simplifies the single particle Hilbert space as the layer index does not couple to the orbital motion. [8]; $\Delta$ is the energy difference between the two layers, controlled by the gate voltage $V_{G}$ in fig. 1. The first term on the right hand side of eq. 1 is the usual interaction term, while the second one is a Zeeman-like term proportional to the density imbalance. In real space the interaction potential depends on the layer indices $V^{\\{\sigma_{i}\\}}(r)=\left\\{\begin{array}[]{cl}\frac{e^{2}}{4\pi r}&\mbox{ if electrons belong to the same layer (i.e. if }\sigma_{1}=\sigma_{2}=\sigma_{3}=\sigma_{4}\text{)}\\\ &\\\ \frac{e^{2}}{4\pi\sqrt{r^{\,2}+d^{\,2}}}&\mbox{ if electrons belong to different layers (i.e. if }\sigma_{1}=\sigma_{3}\neq\sigma_{2}=\sigma_{4}\text{)}\\\ &\\\ 0&\mbox{elsewise}\end{array}\right.$ (2) ## 3 Halperin’s wavefunctions A large variety of FQHS in bilayer systems may be described with the help of a generalization of Laughlin’s wavefunctions [9] to the case of electrons with internal degrees of freedom, initially introduced by Halperin [10]. Those wavefunctions have three integer parameters that characterise the interaction strength between electrons in the $\uparrow$ and $\downarrow$ layers ($m_{\uparrow}$ and $m_{\downarrow}$), and electrons in different layers ($n$). In terms of the complex positions $z_{i}^{\sigma}=x_{i}^{\sigma}-iy_{i}^{\sigma}$, for a system with $N_{\sigma}$ electrons in the $\sigma$-layer, Halperin’s wavefunctions read $\varPsi_{(m_{\uparrow},m_{\downarrow},n)}\big{(}\\{z_{i}^{\uparrow}\\},\\{z_{i}^{\downarrow}\\}\big{)}=\prod_{i<j\leq N_{\uparrow}}(z_{i}^{\uparrow}-z_{j}^{\uparrow})^{m_{\uparrow}}\prod_{i<j\leq N_{\downarrow}}(z_{i}^{\downarrow}-z_{j}^{\downarrow})^{m_{\downarrow}}\prod_{i\leq N_{\uparrow},\,j\leq N_{\downarrow}}(z_{i}^{\uparrow}-z_{j}^{\downarrow})^{n}\quad,$ (3) where the normalization and usual Gaussian factors have been absorbed in a redefinition of the Hilbert space measure $d\mu(\\{z_{i}\\})=\prod_{i}dx_{i}dy_{i}\exp(-\|z_{i}\|^{2}/2l_{B}^{2})/(2\pi l_{B}^{2})$, following Girvin and Jach [11]. Alternatively, we denote the state represented by the Halperin wavefunction (3) the $(m_{\uparrow},m_{\downarrow},n)$ state. Notice that not all Halperin state (3) are capable of describing physically possible FQHS – it has been shown in Ref. [12] within a generalization of Laughlin’s plasma analogy [9] that the exponents must satisfy the condition $m_{\uparrow}m_{\downarrow}\geq n^{2}$ (4) in order to avoid phase separation of the electron species. To compute the filling factor associated with a particular Halperin state, one uses the fact that the maximum value of individual orbital momenta is given by the number of flux quanta $N_{\phi}$ threading the system. The polynomial expansion of eq. (3) yields a maximal exponent $(N_{\sigma}-1)\;m_{\sigma}+N_{-\sigma}\;n$ for the position of the $i$-th electron in the $\sigma$-layer that must equal $N_{\phi}$. This can be written in matrix form as $\begin{pmatrix}N_{\phi}+m_{\uparrow}\\\ N_{\phi}+m_{\downarrow}\end{pmatrix}=\begin{pmatrix}m_{\uparrow}&n\\\ n&m_{-}{\downarrow}\end{pmatrix}\begin{pmatrix}N_{\uparrow}\\\ N_{\downarrow}\end{pmatrix}\quad,$ and after inverting the matrix, one deduces the relationship between the total number of electrons $N$ and $N_{\phi}$, $N=N_{\uparrow}+N_{\downarrow}=\frac{1}{m_{\uparrow}m_{\downarrow}-n^{2}}\Big{[}(m_{\uparrow}+m_{\downarrow}-2n)N_{\phi}+2m_{\uparrow}m_{\downarrow}-n(m_{\uparrow}+m_{\downarrow})\Big{]}\quad.$ (5) The filling factor $\nu$ is defined in the thermodynamic limit, for which we obtain $\nu=\lim_{N,N_{\phi}\rightarrow\infty}\frac{N}{N_{\phi}}=\frac{m_{\uparrow}+m_{\downarrow}-2n}{m_{\uparrow}m_{\downarrow}-n^{2}}\quad.$ (6) We define the polarization as the population difference between the layers $P_{z}=\frac{S_{z}}{N/2}$ (7) in terms of the $z$-component of the total layer pseudospin $S_{z}=\frac{N_{\uparrow}-N_{\downarrow}}{2}=\frac{(m_{\downarrow}-m_{\uparrow})(N_{\phi}-n)}{2(m_{\uparrow}m_{\downarrow}-n^{2})}\quad,$ (8) which is zero for symmetric states (i.e. if $m_{\uparrow}=m_{\downarrow}$). Now we focus on asymmetric Halperin states. More precisely we are interested in the $(5,3,2)$ state, which satisfies the condition (4) since it is the simplest (i.e. lowest correlation factors) asymmetric Halperin state for which the filling factor $\nu_{532}=4/11$ does not belong to the usual composite- fermions sequence $\nu_{\text{CF}}=p/(2sp\pm 1)$ [13]. It may thus describe states the existence of which relies on layer asymmetry. Nevertheless it should be mentioned that $\nu=4/11$ is also the filling factor of a second- generation composite fermion state in monolayer systems [14, 15, 16, 17] generated by interacting composite fermions that has been proposed as a possible explanation for an experimentally observed FQHS at this filling [14]. ## 4 Exact diagonalization In order to check the physical relevance of the $(5,3,2)$ state we calculate, with the help of exact diagonalization [18], two quantities for the Coulomb interaction (2) in the lowest Landau level: the total angular momentum of the ground state and its polarization as a function of the distance $d$ and of the gate-induced energy difference $\Delta$ between the two layers. The calculations are perfomed on the sphere geometry [19]. For the $(5,3,2)$ state, the relation between $N_{\Phi}$ and $N$ is given by $N_{\Phi}=\frac{11}{4}N-\frac{7}{2}$. Finite-size systems induce a constraint on the number of electrons [see eq. (5)]. Only sizes of the form $N=2+4n$, $n\in\mathbb{N}$ can satisfy this constraint. Here we will focus on $N=6$ and $N=10$ which are the only accessible sizes. FQHS correspond to values of $d$ and $\Delta$ for which the gap is finite and the total angular momentum equals zero. Finally we compute the overlap between the $(5,3,2)$ state and the exact ground state for values of $d$ and $\Delta$ which correspond to a FQHS. The polarization of the ground state is shown in fig. 2. For sufficiently high values of $\Delta$ the system is fully polarized, i.e all electrons reside in a single layer and one thus obtains the monolayer system with $\nu=4/11$, studied in Refs. [15, 16, 17]. Conversely for large distances $d$ the polarization tends to be zero. In this situation the potential $V(r)=(r^{\,2}+d^{\,2})^{-1/2}$ between two electrons in different layers is small and the system behaves as two uncorrelated layers with individual filling factors $\nu=2/11$. It is likely that for such a low filling factor the system tends to form a Wigner crystal [20, 21, 22], and FQHS may therefore be ruled out. (a) N=6: $P_{z}^{(532)}=-1/3$ (b) N=10: $P_{z}^{(532)}=-2/5$ Figure 2: Polarization of the bilayer system as a function of the distance $d/l_{B}$ and energy shift $\Delta$ (energies are given in units of $e^{2}/4\pi l_{B}$). The polarization of the $(5,3,2)$ state can be computed directly from eq. (7) and eq. (8) $P_{z}^{(532)}=-\frac{1}{2}+\frac{1}{N}$ (9) and may be compared with that obtained from exact diagonalization for $N=6$ and 10 particles, as a function of the layer separation $d/l_{B}$ and $\Delta/(e^{2}/4\pi l_{B})$ (see fig. 2). Indeed, one finds that in intermediate regimes the polarization of the ground state is the one of the $(5,3,2)$ state (orange regions). We now turn to the characterization of the exact ground state. Figure 3 shows the total angular momentum $L$ of the ground state. This latest is rotationally invariant when $L=0$, as required for an incompressible state. It is zero in the region where the ground-state polarization matches that of the $(5,3,2)$ state, except for large layer separations. As already mentioned in the discussion above, for large distances inter-layer correlations are too weak to allow the system to be in an incompressible state, and an inhomogeneous density state is favored, such as one would for example expect for a Wigner crystal ($d\gtrsim 3.8l_{B}$). Figure 3: Total orbital momentum of the bilayer system for $N=6$ electrons as a function of the distance $d/l_{B}$ and energy shift $\Delta$, in units of $e^{2}/4\pi l_{B}$. The results presented in figs. 2 and 3 indicate that in a certain range of parameters $d/l_{B}$ and $\Delta/(e^{2}/4\pi l_{B})$, exact diagonalization provides a ground state that matches some physical properties of the $(5,3,2)$ state. To corroborate the relevance of this state, we have calculated the overlap between the exact ground state obtained for the interaction (2) and the $(5,3,2)$ state. In order to obtain the $(5,3,2)$ state numerically, we have used exact diagonalization for a model interaction given in terms of the appropriate pseudopotentials [19]. The results are presented in fig. 4 as a function of $d/l_{B}$. The overlap turns out to be high ($>85\%$) for $d\lesssim 2l_{B}$, such that in regions of matching polarizations and for sufficiently short interlayer distances the $(5,3,2)$ state gives a good description of the ground state. Figure 4: Overlap of the $(5,3,2)$ state with the exact ground state as a function of $d/l_{B}$, in the corresponding polarization sector (9). ## 5 Conclusion We have performed an exact-diagonalization study of the bilayer system for various distances between the layers and energy shifts, i.e. the energy difference between electrons in the two layers. The ground state is incompressible over a certain range of parameters, which corresponds to an interlayer distance of roughly three times the magnetic length. Asymmetry between layers does not necessarily destroy the FQHS, and the FQHE at exotic fractions may thus be observed in an asymmetric bilayer system. Asymmetric Halperin’s wavefunctions are good candidates for the accurate description of those states. Also, using the bilayer modeling of a wide quantum well, a similar behavior could be expected in asymmetric wide quantum wells, the asymmetry being induced, for instance, by a back gate voltage that enables one to tune the electronic density of the system. Experimentalists reported a FQHE at $\nu=4/11$ in a wide quantum well ($w\sim 3\;l_{B}$) in 2003 [14]. Since this filling factor corresponds to the one of the $(5,3,2)$ state, one may wonder about its relevance for the explanation of this experimental observation. Within this picture this FQHS would be induced by the shape of the confinement potential, in contrast to the picture of second-generation composite fermions that have been proposed as an alternative explanation for the 4/11 state [14, 15, 16, 17]. ## References * [1] Tsui D C, Stormer H L and Gossard A C 1982 Phys. Rev. Lett. 48(22) 1559–1562 * [2] Eisenstein J P, Boebinger G S, Pfeiffer L N, West K W and He S 1992 Phys. Rev. Lett. 68(9) 1383–1386 * [3] Suen Y W, Santos M B and Shayegan M 1992 Phys. Rev. Lett. 69(24) 3551–3554 * [4] Luhman D R, Pan W, Tsui D C, Pfeiffer L N, Baldwin K W and West K W 2008 Phys. Rev. Lett. 101(26) 266804 * [5] Papić Z, Möller G, Milovanović M V, Regnault N and Goerbig M O 2009 Phys. Rev. B 79(24) 245325 * [6] Shabani J, Gokmen T, Chiu Y T and Shayegan M 2009 Phys. Rev. Lett. 103(25) 256802 * [7] Nuebler J, Friess B, Umansky V, Rosenow B, Heiblum M, von Klitzing K and Smet J 2012 Phys. Rev. Lett. 108(4) 046804 * [8] Girvin S M 1984 Phys. Rev. B 29(10) 6012–6014 * [9] Laughlin R B 1983 Phys. Rev. Lett. 50(18) 1395–1398 * [10] Halperin B I 1983 Helv. Phys. Acta 56 75 * [11] Girvin S M and Jach T 1984 Phys. Rev. B 29(10) 5617–5625 * [12] de Gail R, Regnault N and Goerbig M O 2008 Phys. Rev. B 77(16) 165310 * [13] Jain J K 1989 Phys. Rev. Lett. 63(2) 199–202 * [14] Pan W, Stormer H L, Tsui D C, Pfeiffer L N, Baldwin K W and West K W 2003 Phys. Rev. Lett. 90(1) 016801 * [15] Goerbig M O, Lederer P and Smith C M 2004 Phys. Rev. B 69(15) 155324 * [16] López A and Fradkin E 2004 Phys. Rev. B 69(15) 155322 * [17] Chang C C and Jain J K 2004 Phys. Rev. Lett. 92(19) 196806 * [18] Diagham: http://nick-ux.lpa.ens.fr/diagham/wiki/ * [19] Haldane F D M 1983 Phys. Rev. Lett. 51(7) 605–608 * [20] Lam P K and Girvin S M 1984 Phys. Rev. B 30(1) 473–475 * [21] Jiang H W, Willett R L, Stormer H L, Tsui D C, Pfeiffer L N and West K W 1990 Phys. Rev. Lett. 65(5) 633–636 * [22] Jiang H W, Stormer H L, Tsui D C, Pfeiffer L N and West K W 1991 Phys. Rev. B 44(15) 8107–8114	arxiv-papers	2013-09-28T14:45:52	2024-09-04T02:49:51.634190	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "N. Thiebaut, N. Regnault and M.O. Goerbig", "submitter": "Nicolas Thi\\'ebaut", "url": "https://arxiv.org/abs/1309.7465" }
1309.7473	# A fast map-making preconditioner for regular scanning patterns Sigurd K. Næss [email protected] Thibaut Louis [email protected] Sub-department of Astrophysics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK ###### Abstract High-resolution Maximum Likelihood map-making of the Cosmic Microwave Background is usually performed using Conjugate Gradients with a preconditioner that ignores noise correlations. We here present a new preconditioner that approximates the map noise covariance as circulant, and show that this results in a speedup of up to 400% for a realistic scanning pattern from the Atacama Cosmology Telescope. The improvement is especially large for polarized maps. ## I Introduction As the resolution and sensitivity of Cosmic Microwave Background (CMB) experiments increase, so do the computational resources needed to analyze their data. Because modern detectors are background-limited, the only way to significantly increase sensitivity is to increase the number of detectors. The last decades have seen an increase from tens of detectors to thousands of detectors in experiments like ACT Dünner et al. (2013); Niemack et al. (2010), SPT Carlstrom et al. (2011); Austermann et al. (2012), POLARBEAR Kermish et al. (2012) and Keck Kernasovskiy et al. (2012), and plans already exist for experiments with $10^{5}-10^{6}$ detectors Abazajian et al. (2013). Reducing the data from all these detectors into a coherent map of the sky presents a significant computational challenge, and already with 1000-detector-class experiments this step is the most important bottleneck of the data analysis pipeline Dünner et al. (2013). It is therefore important to investigate ways to speed up this process. Three main classes of map-makers are in popular use: _Maximum likelihood map- makers_ Tegmark (1997); Cantalupo et al. (2010); de Gasperis et al. (2005); Traficante et al. (2011); Dünner et al. (2013); QUIET Collaboration et al. (2012), which are slow, but produce unbiased, optimally noise-weighted maps; faster but slightly less accurate _destripers_ Keihänen et al. (2005); Sutton et al. (2010); and biased and sub-optimal but very fast _naive map-makers_ QUIET Collaboration et al. (2012); Schaffer et al. (2011). The topic of this paper is a method for significantly speeding up maximum likelihood map-makers. Assuming a linear detector response, we can model the time-ordered data $d$ via the the linear system $\displaystyle d=Pm+n,$ (1) where $m$ is the pixelized map of the sky, the pointing matrix $P$ is a sparse111The pointing matrix will be sparse if we solve for a beam-convolved map. For variable beams or asymmetric beams, one may want to reconvolve to a standard beam as part of map-making. This can be done using $P$, at the cost of some of its sparsity. We do not consider this case here. matrix mapping from pixels to samples, and $n$ is the time-domain noise, which we assume to be gaussian with covariance matrix $N$. The maximum likelihood solution for $m$ is given by the map-making equation Tegmark (1997), $\displaystyle(P^{T}N^{-1}P)m$ $\displaystyle=P^{T}N^{-1}d.$ (2) This is of the form $Ax=b$, and while the matrices involved are usually too large to solve by direct inversion, the system is amenable to solution by Preconditioned Conjugate Gradients (PCG) Press et al. (2007) provided a good preconditioner can be found222Without a preconditioner, the number of iterations needed for conjugate gradients is proportional to the condition number of the matrix $A$. By applying a preconditioner $M$, one is effectively solving the system $MAx=Mb$. The goal is then to choose $M$ such that $MA$ is as well-conditioned as possible. This can be acchevied if $M\approx A^{-1}$.. The most commonly used map-making preconditioners are the binned Natoli et al. (2001); Ashdown et al. (2007); Cantalupo et al. (2010) and Jacobi Doré et al. (2001); de Gasperis et al. (2005); Cantalupo et al. (2010); Dünner et al. (2013) preconditioners. The binned preconditioner $M_{\textrm{B}}$ approximates the time covariance matrix $N$ as diagonal (i.e. it ignores correlations), which for pointing matrices where only one pixel is hit per sample results in a diagonal pixel-space covariance matrix. $\displaystyle N^{-1}_{ij}$ $\displaystyle\approx N^{-1}_{ii}\delta_{ij}\textrm{ and }P_{ti}P_{tj}\propto\delta_{ij}\Rightarrow$ (3) $\displaystyle A_{ij}$ $\displaystyle\equiv P_{ti}N^{-1}_{tt^{\prime}}P_{t^{\prime}j}\approx P_{ti}^{2}N^{-1}_{tt}\delta_{ij}\equiv{M_{\textrm{B}}}^{-1}_{ij}$ (4) The Jacobi preconditioner $M_{\textrm{J}}$ simplifies one step further, and assumes that every detector has the same variance, $N=aI$, resulting in $\displaystyle A_{ij}$ $\displaystyle\approx aP_{ti}^{2}\delta_{ij}\equiv{M_{\textrm{J}}}^{-1}_{ij}.$ (5) The proportionality factor $a$ is usually set to 1, as PCG is insensitive to an overall scaling of the preconditioner. $-10^{-4}$ $10^{-4}$ Figure 1: A $4.2^{\circ}$ by $3.2^{\circ}$ subset of a row from the pixel- space inverse correlation matrix from a patch from ACT (each row in the matrix corresponds to a two-dimensional map). In order to highlight the correlation structure, the color scale is capped at $\pm 10^{-4}$. The significantly correlated area has a quite complicated shape, which is driven by the scanning pattern and focal plane layout. $-10^{-3}$ $10^{-3}$ Figure 2: A mosaic of the local correlation structure for a $12^{\circ}$ by $9^{\circ}$ subset of an ACT patch. A set of evenly spaced pixels were chosen as reference points, and for each the correlation in a neighborhood around it was calculated. Each pixel in the map is colored according to its correlation relative to the nearest reference point. The correlation structure is very uniform in the whole central region of the map. The assumption of independent noise is quite inaccurate. All realistic experiments have at least some time-correlation in the noise, and usually also correlations between different detectors. Additionally, filters will also generally introduce correlations. While computing the full, exact $A$ is often too expensive, single rows of it can be computed at the same expense as one CG iteration: $\displaystyle A_{ij}$ $\displaystyle=A_{ik}\delta_{kj}=(A\vec{e}_{i})_{j}$ (6) Here $(\vec{e}_{i})_{j}=\delta_{ij}$ is the pixel-space basis vector corresponding to pixel $i$. An example of what such a row looks like for the Atacama Cosmology Telescope (ACT) Dünner et al. (2013) can be seen in figure 1. It is clear that the approximation of no correlation is quite inaccurate. However, since the correlation structure is driven by the scanning pattern and relative position of the detectors in the focalplane, the correlation structure should be the same for all pixels which are scanned the same way. ACT, which used long-duration, small-amplitude drift scans, pixels at the same declination but different right ascension will be hit by the same phase of the same scanning motion, and should therefore have the same correlation structure.333 Circulant correlation is not a good approximation for every experiment. It is suitable for constant elevation dift scans, as employed by ACT, SPT and POLARBear, but we expect it to work poorly for full-sky scanning patterns. Indeed, that is what measurements show (see figure 2). It may therefore be a good approximation to assume that every point in the map has the same relative correlation structure, i.e. that the correlation between two points on the sky only depends on their relative position. If this is the case, then it is possible to choose a pixelization where the correlation only depends on the difference between pixel numbers, and hence that the pixel correlation matrix is _circulant_ 444 For example, if $\textrm{corr}(\vec{x}_{1},\vec{x}_{2})=f(\vec{x}_{1}-\vec{x}_{2})$, where $\vec{x}$ are coordinates, then a pixelization scheme $\vec{x}=G(p)$, where $p$ is a pixel index and G is a linear function will fulfill $\textrm{corr}(p_{1},p_{2})=f(G(p_{1}-p_{2}))$, resulting in a circulant correlation matrix.. A circulant matrix has the nice property of being diagonal in the frequency domain, which means that it can be computed, stored and applied cheaply, at the cost of a few FFTs. Hence, the constant correlation approximation is promising as a preconditioner for solving the map-making equation. ## II Implementation The inverse pixel covariance matrix $A$ can be decomposed into variance and correlation such that $A=\Sigma^{T}U\Sigma$. Here $\Sigma$ is diagonal (block- diagonal in the case of polarization) in pixel space, and corresponds to a map of the inverse standard deviation per pixel. As per the binned preconditioner, this can be approximated as $\displaystyle(\Sigma^{T}\Sigma)_{ij}$ $\displaystyle=P_{ti}^{2}N^{-1}_{tt}\delta_{ij}.$ (7) The correlation matrix $U$ is in general a dense matrix, but as noted above, it can often be approximated as circulant555The matrix will be circulant provided that the correlation structure is position-independent, that a constant offset in each coordinate corresponds to a constant pixel offset, and provided that indices wrap around at the edges.. Therefore, the constant correlation preconditioner replaces $U$ with a circulant matrix $Q$, such that $\displaystyle A$ $\displaystyle\approx\Sigma^{T}Q\Sigma\equiv{M_{\textrm{C}}}^{-1}$ (8) This relation can be inverted to give us an expression for $Q$ in terms of $A$, $\displaystyle Q=$ $\displaystyle{\Sigma^{T}}^{-1}A\Sigma^{-1}\Rightarrow$ (9) $\displaystyle Q_{ij}=$ $\displaystyle{\Sigma^{T}}^{-1}_{ii}(A\vec{e}_{i})_{j}\Sigma^{-1}_{jj}.$ (10) Since $Q$ is circulant, i.e. $Q_{ij}=Q_{0,j-i}=q_{j-i}$, we have $\displaystyle(FQF^{-1})_{ff^{\prime}}$ $\displaystyle=(Fq)_{-f}\delta_{ff^{\prime}}=(Fq)^{}_{f}\delta_{ff^{\prime}}$ (11) using forward and backwards Fourier transforms $F_{fj}\equiv\mathrm{e}^{-\frac{2\pi ijf}{N}}$ and $F^{-1}_{jf}\equiv\frac{1}{N}\mathrm{e}^{\frac{2\pi ijf}{N}}$, where $N$ is the number of rows in the matrix. With this in hand, the preconditioner can be applied as $\displaystyle M_{\textrm{C}}b$ $\displaystyle=\Sigma^{-1}Q^{-1}{\Sigma^{-1}}^{T}b$ $\displaystyle=\Sigma^{-1}F^{-1}{(Fq)^{}}^{-1}F{\Sigma^{-1}}^{T}b.$ (12) $\Sigma^{-1}$ and ${(Fq)^{}}^{-1}$ can be precomputed, so the cost of applying the preconditioner is simply that of two FFTs and three diagonal matrix multiplications. The choice of the reference pixel at which the correlation is measured is somewhat arbitrary. We used the pixel nearest the center of the map, but other locations not too close to the edge of the map should also work. Figure 3: Example TOD noise power spectrum from the simulation, based on noise behavior from ACT. Low frequencies are dominated by atmospheric noise, while the increase at high frequencies is caused by a Butterworth filter. ## III Regularization While constant correlation is a good approximation for relatively short-scale correlations, it works less well for long-distance correlations, and regions near the edges666Near the edges the telescope must decelerate in order to reverse the scanning direction, which makes the correlation structure different there than in the center. Applying the preconditioner as described above to realistic cases results in the appearance of large scale modes which change extremely slowly during the subsequent CG iteration. A way around this is to artificially limit the range of the correlations that are modeled, by multiplying $q$ by a Gaussian. For ACT, a standard deviation of 20 pixels was found to be effective, but this will depend on the scanning pattern, and some experimentation may be needed to find the optimal number. Figure 4: The residual A-norm Strakos & Tichy (2008) as a function of iteration number for a simple binned preconditioner and the constant correlation preconditioner. The latter converges roughly twice as fast as the former according to this criterion. The horizontal blue line indicates the level at which the maps have mostly stopped changing visually. ## IV Polarization The previous discussion assumed that each pixel only had a single degree of freedom, e.g. temperature-only maps of the sky. In the case of polarization, each pixel has several correlated components, typically the Stokes parameters T, Q and U Zaldarriaga & Seljak (1997), but this can instead be expressed as a larger number of block-correlated single-component pixels. This results in $\Sigma$ being block-diagonal with e.g. one (T,Q,U)-block per physical pixel, while $Fq$ becomes a vector of similar blocks. And instead of a single row of $A$ needing to be measured, all the rows corresponding to a given physical pixel now need to be computed (i.e. $\vec{e}_{i\alpha}$ for all components, where Greek indices indicate polarization components). Hence, eq. (10) becomes $\displaystyle Q_{i\alpha j\beta}=$ $\displaystyle{\Sigma^{T}}^{-1}_{i\alpha i\gamma}(A\vec{e}_{i\gamma})_{j\delta}\Sigma^{-1}_{j\delta j\beta}$ (13) Aside from that, everything works the same. ## V Test setup \| Step 5 \| Step 15 \| Step 45 \| Step 115 \| Step 340 ---\|---\|---\|---\|---\|--- Binned \| \| \| \| \| Const. corr. \| \| \| \| \| Figure 5: Example temperature maps from the CG solution process for the binned (top) and constant correlation (bottom) preconditioners. The rows correspond to steps 5, 15, 45, 115 and 340 from left to right. The steps are chosen such that the binned map in column $n$ is as similar as possible as the constant correlation map in column $n-1$. We see that the constant correlation preconditioner visually converges about 3 times faster than the binned one. The maps have been cropped for compactness of presentation. Step 15 \| Step 70 \| Step 250 ---\|---\|--- \| \| Figure 6: Comparison of the input (blue), binned (red) and constant correlation (green) power spectra at CG steps 15 (left), 70 (middle) and 250 (right). Unlike the map-space plots in figure 5, where only the large-scale convergence is visible, we can here clearly see the convergence at all scales. Small-scale convergence is much slower for EE and BB than for TT, and form the bottleneck for the CG solver if we ignore the $\ell<500$ modes. The constant correlation preconditioner is 3-5 times faster than the binned one here. This is quantified more precisely in figure 9. We tested the preconditioners on a simulated time-ordered data (TOD) based on the scanning pattern for 64 detectors from each of 417 15-minute scans of a subset of ACT’s southern patch centered at $\alpha=56^{\circ},\delta=-53^{\circ}$. Each scan was a constant elevation drift scan with amplitude of $3.5^{\circ}$ in azimuth, and the scan centers were spread over 10 steps in elevation, covering a patch of about $11^{\circ}$ by $8^{\circ}$ degrees. Odd steps in elevation scanned while rising and the even ones when setting. This resulted in most pixels being hit from two directions, and hence the x-shaped correlation pattern seen in figures 1-2. An example of a noise power spectrum used in the simulation can be seen in figure 3. The simulated detectors were polarization-sensitive, with each detector measuring a linear combination $T+\cos(2\psi)Q+\sin(2\psi)U$ of the local radiation field, with each detector having a different, randomly chosen detector angle $\psi$. While an ACT-like noise model, including the effects of atmosphere and inter-detector noise correlations was assumed in the map-making step, no noise was added to the simulated TOD in order to allow the convergence to be studied all the way to the highest multipoles777 This is valid since the convergence rate of PCG is mostly independent of the noise level of the right-hand side after the first few iterationsNatoli et al. (2001). However, with higher noise, higher CG errors also become acceptable, so the number of iterations needed for CG errors to be subdominant will be smaller for realistic noise levels.. For the same reason, the simulated input CMB did not include a beam, and was pixelated at the same resolution as the output map, in order to avoid subpixel noise. We then solved the map-making equation for this data set using PCG, first using the binned preconditioner described in equation (4), and then the constant correlation approximation described in this paper. Each was run for 600 CG iterations, with intermediate maps being output for every 5 steps. ## VI Results The constant correlation preconditioner visually converges roughly 3 times faster than the binned one, as shown for the temperature map in figure 5. Likewise, the residual A-norm Strakos & Tichy (2008) from the CG solver (shown in figure 4) also shows a significant improvement in convergence: roughly a factor 2 according to this metric888The A-norm $\|\|x\|\|_{A}$ of a vector $x$ is defined as $\sqrt{x^{T}Ax}$, where $A$ is $P^{T}N^{-1}P$ in our case. The error A-norm after $i$ CG steps is $\|\|x_{i}-x\|\|_{A}$, where $x_{i}$ is our estimate after $i$ steps, and $x$ is the true map. When the true $x$ is unknown, $\|\|x_{i}-x\|\|_{A}$ can be estimated as $\|\|x_{i}-x\|\|_{A}\approx\sum_{j=i}^{i+d-1}\gamma_{j}\|\|r_{j}\|\|^{2}$, where $r_{j}$ and $\gamma_{j}$ are two internal variables in the CG algorithm at step $j$, and $d$ is is an integer that controls the accuracy of the estimate (4 in our case). Figure 7: An example of the convergence of a single bin in the power spectrum, in this case that centered on $\ell=3200$. This shows the absolute difference between the recovered spectrum $C_{\ell}$ and the spectrum $\hat{C}_{\ell}$ of the input map in units of cosmic variance $\sigma_{\ell}^{\textrm{cv}}$, as a function of the CG step in cosmic variance units. Shown are curves for TT (red), EE (green) and BB (blue) for the binned (solid) and constant correlation (dashed) preconditioners. The trend lines are based on binned averages of the errors for many individual CG steps. This binning reduces the jitter, and makes the trend lines suitable for measuring the time needed to converge to a given level. However, neither of these tests take into account the fact that not all scales in the map are equally interesting. To remedy this, figure 6 compares the binned and constant correlation power spectra with that of the simulated input map. These spectra were computed using the method described in Louis et al. (2013). On large scales ($\ell<500$), this tells the same story as the maps did: The larger the scale, the more slowly it converges, with the constant correlation preconditioner being about 3 times faster than the binned one. Somewhat surprisingly, a similar phenomenon occurs at the small scales. For $\ell>2000$, higher $\ell$ results in slower convergence, and this is especially prominent for the EE and BB power spectra. On all scales, however, the constant correlation preconditioner appears to converge several times faster than the binned one. In order to quantify the convergence more precisely, we consider the time at which the absolute error in a given multipole-bin reaches 0.1 times cosmic variance in that bin. While somewhat arbitrary, this choice ensures that CG errors are guaranteed to be sub-dominant in the power spectrum, regardless of the noise properties of the actual experiment. Figure 7 shows the convergence of TT, EE and BB for both preconditioners for a typical multipole-bin. The overall trend for each component is an initial rapid fall followed by a slower decay, with both being significantly faster for the new preconditioner, particularly for the polarization spectra. Figure 8: The number of CG steps needed for each multipole-bin to converge to 0.1 times cosmic variance, for each of TT (red), EE (green) and BB (blue) for the binned (solid) and constant correlation (dashed) preconditioners. We found that binning the errors in bins of $\sim 50$ conjugate gradients steps and using linear interpolation between these bins resulted in a robust estimate of when each spectrum bin reaches the convergence criterion. The resulting convergence times can be seen in figure 8, and confirm our earlier finding that the largest and smallest scales converge more slowly. The figure also highlights how much trouble the binned preconditioner has with the EE and especially BB spectra, where it performs much more poorly relative the constant correlation preconditioner than we see for the TT spectrum. We speculate that this is due to the X-shaped correlation structure introduced by our scanning pattern. In binned maps, which ignore the correlations, this introduces spurious X-shaped patterns in both Q and U, corresponding to spurious signal in both E and B. With the constant correlation preconditioner, these are partially corrected because some of the correlation structure is taken into account. Figure 9: The ratio of the convergence times to 0.1 times cosmic variance for the binned and constant correlation preconditioners, per multipole bin for each of TT, EE and BB. For TT, the speedup is typically between 20% and 200%, while for BB the speedup ranges from 200% at large scales to about 400% at small scales, with EE being intermediate. Large points here indicate cases where the binned preconditioner did not converge to 0.1 times cosmic variance in time. For these points, the comparison was performed at the lowest threshold where both converged. We summarize the performance characteristics of the new preconditioner in figure 9, which shows its relative speed gain vs. the baseline binned preconditioner. Depending on $\ell$, we have a speedup ranging from 20% to 200% for TT, from 150% to 300% for EE, and from 200% to 400% for BB, with the greatest relative improvement happening at the scales that converge most slowly. ## VII Summary The structure of a CMB map’s pixel covariance matrix is determined by the noise properties of the time-ordered data and the scanning pattern of the telescope. For constant elevation drift scans like those employed by ACT, SPT and POLARBEAR, this results in an approximately circulant covariance. We have developed a new preconditioner for conjugate gradient solutions of the map- making equation which exploit this property by deconvolving the correlations in harmonic space, an operation which is very cheap due to the Fourier representation of a circulant matrices being diagonal. For a realistic scanning pattern and noise model the preconditioner results a speedup of 20% to 200% for temperature and 150% to 400% for polarization compared to a binned preconditioner. Convergence speed might potentially be further improved by allowing the correlation pattern to change slowly across the map, for example by decomposing the map into overlapping tiles, and applying the constant correlation preconditioner separately to each tile, followed by a merging operation. Our preliminary attempts at such a tiled preconditioner have however not been able to beat the simple constant correlation approximation presented here. ###### Acknowledgements. The authors would like to thank Jo Dunkley and Johannes Noller for useful discussion and suggestions, and Jon Sievers for testing the preconditioner in another map-maker. We also thank the ACT collaboration for access to internal ACT data used in the simulations. Computations were performed on the gpc supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. SN and TL are supported by ERC grant 259505. ## References Abazajian et al. (2013) Abazajian, K. N., Arnold, K., Austermann, J., et al. 2013, ArXiv e-prints, arXiv:1309.5383 * Ashdown et al. (2007) Ashdown, M. A. J., Baccigalupi, C., Balbi, A., et al. 2007, A&A, 467, 761 * Austermann et al. (2012) Austermann, J. E., Aird, K. A., Beall, J. A., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8452, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series * Cantalupo et al. (2010) Cantalupo, C. M., Borrill, J. D., Jaffe, A. H., Kisner, T. S., & Stompor, R. 2010, ApJS, 187, 212 * Carlstrom et al. (2011) Carlstrom, J. E., Ade, P. A. R., Aird, K. 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1309.7527	# Structured Spectrum Allocation and User Association in Heterogeneous Cellular Networks Wei Bao and Ben Liang Department of Electrical and Computer Engineering, University of Toronto, Canada Email: {wbao, liang}@comm.utoronto.ca ###### Abstract We study joint spectrum allocation and user association in heterogeneous cellular networks with multiple tiers of base stations. A stochastic geometric approach is applied as the basis to derive the average downlink user data rate in a closed-form expression. Then, the expression is employed as the objective function in jointly optimizing spectrum allocation and user association, which is of non-convex programming in nature. A computationally efficient Structured Spectrum Allocation and User Association (SSAUA) approach is proposed, solving the optimization problem optimally when the density of users is low, and near- optimally with a guaranteed performance bound when the density of users is high. A Surcharge Pricing Scheme (SPS) is also presented, such that the designed association bias values can be achieved in Nash equilibrium. Simulations and numerical studies are conducted to validate the accuracy and efficiency of the proposed SSAUA approach and SPS. ## I Introduction Traditional single-tiered macro-cellular networks provide wide coverage for mobile user equipments (UEs), but they are insufficient to satisfy the exploding demand driven by modern mobile traffic, such as multimedia transmissions and cloud computing tasks. One efficient means to alleviate this problem is to install a diverse set of small-cells (e.g., picocells and femtocells), overlaying the macrocells, to form a multi-tiered heterogeneous cellular network [1]. Each small-cell is equipped with a shorter-range and lower-cost base station (BS), to provide nearby UEs with higher-quality communication links with lower power usage. However, in the presence of multiple tiers of BSs in a cellular network, user association control becomes more challenging. A most direct approach is association by maximum received power, in which UEs are associated with the BS (in any tier) with the largest received power. However, in this case, because small-cell BSs are transmitting at lower power levels, only the UEs very close to them will connect with them, while most other UEs are still crowding in macrocells, leading to degraded performance. An example is shown in Fig. 1(a), in which many UEs are occupying the macrocells, while some small-cells are nearly empty. In order to resolve this issue, a flexible user association approach may be employed [2, 3], in which each tier of BSs is assigned a user association bias value, and a UE is associated with a BS with the largest received power multiplied by the bias value. If small-cell BSs are assigned with larger association bias values, the small-cells are “expanded” accordingly. This can result in a more balanced mobile traffic pattern and thus better network performance. Fig. 1(b) shows an example of flexible user association. However, if the association bias values for small-cell BSs are too large, it will cause improper expansions of small-cells such that UEs at their cell-edge may suffer from inadequate received power. As a consequence, the association bias values should be properly designed so that the overall network performance is optimized. Further complicating the resource management problem in a multi-tier cellular network, the radio spectrum licensed by the network operator needs to be shared by BSs of widely different power and coverage areas. How to optimally allocate spectrum among different tiers is an important open problem. In order to avoid cross-tier interference, and the prohibitive complexity in tracking and provisioning for such interference especially with unplanned deployment of small cells, a disjoint spectrum mode is commonly advocated [4, 5, 6], where different tiers of BSs are allocated non-overlapping portions of the spectrum. Even so, it is still a challenging problem to properly divide the spectrum for optimal network performance. (a) Association by maximum received power. (b) Flexible user association. Figure 1: An example of a three-tier cellular network. Macrocell BSs, picocell BSs, and femtocell BSs are represented by squares, circles, and triangles respectively; UEs are represented by dots; blue lines show cell boundaries. In this work, our objective is to study jointly optimal spectrum allocation and user association in a heterogeneous celluar network with multiple tiers of BSs. First, we develop a stochastic geometric model to study the network performance analytically. A closed-form expression for the average downlink UE data rate is derived, which is then employed as the objective function for jointly optimizing the spectrum allocation among tiers and the user association bias values. This resultant optimization problem is of non-convex programming in nature and cannot be solved with a standard method. Instead, we explore two important structures in solving the problem. Referred to as the density thresholding structure, we show that the problem can be studied separately over sparse UE and dense UE scenarios, divided by a parameter specific threshold. Referred to as the priority ordering structure, we show that a tier with higher BS density should have higher priority in spectrum allocation. Based on these observations, we propose a computationally efficient Structured Spectrum Allocation and User Association (SSAUA) approach to solve the problem optimally in the sparse UE scenario, and near-optimally in the dense UE scenario with a quantified performance bound. Finally, toward practical implementation of SSAUA, we propose a Surcharge Pricing Scheme (SPS), such that the designed association bias values can be achieved in Nash equilibrium. Hence, each UE is incentivized to adopt the proposed design with individual rationality. The rest of the paper is organized as follows. In section II, we discuss the relation between our work and prior works. In Section III, we present the system model. In Sections IV, V, and VI, we present our contributions in UE data rate derivation, SSAUA design, and SPS, respectively. In Section VII, we present numerical results. Finally, conclusions are given in Section VIII. ## II Related Works ### II-A Stochastic Geometry as Analytical Basis Stochastic geometry [7, 8, 9, 10] is a powerful mathematical modeling tool to analyze the performance (e.g., outage probability and data rate) of cellular networks with random spatial patterns of UEs and BSs. In this work, we focus on the downlink user data rate as performance measure. Pioneering works on downlink performance analysis using stochastic geometry include [11], [12], and [13], for either the single-tier or the multi-tier case. None of them considered spectrum allocation or user association. ### II-B Spectrum Allocation and User Association Most prior studies considered either spectrum allocation or user association separately. For example, [6, 14] studied optimal spectrum allocation, in cellular networks limited to two tiers of BSs, without flexible user association. Assuming a fixed number of UEs and BSs and without considering their random spatial patterns, [15, 16, 17, 18, 19] studied optimal user association with deterministic utility optimization. With a stochastic geometric approach, [2] proposed the flexible user association model with bias values, which is adopted in our work. It also derived the coverage probability and UE data rate, considering cross-tier interference, but in non-closed forms. [2] did not provide any design insights to optimize the derived performance metrics. It was later extended in [3] to study optimal user association in a network with two tiers of BSs, without considering spectrum allocation. Joint spectrum allocation and user association was studied in [20]. It was limited to two tiers of BSs. [20] presented a qualitative study on the optimal network performance in terms of coverage probability and data rate, without further providing analytical details in solving the optimization problems. A similar problem was also studied in [21], with frequency reuse instead of tiered spectrum division as the approach for spectrum sharing. It could accommodate more than two tiers, but it provided only conditionally optimal user association given frequency reuse factors or conditionally optimal frequency reuse factors given user association bias values. Joint optimization remained an open problem. Compared with the above studies, we consider multiple tiers of BSs with disjoint spectrum and provide optimal and analytically bounded near-optimal solutions for joint spectrum allocation and user association. ## III System Model ### III-A Multi-tier Cellular Network We consider a heterogeneous cellular network with randomly spatially distributed $K\geq 2$ tiers of BSs. As in conventional stochastic geometric modeling of multi-tier cellular networks [2, 20, 3, 12, 21], each tier of BSs independently forms a homogeneous Poisson point process (PPP) in two dimensional Euclidean space $\mathbb{R}^{2}$. Let $\Phi_{k}$ denote the PPP corresponding to tier-$k$ BSs, with intensity $\lambda_{k}$. Without loss of generality, we assume that $\lambda_{1}<\lambda_{2}\ldots<\lambda_{K}$. (If $\lambda_{i}=\lambda_{j},i\neq j$ in reality, we may approximate $\lambda_{j}=\lambda_{i}+\xi$, where $\xi$ is arbitrarily close to $0$.) UEs are also modeled as a homogeneous PPP $\Psi$ with intensity $\mu$, independent of all BSs. We assume each BS is connected to the core network by separate high-capacity wired or wireless links that have no influence on our performance analysis. In addition, because we focus on downlink analysis, we assume that the downlink and uplink of the system are operated in different spectrum, so that the uplink interference and capacity have no influence on the downlink analysis. ### III-B Power and Pathloss Model We define the tiers of BSs by their transmission power. Let $P_{k}$ be the transmission power of tier-$k$ BSs, which is a given parameter. If $P_{t}(\mathbf{x})$, $P_{t}(\mathbf{x})\in\\{P_{1},P_{2},\ldots,P_{K}\\}$, is the transmission power from a BS at $\mathbf{x}$ and $P_{r}(\mathbf{y})$ is the received power at $\mathbf{y}$, we have $P_{r}(\mathbf{y})=\frac{P_{t}(\mathbf{x})h_{\mathbf{x},\mathbf{y}}}{\alpha\|\mathbf{x}-\mathbf{y}\|^{\gamma}}$, where $\alpha\|\mathbf{x}-\mathbf{y}\|^{\gamma}$ is the propagation loss function with predetermined constants $\alpha$ and $\gamma$ (where the path loss exponent $\gamma>2$ in practice), and $h_{\mathbf{x},\mathbf{y}}$ is the fast fading term. Corresponding to common Rayleigh fading with power normalization, $h_{\mathbf{x},\mathbf{y}}$ is independently exponentially distributed with unit mean. Let $H(\cdot)$ be the cumulative distribution function of $h_{\mathbf{x},\mathbf{y}}$. ### III-C Spectrum Allocation In order to avoid cross-tier interference, different tiers of BSs are allocated separated spectrum. Assume the total spectrum bandwidth is $W$. The network operator allocates $\eta_{k}W$ to each tier-$k$ BS, where $\eta_{k}$ is the spectrum allocation factor and $\sum_{k=1}^{K}\eta_{k}=1$. Let $\bm{\eta}=(\eta_{1},\eta_{2},\ldots,\eta_{K})$. Note that BSs in the same tier are operated on the same spectrum. We additionally consider the possible constraints $\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k}$, for $k=1,2,\ldots,K$. Clearly, we have $\sum_{k=1}^{K}\eta_{\min,k}\leq 1\leq\sum_{k=1}^{K}\eta_{\max,k}$. Furthermore, we assume that $0<\eta_{\min,1}\leq\eta_{\min,2}\ldots\leq\eta_{\min,K}$ and $0<\eta_{\max,1}\leq\eta_{\max,2}\ldots\leq\eta_{\max,K}$, i.e., the network operator is likely (but not necessarily) to allocate more spectrum to a tier with higher BS density. Note that this is a general condition that contains the special case where there is no constraint on $\bm{\eta}$. Given a specific tier-$k$ BS, it is common to assume that all its associated UEs are equally allocated spectrum [21, 2, 6]. Hence, the per-UE assigned spectrum bandwidth is $\beta_{k}=\eta_{k}W/N_{k}$, where $N_{k}$ is a random variable denoting the number of UEs associated with the BS. ### III-D Coverage Probability and UE Data Rate Let $T$ denote the minimum required Signal-to-Interference Ratio (SIR) of UEs. The coverage probability of a UE is defined as the probability that its SIR is no less than $T$ [7]. As in conventional wireless modeling [21, 2, 6], if a UE experiences coverage probability $P^{\prime}$ and is allocated spectrum bandwidth $\beta^{\prime}$, its data rate is $\beta^{\prime}\log(1+T)$ if the SIR is no less than $T$, and its data rate is $0$ if the SIR is less than $T$ (i.e., outage occurs). Thus, the overall data rate of the UE is $\beta^{\prime}\log(1+T)P^{\prime}$. Note that, unless otherwise stated, $\log$ represents $\log_{2}$. Also, the thermal noise is assumed to be negligible compared with interference. ### III-E Flexible User Association Given that a UE is located at $\mathbf{y}$, it associates itself with the BS that provides the maximum biased received power [2, 3, 21] as follows: $\displaystyle\mathcal{BS}(\mathbf{y})=\arg\max_{\mathbf{x}\in\Phi_{k},\forall k}B_{k}P_{k}\|\mathbf{x}-\mathbf{y}\|^{-\gamma},$ (1) where $\mathcal{BS}(\mathbf{y})$ denotes the location of the BS associated with the UE, and $P_{k}\|\mathbf{x}-\mathbf{y}\|^{-\gamma}$ is the received power from a tier-$k$ BS located at $\mathbf{x}$, and $B_{k}$ is the association bias, indicating the connecting preference of a UE toward tier-$k$ BSs. In this case, the resultant cell splitting forms a generalized Dirichlet tessellation, or weighted Poisson Voronoi [22], shown in Fig. 1(b). Note that for $B_{1},B_{2},\ldots,B_{K}$, their effects remain the same if we multiply all of them by the same positive constant. Thus, without loss of generality, we normalize them such that $\sum_{k=1}^{K}B_{k}=1$. Let $\mathbf{B}=(B_{1},B_{2},\ldots,B_{K})$. Let $A_{k}$ denote the probability that a UE associates itself with a tier-$k$ BS, and $\mathbf{A}=(A_{1},A_{2},\ldots,A_{K})$. As derived in [2], we have $\displaystyle A_{k}=\frac{\lambda_{k}(P_{k}B_{k})^{\frac{2}{\gamma}}}{\sum_{j=1}^{K}\lambda_{j}(P_{j}B_{j})^{\frac{2}{\gamma}}},$ (2) and thus $\displaystyle B_{k}=\frac{P_{k}^{-1}(A_{k}/\lambda_{k})^{\frac{\gamma}{2}}}{\sum_{j=1}^{K}P_{j}^{-1}(A_{j}/\lambda_{j})^{\frac{\gamma}{2}}}.$ (3) Hence, there is a one-to-one mapping between $\mathbf{A}$ and $\mathbf{B}$, so we can view them interchangeably. ### III-F Problem Statement We first aim to derive a closed-form expression for the average UE data rate. Then, our objective is to maximize the average UE data rate by jointly optimizing the spectrum allocation factors $\bm{\eta}$ and the user association bias values $\mathbf{B}$ (or equivalently $\mathbf{A}$). Finally, we give a pricing scheme to incentivize each UE to adopt the designed $\mathbf{B}$. ## IV Closed-form Average UE Data Rate In this section, we derive the average UE data rate via stochastic geometric analysis. Consider a reference UE, termed as the typical UE, communicating with its BS, termed as the typical BS. We are interested in the typical UE since the average UE performance in the system is the same as the performance of the typical UE [7]. Furthermore, due to the stationarity of UEs and BSs, throughout this section we will re-define the coordinates so that the typical UE is located at $\mathbf{0}$. First, we study the coverage probability given that the typical UE is associating with a tier-$k$ BS and their distance is $d$. In this case, the overall interference to the typical UE is the sum interference from all tier-$k$ BSs other than the typical BS. Let $I_{k}(d)$ denote such interference. Then $\displaystyle I_{k}(d)=\sum_{\mathbf{x}\in\Phi_{k}^{\prime}}\frac{P_{k}h_{\mathbf{x},\mathbf{0}}}{\alpha\|\mathbf{x}\|^{\gamma}}.$ (4) where $\Phi_{k}^{\prime}$ is the Palm point process corresponding to all tier-$k$ BSs other than the typical BS, given that the typical BS is located at a distance of $d$ from the typical UE. It can be shown that $\Phi_{k}^{\prime}$ is a PPP with intensity $0$ in $\mathcal{B}(\mathbf{0},d)$ and intensity $\lambda_{k}$ in $\mathbb{R}^{2}\backslash\mathcal{B}(\mathbf{0},d)$, where $\mathcal{B}(\mathbf{0},d)$ denotes the disk region centered at $\mathbf{0}$ with radius $d$ [7]. The distribution of $I_{k}(d)$ is derived through its Laplace transform as follows: $\displaystyle\mathcal{L}_{I_{k}}(d,s)=\mathbf{E}\left[\exp\left(-\sum_{\mathbf{x}\in\Phi_{k}^{\prime}}\frac{sP_{k}h_{\mathbf{x},\mathbf{0}}}{\alpha\|\mathbf{x}\|^{\gamma}}\right)\right]$ $\displaystyle=$ $\displaystyle\exp\left(-\lambda_{k}\int_{\mathbb{R}^{2}\backslash\mathcal{B}(\mathbf{0},d)}\left(1-\int_{\mathbb{R}^{+}}e^{-\frac{sP_{k}h}{\alpha\|\mathbf{x}\|^{\gamma}}}H(\mathrm{d}h)\right)\mathrm{d}\mathbf{x}\right)$ (5) $\displaystyle=$ $\displaystyle\exp\left(-\lambda_{k}\int_{\mathbb{R}^{2}\backslash\mathcal{B}(\mathbf{0},d)}\frac{\frac{sP_{k}}{\alpha\|\mathbf{x}\|^{\gamma}}}{\frac{sP_{k}}{\alpha\|\mathbf{x}\|^{\gamma}}+1}\mathrm{d}\mathbf{x}\right)$ (6) $\displaystyle=$ $\displaystyle\exp\left(-2\pi\lambda_{k}\int_{d}^{\infty}\frac{\frac{sP_{k}r}{\alpha}}{\frac{sP_{k}}{\alpha}+r^{\gamma}}\mathrm{d}r\right),$ (7) where (5) is obtained from the Laplace functional of PPP $\Phi_{k}^{\prime}$ [7], (6) is because $h$ is exponentially distributed with unit mean, and (7) is through a transformation to polar coordinates. Let $P_{cover,k}(d)$ denote the conditional coverage probability of the typical UE (given $k$ and $d$). Then $\displaystyle P_{cover,k}(d)=$ $\displaystyle\mathbf{P}\left(\frac{P_{k}h_{\mathbf{x}_{B},\mathbf{0}}}{\alpha d^{\gamma}}\geq TI_{k}(d)\right)$ $\displaystyle=$ $\displaystyle\mathcal{L}_{I_{k}}(d,s)\|_{s=\frac{T\alpha d^{\gamma}}{P_{k}}},$ (8) where $\mathbf{x}_{B}$ is the coordinate of the typical BS, and $\|\mathbf{x}_{B}\|=d$. Substituting (7) into (8), we have $\displaystyle P_{cover,k}(d)=$ $\displaystyle\exp\left(-2\pi\lambda_{k}\int_{d}^{\infty}\frac{Td^{\gamma}r}{Td^{\gamma}+r^{\gamma}}\mathrm{d}r\right)$ $\displaystyle\overset{t=\frac{r^{2}}{T^{2/\gamma}d^{2}}}{=}$ $\displaystyle\exp\left(-\pi\lambda_{k}T^{\frac{2}{\gamma}}d^{2}\int_{(\frac{1}{T})^{\frac{2}{\gamma}}}^{\infty}\frac{1}{1+t^{\frac{\gamma}{2}}}\mathrm{d}t\right).$ (9) Furthermore, the probability density function of the distance between the typical UE and its associated tier-$k$ BS is $\displaystyle f_{k}(d)=$ $\displaystyle\frac{2\pi\lambda_{k}}{A_{k}}d\exp\left(-\pi d^{2}\sum_{j=1}^{K}\lambda_{j}\left(\frac{P_{j}B_{j}}{P_{k}B_{k}}\right)^{\frac{2}{\gamma}}\right)$ (10) $\displaystyle=$ $\displaystyle\frac{2\pi\lambda_{k}}{A_{k}}d\exp\left(-\pi d^{2}\frac{\lambda_{k}}{A_{k}}\right),$ (11) where (10) is derived in [2], and (11) is by substituting (2) into (10). Hence, the coverage probability $P_{cover,k}$ of the typical UE associated with a tier-$k$ BS can be computed as $\displaystyle P_{cover,k}=$ $\displaystyle\int_{0}^{\infty}f_{k}(d)P_{cover,k}(d)\mathrm{d}d$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\frac{2\pi\lambda_{k}}{A_{k}}d\exp\left(-\pi d^{2}\frac{\lambda_{k}}{A_{k}}\right)$ $\displaystyle\qquad\exp\left(-\pi\lambda_{k}\left(T\right)^{\frac{2}{\gamma}}d^{2}\int_{\left(\frac{1}{T}\right)^{\frac{2}{\gamma}}}^{\infty}\frac{1}{1+t^{\gamma/2}}\mathrm{d}t\right)\mathrm{d}d$ $\displaystyle=$ $\displaystyle\frac{1}{A_{k}}\frac{1}{\frac{1}{A_{k}}+C},$ (12) where $C=\left(T\right)^{\frac{2}{\gamma}}\int_{\left(\frac{1}{T}\right)^{\frac{2}{\gamma}}}^{\infty}\frac{1}{1+t^{\gamma/2}}\mathrm{d}t$ is a system-level constant only related to $\gamma$ and $T$. Let $\mathbf{E}_{0}(\beta_{k})$ denote the expected spectrum bandwidth allocated to the typical UE (connecting to a tier-$k$ BS). Following the model in Section III-C, $\mathbf{E}_{0}(\beta_{k})$ equals the spectrum bandwidth allocated to the typical tier-$k$ BS divided by the average number of UEs associated with it conditioned on the typical UE, which is $A_{k}\mu/\lambda_{k}+1$. Hence, $\displaystyle\mathbf{E}_{0}(\beta_{k})=\frac{\eta_{k}W}{A_{k}\mu/\lambda_{k}+1}.$ (13) Then, by Section III-D, the conditional expected data rate of the typical UE, given it is associated with a tier-$k$ BS, can be computed as [2, 21]111By doing so, we slightly underestimate the average data rate because the coverage event and $\beta_{k}$ are not completely independent. Although some efforts have been made to approximate their correlation [3, 23], all of them are inexact but result in tremendous mathematical complexity. In Section VII, we show that the resultant analysis is close to actual performance via simulations. $\displaystyle\overline{R}_{k}=\mathbf{E}_{0}(\beta_{k})\log(1+T)P_{cover,k}.$ (14) Finally, the average data rate of the typical UE, and hence the average data rate per UE in the system, is $\displaystyle\mathbf{F}=$ $\displaystyle\sum_{k=1}^{K}A_{k}\overline{R}_{k}=\sum_{k=1}^{K}A_{k}\mathbf{E}_{0}(\beta_{k})\log(1+T)P_{cover,k}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{K}\frac{\eta_{k}W\log(1+T)}{(A_{k}\mu/\lambda_{k}+1)(\frac{1}{A_{k}}+C)}.$ (15) Note that stochastic geometric analysis often leads to non-closed forms requiring numerical integrations (e.g., [11, 12, 2, 3]), due to the integral form of the Laplace functional or generating functional of PPPs applied in analysis [7, 10]. Fortunately, our derived closed-form expression for the average UE data rate facilitates the tractability of the resultant optimization problem. ## V Joint Optimization Problem and SSAUA We aim to maximize the average UE data rate $\mathbf{F}$ with respect to $\bm{\eta}$ and $\mathbf{B}$. As there is a one-to-one mapping between $\mathbf{A}$ and $\mathbf{B}$, we study the optimization problem over $(\bm{\eta},\mathbf{A})$ instead for analytical convenience. This is formally stated as optimization problem $\mathbf{P}$ as follows: $\displaystyle\underset{\bm{\eta},\mathbf{A}}{\text{maximize}}$ $\displaystyle\mathbf{F}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}M_{k}(A_{k})$ subject to $\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k},\forall k,$ $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad A_{k}\geq 0,\forall k,$ (16) where $M_{k}(A_{k})$ is defined as $\displaystyle M_{k}(A_{k})=\frac{1}{\left(A_{k}\mu/\lambda_{k}+1\right)\left(\frac{1}{A_{k}}+C\right)}.$ (17) Problem $\mathbf{P}$ is non-convex and cannot be solved through a standard method. Instead, we investigate into two important structures of the optimal solution, termed density thresholding and priority ordering, based on which we propose a computationally efficient Structured Spectrum Allocation and User Association (SSAUA) approach to solve the problem. ### V-A Density Thresholding Structure First, we define an important parameter $\displaystyle a_{k}\triangleq\sqrt{\lambda_{k}/(\mu C)}.$ (18) Note that $M_{k}(A_{k})$ is increasing on $[0,a_{k}]$ and decreasing on $[a_{k},\infty)$. We further observe several useful properties of $M_{k}(A_{k})$, which are presented in Appendix -A. Based on these properties, we obtain the following lemma, whose proof is given in Appendix -B. ###### Lemma 1 Consider a potential solution $(\bm{\eta}^{},\mathbf{A}^{})$ to Problem $\mathbf{P}$. If $\exists i\neq j$, such that $A^{}_{i}<a_{i}$ and $A^{}_{j}>a_{j}$, then $(\bm{\eta}^{},\mathbf{A}^{})$ is not an optimal solution. Lemma 1 suggests that, in an optimal solution, every $A_{k}$ must be on the same side of $a_{k}$. This directly leads to the following theorem, which is fundamental to our optimization solution. ###### Theorem 1 (Density Thresholding) Let $(\bm{\eta}^{},\mathbf{A}^{})$ be an optimal solution to Problem $\mathbf{P}$. If $\sum_{k=1}^{K}a_{k}>1$, then $\forall k,A^{}_{k}\leq a_{k}$; if $\sum_{k=1}^{K}a_{k}<1$, then $\forall k,A^{}_{k}\geq a_{k}$; if $\sum_{k=1}^{K}a_{k}=1$, then $\forall k,A^{}_{k}=a_{k}$. ###### Proof: If $\sum_{k=1}^{K}a_{k}>1$, because $\sum_{k=1}^{K}A^{}_{k}=1$, $\exists l$ such that $A^{}_{l}<a_{l}$. This leads to $A^{}_{k}\leq a_{k}$, $\forall k$, according to Lemma 1. The cases where $\sum_{k=1}^{K}a_{k}<1$ and $\sum_{k=1}^{K}a_{k}=1$ are similar. ∎ Note that, the condition $\sum_{i=1}^{K}a_{i}>1$ is equivalent to $\sqrt{\frac{1}{C}}\left(\sum_{i=1}^{K}\sqrt{\lambda_{i}}\right)>\sqrt{\mu}$, implying the density of UEs is sparse (compared with that of BSs). Thus, we refer to the case $\sum_{i=1}^{K}a_{i}>1$ as the _sparse UE scenario_. On the other hand, we refer to the case $\sum_{i=1}^{K}a_{i}<1$, which is equivalent to $\sqrt{\frac{1}{C}}\left(\sum_{i=1}^{K}\sqrt{\lambda_{i}}\right)<\sqrt{\mu}$, as the _dense UE scenario_. If $\sum_{i=1}^{K}a_{i}=1$, Problem $\mathbf{P}$ can be trivially solved and is ignored in the rest of our discussion. Note that because $a_{k}$ can be computed directly from the given parameters, one can judge which scenario Problem $\mathbf{P}$ falls within before solving the problem. Next, the solution to $\mathbf{P}$ will be investigated separately in the sparse UE and dense UE scenarios. ### V-B SSAUA in the Sparse UE Scenario In this case, the original Problem $\mathbf{P}$ becomes Problem $\mathbf{P1}$ as follows: $\displaystyle\underset{\bm{\eta},\mathbf{A}}{\text{maximize}}$ $\displaystyle\mathbf{F}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}M_{k}(A_{k})$ subject to $\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k},\forall k,$ $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad 0\leq A_{k}\leq a_{k},\forall k.$ (19) We first observe an important ordering property of the optimal solution to $\mathbf{P1}$, as shown in the following lemma, whose proof is given in Appendix -C. ###### Lemma 2 (Ordering Property) Let $\mathbf{A}^{}$ be optimal for $\mathbf{P1}$, then $M_{1}(A^{}_{1})\leq M_{2}(A^{}_{2})\leq\ldots\leq M_{K}(A^{}_{K})$. Next, by sequentially computing $\bm{\eta}^{}$ as follows: $\displaystyle\left\\{\begin{array}[]{l}\eta^{}_{K}=\min(1-\sum_{k=1}^{K-1}\eta_{\min,k},\eta_{\max,K}),\\\ \eta^{}_{K-1}=\min(1-\eta^{}_{K}-\sum_{k=1}^{K-2}\eta_{\min,k},\eta_{\max,K-1}),\\\ \ldots,\\\ \eta^{}_{l}=\min(1-\sum_{k=l+1}^{K}\eta^{}_{k}-\sum_{k=1}^{l-1}\eta_{\min,k},\eta_{\max,l}),\\\ \ldots,\\\ \eta^{}_{1}=\min(1-\sum_{k=2}^{K}\eta^{}_{k},\eta_{\max,1}),\\\ \end{array}\right.$ (26) we have the following theorem: ###### Theorem 2 (Priority Ordering) Let $\mathbf{A}^{}$ be optimal for Problem $\mathbf{P1}$, then $(\bm{\eta}^{},\mathbf{A}^{})$, where $\bm{\eta}^{}$ is computed in (26), is an optimal solution to $\mathbf{P1}$. ###### Proof: Consider Problem $\mathbf{P1A}$ as follows: $\displaystyle\underset{\bm{\eta}}{\text{maximize}}$ $\displaystyle\sum_{k=1}^{K}\eta_{k}M_{k}(A^{}_{k})$ subject to $\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min,k}\leq\eta_{k}\leq\eta_{\max,k},\forall k.$ (27) Then $\mathbf{P1A}$ is a simple linear programming problem with ordered linear coefficients in the objective, since $M_{1}(A^{}_{1})\leq M_{2}(A^{}_{2})\leq\ldots\leq M_{K}(A^{}_{K})$ due to Lemma 2. Note that $\bm{\eta}^{}$ does not depend on the exact values of $\mathbf{A}^{}$; it only requires the ordering property as shown in Lemma 2. Also, $\bm{\eta}^{}$ is in the feasible region due to $\sum_{k=1}^{K}\eta_{\min,k}\leq 1\leq\sum_{k=1}^{K}\eta_{\max,k}$. It is easy to verify that, $(\bm{\eta}^{},\mathbf{A}^{})$ is an optimal solution to $\mathbf{P1}$. ∎ Equation (26) indicates the priority ordering structure in spectrum allocation. We see that tier-$K$ has the highest priority in spectrum allocation, followed by tier-$(K-1)$, and so forth. Theorem 2 provides a means to derive an optimal $\bm{\eta}^{}$ regardless of the $\mathbf{A}^{}$ values. We need one further step to derive the corresponding optimal $\mathbf{A}^{}$ by solving the following Problem $\mathbf{P1B}$: $\displaystyle\underset{\mathbf{A}}{\text{maximize}}$ $\displaystyle\sum_{k=1}^{K}\eta^{}_{k}M_{k}(A_{k})$ subject to $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad 0\leq A_{k}\leq a_{k},\forall k.$ (28) Note that $\mathbf{P1B}$ is a convex programming problem, since $M_{k}(A_{k})$ is concave on $[0,a_{k}]$. Thus, $\mathbf{A}^{}$ can be computed by a computationally efficient algorithm, such as the interior point method. Hence the both steps to compute the jointly optimal solution $(\bm{\eta}^{},\mathbf{A}^{})$ have low computational complexity. ### V-C SSAUA in the Dense UE Scenario with Performance Bound In this case, the original Problem $\mathbf{P}$ becomes Problem $\mathbf{P2}$ as follows: $\displaystyle\underset{\bm{\eta},\mathbf{A}}{\text{maximize}}$ $\displaystyle\mathbf{F}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}M_{k}(A_{k})$ subject to $\displaystyle\sum_{k=1}^{K}\eta_{k}=1,\quad\eta_{\min}\leq\eta_{k}\leq\eta_{\max},\forall k,$ $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad A_{k}\geq a_{k},\forall k.$ (29) Problem $\mathbf{P2}$ is more complicated compared with Problem $\mathbf{P1}$, as $M_{k}(A_{k})$ is not concave, but an S-shaped function, in the feasible region. Hence, $\mathbf{P2}$ generally incurs high computational complexity even if an optimal $\bm{\eta}^{}$ is given [24, 25]. Therefore, instead of directly solving $\mathbf{P2}$, we first approximate $M_{k}(A_{k})$ by $\widetilde{M}_{k}(A_{k})$ defined as follows: $\displaystyle\widetilde{M}_{k}(A_{k})=\frac{1}{\left(A_{k}\mu/\lambda_{k}\right)\left(\frac{1}{A_{k}}+C\right)}.$ (30) Note that this approximation is reasonable because $A_{k}\mu/\lambda_{k}$ is much larger than $1$ when $\mu$ is large (i.e., the dense UE scenario). This observation is also supported by the small performance gap as derived in Section V-C2. Some useful properties of $\widetilde{M}_{k}(A_{k})$ are shown in Appendix -D. The approximated problem is referred to as Problem $\mathbf{P2A}$, where we simply replace the objective function of $\mathbf{P2}$ by the following: $\displaystyle\mathbf{F^{\prime}}(\bm{\eta},\mathbf{A})=\sum_{k=1}^{K}\eta_{k}\widetilde{M}_{k}(A_{k}).$ (31) #### V-C1 Solution to $\mathbf{P2A}$ The important ordering property still holds for Problem $\mathbf{P2A}$, as formalized in the following lemma, whose proof is given in Appendix -E. ###### Lemma 3 (Ordering Property) Let $(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})$ be an optimal solution to $\mathbf{P2A}$, then $\widetilde{M}_{1}(\widetilde{A}^{}_{1})\leq\widetilde{M}_{2}(\widetilde{A}^{}_{2})\leq\ldots\leq\widetilde{M}_{K}(\widetilde{A}^{}_{K})$. We observe that with the same ordering property, (26) can again be adopted as an optimal solution to $\mathbf{P2A}$ in the dense UE scenario, leading to the following theorem: ###### Theorem 3 (Priority Ordering) Let $\widetilde{\mathbf{A}}^{}$ be optimal for Problem $\mathbf{P2A}$, then $(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})$, where $\widetilde{\bm{\eta}}^{}$ is computed the same way as $\bm{\eta}^{}$ in (26), is an optimal solution to $\mathbf{P2A}$. ###### Proof: The proof is similar to that of Theorem 2. ∎ Given an optimal $\widetilde{\bm{\eta}}^{}$ for $\mathbf{P2A}$, we find the corresponding optimal $\widetilde{\mathbf{A}}^{}$ for $\mathbf{P2A}$ by solving the following Problem $\mathbf{P2B}$: $\displaystyle\underset{\mathbf{A}}{\text{maximize}}$ $\displaystyle\sum_{k=1}^{K}\widetilde{\eta}^{}_{k}\widetilde{M}_{k}(A_{k})$ subject to $\displaystyle\sum_{k=1}^{K}A_{k}=1,\quad A_{k}\geq a_{k},\forall k.$ (32) Unlike in the sparse UE scenario, here we have an explicit solution, as stated in the following theorem: ###### Theorem 4 Given an optimal $\widetilde{\bm{\eta}}^{}$ for $\mathbf{P2A}$ (computed the same way as $\bm{\eta}^{}$ in (26)), the corresponding optimal $\widetilde{\mathbf{A}}^{}$ can be expressed as follows: $\displaystyle\left\\{\begin{array}[]{ll}\widetilde{A}^{}_{k}=a_{k},&k\geq 2\\\ \widetilde{A}^{}_{1}=1-\sum_{l=2}^{K}\widetilde{A}^{}_{l}.\\\ \end{array}\right.$ (35) ###### Proof: See Appendix -F. ∎ Note that both (26) and (35) can be computed with low computational complexity. #### V-C2 Bounding the Performance Gap Since $(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})$ is optimal for $\mathbf{P2A}$ rather than $\mathbf{P2}$, we next quantify the performance gap between $(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})$ and an optimal solution $(\bm{\eta}^{},\mathbf{A}^{})$ to $\mathbf{P2}$. The performance gap is defined as $\displaystyle E=\mathbf{F}(\bm{\eta}^{},\mathbf{A}^{})-\mathbf{F}(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{}).$ (36) Because $\mathbf{F}(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})\leq\mathbf{F}(\bm{\eta}^{},\mathbf{A}^{})\leq\mathbf{F}^{\prime}(\bm{\eta}^{},\mathbf{A}^{})\leq\mathbf{F}^{\prime}(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})$, we have $\displaystyle E\leq\mathbf{F}^{\prime}(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})-\mathbf{F}(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})\triangleq E^{\prime}.$ (37) Substituting $\widetilde{\bm{\eta}}^{}$ and $\widetilde{\mathbf{A}}^{}$ into $E^{\prime}$, we have $\displaystyle E^{\prime}=$ $\displaystyle\sum_{k=1}^{K}\frac{\widetilde{\eta}^{}_{k}}{\frac{\widetilde{A}^{}_{k}\mu}{\lambda_{k}}\left(\frac{\widetilde{A}^{}_{k}\mu}{\lambda_{k}}+1\right)\left(\frac{1}{\widetilde{A}^{}_{k}}+C\right)}.$ (38) Therefore, the relative performance gap is bounded by $\displaystyle\epsilon\triangleq$ $\displaystyle\frac{E}{\mathbf{F}(\bm{\eta}^{},\mathbf{A}^{})}\leq\frac{E^{\prime}}{\mathbf{F}(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{A}}^{})}$ (39) $\displaystyle=$ $\displaystyle\frac{\sum_{k=1}^{K}\frac{\widetilde{\eta}^{}_{k}}{\frac{\widetilde{A}^{}_{k}\mu}{\lambda_{k}}\left(\frac{\widetilde{A}^{}_{k}\mu}{\lambda_{k}}+1\right)\left(\frac{1}{\widetilde{A}^{}_{k}}+C\right)}}{\sum_{k=1}^{K}\frac{\widetilde{\eta}^{}_{k}}{\left(\frac{\widetilde{A}^{}_{k}\mu}{\lambda_{k}}+1\right)\left(\frac{1}{\widetilde{A}^{}_{k}}+C\right)}}$ (40) $\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}$ $\displaystyle\max_{k}\frac{\lambda_{k}}{\widetilde{A}^{}_{k}\mu}\leq\max_{k}\frac{\lambda_{k}}{a_{k}\mu}$ (41) $\displaystyle=$ $\displaystyle\sqrt{\lambda_{K}C/\mu},$ (42) where inequality (a) is obtained by observing the common factor in the summations in the numerator and denominator of (40). This implies that $\epsilon$ scales as $O\left(\sqrt{\lambda_{K}/\mu}\right)$. Note that because we are considering the dense UE scenario (i.e., $\sqrt{\frac{1}{C}}\left(\sum_{i=1}^{K}\sqrt{\lambda_{i}}\right)<\sqrt{\mu}$), $O\left(\sqrt{\lambda_{K}/\mu}\right)$ is small by definition. ### V-D Estimation for Complexity of Exhaustive Search In this subsection, we briefly discuss the complexity of the exhaustive search approach to solve Problem $\mathbf{P}$. First, as explained in details in our technical report [26], we observe that at least one of the optimal solutions to $\mathbf{P}$, $(\bm{\eta}^{},\mathbf{A}^{})$, has the following property: there is at most one $k\in\\{1,2,\ldots K\\}$ such that $\eta_{\min,k}<\eta^{}_{k}<\eta_{\max,k}$; $\forall j\neq k$, either $\eta^{}_{j}=\eta_{\min,j}$ or $\eta^{}_{j}=\eta_{\max,j}$ (i.e., at the boundary). Thus, the search for $\bm{\eta}^{}$ needs to be performed only at these boundary cases, leading to a complexity of $\Omega(2^{K})$. Furthermore, in the dense UE scenario, a numerical search over all locally optimal $\mathbf{A}$ (at least $2^{K}$ of them) is required, leading to another fold of $\Omega(2^{K})$, i.e., $\Omega(4^{K})$ overall, in complexity. Numerical studies on the computational complexity will be presented in Section VII. ## VI Nash Equilibrium for SSAUA Individual UEs may behave selfishly to derive unfair advantage despite our design of $\mathbf{B}^{}$ (or equivalently $\mathbf{A}^{}$). Thus, in this section, we propose a Surcharge Pricing Scheme (SPS), such that the designed $\mathbf{B}^{}$ is the natural outcome of a Nash equilibrium. Note that the designed spectrum allocation factors $\bm{\eta}^{}$ can be maintained by the network operator and is beyond our concern. We consider a reference individual UE, whose association bias values are $\mathbf{B}^{\prime}=(B_{1}^{\prime},B_{2}^{\prime},\ldots,B_{K}^{\prime})$. Let $\mathbf{A}^{\prime}=(A_{1}^{\prime},A_{2}^{\prime},\ldots,A_{K}^{\prime})$ be its corresponding association probabilities. For the other UEs, suppose they all obey the association bias values $\mathbf{B}^{}$ assigned by the network operator. Similar to the discussions in Section III and IV, the average data rate of the reference UE is $\displaystyle\mathbb{F}=\sum_{k=1}^{K}\frac{\eta^{}_{k}W\log(1+T)}{(A^{}_{k}\mu/\lambda_{k}+1)(\frac{1}{A_{k}^{\prime}}+C)}.$ (43) If the reference UE performs an optimization on $\mathbb{F}$ with respect to $\mathbf{A}^{\prime}$, the resultant optimal $\mathbf{A}^{\prime}=(A^{\prime}_{1},A^{\prime}_{2},\ldots,A^{\prime}_{K})$ is unlikely to be the same as $\mathbf{A}^{}$. Therefore, we add the following Surcharge Pricing Scheme: the network operator applies a surcharge $c_{k}$ to each UE associated with a tier-$k$ BS. Let $\mathbf{c}=(c_{1},c_{2},\ldots,c_{K})$. In this case, the average surcharge for the reference UE is $\sum_{k=1}^{K}c_{k}A_{k}^{\prime}$. Accordingly, the reference UE will perform the following optimization Problem $\mathbf{P3}$: $\displaystyle\underset{\mathbf{A}^{\prime}}{\text{maximize}}$ $\displaystyle\mathbb{F}^{\prime}=\sum_{k=1}^{K}\left(\frac{\eta^{}_{k}W\log(1+T)}{\left(A^{}_{k}\mu/\lambda_{k}+1\right)\left(\frac{1}{A_{k}^{\prime}}+C\right)}-c_{k}A_{k}^{\prime}\right)$ subject to $\displaystyle\sum_{k=1}^{K}A_{k}^{\prime}=1,\quad A_{k}^{\prime}\geq 0.$ (44) Different from $\mathbf{P}$, it can be shown that $\mathbf{P3}$ is a standard convex optimization problem. By the KKT conditions, its optimal solution $\mathbf{A}^{\prime}$ satisfies $\displaystyle\frac{H_{k}}{(1+CA^{\prime}_{k})^{2}}-c_{k}-\nu+\theta_{k}=0,$ (45) $\displaystyle\theta_{k}A^{\prime}_{k}=0,\quad\theta_{k}\geq 0,$ (46) where $H_{k}=\frac{\eta^{}_{k}W\log(1+T)}{A^{}_{k}\mu/\lambda_{k}+1}$, $\theta_{k}$ is a Lagrange multiplier corresponding to the inequality constraint $A_{k}^{\prime}\geq 0$, and $\nu$ is a Lagrange multiplier corresponding to the equality constraint $\sum_{k=1}^{K}A_{k}^{\prime}=1$. Setting $A^{\prime}_{k}=A^{}_{k}$, we have $\displaystyle c_{k}=\begin{cases}\infty,&\textrm{if }A^{}_{k}=0,\\\ \frac{H_{k}}{(1+CA^{}_{k})^{2}}-\nu,&\textrm{otherwise}.\end{cases}$ (47) Note that $\nu$ could be set arbitrarily due to the equality constraint. Without loss of generality, we set $\nu=\min_{k}\frac{H_{k}}{(1+CA^{}_{k})^{2}}$ so that the minimum surcharge among tiers is $0$. As a consequence, a Nash Equilibrium is achieved where every UE adopts the assigned $\mathbf{B}^{}$. ## VII Numerical Study Figure 2: Average UE data rate under different UE density $\mu$. (a) Sparse UE scenario. (b) Dense UE scenario. Figure 3: Average UE data rate under different path loss exponent $\gamma$. Figure 4: Designed association bias values under different UE density $\mu$. Figure 5: Pricing values under different UE density $\mu$. Figure 6: Comparison of run time. In this section, we present numerical studies on the performance of SSAUA. We label the SSAUA solution as $(\widehat{\bm{\eta}}^{},\widehat{\mathbf{B}}^{})$ and $(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{B}}^{})$ in the sparse and dense UE scenarios, respectively. Note that $(\widehat{\bm{\eta}}^{},\widehat{\mathbf{B}}^{})$ is optimal in the sparse UE scenario. We also compare $(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{B}}^{})$ with an optimal solution $(\bm{\eta}^{},\mathbf{B}^{})$ obtained from exhaustive search in the dense UE scenario. First, we study the network performance under different UE density $\mu$. The network parameters are as follows: $K=3$, $\lambda_{1}=1$ units/km2, $\lambda_{2}=5$ units/km2, $\lambda_{3}=10$ units/km2, $P_{1}=56$ dBm, $P_{2}=46$ dBm, $P_{3}=36$ dBm, $\eta_{\min,1}=0.2$, $\eta_{\min,2}=0.25$, $\eta_{\min,3}=0.3$, $\eta_{\max,1}=0.35$, $\eta_{\max,2}=0.4$, $\eta_{\max,3}=0.45$, $\gamma=4$, $W=200$ MHz, and $T=0.2$. In each round of simulation, UEs and BSs are generated on a $10$ km $\times$ $10$ km square, and the UEs in the central $5$ km $\times$ $5$ km square are sampled for performance evaluation (in order to remove the edge effect). Each simulation data point is averaged over all sampled UEs during 100 rounds of simulations. The results are shown in Fig. 2. A vertical line indicates the threshold value of $\mu$, as given in Theorem 1, separating the sparse and dense UE scenarios. For both scenarios, we show results of the analytical and simulated performance of SSAUA, and analytical and simulated performance of a “max power” approach, which employs equal spectrum allocation and user association based on the maximum received power. Since SSAUA is not optimal in the dense UE scenario, we also add two sets of results accordingly: the analytical optimal performance $\mathbf{F}(\bm{\eta}^{},\mathbf{B}^{})$ through exhaustive search and its analytical upper bound $\mathbf{F^{\prime}}(\widetilde{\bm{\eta}}^{},\widetilde{\mathbf{B}}^{})$. Fig. 2 illustrates that the SSAUA achieves near-optimal solution in the dense UE scenario. Furthermore, SSAUA substantially outperforms the max power solution. Finally, the analytical performance is only slightly smaller than the simulated performance, matching our discussions in Section IV. Fig. 3 shows the optimal network performance under different path loss exponent $\gamma$. The network parameters are the same as those used in Fig. 2 except $\mu$ is fixed at $100$ (i.e., sparse UE) and $500$ (i.e., dense UE) units/km2 in Fig. 3 (a) and (b) respectively. This figure further confirms the observations from Fig. 2. Furthermore, it shows that SSAUA is effective for a wide range of path loss conditions. Fig. 4 shows $\widehat{\mathbf{B}}^{}$, $\widetilde{\mathbf{B}}^{}$, and $\mathbf{B}^{}$; and Fig. 5 shows their corresponding prices $\widehat{\mathbf{c}}^{}$, $\widetilde{\mathbf{c}}^{}$, and $\mathbf{c}^{}$, under different $\mu$. We observe that $\widetilde{\mathbf{B}}^{}$ and $\widetilde{\mathbf{c}}^{}$ computed based on SSAUA approach are close to their counterparts $\mathbf{B}^{}$ and $\mathbf{c}^{}$. Finally, a run time experiment is conducted to compare the computational complexity of SSAUA with that of exhaustive search. The experiment is executed by Matlab R2011a on an ASUS PC with Intel i7-3610QM 2.3GHz processor and 4GB RAM. The results are averaged over 1000 runs for SSAUA and 10 runs for exhaustive search (both with randomly generated parameters). Fig. 6 shows that the run time of SSAUA is almost negligible compared with exhaustive search. Note that the $y$-axis is in log scale. When $K$ increases, the run time of exhaustive search exhibits an exponential growth tendency, while SSAUA remains computationally efficient. ## VIII Conclusions In this work, we provide a theoretical framework to study the joint optimization of spectrum allocation and user association in heterogeneous cellular networks. We establish a stochastic geometric model that captures the random spatial patterns of BSs and UEs, and a closed-form expression of the analytical average UE data rate is derived. We then consider the problem of maximizing the average UE data rate by jointly optimizing the spectrum allocation factors $\bm{\eta}$ and user association bias values $\mathbf{B}$, which is non-convex programming in nature. We propose the SSAUA approach to solve this problem with low computational complexity. We show that the SSAUA approach is optimal in the sparse UE scenario and near-optimal in the dense UE scenario, with a quantified tight bound scaling as $O(\sqrt{\lambda_{K}/\mu})$. We also propose the SPS such that the designed association bias values can be achieved in Nash equilibrium. ### -A Useful Properties of $M_{k}(A_{k})$ (M-1) $M_{k}(A_{k})$ is increasing on $[0,a_{k}]$ and decreasing on $[a_{k},\infty)$. (M-2) $M_{k}(A_{k})$ is concave on $[0,a^{\prime}_{k}]$ and convex on $[a^{\prime}_{k},\infty)$, where $a^{\prime}_{k}$ is some threshold value $a^{\prime}_{k}>a_{k}$. $M_{k}(A_{k})$ is concave on $[0,a_{k}]$. (M-3) If $\lambda_{i}<\lambda_{j}$, then $M_{i}(A)<M_{j}(A),\forall A>0$. (M-4) If $\lambda_{i}<\lambda_{j}$, then $M_{j}(A)-M_{i}(A)$ is strictly increasing on $[0,a_{j}]$. ### -B Proof of Lemma 1 ###### Proof: Suppose $\mathbf{A}^{}$ is optimal, $A^{}_{i}<a_{i}$, and $A^{}_{j}>a_{j}$. Consider that we increase $A^{}_{i}$ by a small value $\Delta>0$ and decrease $A^{}_{j}$ by $\Delta$. According to property (M-1), $\eta^{}_{i}M_{i}(A^{}_{i})+\eta^{}_{j}M_{j}(A^{}_{j})<\eta^{}_{i}M_{i}(A^{}_{i}+\Delta)+\eta^{}_{j}M_{j}(A^{}_{j}-\Delta)$. Thus, through replacing $A^{}_{i}$ and $A^{}_{j}$ by $A^{}_{i}+\Delta$ and $A^{}_{j}-\Delta$ respectively, we find a better solution to $\mathbf{P}$, leading to a contradiction. ∎ ### -C Proof of Lemma 2 ###### Proof: Suppose $\exists i<j$ such that $M_{i}(A^{}_{i})>M_{j}(A^{}_{j})$. This implies that $A^{}_{i}>A^{}_{j}$. (Otherwise, suppose $A^{}_{i}\leq A^{}_{j}$, then we have $M_{i}(A^{}_{i})\leq M_{i}(A^{}_{j})<M_{j}(A^{}_{j})$, leading to a contradiction.) A corresponding diagram is shown in Fig. 7. Case 1: $\eta^{}_{i}\leq\eta^{}_{j}$. Let $\widehat{A}^{}_{j}=A^{}_{i}$ and $\widehat{A}^{}_{i}=A^{}_{j}$, then we have $\displaystyle[\eta^{}_{i}M_{i}(\widehat{A}^{}_{i})+\eta^{}_{j}M_{j}(\widehat{A}^{}_{j})]-[\eta^{}_{i}M_{i}(A^{}_{i})+\eta^{}_{j}M_{j}(A^{}_{j})]$ $\displaystyle=$ $\displaystyle[\eta^{}_{i}M_{i}(A^{}_{j})+\eta^{}_{j}M_{j}(A^{}_{i})]-[\eta^{}_{i}M_{i}(A^{}_{i})+\eta^{}_{j}M_{j}(A^{}_{j})]$ $\displaystyle=$ $\displaystyle\eta^{}_{j}[M_{j}(A^{}_{i})-M_{j}(A^{}_{j})]+\eta^{}_{i}[M_{i}(A^{}_{j})-M_{i}(A^{}_{i})]$ $\displaystyle\geq$ $\displaystyle\eta^{}_{i}[M_{j}(A^{}_{i})-M_{j}(A^{}_{j})+M_{i}(A^{}_{j})-M_{i}(A^{}_{i})]>0,$ (48) where (48) is due to property (M-4). As a consequence, if $A^{}_{i}$ and $A^{}_{j}$ are replaced by $\widehat{A}^{}_{i}$ and $\widehat{A}^{}_{j}$ respectively, we obtain a larger $\mathbf{F}$, leading to a contradiction. Case 2: $\eta^{}_{i}>\eta^{}_{j}$. Let $\widehat{A}^{}_{j}=A^{}_{i}$, $\widehat{A}^{}_{i}=A^{}_{j}$, $\widehat{\eta}^{}_{j}=\eta^{}_{i}$, and $\widehat{\eta}^{}_{i}=\eta^{}_{j}$. (Note that because $\eta_{\min,i}\leq\eta_{\min,j}$ and $\eta_{\max,i}\leq\eta_{\max,j}$, $\widehat{\eta}^{}_{j}$ and $\widehat{\eta}^{}_{i}$ are guaranteed to be in the feasible region.) $\displaystyle[\widehat{\eta}^{}_{i}M_{i}(\widehat{A}^{}_{i})+\widehat{\eta}^{}_{j}M_{j}(\widehat{A}^{}_{j})]-[\eta^{}_{i}M_{i}(A^{}_{i})+\eta^{}_{j}M_{j}(A^{}_{j})]$ $\displaystyle=$ $\displaystyle[\eta^{}_{j}M_{i}(A^{}_{j})+\eta^{}_{i}M_{j}(A^{}_{i})]-[\eta^{}_{i}M_{i}(A^{}_{i})+\eta^{}_{j}M_{j}(A^{}_{j})]$ $\displaystyle=$ $\displaystyle\eta^{}_{i}[M_{j}(A^{}_{i})-M_{i}(A^{}_{i})]+\eta^{}_{j}[M_{i}(A^{}_{j})-M_{j}(A^{}_{j})]$ $\displaystyle>$ $\displaystyle\eta^{}_{j}[M_{j}(A^{}_{i})-M_{i}(A^{}_{i})+M_{i}(A^{}_{j})-M_{j}(A^{}_{j})]>0.$ (49) Thus, if $A^{}_{i}$, $A^{}_{j}$, $\eta^{}_{i}$, and $\eta^{}_{j}$ are replaced by $\widehat{A}^{}_{i}$, $\widehat{A}^{}_{j}$, $\widehat{\eta}^{}_{i}$, and $\widehat{\eta}^{}_{j}$ respectively, we can find a larger $\mathbf{F}$, leading to a contradiction. ∎ Figure 7: Diagram of $M_{i}(\cdot)$ and $M_{j}(\cdot)$. ### -D Useful Properties of $\widetilde{M}_{k}(A_{k})$ (M-1’) $\widetilde{M}_{k}(A_{k})$ is a decreasing convex function. (M-2’) If $\lambda_{i}<\lambda_{j}$, $\widetilde{M}_{i}(a_{i})<\widetilde{M}_{j}(a_{j})$. (M-3’) If $\lambda_{i}<\lambda_{j}$, then $\widetilde{M}_{j}(A)-\widetilde{M}_{i}(A)$ is a strictly decreasing function. (M-4’) $\widetilde{M}_{k}(A)-\widetilde{M}_{k}(A+D)>\widetilde{M}_{k}(A^{\prime})-\widetilde{M}_{k}(A^{\prime}+D)$, for any $A^{\prime}>A\geq a_{k}$ and $D>0$. (M-5’) If $\lambda_{i}<\lambda_{j}$, then $\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(a_{j}+D)>\widetilde{M}_{i}(a_{i})-\widetilde{M}_{i}(a_{i}+D)$, for any $D>0$. (M-6’) If $\lambda_{i}<\lambda_{j}$, then $\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(a_{j}+D)>\widetilde{M}_{i}(A^{\prime})-\widetilde{M}_{i}(A^{\prime}+D)$, for any $D>0$ and $A^{\prime}>a_{i}$ (combining (M-4’) and (M-5’)). ### -E Proof of Lemma 3 ###### Proof: Suppose that $\exists i<j$ (i.e., $\lambda_{i}<\lambda_{j}$) such that $\widetilde{M}_{i}(\widetilde{A}^{}_{i})>\widetilde{M}_{j}(\widetilde{A}^{}_{j})$, which also implies that $a_{i}\leq\widetilde{A}^{}_{i}<\widetilde{A}^{}_{j}$. The corresponding diagrams are shown in Fig. 8. Case 1: $\widetilde{\eta}^{}_{i}\leq\widetilde{\eta}^{}_{j}$. Case 1.1: $\widetilde{A}^{}_{i}\geq a_{j}$. Let $\widehat{A}^{}_{j}=\widetilde{A}^{}_{i}$ and $\widehat{A}^{}_{i}=\widetilde{A}^{}_{j}$, then we have $\displaystyle[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widehat{A}^{}_{i})+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(\widehat{A}^{}_{j})]-[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widetilde{A}^{}_{i})+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(\widetilde{A}^{}_{j})]$ $\displaystyle=$ $\displaystyle[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widetilde{A}^{}_{j})+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(\widetilde{A}^{}_{i})]-[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widetilde{A}^{}_{i})+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(\widetilde{A}^{}_{j})]$ $\displaystyle=$ $\displaystyle\widetilde{\eta}^{}_{j}[\widetilde{M}_{j}(\widetilde{A}^{}_{i})-\widetilde{M}_{j}(\widetilde{A}^{}_{j})]+\widetilde{\eta}^{}_{i}[\widetilde{M}_{i}(\widetilde{A}^{}_{j})-\widetilde{M}_{i}(\widetilde{A}^{}_{i})]$ $\displaystyle\geq$ $\displaystyle\widetilde{\eta}^{}_{i}[\widetilde{M}_{j}(\widetilde{A}^{}_{i})-\widetilde{M}_{j}(\widetilde{A}^{}_{j})+\widetilde{M}_{i}(\widetilde{A}^{}_{j})-\widetilde{M}_{i}(\widetilde{A}^{}_{i})]>0,$ (50) where (50) is due to property (M-3’). Thus, if $\widetilde{A}^{}_{i}$ and $\widetilde{A}^{}_{j}$ are replaced by $\widehat{A}^{}_{i}$ and $\widehat{A}^{}_{j}$ respectively, we obtain a larger $\mathbf{F}^{\prime}$, leading to a contradiction. Case 1.2: $\widetilde{A}^{}_{i}<a_{j}$. Let $\widehat{A}^{}_{j}=a_{j}$, $D=\widetilde{A}^{}_{j}-a_{j}$ and $\widehat{A}^{}_{i}=\widetilde{A}^{}_{i}+D$, then we have $\displaystyle[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widehat{A}^{}_{i})+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(\widehat{A}^{}_{j})]-[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widetilde{A}^{}_{i})+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(\widetilde{A}^{}_{j})]$ $\displaystyle=$ $\displaystyle[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widetilde{A}^{}_{i}+D)+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(a_{j})]-[\widetilde{\eta}^{}_{i}\widetilde{M}_{i}(\widetilde{A}^{}_{i})+\widetilde{\eta}^{}_{j}\widetilde{M}_{j}(\widetilde{A}^{}_{j})]$ $\displaystyle=$ $\displaystyle\widetilde{\eta}^{}_{j}[\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(\widetilde{A}^{}_{j})]+\widetilde{\eta}^{}_{i}[\widetilde{M}_{i}(\widetilde{A}^{}_{i}+D)-\widetilde{M}_{i}(\widetilde{A}^{}_{i})]$ $\displaystyle\geq$ $\displaystyle\widetilde{\eta}^{}_{i}[\widetilde{M}_{j}(a_{j})-\widetilde{M}_{j}(\widetilde{A}^{}_{j})+\widetilde{M}_{i}(\widetilde{A}^{}_{i}+D)-\widetilde{M}_{i}(\widetilde{A}^{}_{i})]>0,$ (51) where (51) is due to property (M-6’). Thus, if $\widetilde{A}^{}_{i}$ and $\widetilde{A}^{}_{j}$ are replaced by $\widehat{A}^{}_{i}$ and $\widehat{A}^{}_{j}$ respectively, we obtain a larger $\mathbf{F}^{\prime}$, leading to a contradiction. Case 2: $\widetilde{\eta}^{}_{i}>\widetilde{\eta}^{}_{j}$. Case 2.1: $\widetilde{A}^{}_{i}\geq a_{j}$. Case 2.2: $\widetilde{A}^{}_{i}<a_{j}$. The proof for these cases can be found in our technical report [26]. ∎ (a) Case 1.1 and 2.1. (b) Case 1.2 and 2.2. Figure 8: Diagrams of $\widetilde{M}_{i}(\cdot)$ and $\widetilde{M}_{j}(\cdot)$. ### -F Proof of Theorem 4 ###### Proof: Suppose $\exists k\geq 2$ such that $\widetilde{A}^{}_{k}>a_{k}$. Let $l=1$, $\widehat{A}^{}_{k}=a_{k}$, $D=\widetilde{A}^{}_{k}-a_{k}$, and $\widehat{A}^{}_{l}=\widetilde{A}^{}_{l}+D$. 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Neda, “On the size distribution of Poisson Voronoi cells,” _Physica A: Statistical Mechanics and its Applications_ , vol. 385, no. 2, pp. 518–526, Nov. 2007. * [24] M. Udell and S. Boyd, “Maximizing a sum of sigmoids.” [Online]. Available: http://www.stanford.edu/~udell/doc/max_sum_sigmoids.pdf * [25] M. Chiang, “Nonconvex optimization of communication systems,” _Advances in Mechanics and Mathematics, Special Volumn on Strang’s 70th Birthday_ , pp. 137–196, Oct. 2007. * [26] W. Bao and B. Liang, “Structured spectrum allocation and user association in heterogeneous cellular networks,” University of Toronto, Tech. Rep., 2013. [Online]. Available: http://www.comm.utoronto.ca/~liang/publications/techreport/SSAUA.pdf	arxiv-papers	2013-09-29T03:48:01	2024-09-04T02:49:51.652120	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Wei Bao and Ben Liang", "submitter": "Wei Bao", "url": "https://arxiv.org/abs/1309.7527" }
1309.7563	A closed parameterization of DNA–damage by charged particles, as a function of energy — A geometrical approach Frank Van den Heuvel PhD∗1,2, 1 CRUK/MRC Oxford Institute for Radiation Oncology, Department of Oncology, University of Oxford, Oxford, UK 2 Laboratory for experimental radiotherapy, Department of Oncology, University of Leuven, Leuven, Belgium $\ast$ E-mail: [email protected] ## Abstract Purpose: To present a closed formalism calculating charged particle radiation damage induced in DNA. The formalism is valid for all types of charged particles and due to its closed nature is suited to provide fast conversion of dose to DNA-damage. Methods: The induction of double strand breaks in DNA–strings residing in irradiated cells is quantified using a single particle model. This leads to a proposal to use the cumulative Cauchy distribution to express the mix of high and low LET type damage probability generated by a single particle. A microscopic phenomenological Monte Carlo code is used to fit the parameters of the model as a function of kinetic energy related to the damage to a DNA molecule embedded in a cell. The model is applied for four particles: electrons, protons, alpha–particles, and carbon ions. A geometric interpretation of this observation using the impact ionization mean free path as a quantifier, allows extension of the model to very low energies. Results: The mathematical expression describes the model adequately using a chi–square test ($\chi^{2}/NDF<1$). This applies to all particle types with an almost perfect fit for protons, while the other particles seem to result in some discrepancies at very low energies. The implementation calculating a strict version of the RBE based on complex damage alone is corroborated by experimental data from the measured RBE. The geometric interpretation generates a unique dimensionless parameter $k$ for each type of charged particle. In addition, it predicts a distribution of DNA damage which is different from the current models. ## Introduction The biological effect of ionizing radiation on human cells is believed to be related to the generation of damage in the DNA–molecule located in the cell’s nucleus[1]. The physical mechanism is the ionization of the DNA macro molecule, generating lesions in the molecular structure, either by direct ionization or by the generation of radicals in the vicinity of the DNA which then indirectly damage it. These events (direct or indirect) can create several types of damage to the DNA by combining a number of lesions into a cluster, which can only happen if they occur in close proximity (typically within one turn of the DNA–helix). The most prevalent of these damage types are base damage (2 lesions), followed by single strand breaks (SSB) (3 lesions) , double strand breaks (DSB) (4 lesions), and locally multiple damage sites (LMDS). The latter are clusters of different types of damage occurring close to each other. It is shown that base as well as SSB damage is not likely to be a deciding factor in the destruction of cells, due to the efficient repair mechanisms which exist in the cell[2]. The combination of double strand breaks and LMDS’s is likely to be the root cause for cell kill[3]. To quantify the amount of ionizing interactions in a medium, the physical notion of dose can be used. Dose is defined as the amount of energy deposited in a medium per unit mass and is expressed in Joule(J) per kg or Gray (Gy). In the case of dose deposition by charged particles the Bethe–formalism is used. This describes ionization events in a medium in terms of energy loss of the charged particles in inelastic collisions with the electrons of the medium, through the notion of mass stopping power ($dE/\rho dx$). In his seminal work already in 1930, Bethe showed that there is an intimate relationship between stopping power on the one hand, and energy (i.e. speed), charge, and the medium in which the interaction takes place on the other hand[4]. A further extension taking into account the possibility of the charged particle picking up electrons, thereby changing the stopping power was introduced by Barkas[5], using the concept of an effective charge. In radiation biology, linear energy transfer (LET) is used rather than stopping power. LET is identical to stopping power with the energy delivered to $\delta$–rays (i.e. highly energetic knock on electrons) subtracted. This quantity is called restricted stopping power. As such, LET is a measure for the density of ionization taking place along the track of a charged particle through a medium. Due to its close relationship with stopping power, it follows that there is a close relationship between LET and the kinetic energy of the depositing particle. From observation a dearth of DSB’s and LMDS’s was shown to be related to low LET irradiations, while an increased number of both for the same dose is seen high in hight LET irradiations[1]. Brenner and Ward[6] argued that DSB and LMDS damage was related to multiple interactions by single particles, rather than the combination of single strand breaks generated by single particles. In the field of microdosimetry, this is taken a step further by defining the notion of lineal energy which introduces the amount of energy deposited along lines confined in a convex geometric shape with a given distribution of cord lengths estimating the energy deposited in various shapes, which can be used for measurement (i.e. spheres, cylinders). Extending this, it is natural to propose a model where distance between ionizations along these lines plays a significant role in the generation of DNA–damage. A full listing and treatment of these quantities can be found in the ICRU reports 16, 19, and 36 [7, 8, 9]. To describe the damage impact of charged particles on the DNA–structure, the science community has taken its recourse to using Monte Carlo simulations to quantify the damage introduced[10, 11]. A more fundamental analytical approach is currently lacking, due to the underlying complexity of the DNA molecule, and the paucity of the available experimental data. The data which is available is mainly provided in terms of relative biological effective dose (RBE), a quantity combining physical, spectral, chemical, and biological factors, all of which hamper ab–initio calculations. Monte Carlo calculations are able to predict the induction of simple or complex damage as well as induction of single and double strand breaks in DNA–molecules. These findings are interpreted using the Bethe–Barkas formalism in terms of LET and show that high LET particles indeed introduce more complex damage. In this paper we develop a parameterization using a simple geometrical model, that describes the behavior as calculated by the Monte Carlo codes. We also show that this formalism describes the current knowledge well. ## Methods and Materials ### Theory We use the single charged particle model as proposed by Brenner and Ward, distinguishing three types of interaction results: Low LET, high LET, and intermediate LET mode. The specifics of each mode are explained below. 1. 1. Low LET: A single particle is generally not able to generate lesions close enough together to induce double strand breaks at each interaction. It is clear that DSB’s can be generated but in a limited fashion and that we use the word lesion in liberal fashion to indicate an interactive event which has damage as a consequence. 2. 2. High LET: The particle has the possibility to generate multiple lesions irrespective of any geometrical considerations. We implicitly assume that the double strand break damage is the result of multiple interactions by one particle. How exactly this damage is introduced (direct or indirect) is outside the scope of this article. An implicit assumption however is that ionizing events need to be geometrically close to the DNA structure. 3. 3. Intermediate: In given geometric circumstances it is possible for the charged particle to generate DSB–damage, in a high–LET manner, depending on the angle under which the particle hits the sensitive volume (Fig. 1). As a surrogate to categorize the charged particle in one of the types defined above, we use the mean path length between ionizing interactions in a medium consistent with the atomic make up of a DNA–molecule for the type of particle under consideration. In the remainder, we denote this with $\lambda(E)$, where $E$ is the kinetic energy of the particle. If $\lambda(E)$ is large relative to the sensitive volume, then the lesions on average are too far apart and only damage types related to a few lesions can occur (i.e. SSB and base damage). Charged particles with such energies will be part of the first category. If on the other hand $\lambda(E)$ is small then the probability of lesions creating more complex clusters of damage close together will be higher. Charged particles with this property will be in the high LET category. Finally, charged particles with intermediate distances between ionizing events have the capability of generating DSB and LMDS damage depending on other factors than $\lambda(E)$ alone. In this model we use the geometric direction of the path of the charged particle as a parameter. In Figure 1 a schematic model of this approach is shown. This implies that only a limited amount of directions are available to contribute to the amount of complex damage in the manner as outlined for the high–LET type interactions. This happens when for a particle of a given energy the quantity $\lambda(E)$ is slightly larger than the maximal distance between two DNA–damage lesions to be considered as being in the same cluster (usually about 10 base pairs (bp)). Due to the finite thickness of the sensitive volume it is possible to behave in a high–LET fashion depending on the angle with which the particle’s path crosses the volume. This occurs when the projection of the path is smaller than the previously determined maximum. Figure 1: A schematic model of a source of charged particles with a given mean free path length (i.e. a given energy), which is comparable with the diameter of the sensitive cell volume. As the angle ($\theta$) of the particle’s path with respect to the normal to the axis of the structure increases the chance that more than a single event will occur in the volume. This implies that if the angle ($\theta$) is larger than the one for which the projection of the average length between interactions equals the diameter, more high LET events will be registered. ### Equivalence Principle In the case of irradiation with charged particles all directions of the particle’s paths are possible as are all rotational positions of the DNA–structure. A particle that interacts (i.e. that creates a lesion) at the surface of a given sensitive volume has limited possibilities to interact again given that on average, a specific distance (which depends on the particle energy) has to be travelled before it interacts again. The next interaction’s position is then limited by the constraints outlined above if it is to fall within the sensitive volume. This first interaction can happen anywhere along the volume, but the constraints are relative to the position of that point. This implies that we can invoke an equivalence principle and reduce the problem to that of an isotropic point source positioned at the surface of a sensitive volume. ### Mathematical expression of the equivalence principle We need to calculate what fraction of the paths starting in the given point can interact with the sensitive volume given the fact that there is a length within which this is not likely, provided by $\lambda(E)$, and that there is a maximal distance ($H$) that disqualifies the generated lesion to be registered in the same cluster. We have reduced this problem to that of the distribution of projections of a point source on a line–piece, the solution to which is known as the Cauchy–distribution[12], and is described by the Lorenz function $f(x)$ with $x\in\Re$ expressed as follows: $f(x)~{}=~{}\frac{1}{1+(\frac{x}{r})^{2}}$ (1) Figure 2: The abstracted version of Figure 1 describes the distribution of horizontal distances at which a line segment tilted at a random angle $\theta$ cuts the x–axis. Only particles with large angles contribute to double strand break events by combining damage generated by a single particle. The red line indicates the “forbidden” area as (on average) this distance between the ionization events is observed. From Figure 2, it follows that the contribution $P$ for a given energy of charged particles to high LET events is proportional to: $P~{}\sim~{}2\int_{H-\lambda(E)}^{H}\frac{1}{1+(\frac{x}{r})^{2}}dx$ (2) Performing the calculation, we obtain: $\displaystyle P~{}$ $\displaystyle\sim~{}\frac{2}{\pi}[\tan^{-1}(\frac{x}{r})\biggm{\|}_{H-\lambda(E)}^{H}]+C$ (3) $\displaystyle\sim~{}\frac{2}{\pi}[\tan^{-1}(\frac{H}{r})-\tan^{-1}(\frac{H-\lambda(E)}{r})]+C$ (4) $\displaystyle\sim~{}\frac{2}{\pi}[\tan^{-1}(\frac{\lambda(E)-H}{r})+\tan^{-1}(\frac{H}{r})]+C$ (5) This implies that the amount of DSB–damage for a given dose and given energy of the charged particle is governed by the following expression. $F_{cd}(E)~{}=~{}(a-b)\frac{2}{\pi}[\tan^{-1}(\frac{\lambda(E)-H}{r})]+b$ (6) The change from a low to a high LET regimen occurs over a small energy interval. In such a small interval the average distance dependence on the energy of the particle can be approximated with a linear function. Therefore, we expect the energy dependence of the contribution of complex damage to follow the same form as in Equation 6 yielding the following expression, with H=$\lambda(E_{0})$ and r=$\lambda(\Gamma/2)$, $E_{0}$ being the energy, where the change in DSB is maximized and $\Gamma$ a measure for the width of the slope (i.e. the full width at half maximum in differential energy space). $F_{cd}(E)~{}=~{}(a-b)\frac{2}{\pi}[\tan^{-1}(\frac{E-E_{0}}{\Gamma/2})]+b$ (7) With the parameters $a$ and $b$ related to the levels 1 and 2 as outlined above. From boundary conditions we find that at very large energies (i.e. $E>>E_{0}$) the expression is reduced to minimal number of double interactions ($D_{min}$) which is equal to $a$. The value of $b$ is related to the maximal number of double interactions ($D_{max}$) as follows: $D_{max}=D_{min}+(1-\tan^{-1}(-\frac{E_{0}}{\Gamma/2}))b.$ (8) The formalism using energy alone allows us to forego specific assumptions regarding the dimensions of the DNA–molecule. Furthermore, it also allows us to apply this technique to particles where the values for $\lambda(E)$ are less well known. In addition, it allows us to test this formalism using experimentally available data which is available as a function of energy. ### Validation using Monte Carlo Simulations The use of microdosimetric calculations has provided important insight into the mechanisms and effects of radiation deposition. In the past, Monte Carlo simulations of charged particle deposition by various modalities were used to quantify and typify the kinds of damage introduced by the different modalities[10]. The Monte Carlo Damage Simulation code (MCDS) developed by Semenenko and Stewart, generates spatial maps of the damaged nucleotides forming many types of clustered DNA lesion, including single-strand breaks (SSB), double strand breaks (DSB), and individual or clustered base damages[11]. This approach has been shown to yield a linear relationship of the number of generated DSB’s up to a high dosage. It follows that this parameterization also provides the possibility to link dose to damage. In this paper, MCDS version 3.0 was used with the parameters described below. The DNA length which was chosen to be 1Gbp (Giga base pairs) and a nucleus diameter of 5$\mathrm{\mu m}$. In the MCDS software, the geometry of the DNA–molecule is not an explicit parameter. Here four parameters are used: 1) the DNA–segment length $n_{seg}$, which is an ad hoc parameter expressed as base pairs $Gy^{-1}cell^{-1}$, 2) the number of strand breaks generated $\sigma_{Sb}$, 3) the number of base pair damages generated $\sigma_{Bb}$ by defining $f=\sigma_{Bb}/\sigma_{Sb}$, and 4) a parameter $N_{min}$ (bp) describing the minimal separation for damage to be apart not to be counted as being in the same cluster. The values of these parameters is determined on the basis of other simulations and measurements. For a more in–depth treatment of these parameters we refer to the work by Semenenko and Stewart[13]. Variable input parameters MCDS were; the modality (i.e. energy depositing particle (electron, proton,…)), the energy (in MeV), and the oxygen concentration in %. In the implementation described here we chose to omit any oxygen enhancement as this could be a confounding factor and is the subject of another study. In this study it was found that oxygen only changed the amount of damage in the low LET regimen, leaving the formalism unchanged (data not shown). Therefore, a concentration of 0% oxygen was used. For every particle type at the relevant kinetic energies, all complex damage was noted per Gy, per cell and per kinetic energy. ### Fitting procedure The ultimate goal was to fit the complex damage function to the data obtained by the Monte Carlo simulation. The parameters that need fitting are the energy position ($E_{0}$) the width of the underlying Cauchy distribution ($\Gamma$) and the parameters $a$ and $b$. If a regular fit (i.e. all parameters fit at the same time) is performed we see strong co–variances between the parameters. To come to meaningful results we opted to perform a two step procedure: First, we eliminate the parameters $a$ and $b$ by fitting the differential, thereby reducing expression 6 to the Lorenz function. $\frac{dF_{cd}}{dE}=\frac{\Gamma^{2}/4}{(E-E_{0})^{2}+\Gamma^{2}/4}$ (9) This is also mathematically equivalent to the fit of a Breit–Wigner resonance in high energy physics[14]. In a second fit–procedure, the remaining variables $a$ and $b$ are fit using the cumulative Cauchy function. The fitting procedures were performed in the gnuplot111http://www.gnuplot.info–software using a Levenberg–Marquardt minimization routine. ## Results In Figure 3, the Lorenz expression as outlined in Equation 6 together with a normalization factor, is used to fit the energy differential probability for the generation of DSBs. The fit is performed to minimize the $\chi^{2}$–value. In all cases, the resulting $\chi^{2}/NDF$ (NDF = Number of Degrees of Freedom) are lower than 1. The values of the parameters are provided in Table 1. (a) Electrons (b) Protons (c) $\alpha$–particles (d) Carbon Ions Figure 3: Fitting the Cauchy expression to the energy differential probability of generating DSB’s denoted $\frac{d\sigma}{dE}$ Particle \| $\Gamma$ \| $E_{0}$ \| $a$ \| $b$ ---\|---\|---\|---\|--- $e^{-}$ \| (2.854$\pm$0.051)$10^{-04}$ MeV \| (1.05736$\pm$0.036)$10^{-04}$ MeV \| 2.9061 \| 21.460 $p^{+}$ \| 0.5575$\pm$0.0094 MeV \| 0.1642 $\pm$ 0.0037 MeV \| 2.89068 \| 21.4273 $\alpha^{++}$ \| 8.20$\pm$0.17 MeV \| 3.1850$\pm$0.056 MeV \| 3.0856 \| 20.7933 C6+ \| 201.7$\pm$8.4 MeV \| 95.4$\pm$2.5 MeV \| 3.01459 \| 21.8489 Table 1: The different values for $\Gamma$ and $E_{0}$ as defined by Equation 9 and obtained from a fitting procedure together with the asymptotic standard error of the fitted parameter. All fits exhibited minimal values of $\chi^{2}/NDF$ (NDF = Number of Degrees of Freedom). The columns $a$ and $b$ are the parameters indicating the levels of DSB at low, resp. high LET. Note that even in low LET the number of DSB’s is not zero as complex damage can occur due to the combination of simple damage events. All fits are completely satisfactory at energies higher than $E_{0}$. On the lower energy side some discrepancies can be observed depending on the incoming particles, particularly in the case of electrons and carbon ions. We refer the reader to the discussion section. For protons we see a satisfactory fit over the full energy range. Figure 4 shows the final results with all parameters fit. Again, all fits have $\chi^{2}$–values commensurate with a positive goodness of fit. The final values and the standard errors for the fitted parameters are listed in Table 1. Note, that the noise in the differential curves increases as the particles become heavier. The random-seeming errors in the estimates of the derivative arise in part from the Monte Carlo estimates of the mean number of DSB per Gy per Gbp and from numerical instabilities associated with the calculation of the derivative using finite difference methods. (a) Electrons (b) Protons (c) $\alpha$–particles (d) Carbon Ions Figure 4: The prediction of the number of double strand breaks or more complex damage as a function of energy for 4 relevant charged particles. This provides the number of Double Strand breaks (DSB) per Gy, Gbp and per cell. The prediction for protons and alpha particles is almost perfect. For electrons and carbon ions some discrepancies exist at lower energies. ### Geometric approach Now is the time to investigate the geometric interpretation further. To quantify the function $\lambda(E)$ we can use the inelastic mean free path as measure (IMFP). Values for IMFP for electrons are well known in the literature, not in the least as they are important in solid state physics and electron microscopy. They can be found in freely available databases for a variety of elements and compounds, even for organic molecules like DNA[15]. Proton values can be found in a publication by Zhen–Yu and colleagues[16]. For heavier particles such as $\alpha$–particles and carbon–ions, the data is more difficult to find. We therefore opt not to use the data for these particles and restrict ourselves to electrons and protons in this further treatment. In all current microdosimetric codes, the Bethe formalism is used which is valid for higher energies (i.e. above 500eV for electrons). This implies that changes in IMFP, denoted by $\lambda$, which impact the damage calculated using these codes, also reflect the limitations of the Bethe formalism. From the theory the following expression is used: $\lambda(E)=\frac{E}{A\log(E/E_{0})+B}$ (10) Particle \| A \| B ---\|---\|--- Electrons \| 69.200 eV/nm \| -153.94 eV/nm Protons \| 115.231 keV/nm \| -301.45 keV/nm Table 2: Parameters obtained by fitting Eq. 10 to data obtained from NIST (electrons) and Zhen Yu et al. (protons) In this work the parameters $H$ and $r$ have thus far not been linked to any physical property but were fit. An interesting proposition could be to link these to dimensions of the target structure. Indeed, the choice of a cylinder as a geometric representation is not an accident. It is natural to use the diameter of a DNA–molecule as a measure of the cylinder’s diameter. The length of the cylinder is then related to the maximal distance we allow to classify two damage events, being part of the same cluster of complex damage. Both values can readily be found in the literature and text books[17]. For the most prevalent form of cellular DNA (B–DNA), the values are $3.4$nm (i.e. the height of a spiral of 10 base pairs), and $2.37$nm as the diameter. We now define a dimensionless quantity $k$ which is specific to the type of charged particle used. It is clear that this parameter acts as a scaling parameter but also depends on the ratio of both fixed parameters. Equation 6 now reads as follows: $F_{d}(E)~{}=~{}(a-b)\frac{2}{\pi}[\tan^{-1}(\frac{k\lambda(E)-3.4}{2.37})]+b$ (11) This reduces the impact of the charged particle’s energy on the induction of complex damage in a DNA–molecule to three parameters $a$, $b$, and $k$. Figure 5 illustrates the use of these parameters and shows that comparable results to the energy–based formalism can be obtained. It follows that we can repeat the fitting procedure keeping $a$ and $b$ from the expression based on energy (Eq 7). We find values of k=5.18 for electrons and k=4.82 for protons. ### Extending the model In the work presented above as well as in the used Monte Carlo simulations, the Bethe–Barkas formalism together with its flawed approach in the lower energy regions has always been used. It is well established that the IMFP does not follow the expression outlined in Equation 10, where $\lambda(E)$ keeps diminishing as the energy diminishes. Indeed, when the energy is lower than 200eV an increase in IMFP is observed due to plasmonic effects[18]. Ziaja et al[19] showed that it is possible to describe this behavior analytically by extending Equation 10 with a second term as follows: $\lambda(E)_{Z}~{}=~{}\frac{\sqrt{E}}{A_{1}(E-E_{th})^{B_{1}}}~{}+~{}\frac{E-E_{0}\exp(-B/A)}{A\log(E/E_{0})+B}$ (12) In this equation the parameter $E_{th}$ serves as a threshold separating the behavior as described by Bethe from the plasmonic interactions. Using the data provided in the work from Zhen–Yu and colleagues[16] it is straightforward to obtain parameters for the behavior of protons. These are presented in Table 3. Particle \| $A_{1}$ \| $B_{1}$ \| $E_{th}$ \| $A$ \| $B$ \| $E_{0}$ ---\|---\|---\|---\|---\|---\|--- Electrons \| 0.6560 \| 1.0100 \| 24.2838 \| 65.898 \| -128.23 \| 1.0 Protons \| 0.681 \| 1.249 \| 42.38 \| 117.01 \| -318.7 \| $1.0\times 10^{3}$ Table 3: Parameters as in Table 2 with added lower energy factors. The fitting was performed using Eq. 12 To extend our model to incorporate the behavior of very low energy particles it is sufficient to replace the expression $\lambda(E)$ by $\lambda(E)_{Z}$ in equation 11. In Figure 5, the modified curves show the difference with the calculations based on the Bethe formalism only. This also shows that there is an upper limit to the increase in DSB’s which depends on the type of particle. It is conceivable that this approach also works for the heavier particles which can be seen when using the IFMP’s in water for these (not shown). Figure 5: Using the quantities for $H$ and $r$, the dimensionless constant $k$ for electrons (left) and protons (right) is determined. Using both the limited expression for $\lambda(E)$ and the more accurate estimate $\lambda_{Z}(E)$. The former provides a fit to the Monte Carlo data comparable with the results obtained using the energy–based formalism. The second approach provides a maximal complex damage yield which differs for electrons and protons. ## Discussion We developed an approach to predict damage in complicated situations where fields of different charged particles and their respective energy spectra impact on living cells. The approach, due to its analytical nature allows very fast calculation of damage in otherwise long simulations. In the derivation of this approach using energy alone there are no assumptions on the mechanics with which DNA–damage is caused by the charged particles. The only assumption is that there is a sensitive volume where, if ionizations take place, damage is introduced in the DNA. How exactly this damage is caused is not specified. In the remainder of the text a parameter is identified, the average distance between ionizations for the given charged particle in the medium ($\lambda$). We show that this approach adequately quantifies the results from Monte Carlo simulations based on phenomenological data and reduces these to a closed analytical expression whereby the type of charged particle is expressed by a single parameter ($k$). On the other hand we should be aware that issues like repair mechanisms and oxygen effects are not present in the model, making its applicability limited. However, if all things are identical (i.e. the type of cells, oxygenation, etc…) and the only thing different is the type of charged particle and its energy, then the original damage introduced in the DNA structure should correlate with the outcome. An underlying assumption here is that the repair processes are somehow independent from the modality with which the cell is irradiated. The results of this approach can be applied to determine the biological impact of radiation in mixed environments, as in the case of proton therapy, where protons, electrons and heavier ions (due to neutrons), deposit energy. Other approaches have been proposed to try to predict outcomes from mixed fields, which are based on available clinical response data. Most notably, an approach based on the local effect model (LEM), where macroscopic response data in the form of dose–effect curves is used to quantify the relative effect of the dose delivered[20]. The parameterization, however, of the latter approach is extensive due to the fact that every effect curve has two parameters for a given $\alpha/\beta$–value, making the model over–parameterized. As such, it is possible to have this model reflect the current knowledge of dose and modality response adequately, which forms an important, albeit controversial tool[21, 22]. Its power to predict the behavior outside of the current knowledge therefore seems to be limited. Cucinotta et al. attempted to incorporate the volumetric properties of the dose deposition[23] to account for differences in track structure. They observed that: “LET is a poor descriptor of energy deposition in small volumes because of the diffusion of secondary electrons out of the volume and contribution of $\delta$–rays that pass outside of the volume”. To address this problem a quantification of the energy distribution of generated secondary particles, or $\delta$–rays was proposed. Such a secondary charged particle indirectly changes the behavior with respect to the DNA damage induced. Indeed, depending on the median energy of the spectrum the DNA damage changes accordingly if the dose is kept constant. In the paper presented here this behavior could be easily incorporated by considering the DNA damage for all the particles (i.e. ions and $\delta$–rays) separately using a methodology modeled on the use of the electronic equilibrium concept in photon cavity theory. Currently this behavior is hidden in the $k$ parameter and it would be interesting to see if such an approach will lead to a convergence of all $k$–values for all particles. To take these actions fully into account an approach to provide a more detailed model of the biological effect directly in the Monte Carlo simulation is proposed by Sato et al.[24] This would, in theory, allow a direct calculation of the effect in terms of energy deposited. However, as outlined by Cucinotta this is not without problems as the behavior of low energy electrons needs to be adequately modelled. This work predicts that the current knowledge using the Bethe formalism, might not be suitably extended. The results from the geometric interpretation indicate that the overall behavior of the DNA damage induction is identical for all types of charged particles. The only difference is in the dimensionless parameter $k$. The latter seems to change as the ion used is heavier. Preliminary calculations using the IMFP in water indicate that the value of $k$ diminishes as the charged particles used are heavier (or more charged, data not shown). A possible reason for this is that the track structure can be quite different for different charged particles. This fact could also be an explanation for the discrepancy found at very low energies for carbon–ions. Indeed, allowing the parameter $k$ to be covariant with the other parameters, does provide a more adequate fit (data nor shown). The results for the electrons also shows a discrepancy with regard to the generation of complex damage at lower energies. For electrons, the data on very low energy electrons are not available in terms of energy deposition. Indeed, the model proposed here shows a much lower incidence of complex damage due to plasmonic effects in that region. In summary, the model proposed here allows extension to very low energies for electrons and protons.The fact that there are indications that the induction of DSB’s varies linearly with dose, provides an easy implementation to dose planning systems, given the knowledge of dose deposition spectra in a treatment beam. An example of such implementation is provided in the Appendix. ## Appendix: Implementation in dose deposition calculations ### Mono–energetic treatment In dose calculations a dose matrix is obtained on a dose grid Let $\mathbf{D}=D[i,j,k]$ be the dose matrix provided. Then we can write the amount of complex damage incurred by particles with an energy ($E$) as a damage matrix, denoted as ($M_{cd}$). as follows: $\displaystyle\mathbf{M_{cd}}=M_{cd}[i,j,k]$ $\displaystyle=\mathbf{D}\times F_{cd}(E)$ (13) $F_{cd}(E)$ then denotes a response function converting dose to damage. ### Poly–energetic treatment Dose deposition spectra rarely consist of a field of mono–energetic electrons. For a photon source with a given photon spectrum, an energy depositing electron fields exists, which is roughly constant throughout the target volume. Using Monte Carlo simulations it is possible to calculate this field and its spectrum $\Psi(E)$. It then becomes possible to include the spectrum in the calculation of the damage matrices. This approach has been used already by different authors [25, 26]. $\mathbf{M_{cd}}~{}=~{}\mathbf{D}\times\frac{\int_{0}^{E_{max}}\Psi(E)F_{cd}(E)dE}{\int_{0}^{E_{max}}\Psi(E)dE}$ (14) In the case of charged particle treatment, the particles are moderated and the energy spectrum changes depending on the position of the point where the dose is being deposited. It is therefore necessary to apply Equation 14 to each point separately with knowledge of the depositing energy spectrum in that point. Due to the closed nature of the formalism developed in this paper, it becomes feasible to use off the shelf computing equipment. ### Application: Proton treatment Recently, the coupling of Monte Carlo simulations in dose deposition to micro- dosimetric code has been proposed and applied by several groups[26, 25]. Here a two step approach is followed; 1) a general purpose Monte Carlo code (MCNPX 2.7b)[27] is used to estimate the spectrum of all different dose contributing particles, 2) a micro dosimetric code[13] is used to determine the biological damage. The framework for conversion of dose to biological effect is implemented on a simulation of a pristine 200MeV proton beam, taking into account the changing proton spectrum. The proton simulation is performed using MCNPX. Figure 6(a) shows the variation of the number of complex damage events as a function of energy of the proton. In addition, the spectrum of depositing protons is shown at a position before the Bragg peak and at the Bragg peak. In Figure 6(b) the effect on the dose deposition is shown together with the $RBE_{cd}$ calculated as the complex damage yield generated by the protons at that particular position, divided by the complex damage induced by a 6MV photon beam with the same spatial characteristics. Note, that the $RBE_{cd}$ is of the order of 1.1 with larger value of 2 a few mm distal from the Bragg peak. This is commensurate with the cell data reported by Paganetti et al.[28] and Chaudhary et al.[29], who showed that the radiobiological effect at the distal end of a spread out bragg peak increases, a fact predicted by Goitein[30]. Currently, data of direct measurement of DNA–damage in–vitro along a proton beam are scarce. The advent of $\gamma$–H2AX measurements, as a marker for DSB–damage is promising in this regard and has been used to investigate anti–protons[31]. (a) Impact of the spectrum (b) Proton dose deposition Figure 6: In Fig 6(a) the spectrum at the beginning of the Bragg peak (scaled by 0.1) is completely within the low LET regimen. While at the end of the Bragg peak a significant part of the dose depositing protons exhibits high LET characteristics. Fig.6(b) shows the $\mathrm{RBE_{cd}}$ (red line) together with the damage induced by a mono–energetic proton beam. ## Acknowledgements I am indebted to Rob Stewart, not only for generously providing the MCDS–code for everyone to use, but also for providing much needed input on the biological aspects of the MCDS implementation. Both Ricardo Raabe and Mike Partridge reviewed some of the physics and made it a more rigorous paper. Sandra Nuyts and Dirk De Ruysscher’s clinical input was also highly appreciated. ## References * 1. Hall EJ (1978) LET and RBE. In: Radiobiology for the radiologist, Philadelphia: Harper & Row, Publishers. * 2. Caldecott KW (2008) Single-strand break repair and genetic disease. Nat Rev Genet 9: 619–631. * 3. Ward JF (1985) Biochemistry of dna lesions. Radiation Research Supplement 8: pp. S103-S111. * 4. Bethe H (1930) Zur theorie des durchgangs schneller korpuskularstrahlen durch materie. Ann Phys 5. * 5. 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(2002) Relative biological effectiveness (RBE) values for proton beam therapy. International journal of radiation oncology, biology, physics 53: 407–421. * 29. Chaudhary P, Marshall TI, Perozziello FM, Manti L, Currell FJ, et al. (2014) Relative biological effectiveness variation along monoenergetic and modulated bragg peaks of a 62-mev therapeutic proton beam: A preclinical assessment. Int J Radiat Oncol Biol Phys 90: 27–35. * 30. Goitein M (2008) Radiation Oncology: A Physicist’s-Eye View. Springer Verlag. * 31. Kavanagh JN, Currell FJ, Timson DJ, Savage KI, Richard DJ, et al. (2013) Antiproton induced dna damage: proton like in flight, carbon-ion like near rest. Sci Rep 3: –.	arxiv-papers	2013-09-29T09:53:41	2024-09-04T02:49:51.663462	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Frank Van den Heuvel", "submitter": "Frank Van den Heuvel", "url": "https://arxiv.org/abs/1309.7563" }
1309.7676	# An upper bound on prototype set size for condensed nearest neighbor Eric Christiansen ###### Abstract The condensed nearest neighbor (CNN) algorithm is a heuristic for reducing the number of prototypical points stored by a nearest neighbor classifier, while keeping the classification rule given by the reduced prototypical set consistent with the full set. I present an upper bound on the number of prototypical points accumulated by CNN. The bound originates in a bound on the number of times the decision rule is updated during training in the multiclass perceptron algorithm, and thus is independent of training set size. ## 1 Introduction The nearest neighbor (NN) rule assigns to an unclassified point the class of a closest point from a set of prototypical points. The NN algorithm stores every training point as a prototypical point and classifies new points according to the NN rule. A nice property is that, for arbitrary class distributions, as the number of training points goes to infinity, the error of the rule produced by the NN algorithm converges to within twice the Bayes error [4]. Unfortunately, storing every training point as a prototypical point can be impractical for huge training sets in terms of both memory complexity and the time complexity of classifying according to the NN rule. As a result, many techniques exist for reducing the size of the set of prototypical points. See [8] and [6] for an overview. It is suggested in [1] that using a smaller set with the NN rule may be preferable to using the entire training set, because the VC dimensionality of a NN classifier is given by the number of its prototypical points. However, experimental results in [8] suggest generalization performance is best when the entire training set is used, illustrating a tradeoff between computational complexity and generalization performance. This paper focuses on an algorithm for finding a consistent subset of the training data, where a set of prototypes is consistent if it classifies the training set correctly using the NN rule. Finding a consistent subset of minimum cardinality is intractable [7], but several heuristic approaches exist. One approach is the condensed nearest neighbor (CNN) algorithm, a simple method introduced by Hart [5] that has met with empirical success [1][8]. In this paper, I point out a striking similarity between the CNN algorithm and the multiclass perceptron algorithm described by Collins in [3]. I develop that connection to derive an upper bound on the number of prototypical points accumulated by CNN. The existence of this bound may help explain CNN’s success in [1] and [8]. ## 2 Condensed nearest neighbor We will first need some definitions. Let $T=\\{({\bf x}_{i},c_{i})\\}$ be a set of classified points, where each point ${\bf x}_{i}\in\mbox{${\mathbb{R}}$}^{d}$ and each label $c_{i}$ comes from a finite set of classes $C$. Let $N_{T}({\bf x})\in\mbox{${\mathbb{R}}$}^{d}$ be the point part of the element in $T$ minimizing Euclidean distance to ${\bf x}$; so $N_{T}({\bf x})$ is ${\bf x}$’s nearest neighbor in $T$.111To simplify analysis, we shall assume $N_{T}$ is always well defined, meaning that every point we consider has exactly one nearest neighbor in $T$. We will also assume there do not exist $({\bf x},c),({\bf x}^{\prime},c^{\prime})\in T$ such that ${\bf x}={\bf x}^{\prime}$ and $c\neq c^{\prime}$; in other words, there are no conflicting labels. Let $C_{T}({\bf x})\in C$ be the class of $N_{T}({\bf x})$. If $T=\emptyset$, then neither $N_{T}$ nor $C_{T}$ exist, and it is understood they cannot be said to equal anything. A set $P$ is said to be consistent with $T$ if $C_{P}({\bf x})=c$ for every $({\bf x},c)\in T$. Suppose $T=\\{({\bf x}_{i},c_{i})\\}$ is our training set. Then the condensed nearest neighbor (CNN) algorithm builds a consistent subset of prototypical points $P$ [5]. Its definition is given in Algorithm 1. Here the for loop selects points in arbitrary order, but without repetition, from $T$. Algorithm 1 Condensed nearest neighbor algorithm $P\leftarrow\emptyset$ flag $\leftarrow$ True while flag is True do flag $\leftarrow$ False for $({\bf x},c)$ in $T$ do if $C_{P}({\bf x})\neq c$ then $P\leftarrow P\cup\\{({\bf x},c)\\}$ flag $\leftarrow$ True end if end for end while For finite $T$, this algorithm is guaranteed to terminate; in the worst case, it stops after adding every element of $T$ to $P$. It is also easy to see that after termination, $P$ will be consistent with $T$. In practice, CNN often finds a prototypical set far smaller than the original training set [1][8], though this improvement is obviously more pronounced when the training set is much larger than it needs to be. CNN can be naturally modified for the online context; the while loop is dropped and it is understood the for loop becomes an iteration over an infinite stream of training data. However, in the case of overlapping class distributions (non-zero Bayes’ error), the expected size of $P$ will grow linearly with the number of observed points, perhaps making CNN unsuitable. ## 3 Multiclass perceptron The multiclass perceptron (MP) algorithm [3] is listed as Algorithm 2. It can be thought of as a generalization of the kernelized perceptron algorithm ([2], pages 192-196) to handle multiple classes. Here, $\phi:\mbox{${\mathbb{R}}$}^{d}\times C\rightarrow H$ is an arbitrary feature function, where $H$ is a Hilbert space. The algorithm builds a decision rule through iterative updates to a vector ${\bf w}$. Note a nonlinear decision rule can be obtained through the use of a nonlinear feature function. Algorithm 2 Multiclass perceptron algorithm ${\bf w}\leftarrow{\bf 0}$ flag $\leftarrow$ True while flag is True do flag $\leftarrow$ False for $({\bf x},c)$ in $T$ do $y\leftarrow\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x},y)$ if $y\neq c$ then ${\bf w}\leftarrow{\bf w}+\phi({\bf x},c)-\phi({\bf x},y)$ flag $\leftarrow$ True end if end for end while I will say a set $T=\\{({\bf x}_{i},c_{i})\\}$ is separable with margin $\delta>0$ if there exists a unit vector ${\bf w}^{}\in H$ such that for every $({\bf x},c)\in T$ and for every $y\in C-\\{c\\}$ we have ${\bf w}^{}\cdot\phi({\bf x},c)-{\bf w}^{}\cdot\phi({\bf x},y)\geq\delta$. We will say $T$ has radius at most $R$ if for every $({\bf x},c)\in T$ and for every $y\in C-\\{c\\}$ we have $\|\|\phi({\bf x},c)-\phi({\bf x},y)\|\|\leq R$. Note both margin and radius depend on the feature function $\phi$ as well as the data $T$. From [3] we have Theorem 1. ###### Theorem 1. Let $T$ be separable with margin $\delta$ and radius at most $R$. Then the MP algorithm updates ${\bf w}$ at most $R^{2}/\delta^{2}$ times. Note the number of updates to ${\bf w}$ is the same as the number of points $({\bf x},c)\in T$ that are misclassified during the construction of the decision rule. Also note if $T$ is not separable, then it will not have a margin $\delta>0$. In fact, the MP algorithm will never terminate. ## 4 An upper bound on prototype set size for condensed nearest neighbor Here I develop the connection between the condensed nearest neighbor (CNN) and multiclass perceptron (MP) algorithms. The motivating insight comes from support vector machine (SVM) lore, which asserts that an SVM with a Gaussian kernel is like a smoothed nearest neighbor (NN) classifier. In a similar vein, I will show that for certain $\phi$s and ${\bf w}$s, we can use ${\bf w}\cdot\phi({\bf x},y)$ as a proxy for the nearness of ${\bf x}$ and its nearest neighbor of class $y$, making $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x},y)$ the class of ${\bf x}^{\prime}s$ nearest neighbor. First we need some definitions. Let $T=\\{({\bf x}_{i},c_{i})\\}$ be a training set, and let $P\subseteq T$. Let $o:\mbox{${\mathbb{R}}$}^{d}\rightarrow C$ be any multifunction, where for every $({\bf x},c)\in T$, we have $o({\bf x})\neq c$. So $o({\bf x})$ can be any class, except the true class of ${\bf x}$. I say the set of all ${\bf w}$s of the form given in (1) are the restricted ${\bf w}$s corresponding to $P$. $\displaystyle{\bf w}$ $\displaystyle=\sum_{({\bf x},c)\in P}\phi({\bf x},c)-\phi({\bf x},o({\bf x}))$ (1) Note in (1), ${\bf w}$ has exactly the form one might expect the ${\bf w}$ from the MP algorithm to have after termination, except that every summand in a restricted ${\bf w}$ has the unitary coefficient. Now I define a condition on $\phi$ that makes $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x},y)$ behave like the NN rule. I will say $\phi$ is neighborly (with respect to $T$) if for every $P\subseteq T$ and every corresponding restricted ${\bf w}$, as well as for every training point $({\bf x},c)\in T$, we have $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x},y)=C_{P}({\bf x})$. When $\|T\|$ is finite, we can always find uncountably many neighborly $\phi$s; see Appendix A for a construction using the Gaussian kernel. By Lemma 1, when $\phi$ is neighborly, the MP algorithm will never misclassify the same point twice. So we will only need to consider restricted ${\bf w}$s when analyzing the MP algorithm. ###### Lemma 1. Let $\phi$ be neighborly. Then at all times during the execution of Algorithm 2, ${\bf w}$ will be restricted. ###### Proof. This is a proof by induction. Note ${\bf w}$ is initially restricted. Further, if ${\bf w}$ is restricted before the $n^{\text{th}}$ update, then it will be restricted after the $n^{\text{th}}$ update. For let $({\bf x},c)\in T$ be the point causing the $n^{\text{th}}$ update. So $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x},y)\neq c$. But $\phi$ is neighborly, and ${\bf w}$ is restricted before this update, so $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x},y)=C_{P}({\bf x})$. So $C_{P}({\bf x})\neq c$. So $({\bf x},c)\not\in P$, and thus ${\bf w}$ is restricted after the $n^{\text{th}}$ update. ∎ The CNN algorithm is essentially the MP algorithm 2 in the special case $\phi$ is neighborly, so we should be able to apply to the CNN algorithm theorems that apply to the MP algorithm. In that spirit, Lemma 2 says we can apply Theorem 1 to the CNN algorithm. ###### Lemma 2. Let $\phi$ be neighborly, and suppose $T$ is separable with margin $\delta$ and radius at most $R$. Then the CNN algorithm updates $P$ at most $R^{2}/\delta^{2}$ times. ###### Proof. The idea is to morph the MP algorithm into the CNN algorithm without changing the number of misclassifications, which are in one-to-one correspondence with the number of updates to ${\bf w}$ in the MP algorithm, and are in one-to-one correspondence with the number of updates to $P$ in the CNN algorithm. We begin by inserting two lines of code into the MP algorithm, obtaining Algorithm 3. The new lines in Algorithm 3 are lines 2 and 10. These lines clearly have no effect on the number of updates to ${\bf w}$, but note in Algorithm 3 the number of updates to $P$ equals the number of updates to ${\bf w}$. Algorithm 3 Hybrid multiclass perceptron algorithm 1: ${\bf w}\leftarrow{\bf 0}$ 2: $P\leftarrow\emptyset$ 3: flag $\leftarrow$ True 4: while flag is True do 5: flag $\leftarrow$ False 6: for $({\bf x}_{i},c_{i})$ in $T$ do 7: $y_{i}\leftarrow\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x}_{i},y)$ 8: if $y_{i}\neq c_{i}$ then 9: ${\bf w}\leftarrow{\bf w}+\phi({\bf x}_{i},c_{i})-\phi({\bf x}_{i},y_{i})$ 10: $P\leftarrow P\cup\\{({\bf x}_{i},c_{i})\\}$ 11: flag $\leftarrow$ True 12: end if 13: end for 14: end while We then use Lemma 1, together with the assumption $\phi$ is neighborly, to replace $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x}_{i},y)$ with $C_{P}({\bf x}_{i})$ in Algorithm 3, yielding Algorithm 4. Algorithm 4 Hybrid condensed nearest neighbor algorithm 1: ${\bf w}\leftarrow{\bf 0}$ 2: $P\leftarrow\emptyset$ 3: flag $\leftarrow$ True 4: while flag is True do 5: flag $\leftarrow$ False 6: for $({\bf x}_{i},c_{i})$ in $T$ do 7: if $C_{P}({\bf x}_{i})\neq c_{i}$ then 8: ${\bf w}\leftarrow{\bf w}+\phi({\bf x}_{i},c_{i})-\phi({\bf x}_{i},C_{P}({\bf x}_{i}))$ 9: $P\leftarrow P\cup\\{({\bf x}_{i},c_{i})\\}$ 10: flag $\leftarrow$ True 11: end if 12: end for 13: end while Note Algorithm 4 updates $P$ exactly as many times as the MP algorithm updated ${\bf w}$, and from Algorithm 4 we can simply delete both lines referencing ${\bf w}$ (lines 1 and 8). This yields the CNN algorithm, without changing the number of updates to $P$. So by Theorem 1, we are done. ∎ Since Lemma 2 gives a bound for any neighborly feature function, the set of all neighborly feature functions yields a set of bounds. It is natural to choose the best bound, which is the essence of Theorem 2. Note the number of updates to $P$ is exactly the ultimate size of $P$, or equivalently the number of points accumulated by the CNN algorithm. ###### Theorem 2. Let $\Phi=\\{\phi:\phi\text{ is neighborly}\\}$ and suppose, for each $\phi\in\Phi$, $T$ is separable with margin $\delta_{\phi}$ and radius at most $R_{\phi}$. Then the CNN algorithm accumulates at most $\inf_{\phi\in\Phi}R_{\phi}^{2}/\delta_{\phi}^{2}$ prototypical points. ###### Proof. By Lemma 2, for any neighborly $\phi$, the CNN algorithm accumulates at most $R_{\phi}^{2}/\delta_{\phi}^{2}$ representative points. Thus since $\Phi$ is the set of all neighborly $\phi$s, the CNN algorithm accumulates at most $\inf_{\phi\in\Phi}R_{\phi}^{2}/\delta_{\phi}^{2}$ representative points. ∎ So $\|P\|$ is bounded above by the best bound over all neighborly $\phi$s. ## 5 Conclusion In Theorem 2 I presented a bound on the number of prototypical points accumulated by the condensed nearest neighbor (CNN) algorithm. This bound came from a bound on the number of updates to the decision rule in the multiclass perceptron algorithm. Unfortunately, as with the multiclass perceptron bound presented in [3], estimating the bound is likely to be too expensive to be practical. Fortunately, as with the multiclass perceptron bound, this bound is independent of the size of the training set used as input to the CNN algorithm. Perhaps the existence of this bound may help explain CNN’s empirical success. ## Appendix A Neighborly $\phi$s Let $T=\\{({\bf x}_{i},c_{i})\\}$ be a finite training set, where ${\bf x}_{i}\in\mbox{${\mathbb{R}}$}^{d}$, $c_{i}\in C$, and $\|C\|$ is finite. I will show there exist uncountably many neighborly feature functions $\phi:\mbox{${\mathbb{R}}$}^{d}\times C\rightarrow H$, where $H$ is a Hilbert space. In other words, there exist uncountably many functions $\phi:\mbox{${\mathbb{R}}$}^{d}\times C\rightarrow H$ such that for any $P\subseteq T$ and restricted ${\bf w}$ corresponding to $P$, as well as for any $({\bf x},c)\in T$, we have $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi({\bf x},y)=C_{P}({\bf x})$. The idea is to construct an uncountable family of such functions using the Gaussian kernel. Let $\psi_{\sigma}:\mbox{${\mathbb{R}}$}^{d}\rightarrow H$ correspond to the Gaussian kernel, so that $\psi_{\sigma}({\bf x})\cdot\psi_{\sigma}({\bf x}^{\prime})=e^{\frac{-1}{2\sigma^{2}}\|\|{\bf x}-{\bf x}^{\prime}\|\|^{2}}.$ We know such a $\psi_{\sigma}$ exists ([2], pages 294-299). Let $\phi_{\sigma}({\bf x},c)\in H$ be a vector such that the $i^{\text{th}}$ element $\phi_{\sigma}({\bf x},c)_{i}$ of $\phi_{\sigma}({\bf x},c)$ is $\phi_{\sigma}({\bf x},c)_{i}=\left\\{\begin{tabular}[]{ c l}$\psi_{\sigma}({\bf x})_{j}$&if there exists a $j$ such that $i=(j-1)\|C\|+c$\\\ $0$&otherwise\\\ \end{tabular}\right..$ The idea behind the above definition is to make $\phi_{\sigma}({\bf x},c)\cdot\phi_{\sigma}({\bf x}^{\prime},c^{\prime})=I(c=c^{\prime})\psi_{\sigma}({\bf x})\cdot\psi_{\sigma}({\bf x}^{\prime}),$ where $I$ is the indicator function on Boolean inputs given by $I(\texttt{true})=1$ and $I({\texttt{false}})=0$. I claim $\phi_{\sigma}$ will be neighborly if we choose $\sigma>0$ sufficiently small. First we need the following result. ###### Lemma 3. Let $P\subseteq T$ and let ${\bf w}$ be restricted corresponding to $P$. Let $({\bf x}^{\prime},c^{\prime})\in T$. Then there exists a $\sigma^{}>0$ such that for every $0<\sigma<\sigma^{}$, we have $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi_{\sigma}({\bf x}^{\prime},y)=C_{P}({\bf x}^{\prime})$. ###### Proof. Note ${\bf w}$ is of the form $\displaystyle{\bf w}$ $\displaystyle=\sum_{({\bf x},c)\in P}\phi_{\sigma}({\bf x},c)-\phi_{\sigma}({\bf x},o({\bf x})).$ So $\displaystyle{\bf w}\cdot\phi_{\sigma}({\bf x}^{\prime},y)$ $\displaystyle=\sum_{({\bf x},c)\in P}I(c=y)\psi_{\sigma}({\bf x})\cdot\psi_{\sigma}({\bf x}^{\prime})-I(o({\bf x})=y)\psi_{\sigma}({\bf x})\cdot\psi_{\sigma}({\bf x}^{\prime})$ (2) $\displaystyle=\sum_{({\bf x},c)\in P}\left[I(c=y)-I(o({\bf x})=y)\right]e^{\frac{-1}{2\sigma^{2}}\|\|{\bf x}-{\bf x^{\prime}}\|\|^{2}}.$ (3) I want to show that as $\sigma\rightarrow 0$, the summation in (3) will be dominated by the term corresponding to the nearest neighbor of ${\bf x}^{\prime}$ in $P$. The intuition behind this is that the summands of (3) correspond to individual Gaussians in a mixture of Gaussians, where the typical constraints on the mixing coefficients are dropped. For large $\sigma$, the value of the mixture at a point might depend significantly on several nearby Gaussians, but as we uniformly shrink the variance of the Gaussians, the value of the mixture at a point will depend almost entirely on the Gaussian nearest that point, as is borne out in the following analysis. First, I divide through by $e^{\frac{-1}{2\sigma^{2}}\|\|N_{P}({\bf x}^{\prime})-{\bf x^{\prime}}\|\|^{2}}$, giving us $\displaystyle(\ref{eqn:appndxdom})$ $\displaystyle\propto\sum_{({\bf x},c)\in P}\left[I(c=y)-I(o({\bf x})=y)\right]e^{\frac{1}{2\sigma^{2}}\left[\|\|N_{P}({\bf x}^{\prime})-{\bf x^{\prime}}\|\|^{2}-\|\|{\bf x}-{\bf x^{\prime}}\|\|^{2}\right]}$ (4) $\displaystyle=\left[I(C_{P}({\bf x}^{\prime})=y)-I(o(N_{P}({\bf x}^{\prime}))=y)\right]$ $\displaystyle\quad+\sum_{({\bf x},c)\in P:{\bf x}\neq N_{P}({\bf x}^{\prime})}\left[I(c=y)-I(o({\bf x})=y)\right]e^{\frac{1}{2\sigma^{2}}\left[\|\|N_{P}({\bf x}^{\prime})-{\bf x^{\prime}}\|\|^{2}-\|\|{\bf x}-{\bf x^{\prime}}\|\|^{2}\right]}.$ (5) Let $S$ be the summation given in (5). Note for every $({\bf x},c)\in P$ such that ${\bf x}\neq N_{P}({\bf x}^{\prime})$, we have $\|\|N_{P}({\bf x}^{\prime})-{\bf x^{\prime}}\|\|^{2}-\|\|{\bf x}-{\bf x^{\prime}}\|\|^{2}<0.$ Thus as $\sigma\rightarrow 0$, every term in S will go to zero. So by the definition of a limit, there is a $\sigma^{}>0$ such that $\sigma<\sigma^{}$ implies $\|S\|<\frac{1}{2}$. Let $\sigma<\sigma^{}$. Then if $y=C_{P}({\bf x}^{\prime})$, we have $(\ref{eqn:appndxsep})\geq 1-\|S\|>\frac{1}{2}$. If $y\neq C_{P}({\bf x}^{\prime})$, we have $(\ref{eqn:appndxsep})\leq 0+\|S\|<\frac{1}{2}$. So $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi_{\sigma}({\bf x}^{\prime},y)=C_{P}({\bf x}^{\prime})$. ∎ With Lemma 3 I can prove Theorem 3, establishing the desired result. ###### Theorem 3. For any finite $T$, there exist uncountably many neighborly feature functions. ###### Proof. By Lemma 3, for every $P\subseteq T$, restricted ${\bf w}$, and $({\bf x}^{\prime},c)\in T$, we can find a $\sigma^{}>0$ such that $\sigma<\sigma^{}$ implies $\operatornamewithlimits{argmax}_{y\in C}{\bf w}\cdot\phi_{\sigma}({\bf x}^{\prime},y)=C_{P}({\bf x}^{\prime})$. Let $\Sigma^{}$ be the set of all such $\sigma^{}$s. Since there are finitely many subsets $P$ of $T$, finitely many restricted ${\bf w}$s corresponding to any given $P$, and finitely many elements in $T$, the set $\Sigma^{}$ is finite and thus has a minimum element $\sigma^{\text{min}}$. Thus for any of the uncountably many $\sigma$s such that $\sigma<\sigma^{\text{min}}$, there exists a neighborly feature function $\phi_{\sigma}$. Thus the set of all neighborly feature functions contains a subset of uncountable cardinality, and is thus uncountable. ∎ ## Acknowledgements I would like to thank Mehran Bozorgi, Charles Elkan, and Matus Telgarsky for acting as sounding boards for this idea, and I would like to thank Charles Elkan and Nicolaus Hepler for constructively criticizing the drafts of this paper. This work was funded by NSF grant #SBE-0542013 to the Temporal Dynamics of Learning Center, an NSF Science of Learning Center. ## References [1] Fabrizio Angiulli. Fast condensed nearest neighbor rule. In ICML ’05: Proceedings of the 22nd international conference on Machine learning, pages 25–32, New York, NY, USA, 2005. ACM. * [2] Christopher M. Bishop. Pattern Recognition and Machine Learning (Information Science and Statistics). Springer, August 2006. * [3] Michael Collins. Discriminative training methods for hidden Markov models: theory and experiments with perceptron algorithms. In EMNLP ’02: Proceedings of the ACL-02 conference on empirical methods in natural language processing, pages 1–8, Morristown, NJ, USA, 2002\. Association for Computational Linguistics. * [4] Luc Devroye. On the inequality of Cover and Hart in nearest neighbor discrimination. IEEE Transactions on Pattern Analysis and Machine Intelligence, 3:75–78, 1981. * [5] P. Hart. The condensed nearest neighbor rule (corresp.). Information Theory, IEEE Transactions on, 14(3):515–516, May 1968\. * [6] Godfried Toussaint. Proximity graphs for nearest neighbor decision rules: recent progress. In Proceedings of the 34th Symposium on the INTERFACE, pages 17–20, 2002. * [7] Gordon Wilfong. Nearest neighbor problems. In SCG ’91: Proceedings of the seventh annual symposium on Computational geometry, pages 224–233, New York, NY, USA, 1991. ACM. * [8] D. Randall Wilson and Tony R. Martinez. Reduction techniques for instance-based learning algorithms. In Machine Learning, pages 257–286, 2000.	arxiv-papers	2013-09-29T23:45:59	2024-09-04T02:49:51.682533	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Eric Christiansen", "submitter": "Eric Christiansen", "url": "https://arxiv.org/abs/1309.7676" }
1309.7740	# The quantum approach to human reasoning does explain the belief-bias effect E. D. Vol [email protected] B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47, Lenin Ave., Kharkov 61103, Ukraine. ###### Abstract Based on the ideas of quantum physics and dual-process theory of human reasoning that takes into account two primary mechanisms of reasoning : 1) deductive rational thinking and 2) intuitive heuristic judgment, we proposed the ”quantum” approach to practical human logic that allows one to specify the most distinctive peculiarities in activity of two reasoning systems mentioned above and in addition to describe phenomenologically well-established experimentally belief-bias effect . ###### pacs: 05.40.-a ## I Introduction The idea that some essential human values and concepts may be incompatable with each other had originated long before the beginning of scientific psychology. By distinct ways this idea was justified by such outstanding philosophers and thinkers as G.W. Leibniz, N.Machiavelli and I. Kant. The interested reader can find detail account of the history of this idea with relevant references in 1s . However only in the XX century with the rise of quantum theory this idea has received adequate scientific expression in the language of the Bohr’s Complementarity Principle. We shall give here only two distinctive quotations of founding fathers of quantum mechanics that are clearly demonstrating their profound understanding of the inconsistancy of some basic concepts relating to human psychology. So, in the paper of 1948 ”On the Notions of Causality and Complementarity” N. Bohr wrote:”Recognition of complementary relationship is not least required in psychology, where the conditions for analysis and synthesis of experience exhibit striking analogy with the situation in atomic physis. In fact, the use of words like thoughts, and sentiments, equally indispensable to illustrate the diversity of psychical experience, pertain to mutually exclusive situations characterized by a different drawing of the line of separation between subject and object. In particular, the place left for the feeling of volition is afforded by the very circumstance that situations where we experience freedom of will are incompatible with psychological situations where causal analysis is reasonably attempted. In other words, when we use the phrase ”I will” we renounce explanatory argumentation. In fact, the use which we make of words like ”thought” and ”feeling,” or ”instinct” and ”reason” to describe psychic experiences of different types, shows the existence of characteristic relationships of complementarity conditioned by the peculiarity of introspection” 2s . On the other hand W.Pauli drew particular attention to the problem of relation between complementarity of mental and physical aspects of the same reality. In his inspiring paper ”The influence of archetypal ideas on the scientific theories of Kepler” 2s he wrote:” “The general problem of the relationship between psyche and physics, between inside and outside, will hardly be solved with the notion of a ‘psychophysical parallelism, put forward in the past century. However, modern science has perhaps brought us closer to a more satisfying conception of this relationship insofar as it introduced the concept of complementarity within physics. It would be most satisfactory if physis and psyche could be conceived as complementary aspects of the same reality.”.Unfortunately at the time these deep ideas are not influenced the development of experimental psychology. All the more remarkable that modern cognitive psychology irrespectively came in fact to the similar conclusions.In particular numerous experts in so different areas of cognitive psychology as attention,memory,decision making, learning with one accord believe that dual processes and dual systems play fundamental role for nearly all basic cognitive mechanisms in human mind (see e.g. 4s for brief review of dual- process theory in reasoning with the list of necessary references).In what follows we are interested only in human reasoning where two primary dual systems of interest can be specified. One of these systems we will call it below as deductive reasoning system (DRS) is rational, sequental and consistent but acts relatively slow while the other - we will call it further as heuristic reasoning system (HRS) is intuitive, fast, automatic, but to a large extent influenced by emotions and last unconsious experience.Numerous researches and experiments conclusively proved that there is hidden interaction between these two cognitive systems such that a reasoning subject is not aware of this.The belief-bias effect is the most striking manifestation of such interaction. Roughly speaking the belief-bias effect is the innate tendency of reasoning subjects to be more likely to accept reasons ang arguments if they find them believable and to pay less attention of their logical validity. The main goal of present paper is based on quantum ideas of complementarity and dual-process theory in human reasoning to describe the belief-bias effect phenomenologically by purely logical tools. To this end we will use also the simplified version of discrete-continuous logic that was formulated earlier in author preprint 5s . The remainder of the paper is organized as follows. In chapter 2 we briefly remind basic facts relating to discrete-continuous logic (DCL) that are necessary for the understanding of the present paper.The main contribution of this chapter is the interpretation of the general propositions in DCL as the integral mental structures that consist both of logical and heuristic constituents. Under such interpretation these two constituents of the proposition can be considered as complementary to each other exactly like two noncommuting observables in quantum mechanics.In chapter 3 we state the uncertainty relation that just reflects the complementary nature of such concepts as logical rigour and the heuristic grasp. And finally in chapter 4 using only logical tools we give the phenomenological explanation of the belief-bias effect. Now let us go to the details. ## II Preliminiries In this part we briefly remind for the reader convenience the necessary facts relating to the discrete -continuous logic that were outlined more detail in author preprint 5s . So, we will consider as the primary objects of our study the set of general propositions (GP) -$\left\\{A_{j}\right\\}_{\text{ }}$that may be represented by $2\times 2$ positive definite matrices with unit trace of the following form: $A_{j}=\begin{pmatrix}p_{j}&i\alpha_{j}\\\ -i\alpha_{j}&1-p_{j}\end{pmatrix},$ (1) (where $i$ is imaginary unit). In this case the negation of such proposition \- ($not$ $A_{j}$) may be defined as $(not$ $A)=\begin{pmatrix}1-p_{j}&-i\alpha_{j}\\\ i\alpha_{j}&p_{j}\end{pmatrix}$. It turns out that in addition to negation another but already two place operation -$\bigtriangleup$ (which is the analogue of strong disjunction in ordinary Boolean logic) can be introduced in DCL according to the next definition: Let $A=\begin{pmatrix}p&i\alpha\\\ -i\alpha&1-p\end{pmatrix}$ and $B=\begin{pmatrix}q&i\beta\\\ -i\beta&1-q\end{pmatrix}$ then $(A\bigtriangleup B)=\begin{pmatrix}R&i\gamma\\\ -i\gamma&1-R\end{pmatrix},$ (2) where $R=p+q-2pq+2\alpha\beta$, $\gamma=\alpha\left(1-2q\right)+\beta\left(1-2p\right)$. Comparing representation Eq. (1) with standard form of density matrix of the mixed state of two-level quantum system that looks as $\rho=\begin{pmatrix}\frac{1+P_{z}}{2}&\frac{P_{x}-iP_{y}}{2}\\\ \frac{P_{x}+iP_{y}}{2}&\frac{1-P_{z}}{2}\end{pmatrix}$ (where $P=(P_{x},P_{y},P_{z})$ is the Bloch vector of the state ) we see that GP may be represented by the similar way but in this case $x-$ component of the Bloch vector is equal to zero. In the rest of the paper we will use such reduced Bloch representation for the arbitrary proposition $A,$ that is: $\ A=\begin{pmatrix}\frac{1+P_{z}}{2}&\frac{-iP_{y}}{2}\\\ \frac{iP_{y}}{2}&\frac{1-P_{z}}{2}\end{pmatrix}$ with $P=(P_{y},P_{z})$. In this case it is convenient to introduce the complex vector $P=P_{z}-iP_{y}$ which we call further as representating vector (RV) of proposition $A$. It is easy to verify directly that the RV of proposition $(notA)$ is equal to $(-P)$ and RV of proposition $(A\bigtriangleup B)$ is equal to $(-PQ)$ (where $Q$ is RV of $B$ ).Note also the useful relation connecting negation with operation $\bigtriangleup$: $not(A\bigtriangleup B)\equiv(A\overline{\bigtriangleup}B)=(notA)\bigtriangleup B=A\bigtriangleup(notB).$ (3) It should be noted that unlike of ordinary Boolean logic in DCL it is possible to define the whole one-parameter group of continuous logical operations (logical rotations of propositions in the plane $P_{y}-P_{z}$) according to the following rule: if proposition $A$ has the RV -$P$ then rotated at an angle $\Phi$ proposition $A^{{{}^{1}}}$ has RV - $P{{}^{1}}$ with components: $\displaystyle P_{y}^{{{}^{1}}}$ $\displaystyle=$ $\displaystyle P_{y}\cos\phi+P_{z}\sin\Phi$ $\displaystyle P_{z}^{{{}^{1}}}$ $\displaystyle=$ $\displaystyle P_{z}\cos\Phi-P_{y}\sin\Phi$ (4) It is easy to see that the negation of any proposition coincides with logical rotation of it at an angle $\pi$ and in addition that if one rotates the GP $A$ at an angle $\Phi_{1}$ and the other proposition $B$ at an angle $\Phi_{2}$ then the proposition $(A\bigtriangleup B)$ will be rotated at an angle $\Phi_{1}+\Phi_{2}$. Thus all logical operations in DCL obtain quite clear geometric meaning. Now after describing the syntax of DCL we can pass to the more difficult task: clarification of its semantics that is the interpretation both the meaning of general propositions and logical operations with them. It should be noted that interpretation that we are going to propose here is not the only possible but it is appropriate for our ultimate goal namely to explain the beliefe-bias effect in human reasoning from pure logical point of view. So, as before we will assume that diagonal elements of representing matrix for arbitrary GP describes degree of its logical validity (from DRS point of view) while its nondiagonal elements we will interpret as the believability of the same proposition inspired by the heuristic reasoning system (HRS). This interpretation can be expressed more precisely as follows. Let us introduce two projection operators:$P_{1}$ and $P_{2}$ $\left(P_{1}^{2}=P_{1}\text{, }P_{2}^{2}=P_{2}\right)$ according to the definition: $P_{1}=\frac{1+\sigma_{z}}{2}$ and $P_{2}=\frac{1+\sigma_{y}}{2}$. It is easy to see that average values of these operators in the state whose density matrix coincides with representating matrix of proposition $A=\begin{pmatrix}p&i\alpha\\\ -i\alpha&1-p\end{pmatrix}$ give us the propabilities of its logical plausability $p_{t}$ and its believability $p_{b\text{ }}$respectively.Thus we obtain $\displaystyle p_{t}$ $\displaystyle=$ $\displaystyle\left\langle P_{1}\right\rangle=Sp(P_{1}A)=p$ (5) and $\displaystyle p_{c}$ $\displaystyle=$ $\displaystyle\left\langle P_{2}\right\rangle=Sp(P_{2}A)=\frac{1-2\alpha}{2}$ In connection with above interpretation we want to point out two important marginal GP: 1) $T=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$-true proposition, and 2) $B=\begin{pmatrix}\frac{1}{2}&-\frac{i}{2}\\\ \frac{i}{2}&\frac{1}{2}\end{pmatrix}$ \- highest possible believable proposition and their negations: $F=(notT)-$false proposition and $U=(notB)$-unbelievable proposition. Note in addition that noncommutativity of operators $P_{1}$and $P_{2}$ implies that main predicates of arbitrary GP (plausibility and belief) may be considered as complementary (in the sence of quantum theory) aspects of the same proposition. This important fact implies specific uncertainty relation for the observables $P_{1}$and $P_{2}$ connected with any GP. The simple derivation of these relation is the subject of the next section of the presenr paper. ## III The Uncertainty Relation between predicates plausibility and believability in DCL. To derive the required uncertainty relation it is convenient to represent any GP $A$ in the Bloch form: $A=\begin{pmatrix}\frac{1+P_{z}}{2}&\frac{-iP_{y}}{2}\\\ \frac{iP_{y}}{2}&\frac{1-P_{z}}{2}\end{pmatrix}.$According to definition the uncertainty of logical truth for the proposition $A$ can be written with the help of operator $P_{1}=\frac{1+\sigma_{z}}{2}$ as: $\bigtriangleup p_{t}^{2}\equiv\overline{\left(\frac{1+\sigma_{z}}{2}-\overline{\frac{1+\sigma_{z}}{2}}\right)^{2}}=\frac{1}{4}\left(1-\overline{\sigma_{z}}^{2}\right)=\frac{1-P_{z}^{2}}{4}$.In the similar manner the uncertainty of believability of the same proposition is equal to : $\bigtriangleup p_{c}=\frac{1}{4}\left(1-\overline{\sigma_{y}}^{2}\right)=\frac{1}{4}\left(1-P_{y}^{2}\right).$By adding these two expressions we obtain: $\bigtriangleup p_{A}^{2}\equiv\bigtriangleup p_{t}^{2}+\bigtriangleup p_{c}^{2}=\frac{1}{4}\left(2-P_{y}^{2}-P_{z}^{2}\right).$Finally taking into accout that $P_{y}^{2}+P_{z}^{2}\leqslant 1$ we get the desired relations: $\frac{1}{4}\leqslant\bigtriangleup p_{A}^{2}\leqslant\frac{1}{2}.$ (6) The notable fact should be mentioned here:if one takes two propositions $A$ and $B$ with RV $P$ and $Q$ respectively then according above calculation one can write two equations 1) $\bigtriangleup p_{A}^{2}=\frac{\left(2-P^{2}\right)}{4}$ and 2)$\bigtriangleup p_{B}^{2}=\frac{\left(2-Q^{2}\right)}{4}.$ On the other hand as we marked earlier the proposition $\left(A\bigtriangleup B\right)$ has RV $(-PQ)$ and hence its uncertainty is equal to $\bigtriangleup$ $p_{(A\bigtriangleup B)}^{2}=\frac{\left(2-P^{2}Q^{2}\right)}{4}$ .As long as $P^{2}Q^{2}\leqq P^{2},Q^{2}$ one can conclude that $\bigtriangleup p_{(A\bigtriangleup B)}^{2}\geqslant(\bigtriangleup p_{A}^{2},\bigtriangleup p_{B}^{2})$ and hence as a final result of logical operation $\bigtriangleup$ the ending uncertainty of proposition can only increases.We would like to hope that properly organized experiments with specially selected reasoning tasks will be able to confirm (or may be disprove) the proposed uncertainty relations $\left(6\right)$.Now we come back to the main goal of present paper: the explanation of the belief-bias effect in human reasoning. ## IV Many valued probabilistic logic and the Belief-Bias effect. In this part we will try to describe (phenomenologically) the belief-bias effect in human reasoning by purely logical tools.For this purpose it is convenient to use some version of probabilistic many-valued logic that in some sense can be considered as simplified version of original DCL.Really if in original version of DCL we restrict ourselves only by discrete set of logical rotations with angles: $0,\frac{2\pi}{N}...\frac{2\pi}{N}\left(N-1\right)$ we obtain the closed logic with N marginal propositions which possess representing matrices: $A_{0},A_{1}....A_{N-1}$ (where $A_{0}=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix},...$ $A_{k}=\begin{pmatrix}\frac{1+\cos\frac{2\pi}{N}}{2}&-\frac{i}{2}\sin\frac{2\pi}{N}\\\ \frac{i}{2}\sin\frac{2\pi}{N}&\frac{1-\cos\frac{2\pi}{N}}{2}\end{pmatrix}(k=1..N-1)$). In the case when we are not interested in the ”quantum correlations” between these marginal propositions we can consider them as approximately independent quantities and associate with these propositions the logical basis consisting of N distinct logical alternatives.Acting in this manner one can pass from original DCL to standard many- valued probabilistic Boolean logic. After this remark we will examine further four valued probabilistic logic every proposition of which can be represented as $4\times 4$ diagonal matrix : $A=diag\left(P_{1},P_{2},P_{3},P_{4}\right)$.Here we mean that the space of these propositions is a tensor product of two spaces with $2\times 2$ diagonal matrices, that is: $A=\sum\limits_{i}a_{i}T_{i}\otimes B_{i},$ (7) where $T_{i}=\begin{pmatrix}p_{i}&0\\\ 0&1-p_{i}\end{pmatrix},$ $B_{i}=\begin{pmatrix}q_{i}&0\\\ 0&1-q_{i}\end{pmatrix}$and $\sum\limits_{i}a_{i}=1$. In addition we assume that matrices $T_{i}$ in the decomposition Eq. (7) are associated with the activity of deductive cognitive subsystem (DRS),while matrices $B_{i}$ are connected with its heuristic subsystem (HRS).Thus the basis of this logic consists of four propositions:$1)$ truth-believable $TB=diag(1,0,0,0),2)$ truth-unbelievable $TU=diag(0,1,0,0),3)$false-believable $FB=diag(0,0,1,0)$ and $4)$false- unbelievable $FU=diag(0,0,0,1).$Our next step is to define basic logical operations that can be implemented with such propositions. The interpetation that we have adopted above implies that the negation of proposition $A$ must be defined as $\left(notA\right)=diag\left(P_{4},P_{3},P_{2},P_{1}\right)$. The certain dilemma arises however when we want to define the conjunction of two propositions $A=diag\left(P_{1},P_{2},P_{3},P_{4}\right)$ and $B=diag\left(Q_{1},Q_{2},Q_{3},Q_{4}\right)$. We have proposed here the following definition: $C\equiv\left(AandB\right)=diag\left(C_{1},C_{2},C_{3},C_{4}\right),$ (8) where $C_{1}=P_{1}\left(Q_{1}+Q_{2}\right)+P_{2}Q_{1}$, $C_{2}=P_{2}Q_{2}$, $C_{3}=P_{1}\left(Q_{3}+Q_{4}\right)+P_{2}Q_{3}+P_{3}+P_{4}\left(Q_{1}+Q_{3}\right)$, $C_{4}=P_{2}Q_{4}+P_{4}\left(Q_{2}+Q_{4}\right)$. This definition of conjuction namely Eq. (8) certainly needs to be explained.First of all we note that definition Eq. (8) satisfies to the necessary symmetry condition : $\left(AandB\right)=\left(BandA\right)$as it should be.In addition if one takes the projection of conjunction Eq. (8) in DRS (first reasoning subsystem) the result is:$\left(AandB\right)_{1}=\begin{pmatrix}pq&0\\\ 0&1-pq\end{pmatrix}\equiv\left(A_{1}andB_{1}\right)$ where $p=P_{1}+P_{2}$ and $q=Q_{1}+Q_{2}$. This result obviously consistent with definition of conjunction in ordinary probabilistic Boolean logic. On the other hand if one takes the projection of Eq. (8) in HRS (second reasoning subsystem) the obtained result reads as: $\left(AandB\right)_{2}=\begin{pmatrix}1-\left(P_{2}+P_{4}\right)\left(Q_{2}+Q_{4}\right)&\\\ \left(P_{2}+P_{4}\right)\left(Q_{2}+Q_{4}\right)&\end{pmatrix}.$ (9) We see that conjunction in heuristic system differs from standatd logical conjunction .In our opinion this distinction explicitly reflects (from phenomenological point of view) the essential difference existing between two reasoning systems when they operate jointly. In particular the definition Eq. (9) implies for two basic marginal propositions in second reasoning subsystem: $B=\begin{pmatrix}1&0\\\ 0&0\end{pmatrix}$ responding to the statement of unconditional belief and $D=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ -which is the most doubtful statement, the next conjunction relations:$\left(BandB\right)=B$, $\left(BandD\right)=\left(DandB\right)=B$ and $\left(DandD\right)=D$.Thus we obtain that the unconditional belief when it conflicts with certain doubtful one always overcomes it. Now if one takes the expression Eq. (8) for granted then he (she)can define another logical operations (in particular implication that we especially interested in ) without any obstacles.To this end one should be guided by two relations of ordinary logic which as we assume continue to be valid in our case as well: 1) $\left(AorB\right)=not\left[\left(notA\right)and\left(notB\right)\right]$ and 2) $(A\Longrightarrow B)=\left(notA\right)orB$ .Acting in this manner we obtain for the implication $(A\Longrightarrow B)$ the required relation: $I\equiv(A\Longrightarrow B)=diag\left(I_{1},I_{2},I_{3},I_{4}\right),$ (10) where $I_{1}=p_{4}\left(q_{1}+q_{3}\right)+p_{2}q_{1}$, $I_{2}=p_{3}+q_{2}\left(1-p_{3}\right)+p_{1}q_{1}+p_{4}q_{4}$, $I_{3}=p_{2}q_{3}$, $I_{4}=p_{1}\left(q_{3}+q_{4}\right)+p_{2}q_{4}$. The expression Eq. (10) for the implication of two probabilistic propositions in four- valued logic is the foundation for our following explanation of bias- belief effect.Note that here we are going to demonstrate only the simplest case of the application of the approach proposed. The detail quantitative analysis of numerous possible situations connected with the interaction between DRS and HRS will be realized by us at length in separate publication. So, let us take the proposition $B$\- (consequent of the implication) in the form: $B=diag\left(1,0,0,0\right)$, that means that consequent is both true and believable proposition. Then the expression Eq. (10) implies that matrix $\left(A\Longrightarrow B\right)$ has the form: $(A\Longrightarrow B)=diag\left(p_{2}+p_{4},p_{1}+p_{3},0,0\right),$ (11) and hence its projections in DRS (1) and HRS (2) systems are respectively : $\left(A\Longrightarrow B\right)_{1}=\begin{pmatrix}1&\\\ &0\end{pmatrix}$ , and $\left(A\Longrightarrow B\right)_{2}=\begin{pmatrix}p_{2}+p_{4}&\\\ &p_{1}+p_{3}\end{pmatrix}$ On the other hand if one choose the consequent $B$ in the form $B=diag\left(0,1,0,0\right)$ that means that consiquent $B$ is true but unbelievable proposition then according to expression $\left(10\right)$ one obtain for the implication $\left(A\Longrightarrow B\right)$ the relation: $\left(A\Longrightarrow B\right)=diag\left(0,1,0,0\right),$ (12) and hence the projections of this proposition in two cognitive systems are:$\left(A\Longrightarrow B\right)_{1}=\begin{pmatrix}1&\\\ &0\end{pmatrix}$ and $\left(A\Longrightarrow B\right)_{2}=\begin{pmatrix}0&\\\ &1\end{pmatrix}.$ Now if we make the natural assumption that after the first ( unconscious) stage of reasoning, when two cognitive systems operate jointly, at the second stage the conscious evaluation of the validity of a conclusion $V$ occurs in accordance with the simple rule: $V=aP_{t}+\left(1-a\right)P_{b},$ (13) (where $a$ $(0\leqslant a\leqslant 1)$ is certain number coefficient depending on age,intellect,training of the subject and possibly some other factors).Note that this assumption in fact coincides with similar rule which was used in the paper 4s . Now returning to the above example of interest we result in that the magnitude of the bias-belief effect $V$ can be evaluate quantitatively as $V\equiv V_{1}-V_{2}=\left(1-a\right)\left(p_{2}+p_{4}\right)$.We believe that although the value of coefficient $a$ is unknown in advance nevertheless the validity of the Eq. (13) can be explicitly verified in seria of properly organized psychological experiments with various subjects using the identical cognitive tasks . In conclusion of our study let us formulate once more the central results of the present paper: 1)We introduced the novel version of DCL with both discrete and continuous logical operations between generalized propositions . 2)We proposed the concrete interpretation of propositions in DCL as integral mental structures that include both logical and heuristic constituents. 3)We stated the specific uncertainty relation between logic rigour and heuritic grasp that reflect complementary aspects of human reasoning process. 4)We proposed phenomenological model of human reasoning based on simplified version of DCL and demonstrated that it is able to explain belief-bias effect qualitatively and possibly quantitatively as well. All these conclusions we hope to discuss more detail in our further publications. ## References * (1) Stent GS.Proc Am Philos Soc. 2004 Jun;148(2): 205-12. * (2) N. Bohr, Science, New Series, Vol. 111, No. 2873 (Jan. 20, 1950), pp. 51-54 * (3) Enz, P.; von Meyenn, Karl (editors); Schlapp, Robert (translator) Wolfgang Pauli, Writings on physics and philosophy, pp218 - 279, Berlin: Springer Verlag,(1994) * (4) Jonathan St. B. T. Evans, Thinking & Reasoning, V13, Issue 4, pp321-339, (2007) * (5) E. D. Vol, physics.soc-ph 1306.2433, (2013)	arxiv-papers	2013-09-30T07:11:05	2024-09-04T02:49:51.690984	{ "license": "Public Domain", "authors": "E. D. Vol", "submitter": "Evgenii D. Vol", "url": "https://arxiv.org/abs/1309.7740" }
1309.7742	# Emergent of Majorana Fermion mode and Dirac Equation in Cavity Quantum Electrodynamics Sujit Sarkar Poornaprajna Institute of Scientific Research, 4 Sadashivanagar, Bangalore 5600 80, India. ###### Abstract We present the results of low lying excitation of coupled optical cavity arrays. We derive the Dirac equation for this system and explain the existence of Majorana fermion mode in the system. We present quite a few analytical relations between the Rabi frequency oscillation and the atom-photon coupling strength to achieve the different physical situation of our study and also the condition for massless excitation in the system. We present several analytical relations between the Dirac spinor field, order and disorder operators for our systems. We also show that the Luttinger liquid physics is one of the intrinsic concept in our system. PACS: 42.50.Pq, 03.65.Vf, 42.50.-p Keywords: Cavity Quantum Electrodynamics , Phases: Geometric, Dynamic or Topological , Quantum Optics Introduction: The recent experimental success in engineering strong interaction between the photons and atoms in high quality micro-cavities opens up the possibility to use light matter system as quantum simulators for many body physics [1-21]. Many interesting results are coming out to understand the complicated quantum many body system. The Bose-Hubbard model, quantum spin model and the other exotic quantum phases of the quantum many body system have already been studied [3-7]. The further application of the basic principle of cavity-QED system is the circuit QED [22-26]. When a qubit (qubits) coupled to the high quality LC circuits, it presents the same physical picture with few extra achievement over the conventional cavity-QED system. A focus on the coupled cavities is one of the most potential candidate for an efficient quantum simulator due to the control of the microcavities parameters and success of fabrication of large scale cavity arrays [25-26]. This is the very brief discussion of the presence status of the cavity QED system. In the present study one of our goal is to predict the presence of Majorana fermions in our model system. Before we proceed further , we would like to describe very briefly about the appreance of Majorana fermions in quantum condensed matter system. Majorana had introduced a special kind of fermions which are their own antiparticle, i.e., the neutral particle majo ; wil . He had introduced this particle to describe neutrions. In recent years, there are several candidates of Majorana fermions in quantum condensed matter system like quantum Hall system with filling fraction $5/2$ read ; read2 . Kitaev at first found the existence of Majorana fermion mode in one dimensional model kitaev . Many research group have already been proposed the physically existence of MFs at the edge state of 1D system like electrostatic defects lines in superconductor, quasi-one dimensional superconductor and cold atom trapped in one dimension wimmer ; kop . Majorana fermions are obey non-Abelian statics both in 2D and 1D, allowing of certain gate operation required in quantum computation nayak . Due to the non-local character, the qubit built out of Majorana fermions are insensitive to local parity conserving perturbation nayak ; lut ; oreg ; potter ; jiang . The search for experimentally accessible systems that are described by a Dirac equation has received much attention in recent years nayak ; lut ; oreg ; potter ; jiang . In this research paper, we present an extensive derivation of Dirac equation and also the existence of Majorana fermions mode in an optical cavity array. We also present the analytical relation between the Rabi frequency oscillation and the atom-photon coupling strengths to mimic the transverse Ising model, Dirac equation, magnetic ordered state, quantum paramagnetic state and massless excitation. Quantum state engineering of the optical cavity array system is in the state of art due to the rapid technical development of this field [1] therefore one can achieve these quantum phases in the laboratory. The Model Hamiltonian and Majorana Fermion Modes: The Hamiltonian of our present study consists of three parts: $H~{}=~{}{H_{A}}~{}+~{}{H_{C}}~{}+~{}{H_{AC}}$ (1) The Hamiltonians are the following ${H_{A}}~{}=~{}\sum_{j=1}^{N}{{\omega}_{e}}\|e_{j}><e_{j}\|~{}+~{}{\omega}_{ab}\|b_{j}><b_{j}\|$ (2) where $j$ is the cavity index. ${\omega}_{ab}$ and ${\omega}_{e}$ are the energies of the state $\|b>$ and the excited state respectively. The energy level of state $\|a>$ is set as zero. $\|a>$ and $\|b>$ are the two stable state of a atom in the cavity and $\|e>$ is the excited state of that atom in the same cavity. The following Hamiltonian describes the photons in the cavity, ${H_{C}}~{}=~{}{{\omega}_{C}}\sum_{j=1}^{N}{{a_{j}}}^{\dagger}{a_{j}}~{}+~{}{J_{C}}\sum_{j=1}^{N}({{a_{j}}}^{\dagger}{a_{j+1}}+h.c),$ (3) where ${a_{j}}^{\dagger}({a_{j}})$ is the photon creation (annihilation) operator for the photon field in the $j$’th cavity, ${\omega}_{C}$ is the energy of photons and $J_{C}$ is the tunneling rate of photons between neighboring cavities. The interaction between the atoms and photons and also by the driving lasers are described by ${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+{g_{a}}{a_{j}})\|e_{j}><a_{j}\|+h.c]+[a\leftrightarrow b].$ (4) Here ${g_{a}}$ and ${g_{b}}$ are the couplings of the cavity mode for the transition from the energy states $\|a>$ and $\|b>$ to the excited state. ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the lasers with frequencies ${\omega}_{a}$ and ${\omega}_{b}$ respectively. The authors of Ref. hart1 ; hart2 ; sujop have derived an effective spin model by considering the following physical processes: A virtual process regarding emission and absorption of photons between the two stable states of neighboring cavity yields the resulting effective Hamiltonian as ${H_{xy}}=\sum_{j=1}^{N}B{{\sigma}_{j}}^{z}~{}+~{}\sum_{j=1}^{N}(\frac{J_{1}}{2}{{\sigma}_{j}}^{\dagger}{{\sigma}_{j+1}}^{-}~{}+~{}\frac{J_{2}}{2}{{\sigma}_{j}}^{-}{{\sigma}_{j+1}}^{-}+h.c)$ (5) When $J_{2}$ is real then this Hamiltonian reduces to the XY model. Where ${{\sigma}_{j}}^{z}=\|b_{j}><b_{j}\|~{}-~{}\|a_{j}><a_{j}\|$, ${{\sigma}_{j}}^{+}=\|b_{j}><a_{j}\|$, ${{\sigma}_{j}}^{-}=\|a_{j}><b_{j}\|$ . $\displaystyle H_{xy}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N}(B{{\sigma}_{i}}^{z}~{}+~{}{J_{1}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}+{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y})$ (6) $\displaystyle+{J_{2}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}-{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}))$ $\displaystyle=\sum_{i=1}^{N}B({{\sigma}_{i}}^{z}~{}+~{}{J_{x}}{{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}~{}+~{}{J_{y}}{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}).$ With ${J_{x}}=(J_{1}+J_{2})$ and ${J_{y}}=(J_{1}-J_{2})$. We follow the references james ; hart1 , to present the analytical expression for the different physical parameters of the system. $B=\frac{\delta_{1}}{2}-\beta$ (7) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}({\Delta}_{b}-\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}-\frac{{\|{\Omega_{b}}\|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-{\gamma_{b}}{g_{b}}^{2}-{\gamma_{1}}{g_{a}}^{2}$ (8) $\displaystyle+{\gamma_{1}}^{2}\frac{{g_{a}}^{4}}{{\Delta_{b}}}-(a\leftrightarrow b)]$ ${J_{1}}=\frac{\gamma_{2}}{4}(\frac{{\|{\Omega_{a}}\|}^{2}{g_{b}}^{2}}{{{\Delta}_{a}}^{2}}+\frac{{\|{\Omega_{b}}\|}^{2}{g_{a}}^{2}}{{{\Delta}_{b}}^{2}}),{J_{2}}=\frac{\gamma_{2}}{2}(\frac{{\Omega_{a}}{\Omega_{b}}g_{a}g_{b}}{{\Delta}_{a}{\Delta_{b}}}).$ (9) Where $\gamma_{a,b}=\frac{1}{N}\sum_{k}\frac{1}{{\omega}_{a,b}-{\omega}_{k}}$ $\gamma_{1}=\frac{1}{N}\sum_{k}\frac{1}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$ and $\gamma_{2}=\frac{1}{N}\sum_{k}\frac{e^{ik}}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$ ${\delta_{1}}={\omega}_{ab}-({\omega}_{a}-{\omega}_{b})/2$, ${\Delta}_{a}={\omega}_{e}-{\omega}_{a}$. ${\Delta}_{b}={\omega}_{e}-{\omega}_{a}-({\omega}_{ab}-{\delta_{1}})$. ${{\delta}_{a}}^{k}={\omega}_{e}-{\omega}_{k}$, ${{\delta}_{b}}^{k}={\omega}_{e}-{\omega}_{k}-({\omega}_{ab}-{\delta_{1}})$, ${\omega}_{k}={\omega}_{c}+J_{c}\sum_{k}cosk$. $g_{a}$ and $g_{b}$ are the couplings of respective transition to the cavity mode, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequency of laser with frequency $\omega_{a}$ and $\omega_{b}$. The system reduces to Ising model with transverse field at $J_{1}=J_{2}$, i.e., $J_{x}$ become $J_{1}+J_{2}$ and $J_{y}=0$. The effective Hamiltonian become the transverse Ising model which studied in the previous literature ss ; mussardo ; druff . Here our main motivation is to use some of important results of this model Hamiltonian to discuss the relevant physics of array of cavity QED system. Before we proceed further, we would like to discuss in detail the analytical relation between the different coupling constants of cavity QED system to achieve this Hamiltonian. In the microcavity array, the condition for $J_{1}=J_{2}$ achieve when ${{\Omega}_{a}}^{2}{g_{b}}^{2}{\Delta_{b}}^{2}+{{\Omega}_{b}}^{2}{g_{a}}^{2}{\Delta_{a}}^{2}=2{\Omega}_{a}{\Omega}_{b}g_{a}g_{b}{\Delta}_{a}{\Delta}_{b}.$ (10) The above condition implies that ${\Omega}_{a}={\Omega}_{b}\frac{g_{a}\Delta_{a}}{g_{b}\Delta_{b}}$. The only constraint is that ${\Delta}_{a}\neq{\Delta}_{b}$, the magnetic field diverge when ${\Delta}_{a}={\Delta}_{b}$. At the same time, ${\Omega}_{a}={\Omega}_{b}$ and $g_{a}=g_{b}$ are also not possible because this limit also leads to the condition ${\Delta}_{a}={\Delta_{b}}$. Suppose we consider, ${\Omega}_{a}=\alpha_{1}{\Omega}_{b}$, $g_{a}=\alpha_{2}g_{b}$ and ${\Delta}_{a}=\alpha_{3}{\Delta}_{b}$. These relations implies that ${\alpha_{1}}^{2}+{\alpha_{2}}^{2}{\alpha_{3}}^{2}=2{\alpha}_{1}{\alpha}_{2}{\alpha}_{3}$. ${\alpha}_{1}={\alpha}_{2}{\alpha}_{3}$, ${\alpha}_{1},{\alpha_{2}}$ and ${\alpha_{3}}$ are the numbers. These analytical relations help to implement the transverse Ising model Hamiltonian but $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ should not be equal to 1. The quantum state engineering of cavity QED is in the state of art due to the rapid progress of technological development of this field [1]. Therefore one can achieve this limit to get the desire quantum state. $H_{T}=B\sum_{j=1}^{N}({{\sigma}_{z}}(j)+\lambda{{\sigma}_{x}}(j){{\sigma}_{x}}(j+1)),$ (11) where $\lambda=\frac{J_{1}+J_{2}}{B}$. The transverse Ising model was studied widely in the literature and also exhibit a quantum phase transition between the magnetically ordered state to the quantum paramagnetic phase for $\lambda>1$ and $\lambda<1$ respectively ss ; mussardo ; druff . Now we express the condition for the magnetic order phase and quantum paramagnetic phase in terms of the physical parameters of the optical cavity QED system which gives us the relevant physics of the system. The condition for the magnetic ordered system can be expressed as $\frac{\gamma_{2}}{4}(\frac{{\Omega}_{a}{g_{b}}^{2}}{{\Delta_{a}}^{2}}+\frac{{\Omega}_{b}{g_{a}}^{2}}{{\Delta_{b}}^{2}})+\frac{\gamma_{2}}{2}(\frac{\Omega_{a}\Omega_{b}g_{a}g_{b}}{\Delta_{a}\Delta_{b}})>\omega_{ab}-\frac{\omega_{a}-\omega_{b}}{2}-2\beta$ (12) The condition for the quantum paramagnetic phase is $\frac{\gamma_{2}}{4}(\frac{{\Omega}_{a}{g_{b}}^{2}}{{\Delta_{a}}^{2}}+\frac{{\Omega}_{b}{g_{a}}^{2}}{{\Delta_{b}}^{2}})+\frac{\gamma_{2}}{2}(\frac{\Omega_{a}\Omega_{b}g_{a}g_{b}}{\Delta_{a}\Delta_{b}})<\omega_{ab}-\frac{\omega_{a}-\omega_{b}}{2}-2\beta$ (13) When the applied magnetic field is absent, the effective Ising model has two degenerate ground states. The ground states are $\|A>=\Pi_{j}{\|\rightarrow>}_{j}$, $\|B>=\Pi_{j}{\|\leftarrow>}_{j}$. For a finite magnetic field but less than $J_{1}+J_{2}$, the system has a tendency to flip the pseudo spin. At that phase one can write down the true eigen state, $\|{\psi}_{A}>=\frac{1}{\sqrt{2}}(\|A>+\|B>)$ , $\|{\psi}_{B}>=\frac{1}{\sqrt{2}}(\|A>-\|B>)$. Now our main intention is to recast this spin model in spinless fermion model through Jordon-Wigner transformation which relate the spin operators to the spinless fermion operators. We use the following relation: ${\sigma}_{z}=2c^{\dagger}(j)c(j)-1$, ${\sigma}_{x}(j){\sigma}_{x}(j+1)=(c^{\dagger}(n)-c(n))(c^{\dagger}(n+1)-c(n+1))$. One can write the Hamiltonian after the Jordon-Wigner transformation as $H=2\sum_{j=1}^{N}{c}^{\dagger}(j)c(j)+\lambda(c^{\dagger}(j)-c(j))(c^{\dagger}(j+1)-c(j+1))$ (14) We solve this Hamiltonian, to get the energy spectrum by taking the Fourier transform. $c(j)=\frac{1}{\sqrt{N}}\sum_{k}e^{-ika}$, $c^{\dagger}(j)=\frac{1}{\sqrt{N}}\sum_{k}e^{ika}$. Where $c_{k}$ and ${c_{k}}^{\dagger}$ are the fermionic annihilation and creation operator in momentum space. The Hamiltonian reduce to $\displaystyle H$ $\displaystyle=$ $\displaystyle 2\sum_{k>0}(1+\lambda cosk)({c_{k}}^{\dagger}{c_{k}}+{c_{-k}}^{\dagger}{c_{-k}})$ (15) $\displaystyle+2i\lambda\sum_{k>0}sink({c_{k}}^{\dagger}{c_{-k}}^{\dagger}+{c_{k}}{c_{-k}})$ Now our main task is to express the Hamiltonian in the diagonalized form. We follow the Bogoliubov transformation. $\eta_{k}=\alpha_{k}c_{k}+i{\beta}_{k}{c_{-k}}^{\dagger}$ and $\eta_{-k}=\alpha_{k}c_{-k}-i{\beta}_{k}{c_{k}}^{\dagger}$, $k>0$. The operator ${\eta}_{k}$ and ${{\eta}_{k}}^{\dagger}$ are the fermionic operators. We use the following relations, $\\{{\eta}_{k},{{\eta}_{p}}^{\dagger}\\}=\delta_{k,p}$, $\\{{\eta}_{k},{{\eta}_{p}}\\}=0$, $\\{{{\eta}_{k}}^{\dagger},{{\eta}_{p}}^{\dagger}\\}=0$. This relation implies, ${{\alpha}_{k}}^{2}+{{\beta}_{k}}^{2}=1$. One can also revert the relation between $c_{k}$ and ${\eta}_{k}$. We also parameterize $\alpha_{k}=cos\theta_{k}$ and $\beta_{k}=sin\theta_{k}$. One can express the transformed Hamiltonian in two parts $H=H_{A}+H_{B}$ (16) $\displaystyle H_{A}$ $\displaystyle=$ $\displaystyle\sum_{k>0}[-2(1+\lambda cosk)({{\alpha}_{k}}^{2}-{{\beta}_{k}}^{2})+4\lambda sink{\alpha}_{k}{\beta}_{k}]$ (17) $\displaystyle({{\eta}_{k}}^{\dagger}{\eta_{k}}{{\eta}_{-k}}^{\dagger}{\eta_{-k}})$ $\displaystyle H_{B}$ $\displaystyle=$ $\displaystyle\sum_{k>0}[4i(1+\lambda cosk){\alpha_{k}}{\beta_{k}}+2i\lambda sink({{\alpha}_{k}}^{2}-{\beta_{k}}^{2})]$ (18) $\displaystyle({{\eta}_{k}}^{\dagger}{\eta_{-k}}^{\dagger}{{\eta}_{k}}{\eta_{-k}})$ To express this Hamiltonian in the diagonal form, we find the following relation $4(B+\frac{\gamma_{2}{\Omega_{b}}^{2}{g_{a}}^{2}}{{\Delta_{b}}^{2}}cosk){\alpha_{k}}{\beta_{k}}+2\frac{\gamma_{2}{\Omega_{b}}^{2}{g_{a}}^{2}}{{\Delta_{b}}^{2}}sink({\alpha_{k}}^{2}-{\beta_{k}}^{2})=0$. Finally this gives the condition, $tan2{\theta_{k}}=\frac{2\gamma_{2}{\Omega_{b}}^{2}{g_{a}}^{2}}{2{\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}cosk-({\delta_{1}}-2\beta){\Delta_{b}}^{2}}$ (19) $2{\alpha}_{k}{\beta}_{k}=sin2{\theta_{k}}$, ${\alpha_{k}}^{2}-{\beta_{k}}^{2}=cos2\theta_{k}$. Now we analysis the spectrum: $H=2\sum_{k}\Omega_{K}{\eta_{k}}^{\dagger}{\eta_{k}}$ (20) We can express this energy spectrum in terms of Rabi frequency oscillation, atom-photon coupling strength, $\Omega_{k}=\frac{2}{{\delta_{1}}/2-\beta}\sqrt{{({\delta_{1}}/2-\beta)}^{2}+{\gamma_{2}}^{2}\frac{{{\Omega_{b}}}^{4}{g_{a}}^{4}}{{\Delta_{a}}^{4}}+2({\delta_{1}}/2-\beta){\gamma_{2}}\frac{{{\Omega_{b}}}^{2}{g_{a}}^{2}}{{\Delta_{b}}^{2}}}$ (21) The minimum occurs at $k=\pm\pi$, $\Omega_{k=\pm\pi}=2\|1-\lambda\|=2\|1-\frac{2\gamma_{2}}{\delta_{1}-2\beta}\frac{{{\Omega}_{b}}^{2}{g_{a}}^{2}}{{\Delta_{b}}^{2}}\|$. We are interested in the continuum limit and also restore the lattice spacing $\alpha$ and measure the momentum w.r.t the minimum value, $k=\pi+k^{{}^{\prime}}\alpha$ . The energy expression which contains the physical dimension of energy is $E(k^{{}^{\prime}})=\frac{\Omega_{k}}{2\alpha}$. In this limit, $E(k^{{}^{\prime}})=\sqrt{{(\frac{1-\lambda}{\alpha})}^{2}+\lambda{k^{{}^{\prime}}}^{2}}$. If $\lambda$ is close to a critical value $\lambda\sim 1$, we then have the dispersion of a particle with mass $m=\frac{1-\lambda}{\alpha}$. If $\lambda=1$, it becomes the massless particle $E(k^{{}^{\prime}})\sim k^{{}^{\prime}}$. In the cavity QED system, we can express the condition for massless excitation of the system as $({{\delta_{1}}/2-\beta}){\Delta_{b}}^{2}={\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}$. In terms of fields, $\eta(a)=\frac{1}{\sqrt{N}}\sum_{k}e^{ika}{\eta}_{k}$. ${\eta(a)}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{k}e^{-ika}{{\eta}_{k}}^{\dagger}$. ${\chi}_{1}(a)=(1/2)(\eta(a)+{\eta(a)}^{\dagger})$, ${\chi}_{2}(a)=(1/2i)(\eta(a)-{\eta(a)}^{\dagger})$ ${{\chi}_{1}(a)}^{\dagger}={\chi}_{1}(a)$, ${{\chi}_{2}(a)}^{\dagger}={\chi}_{2}(a)$. $\\{{\chi_{1}}(x1),{\chi_{2}}(x2)\\}={\delta}_{x1,x2}{\delta}_{1,2}$. Therefore, ${\chi}_{1}$ and ${\chi}_{2}$ are satisfying the all properties of neutral fermionic fields what Majorana proposed. The authors of choi2 have investigated the low lying excitation of one dimensional array of circuit QED (cktQED) with each cktQED being in the ultra strong coupling regime and they have found the Majorana bound state. But the starting Hamiltonian of our system is completely different. Now we calculate the energy density of the system using Eq. 21. We would like to integrate the dispersion spectrum ${\Omega}_{k}$ to get the energy density. The analytical expression for energy density is ${\epsilon}_{0}=\frac{2}{\pi}~{}(1+\frac{2\gamma_{2}}{\delta_{1}-2\beta}\frac{{{\Omega}_{b}}^{2}{g_{a}}^{2}}{{\Delta_{b}}^{2}})~{}E(\frac{\pi}{2},\sqrt{1-{\gamma}^{2}})$ (22) Where ${\gamma}=\|\frac{(\delta_{1}-2\beta){\Delta_{b}}^{2}-2{\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}}{(\delta_{1}-2\beta){\Delta_{b}}^{2}+2{\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}}\|$, $E(\frac{\pi}{2},\sqrt{1-{\gamma}^{2}})$ is the complete Elliptic integral of 2nd kind. After a little bit of calculation, we obtain the analytical expression in the asympototic limit for energy density. $\displaystyle{\epsilon}_{0}$ $\displaystyle=$ $\displaystyle 1+1/2~{}(ln\|4\frac{(\delta_{1}-2\beta){\Delta_{b}}^{2}+2{\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}}{(\delta_{1}-2\beta){\Delta_{b}}^{2}-2{\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}}\|-1/2){\gamma}^{2}$ (23) $\displaystyle+3/16(ln\|4\frac{(\delta_{1}-2\beta){\Delta_{b}}^{2}+2{\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}}{(\delta_{1}-2\beta){\Delta_{b}}^{2}-2{\gamma_{2}}{\Omega_{b}}^{2}{g_{a}}^{2}}\|-13/12){\gamma}^{4}$ Derivation of Dirac Equation and Condition of Massless Excitations: During the derivation of Dirac equation we rotate the y-axis of the spin basis through $\pi/2$. Through this rotation z-axis become x-axis and x-axis become -z-axis. The physics of the system remain the same. The starting Hamiltonian of our system now becomes $H=\sum_{n}[\lambda s_{z}(n)s_{z}(n+1)+s_{x}(n)].$ (24) We recast the Hamiltonian in the following form because it will help us to use the order and disorder operator directly to the derivation of the equation of motion and finally the Dirac equation. Here we present an extensive derivation of Dirac equation for our model system. Kramers-Wannier symmetry for two dimensional Ising model reveals for the case of one-dimensional quantum Ising chain through the dual lattice to the original one (site index n). Here we introduce the order and disorder operator following mussardo ; druff . These operators are defining the sites of the dual lattice, i.e., we define the operator between the nearest-neighbor site of the original lattice. The analytical relation between the Pauli operators and $\mu$ operators are the following: ${{\mu}_{z}}^{2}=1={{\mu}_{x}}^{2},$ (25) ${{\mu}_{z}}(n-1/2){{\mu}_{z}}(n+1/2)={\sigma}_{x}(n).$ (26) ${{\mu}_{x}}(n+1/2)={\sigma}_{z}(n){\sigma}_{z}(n+1),$ (27) ${{\mu}_{z}}(n+1/2)=\Pi_{j=1}^{n}{\sigma_{x}}(j).$ (28) ${{\sigma_{z}}}(n)=\Pi_{j=0}^{n-1}{\mu_{x}}(j+1/2),$ (29) $[{\mu}_{x}(n+1/2),{\mu}_{z}(n^{{}^{\prime}}+1/2)]=2\delta_{n,n^{{}^{\prime}}}$ (30) $[{\mu}_{z}(n+1/2),{\mu}_{z}(n^{{}^{\prime}}+1/2)]=0,$ (31) $[{\mu}_{z}(n+1/2),{\sigma}_{x}(n^{{}^{\prime}})]=0$ (32) The operator ${\mu}_{z}(n+1/2)$ acting on the original spin of the lattice makes a spin flip of all those spin placed on the left hand side of spin at the site n. Therefore ${\mu}_{z}(n+1/2)$ is a kink operator, it introduce the disorder in the system. It is very clear from the above analytical relation of the operators that ${\mu}_{x}$ is related with the allinegment of the spin operator. Here we define the Dirac spinor, $\chi_{1}(n)=\sigma_{z}(n)\mu_{z}(n+1/2)$ and $\chi_{2}(n)=\sigma_{z}(n)\mu_{z}(n-1/2)$. Now our main task is to find the equation of motion for the operators, ${\sigma}_{3}(n)$ and ${\mu}_{3}(n)$ which help us to build the Dirac equation. The equation of motion for the ${\sigma}_{z}(n)$ is the following: $\frac{\partial\sigma_{z}(n)}{\partial\tau}=[H,\sigma_{z}(n)]={\sigma_{x}(n)\sigma_{z}(n)}$ (33) The equation of motion for $\mu_{z}(n+1/2)$ is the following: $\displaystyle\frac{\partial\mu_{z}(n+1/2)}{\partial\tau}$ $\displaystyle=$ $\displaystyle\lambda{\mu_{x}(n+1/2)\mu_{z}(n+1/2)}$ (34) $\displaystyle=\lambda\sigma_{z}(n)\sigma_{z}(n+1/2)\mu_{z}(n+1/2)$ Now we use the properties of the $\sigma$ and $\mu$ operators to derive the equation of motion for the Majorana fields $\chi_{1}(n)$ and $\chi_{2}(n)$. $\frac{\partial\chi_{1}(n)}{d\tau}=\frac{\partial\sigma_{z}(n)}{\partial\tau}\mu_{z}(n+1/2)+\sigma_{z}(n)\frac{\partial\mu_{z}(n)}{\partial\tau}.$ (36) $\frac{\partial\chi_{1}(n)}{d\tau}=\sigma_{x}(n)\sigma_{z}(n)\mu_{z}(n+1/2)+\lambda\sigma_{z}(n)\sigma_{z}(n)\sigma_{z}(n+1)\mu_{z}(n+1/2).$ (37) $\displaystyle\frac{\partial\chi_{1}(n)}{d\tau}$ $\displaystyle=$ $\displaystyle-\sigma_{z}(n)\mu_{z}(n-1/2)\mu_{z}(n+1/2)\mu_{z}(n+1/2)$ (38) $\displaystyle+\lambda\sigma_{z}(n)\sigma_{z}(n)\sigma_{z}(n+1)\mu_{z}(n+1/2).$ $\frac{\partial\chi_{1}(n)}{d\tau}=-\chi_{2}(n)+\lambda\chi_{2}(n+1).$ (39) Now the equations of motion for $\chi_{2}(n)$ are $\displaystyle\frac{\partial\chi_{2}(n)}{d\tau}$ $\displaystyle=$ $\displaystyle\frac{\partial\sigma_{z}(n)}{\partial\tau}\mu_{z}(n-1/2)$ (40) $\displaystyle+\sigma_{z}(n)\frac{\partial\mu_{z}(n-1/2)}{\partial\tau}$ $\displaystyle\frac{\partial\chi_{2}(n)}{d\tau}$ $\displaystyle=$ $\displaystyle\sigma_{x}(n)\sigma_{z}(n)\mu_{z}(n-1/2)$ (41) $\displaystyle+\lambda\sigma_{z}(n)\sigma_{z}(n-1)\sigma_{z}(n)\mu_{z}(n-1/2)$ $\displaystyle\frac{\partial\chi_{2}(n)}{d\tau}$ $\displaystyle=$ $\displaystyle\mu_{z}(n-1/2)\mu_{z}(n+1/2)\sigma_{z}(n)\mu_{z}(n-1/2)$ (42) $\displaystyle+\lambda\sigma_{z}(n-1)\mu_{z}(n-1/2).$ After a little bit of calculations and using the relation between the disorder operators (Eq. 23-30), we finally arrive the equation of motion of $\chi_{2}(n)$ as, $\frac{\partial\chi_{2}(n)}{d\tau}=-\chi_{1}(n)+\lambda\chi_{1}(n-1).$ (43) These two fields, $\chi_{1}(n)$ and $\chi_{2}(n)$ satisfy the following relations, $\\{\chi_{1}(n1),\chi_{2}(n2)\\}=2\delta_{n1,n2}$. One can write down the above equation in the following compact form, $({\gamma}^{0}\frac{\partial}{\partial t}+{\gamma}^{3}\frac{\partial}{\partial r}+m)\chi=0.$ (44) where ${\chi}^{\dagger}=(\chi_{1},\chi_{2})$ and $m=\frac{1-\lambda}{\alpha}$, ${\gamma}^{0}=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),{\gamma}^{3}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$ . Now we prove the presence of Luttinger liquid physics is intrinsic to the optical cavity array system by comparing the analytical relation of Majorana fermion operators with the order and disorder operator with the free Dirac field in Abelian bosonization theory. Here we express the analytical relation between the Majorana operators and the disorder and Pauli operators: ${\chi}_{1}(n)={\sigma}_{z}(n){\mu}_{z}(n+1/2)=-{\mu}_{z}(n+1/2){\sigma}_{z}(n)$ (45) ${\chi}_{2}(n)={\sigma}_{z}(n){\mu}_{z}(n-1/2)={\mu}_{z}(n-1/2){\sigma}_{z}(n)$ (46) ${\sigma}_{z}(n){\chi}_{2}(n)={\mu}_{z}(n-1/2)={\chi}_{2}(n){\sigma}_{z}(n)$ (47) ${\sigma}_{z}(n){\chi}_{1}(n)={\mu}_{z}(n+1/2)=-{\chi}_{1}(n){\sigma}_{z}(n)$ (48) ${\sigma}_{z}(n)={\mu}_{z}(n-1/2){\chi}_{2}(n)={\chi}_{2}(n){\mu}_{z}(n-1/2)$ (49) ${\mu}_{z}(n+1/2){\chi}_{1}(n)=-{\chi}_{1}(n){\mu}_{z}(n+1/2)={\sigma}_{z}(n)$ (50) The above relations can be extended to account for arbitrary space separation between different operators. Then one obtains the following sets of commutation relations. ${\sigma}_{z}(x1){\mu}_{z}(x2)={\mu}_{z}(x2){\sigma}_{z}(x1)sign(x1-x2)$ (51) ${\sigma}_{z}(x1){\chi}(x2)={\chi}(x2){\sigma}_{z}(x1)sign(x1-x2)$ (52) ${\mu}_{z}(x1){\chi}(x2)=-{\chi}(x2){\sigma}_{z}(x1)sign(x1-x2)$ (53) It is very clear from the above analytical relations that ${{\chi}_{1}}^{\dagger}={\chi}_{1}$ and ${{\chi}_{2}}^{\dagger}={{\chi}_{2}}$. The above relation has similarity with the free Dirac field in Abalian bosonization theory, where Dirac field operator is a local product of two phase exponential depending on the scalar field and its dual gia ; gogo , as one study the Luttinger liquid physics in Abelian bosonization theory. Therefore the Luttinger liquid physics is the intrinsic to the optical microcavity system. Conclusions We have presented an extensive derivation of Dirac equation and the existence of Majorana fermion modes for the optical cavity array with the relation between Rabi frequency oscillation and the atom photon coupling strength. We have presented the condition for massless excitation. We have also presented several analytical relations between the Majorana field, order and disorder operator. Acknowledgement: The author would like to acknowledge the discussions with Prof. S. Girvin during the international workshop/school on Dirac Materials and Chandrashekar lecture at ICTS (December, 2012) and the library of Raman Research Institute (Mr. Manjunath). Finally, the author would like to thank Dr. P. K. Mukherjee for reading the manuscript carefully. ## References * (1) * (2) G. Agarwal, ”Quantum Optics”, (Cambridge University Press, Delhi 2013). * (3) Greentree. 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Giamarchi, ”Quantum Physics in One Dimension”. * (43) A.O. Gogolin, A. A. Nersesyan and A. M. Tsvelik, ”Bosonization and Strongly Correlated Systems”, Cambridge University Press (New York, 1998).	arxiv-papers	2013-09-30T07:28:42	2024-09-04T02:49:51.698003	{ "license": "Public Domain", "authors": "Sujit Sarkar", "submitter": "Sujit Sarkar", "url": "https://arxiv.org/abs/1309.7742" }
1309.7777	# Quantum phase transition of light in coupled optical cavity arrays: A renormalization group study Sujit Sarkar Poornaprajna Institute of Scientific Research, 4 Sadashivanagar, Bangalore 5600 80, India. ###### Abstract We study the quantum phase transition of light of a system when atom trapped in microcavities and interacting through the exchange of virtual photons. We predict the quantum phase transition between the photonic Coulomb blocked induce insulating phase and anisotropic exchange induced photonic superfluid phase in the system due to the existence of two Rabi frequency oscillations. The renormalization group equation shows explicitly that for this system there is no self-duality. The system also shows two Berezinskii-Kosterlitz-Thouless (BKT) transitions for the different physical situation of the system. The presence of single Rabi frequency oscillation in the system leads to the BKT transition where system shows the quantum phase transition from photonic metallic state to the Coulomb blocked induced insulating phase. For the other BKT transition when the z-component of exchange interaction is absent, the system shows the transition from the photonic metallic state to the photonic superfluid phase. We also predict the commensurate to incommensurate transition under the laser field detuning. Pacs: 42.50.Dv, 42.50.Pq, 03.67.Bg, 75.10.Jm Keywords: Cavity Quantum Electrodynamics, Renormalization Group Theory, Quantum Optics and Quantum Spin Model Introduction: The recent experimental success in engineering strong interaction between the photons and atoms in high quality micro-cavities opens up the possibility to use light matter system as quantum simulators for many body physics [1-18]. The authors of Ref. (hart1 ,hart2 , suj2 ) have shown that effective spin lattice can be generated with individual atom in the microcavities that are coupled to each other via exchange of virtual photons. The two states of spin polarization are represented by the two long lived atomic levels in the system. A Many body Hamiltonians can be created and probed in coupled cavity arrays. In our previous study, we have explained explicitly the basic physics of the formation of micro-optical cavity suj2 . Atoms in the cavity are used for detection and also for generation of interaction between photons in the same cavity. This artificial system can act as a quantum simulator [1-6]. This micro-cavity system shows the different quantum phases and quantum phase transition (QPT) of photonic states of the system. QPT associate with the fundamental changes that occurs in the macroscopic nature of the matter at zero temperature due to the variation of some external parameter. Quantum phase transitions are characterized by the drastic change in the ground state properties of the system driven by the quantum fluctuations subir ; indrani . In this paper, we derive the renormalization group equation for the continuum field theoretical study of the model Hamiltonian. We predict with the existence of photonic metallic state, the anisotropic exchange induces the photonic superfluid phase and photon blocked induces the insulating phase. The transition from the photonic superfluid phase to the photon blocked induced insulating phase is the second order quantum phase transition. Whereas the transition from photonic metallic state to the photonic insulating state is the Berezinski-Kosterlitz-Thouless (BKT) transition subir ; gia ; druf where the anisotropic exchange interaction is absent. The other BKT transition occurs when we consider the transition from photonic metallic state to the photonic superfluid state when the z-component interaction is absent. The quantum state of engineering of the cavity QED system is in the state of art due to the rapid technological development in this field [1]. Therefore one can achieve the all states through the proper manipulation of laser frequencies, detuning field, Rabi frequency oscillations. In our study, the photonic metallic state is the Luttinger liquid phase of the microcavities array system. The photonic superfluid phase is originated from the anisotropic exchange interactions which discuss in Ref.jor . To the best of our knowledge, the explicit study of the different quantum phases through the RG equations and the proposal of photonic metallic state is the first in the literature for the Cavity QED system. Now we discuss in detail about the photon blocked induce insulating phase. The photon blocked is a dressed state of strongly coupled atom-cavity system. In this phase the inhivitation of resonant absorption of a second photon if one photon is already resonantly absorbed. This phenomena was first observed by Birnbaum $et~{}al.$ birn . In this studies they have predicted the photon blocked in the single atom in the cavity birn . Here we predict this phenomena for the array of cavities, where the atoms in different cavities interacting through virtual photon emission/absorption process. We will see after the continuum field theoretical study that our model Hamiltonian contains two strongly relevant and mutually nonlocal perturbation over the Gaussian (critical) theory. In such a situation the strong coupling fixed point is usually determined by the most relevant perturbation whose amplitude grows up according to its Gaussian scaling dimensions and it is not much affected by the less relevant coupling terms. However, this is not the general rule if the two operators exclude each other, i.e., if the field configurations which minimize one perturbation term do not minimize the other. In this case interplay between the two competing relevant operators can produce a novel quantum phase transition through a critical point or a critical line. Therefore, we would like to study the RG equation to interpret the quantum phases of the system. Model Setup and Motivation We would like to solve the quantum phases and phase boundaries of coupled cavity arrays by establishing the quantum spin chain model of the system hart1 ; hart2 ; suj2 . At first we discuss the relation between the spin operators and the atom-photon system. Our starting point is the Janes-Cummings Hamiltonian, $H~{}=g({\sigma}a^{\dagger}+h.c)$. Cavity mode represent by the bosonic operators ($a,{a}^{\dagger}$ ) and atomic mode represent by the fermionic operator (${\sigma},{\sigma}^{\dagger}$). Where $\sigma$ is the Pauli matrix which transform one excitation from the radiation field to the atomic field. Therefore, one can write ${\sigma}^{\dagger}=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right)$ , similarly one can write for ${\sigma}^{-}$ with $<1\|=(1,0)$ and $<0\|=(0,1)$. The excitation in this system only transfer between atom and photon in the cavity. For a fixed number ($n$) of total excitation, one can express the manifold: ${H_{n}}=\\{\|0,n>,\|1,n-1>\\}$ provided $n\geq 1$. Here $\|0,n>$ and $\|1,n-1>$ are represent atom in the ground state with $n$ photon and excited states of the atom with ($n-1$) photon respectively. We consider the initial state $\|e,n-1>$, we obtain the state $\|g,n>$ by the following operation, $\|g,n>={\sigma}a^{+}\|e,n-1>$. Therefore, we may write the following relation based on the conservation of the number of excitation. ${\sigma}^{\dagger}a\|e,n-1>=~{}0~{}={\sigma}a^{\dagger}\|g,n>$ and ${\sigma}^{\dagger}a\|g,n>=\|e,n-1>$. These relations are nothing but the properties of spin operators acting on the spinors in the z basis. Micro-cavities of a photonic crystal are coupled through the exchange of photons. Each cavity consists of one atom with three levels in the energy spectrum, two of them are long lived and represent two spin states of the system and the other represent excited states (fig. 1 and fig. 2 of Ref. hart1 ; hart2 ). Externally applied laser and cavity modes couple to each atom of the cavity. It may induce the Raman transition between these two long lived energy levels. Under a suitable detuning between the laser and the cavity modes, virtual photons are created in the cavity which mediate interactions with another atom in a neighboring cavity. One can eliminate the excited states by choosing the appropriate detuning between the applied laser and cavity modes. Then one can achieve only two states per atom in the long lived state and the system can be described by a spin-1/2 Hamiltonian hart1 ; hart2 . The Hamiltonian of the system consists of three parts: $H~{}=~{}{H_{A}}~{}+~{}{H_{C}}~{}+~{}{H_{AC}}$ (1) Hamiltonians are the following ${H_{A}}~{}=~{}\sum_{j=1}^{N}{{\omega}_{e}}\|e_{j}><e_{j}\|~{}+~{}{\omega}_{ab}\|b_{j}><b_{j}\|$ (2) where $j$ is the cavity index. ${\omega}_{ab}$ and ${\omega}_{e}$ are the energies of the state $\|b>$ and the excited state respectively. The energy level of state $\|a>$ is set as zero. $\|a>$ and $\|b>$ are the two stable state of a atom in the cavity and $\|e>$ is the excited state of that atom in the same cavity. The following Hamiltonian describes photons in the cavity, ${H_{C}}~{}=~{}{{\omega}_{C}}\sum_{j=1}^{N}{{a_{j}}}^{\dagger}{a_{j}}~{}+~{}{J_{C}}\sum_{j=1}^{N}({{a_{j}}}^{\dagger}{a_{j+1}}+h.c),$ (3) where ${a_{j}}^{\dagger}({a_{j}})$ is the photon creation (annihilation) operator for the photon field in the $j$’th cavity, ${\omega}_{C}$ is the energy of photons and $J_{C}$ is the tunneling rate of photons between neighboring cavities. Interaction between the atoms and photons and also by the driving lasers are described by ${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+{g_{a}}{a_{j}})\|e_{j}><a_{j}\|+h.c]+[a\leftrightarrow b].$ (4) Here ${g_{a}}$ and ${g_{b}}$ are the couplings of the cavity mode for the transition from the energy states $\|a>$ and $\|b>$ to the excited state. ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the lasers with frequencies ${\omega}_{a}$ and ${\omega}_{b}$ respectively. The authors of Ref. hart1 ; hart2 have derived an effective spin model by considering the following physical processes: A virtual process regarding emission and absorption of photons between the two stable states of neghbior cavity yields the resulting effective Hamiltonian as ${H_{xy}}=\sum_{j=1}^{N}B{{\sigma}_{j}}^{z}~{}+~{}\sum_{j=1}^{N}(\frac{J_{1}}{2}{{\sigma}_{j}}^{\dagger}{{\sigma}_{j+1}}^{-}~{}+~{}\frac{J_{2}}{2}{{\sigma}_{j}}^{-}{{\sigma}_{j+1}}^{-}+h.c)$ (5) When $J_{2}$ is real then this Hamiltonian reduces to the XY model. Where ${{\sigma}_{j}}^{z}=\|b_{j}><b_{j}\|~{}-~{}\|a_{j}><a_{j}\|$, ${{\sigma}_{j}}^{+}=\|b_{j}><a_{j}\|$, ${{\sigma}_{j}}^{-}=\|a_{j}><b_{j}\|$ $\displaystyle H_{xy}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N}(B{{\sigma}_{i}}^{z}~{}+~{}{J_{1}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}+{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y})$ (6) $\displaystyle+{J_{2}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}-{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}))$ $\displaystyle=\sum_{i=1}^{N}B({{\sigma}_{i}}^{z}~{}+~{}{J_{x}}{{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}~{}+~{}{J_{y}}{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}).$ with ${J_{x}}=(J_{1}+J_{2})$ and ${J_{y}}=(J_{1}-J_{2})$. Here we present the analytical expression of $B$, $J_{1}$ and $J_{2}$ in terms of different physical parameters of the system. We follow the references hart1 ; james . $B=\frac{\delta_{1}}{2}-\beta$ (7) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}({\Delta}_{b}-\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}-$ (8) $\displaystyle\frac{{\|{\Omega_{b}}\|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-{\gamma_{b}}{g_{b}}^{2}-{\gamma_{1}}{g_{a}}^{2}+{\gamma_{1}}^{2}\frac{{g_{a}}^{4}}{{\Delta_{b}}}-(a\leftrightarrow b)]$ ${J_{1}}=\frac{\gamma_{2}}{4}(\frac{{\|{\Omega_{a}}\|}^{2}{g_{b}}^{2}}{{{\Delta}_{a}}^{2}}+\frac{{\|{\Omega_{b}}\|}^{2}{g_{a}}^{2}}{{{\Delta}_{b}}^{2}})$ (9) ${J_{2}}=\frac{\gamma_{2}}{2}(\frac{{\Omega_{a}}{\Omega_{b}}g_{a}g_{b}}{{\Delta}_{a}{\Delta_{b}}}).$ (10) where $\gamma_{a,b}=\frac{1}{N}\sum_{k}\frac{1}{{\omega}_{a,b}-{\omega}_{k}}$ $\gamma_{1}=\frac{1}{N}\sum_{k}\frac{1}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$ and $\gamma_{2}=\frac{1}{N}\sum_{k}\frac{e^{ik}}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$ ${\delta_{1}}={\omega}_{ab}-({\omega}_{a}-{\omega}_{b})/2$, ${\Delta}_{a}={\omega}_{e}-{\omega}_{a}$. ${\Delta}_{b}={\omega}_{e}-{\omega}_{a}-({\omega}_{ab}-{\delta_{1}})$. ${{\delta}_{a}}^{k}={\omega}_{e}-{\omega}_{k}$, ${{\delta}_{b}}^{k}={\omega}_{e}-{\omega}_{k}-({\omega}_{ab}-{\delta_{1}})$, $g_{a}$ and $g_{b}$ are the couplings of respective transition to the cavity mode, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequency of laser with frequency $\omega_{a}$ and $\omega_{b}$. Here we discuss very briefly about an effective $z$-component of interactions (${{\sigma}_{i}}^{z}{{\sigma}_{i+1}}^{z}$) in such a system. The authors of Ref.hart1 ; hart2 have proposed the same atomic level configuration but having only one laser of frequency ${\omega}$ that mediates the atom-atom coupling through virtual photons. Another laser field with frequency $\nu$ is used to tune the effective magnetic field. In this case the Hamiltonian ${H_{AC}}$ changes but the Hamiltonians $H_{A}$ and $H_{C}$ are the same. $\displaystyle{H_{AC}}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{N}[(\frac{{\Omega}}{2}e^{-i{{\omega}}t}+\frac{{\Lambda}}{2}e^{-i{{\nu}_{a}}t}{g_{a}}{a_{j}})\|e_{j}><a_{j}\|+h.c]$ (11) $\displaystyle+[a\leftrightarrow b].$ Here, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the driving laser with frequency ${\omega}$ on transition $\|a>\rightarrow\|e>$ , $\|b>\rightarrow\|e>$, whereas ${\Lambda}_{a}$ and ${\Lambda}_{b}$ are the driving laser with frequency ${\nu}$ on transition $\|a>\rightarrow\|e>$ , $\|b>\rightarrow\|e>$. One can eliminate adiabatically the excited atomic levels and photons by considering the interaction picture with respect to $H_{0}=H_{A}~{}+~{}H_{C}$ [6,7]. They have considered the detuning parameter in such a way that the Raman transitions between two level are suppressed and also chosen the parameter in such a way that the dominant two-photon processes are those that involve one laser photon and one cavity photon but the atom makes no transition between levels a and b. Whenever two atoms exchange a virtual photon both of them experience a Stark shift and play the role of an effective ${{\sigma}^{z}}{{\sigma}^{z}}$ interaction hart1 ; hart2 ; suj2 . Then the effective Hamiltonian reduces to ${H_{zz}}~{}=~{}\sum_{j=1}^{N}({B_{z}}{{\sigma}_{j}}^{z}~{}+~{}{J_{z}}{{\sigma}_{j}}^{z}{{\sigma}_{j+1}}^{z})$ (12) These two parameters can be tuned independently by varying the laser frequencies. Finally, they have obtained an effective model by combining Hamiltonians $H_{xy}$ and $H_{zz}$ by using Suzuki-Trotter formalism. The effective Hamiltonian simulated by this procedure is $H_{spin}~{}=~{}\sum_{j=1}^{N}(B_{tot}{{\sigma}_{j}}^{z}~{}+~{}\sum_{{\alpha}=x,y,z}J_{\alpha}{{\sigma}_{j}}^{\alpha}{{\sigma}_{j+1}}^{\alpha})$ (13) where $B_{tot}=B+{B_{z}}$. It has been shown in Ref. hart2 that $J_{y}$ is less than $J_{x}$. From the analytical expressions of $J_{x}$ and $J_{y}$, it is clear that the magnitudes of ${J_{1}}$ and $J_{2}$ are different. The result of numerical simulations trigger us also to define a model to study the quantum phases of this system. In the next section, we present the RG study of this model Hamiltonian to extract quantum phases and transitions between them. Here we present the analytical expression of $B_{tot}$, $J_{z}$ in terms of different physical parameters of the system. We follow the references hart1 ; james ; suj2 . ${J_{z}}={\gamma}_{2}{\|\frac{{{\Omega}_{b}}^{}g_{b}}{4\Delta_{b}}-\frac{{{\Omega}_{a}}^{}g_{a}}{4\Delta_{a}}\|}^{2}$ $B_{tot}=-\frac{1}{2}[\frac{{\|{\Lambda}_{b}\|}^{2}}{16{\tilde{\Delta_{b}}}^{2}}(4\tilde{\Delta_{b}}-\frac{{\|{\Lambda_{a}}\|}^{2}}{4({\tilde{\Delta}_{a}}-{\tilde{\Delta}_{b}})}-\frac{{\|{\Lambda_{b}}\|}^{2}}{{\tilde{\Delta}_{b}}}-{\beta_{2}})-{\beta_{3}}].$ ${\beta_{2}}=\sum_{j=a,b}\frac{{\|{\Omega_{j}}\|}^{2}}{4({\Delta}_{j}-\tilde{{\Delta}_{b}})}.4\tilde{\gamma_{jb}}{g_{j}}^{2}$ $\displaystyle\beta_{3}$ $\displaystyle=$ $\displaystyle[\frac{{\|{\Omega_{b}}\|}^{2}}{16{{\Delta}_{b}}^{2}}(4{\Delta}_{b}-\frac{{\|{\Omega_{a}}\|}^{2}}{4{\Delta}_{b}}-$ (14) $\displaystyle\frac{{\|{\Omega_{b}}\|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-\frac{{\|{\Omega_{b}}\|}^{2}}{\Delta_{b}}-\sum_{j=a,b}\frac{{\|{\Lambda_{j}}\|}^{2}}{4({\Delta}_{j}-\tilde{{\Delta}_{b}})}.4{\gamma_{jb}}{g_{j}}^{2})+{\gamma_{bb}}^{2}\frac{{g_{b}}^{4}}{{\Delta_{b}}}-(a\leftrightarrow b)]$ Here ${\gamma}_{1}=\frac{1}{N}\sum_{k}\frac{1}{\omega-{\omega}_{k}}$, ${\gamma}_{2}=\frac{1}{N}\sum_{k}\frac{e^{ik}}{\omega-{\omega}_{k}}$, ${\gamma}_{aa}={\gamma}_{bb}=\frac{1}{N}\sum_{k}\frac{1}{\omega-\omega_{k}}$. ${\gamma}_{ab}={\gamma}_{ba}=\frac{1}{N}\sum_{k}\frac{1}{\omega\pm\omega_{ab}-{\omega}_{k}}$ $\tilde{{\gamma}}_{ab}=\tilde{{\gamma}}_{ba}=\frac{1}{N}\sum_{k}\frac{1}{\nu\pm\omega_{ab}-{\omega}_{k}}$ $\tilde{{\gamma}}_{aa}=\tilde{{\gamma}}_{bb}=\frac{1}{N}\sum_{k}\frac{1}{\nu-\omega_{k}}$. Analytical Derivation and Analysis of RG Equations: To study the different quantum phases of the system described by the Hamiltonian (Eq. 13), we express this Hamiltonian in more explicit way, $\displaystyle H_{2}~{}=~{}\sum_{n}~{}[$ $\displaystyle(1+a)~{}S_{n}^{x}S_{n+1}^{x}~{}+~{}(1-a)~{}S_{n}^{y}S_{n+1}^{y}$ (15) $\displaystyle+~{}\Delta~{}S_{n}^{z}S_{n+1}^{z}~{}+~{}h~{}S_{n}^{z}~{}]~{},$ where $S_{n}^{\alpha}$ are the spin-1/2 operators. We assume that the $XY$ anisotropy $a$ and the $zz$ coupling $\Delta$ satisfy the condition $-1\leq\Delta\leq 1$, and $0<a\leq 1$ and magnetic field strength is $h\geq 0$. The parameters correspondence between the micro cavities and spin chain are the following, $h\sim B_{tot}$, ${\Delta=J_{z}}$, ${J_{1}=1}$ and ${J_{2}}=a$. The $XY$ anisotropy breaks the in plane rotational symmetry. The study of the quantum phases from the perspective of quantum spin system and magnetism is not entirely a new one suj ; ric ; zamo . Here our main aim is to study the quantum phases of microcavities array through the RG analysis of this model Hamiltonian. Spin operators can be recast in terms of spinless fermions through Jordan- Wigner transformation and then finally one can express the spinless fermions in terms of bosonic fields gia . We recast the spinless fermions operators in terms of field operators by this relation. ${\psi}(x)~{}=~{}~{}[e^{ik_{F}x}~{}{\psi}_{R}(x)~{}+~{}e^{-ik_{F}x}~{}{\psi}_{L}(x)]$ , where ${\psi}_{R}(x)$ and ${\psi}_{L}(x)$ describe the second-quantized fields of right- and the left-moving fermions respectively, and $k_{F}$ is the Fermi wave vector. We express the fermionic fields in terms of bosonic field by the relation ${{\psi}_{r}}(x)~{}=~{}~{}\frac{U_{r}}{\sqrt{2\pi\alpha}}~{}~{}e^{-i~{}(r\phi(x)~{}-~{}\theta(x))},$ where $r$ denotes the chirality of the fermionic fields, right (1) or left movers (-1). The operators $U_{r}$ is the Klein factor to preserve the anti- commutivity of fermions. $\phi$ field corresponds to the quantum fluctuations (bosonic) of spin and $\theta$ is the dual field of $\phi$. They are related by the relations ${\phi}_{R}~{}=~{}~{}\theta~{}-~{}\phi$ and ${\phi}_{L}~{}=~{}~{}\theta~{}+~{}\phi$. Hamiltonian $H_{0}=\frac{v}{2}~{}\int~{}dx~{}[~{}(\partial_{x}\theta)^{2}~{}+~{}(\partial_{x}\phi)^{2}~{}]$ is non-interacting part of $H_{XYZ}$. Here $v$ is the velocity of the low- energy excitations. It is one of the Luttinger liquid parameters and the other is $K$, which is related to $\Delta$ by gia ; ric $K~{}=~{}\frac{\pi}{\pi+2\sin^{-1}(\Delta)}~{}.$ (16) where $K$ takes the values 1 and 1/2 for $\Delta=0$ (free field), and $\Delta=1$ (isotropic anti-ferromagnet), respectively. The relation between $K$ and $\Delta$ is not preserved under the renormalization, so this relation is only correct for the initial Hamiltonian. The analytical form of the spin operators in terms of the bosonic fields are: $S_{n}^{x}~{}=~{}[~{}c_{2}\cos(2{\sqrt{\pi K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\cos({\sqrt{\frac{\pi}{K}}}\theta)$; $S_{n}^{y}~{}=~{}-[~{}c_{2}\cos(2{\sqrt{\pi K}}\phi)~{}+~{}(-1)^{n}c_{3}~{}]~{}\sin({\sqrt{\frac{\pi}{K}}}\theta)$, and $S_{n}^{z}~{}=~{}{\sqrt{\frac{\pi}{K}}}~{}\partial_{x}\phi~{}+~{}(-1)^{n}c_{1}\cos(2{\sqrt{\pi K}}\phi)~{}$ where $c_{i}$’s are constants as given in Ref. zamo . The Hamiltonian $H_{2}$ in terms of bosonic fields is the following, $\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle H_{0}+\frac{a}{2\pi\alpha}\int\cos(2{\sqrt{\frac{\pi}{K}}}\theta(x))dx$ (17) $\displaystyle+\frac{\Delta}{{2\pi\alpha}^{2}}\int\cos(4{\sqrt{\pi K}}\phi(x))dx+\frac{h\sqrt{K}}{\pi\alpha}\int{{\partial}_{x}}{\phi(x)}dx$ One can also write the above Hamiltonian in the following form $\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle H_{01}+\frac{a}{2\pi\alpha}\int\cos(2{\sqrt{{\pi}}}\theta(x))dx$ (18) $\displaystyle+\frac{\Delta}{{2\pi\alpha}^{2}}\int\cos(4{\sqrt{\pi}}\phi(x))dx+\frac{h}{\pi\alpha}\int{{\partial}_{x}}{\phi(x)}dx$ Where $H_{01}$, $\displaystyle H_{01}=\frac{1}{2\pi}\int dx[uK{({\nabla\theta(x)})}^{2}+(u/K){({\nabla\phi(x)})}^{2}]$ (19) One can get the $H_{XY}$ Hamiltonian by simply putting $\Delta=0$ in the above Hamiltonian. In this derivation, different powers of coefficients $c_{i}$ have been absorbed in the definition of $a,h$ and $\Delta$. The integration of the oscillatory terms in the Hamiltonian yield negligible small contributions and the origin of the oscillatory terms occur due the spin operators. So it’s a reasonably good approximation to keep only the non-oscillatory terms in the Hamiltonian. The Gaussian scaling dimension of these coupling terms, $a$ and $\Delta$ are $1/K$ and $4K$ respectively. The third term ($\Delta$) of the Hamiltonian tends to order the system into density wave phase , whereas the second term ($a$) of the Hamiltonian favors the staggered order in the $XY$ plane. Two sine-Gordon coupling terms are from two dual fields. Therefore, the model Hamiltonian consists of two competing interactions. This Hamiltonian contains two strongly relevant and mutually nonlocal perturbation over the Gaussian (critical) theory. In such a situation the strong coupling fixed point is usually determined by the most relevant perturbation whose amplitude grows up according to its Gaussian scaling dimensions and it is not much affected by the less relevant coupling terms. However, this is not the general rule if the two operators exclude each other, i.e., if the field configurations which minimize one perturbation term do not minimize the other. In this case interplay between the two competing relevant operators can produce a novel quantum phase transition through a critical point or a critical line. Therefore, we would like to study the RG equation to interpret the quantum phases of the system. In the RG theory, we not only able to predict the weak coupling limit but also the the intermediate values of the coupling. The RG theory is a perturbative theory and it ceases to be valid when the coupling constant $g(l)\sim 1$. We now study how the parameters $a$, $\Delta$ and $K$ flow under RG. The operators in Eq. (17) are related to each other through the operator product expansion. So the RG equations for their coefficients therefore are coupled to each other. We use operator product expansion to derive these RG equations which is independent of boundary condition cardy . In our derivation, we consider two operators, $X_{1}=e^{(ia_{1}\phi+ib_{1}\theta)}$ and $X_{2}=e^{(ia_{2}\phi+ib_{2}\theta)}$. In the RG procedure, one can write these two field operators as a sum of fast and slow mode fields. In the fast field, the momentum range is $\Lambda e^{-dl}<K<\Lambda$ and for the slow field $K<\Lambda e^{-dl}$, where $\Lambda$ is the momentum cut-off, $dl$ is the change in the logarithmic scale. The next step is the integration of the fast field for the operators $X_{1}$ and $X_{2}$, it yields a third operator at the same space time point, $X_{3}=e^{i(a_{1}+a_{2})\phi+i(b_{1}+b_{2})\theta)}$. The prefactor of $X_{3}$ can be found by the relation, ${X_{1}}{X_{2}}\sim e^{-(a_{1}a_{2}+b_{1}b_{2})}\frac{dl}{2\pi}X_{3}$. Our Hamiltonian consists of two operators, if we consider $l_{1}$ and $l_{2}$ as the coefficient of the operators $X_{1}$ and $X_{2}$ respectively. Then the RG expressions for $\frac{dX_{3}}{dl}$ contains the term $(a_{1}a_{2}+b_{1}b_{2})\frac{l_{1}l_{2}}{2\pi}$. This is the procedure to derive these RG equations. In the RG process, one can write RG equations themselves are established in a perturbative expansion in coupling constant ($g(l)$). They cease to be valid beyond a certain length scale, where $g(l)\sim 1$ gia . The RG equations for the coefficients of Hamiltonian $H_{XYZ}$ are $\displaystyle\frac{da}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}(2-\frac{1}{K})a,$ $\displaystyle\frac{d{\Delta}}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}(2-4K){\Delta}$ $\displaystyle\frac{dK}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}\frac{a^{2}}{4}~{}-~{}K^{2}{\Delta}^{2}~{},$ (20) We have followed Ref.suj during the derivation of these RG equations. These RG equations have trivial (${a^{}}=0={{\Delta}^{}}$) fixed points for any arbitrary $K$. Apart from that these RG equations have also two non-trivial fixed lines, $a=\Delta$ and $a=-{{\Delta}}$ for $K=1/2$. The above RG equations show that there is no duality in flow diagram. Here we mean, duality, that if $\theta$ and $\phi$ interchange $\theta\leftrightarrow\phi$, $K\leftrightarrow K^{-1}$ and $\Delta\leftrightarrow a$ will not produce the same set of RG equations. For $K>1/2$, the sine-Gordon coupling term correspond to the anisotropic exchange coupling become relevant and the system flows to the photonic superfluid phase. For $K<1/2$, the sine-Gordon coupling term correspond to the z-component exchange interaction become relevant and the system flowing off to the photon blocked induced insulating phase. Here we explain the physical significance of different quantum phases of the atom-cavity system what we find in our study. (1). When both the anisotropic exchange interaction and the z-component of exchange interaction is absent then there are no sine-Gordon coupling terms in the Hamiltonian. Then the system is in the mass less Luttinger liquid phase, i.e., the system is in the photonic metallic state. The other source of photonic metallic state is that when the system shows the BKT transition which we will discuss in the next section. (2). The photonic insulator state of the atom-cavity array system corresponds to the insulating state of the system where there is no transmission of photon between the microcavities in the array due to the interaction between photons. The appearance of this phase has already discussed in the analysis of three RG equations. This phase will occurs when the system shows the BKT transition, which we will discuss in the next section. (3). Photonic superfluid state of the atom-cavity array system corresponds to the gapless excitations of the system where the photon transmit from one cavity to the other without any blocking . For this one dimensional cavity QED system where there is no order parameter, one can only discriminate between the photonic metallic state and photonic superfluid state by only finding the difference of fluctuation in photon number in every sites of the array. This photonic superfluid phase corresponds to dissipitionless flow of photon in the system. We have already discussed about the appearance of photonic superfluid phase from the analysis of three RG equations. The other source of the appearance of photonic superfluid phase is the BKT transition which we will discuss in the next section. Berezinskii-Kosterlitz-Transition Physics in Cavity QED System. The physics of BKT transition has found in different one dimensional and two dimensional (classical system) and it has discussed extensively in different context in the Ref. subir ; gia ; druf . Before we start to discuss the appearance of BKT transition in our system, we would like to discuss very briefly why it is necessary to study the BKT transition. Here we study two different situations of our model Hamiltonian. For the first case the exchange anisotropy is absent ($J_{2}=0$) and for the second case z-component of exchange interaction is absent ($J_{z}=0$). For both of these cases only one of the sine-Gordon coupling term is present, therefore, there is no competition between the two mutually non local perturbation. Therefore one can think that there is no need to study the RG to extract the quantum phases and phase boundaries. But we still apply RG method for the following reason. Each of these Hamiltonians consist of two part, the first one ($H_{01}$ ) is the non-interacting where the $\phi$ and $\theta$ fields show the quadratic fluctuations and the other part of these Hamiltonians are the sine-Gordon coupling terms which of either $\theta$ or $\phi$ fields. The sine-Gordon coupling term lock the field either $\theta$ or $\phi$ in the minima of the potential well. Therefore the system has a competition between the quadratic part of the Hamiltonian and the sine-Gordon coupling term and this competition will govern the low energy physics of these Hamiltonians in different limit of the system. The RG process (BKT transition) will predict the quantum phases of these system correctly. It is very clear from the analytical expression of ${J_{1}},J_{2}$ and $J_{z}$ that one can control these parameters in the laboratory. The quantum state of engineering of cavity QED system is in the state of art due to the rapid technological development of this system [1]. In this study we consider the situations where the $J_{2}$ is absent, i.e., the system with a single Rabi frequency oscillation. We also consider the situation where the z-component of exchange interaction is absent. These two situations lead to the two different set of RG equations which show BKT transition. For the first case, there is no anisotropic exchange coupling. In this situation, the three RG equations of the previous section reduce to two RG equations, which are the following $\displaystyle\frac{d{\Delta}}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}(2-4K){\Delta}$ $\displaystyle\frac{dK}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}-~{}K^{2}{\Delta}^{2}~{},$ (21) Now we express these RG equations in the form of BKT transition form. As we understand from the RG equations that the transition occurs at $K=1/2$. To study the flow of the RG equation around this transition point, we recast the RG equation in suitable form. Here we follow the following transformation, $K=1/2+\frac{y_{\|\|}}{4}$ and $\Delta\rightarrow\Delta/2$. The above equation reduce to the standard BKT equation. $\displaystyle\frac{dy_{\|\|}}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}-{\Delta}^{2}$ $\displaystyle\frac{d\Delta}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}-~{}y_{\|\|}{\Delta}~{},$ (22) In our case, $\Delta\frac{d\Delta}{dl}=y_{\|\|}\frac{dy_{\|\|}}{dl}$. Therefore $X^{2}={y_{\|\|}}^{2}-{\Delta}^{2}$ is a constant of motion. Here we discuss the relevant physics and the quantum phase transition between the photonic metallic state and photonic Coulomb blocked state based on these equations. Here we consider the following situations based on these equations: (1) When $X>0$ and $y_{\|\|}>0$, the sine-Gordon coupling term corresponding to $\Delta$ term is irrelevant, the fixed point, ${\Delta}^{}=0$ and ${y_{\|\|}}^{}=X$, close to this fixed point, we can write the RG equations as $\frac{d\Delta}{dl}=2(1-2K^{}){\Delta(l)}$ and $\frac{dy_{\|\|}}{dl}=0$. Using the flow equation and constant of motion for ${y_{\|\|}}>{\Delta}$. One can write the solution of coupling terms as $y_{\|\|}(l)=\frac{X}{tanh(Xl+atanh(\frac{X}{{y_{\|\|}}^{0}})}$ (23) $\Delta(l)=\frac{X}{sin(Xl+atanh(\frac{X}{{y_{\|\|}}^{0}})}$ (24) The condition at the line of seperatix is ${y_{\|\|}}(l)={\Delta(l)}=\frac{{\Delta}^{0}}{1+{\Delta}^{0}l}$. There is no mass gap excitation in the elementary excitation of the system, i.e, there is no photon blocked induced insulating phase. Hence the system is in the photonic metallic state. To the best of our knowledge for the first time in the literature we predict the existence of photonic metallic state for the array of Cavity QED system. Here we would like to present the basic origin of photonic-metallic state explicitly. If we do the Jordan-Wigner transformation of our model Hamiltonian as we present in Ref. 22. It is very clear from the Eq. 32 of Ref. 22 that the first term represent the photon hopping term across the lattice of the cavity QED array. The second and third term of the Hamiltonian represent the photonic pair correlation and photonic density wave respectively in the system. When the second and third term in the Hamiltonian either are absent or irrelevant in the RG sense at that situation the system shows the photonic metallic state. (2) When ${\Delta}>y_{\|\|}$. The RG equation for the coupling $\Delta(l)$ flowing off to the strong coupling regime. A perturbative expansion in $\Delta$ cease to be valid beyond a certain length scale for which $\Delta(l)\sim 1$. The analysis of the RG equation is not valid beyond this length scale. The analytical relation between the coupling constant is $arctan(\frac{{{y}_{\|\|}}^{0}}{\sqrt{{{\Delta}_{0}}^{2}-{{y_{\|\|}}^{0}}^{2}}})-arctan(\frac{{{y}_{\|\|}}}{\sqrt{{{\Delta}_{0}}^{2}-{{y_{\|\|}}^{0}}^{2}}})=\sqrt{{{\Delta}_{0}}^{2}-{{y_{\|\|}}^{0}}^{2}}.$ (25) In this limit, the RG flowing off to the strong coupling phase. Here we discuss the relevant physics of the phase. The sine-Gordon coupling term is $\frac{\Delta_{1}u}{2\pi{\alpha}^{2}}\int dxcos(4\phi(x)).$ where $\Delta_{1}=\frac{\Delta}{\pi u}$. The $\phi(x)$ field locks into one of the minima of the cosine potential. Now we expand the potential for large $\Delta_{1}$ in the spirit of usual RG method subir ; gia ; druf . We can write the effective Hamiltonian near to the minima $H=H_{0}+\frac{4\Delta_{1}u}{\pi{\alpha}^{2}}{\phi}^{2}(x)$ . The total action of the Hamiltonian of the system can be written as $S=\frac{1}{2\pi K}\frac{1}{\beta\Omega}\sum_{k,{\omega}_{n}}[\frac{{{\omega}_{n}}^{2}}{u}+uk^{2}+\frac{8K\Delta_{1}u}{{\alpha}^{2}}]{\phi}^{}(k,\omega_{n}){\phi}(k,\omega_{n})$ (26) The excitations of the system which cost a finite energy even at $k=0$. In this situation the field, $\phi(x)$ is massive. The system posses phononic type mode. This phononic mode is the small oscillation of the field ${\phi}(x)$ around the minima of the cosine potential. Suppose we consider the RG equation up to the point where ${\Delta_{1}}(l)\sim 1$. This excitations gap in the spectrum has the dimension of an energy and the renormalization relation is ${\Omega}_{M}(l)=e^{l}{\Omega}_{M}(l=0)$, for the case, ${\Delta_{1}}(l^{})\sim 1$. We can write the expression for the gap following the action as, ${\Omega}_{M}(l^{})\sim\sqrt{{\Delta_{1}(l^{})}}u/{\alpha}$. The true gap of the system is ${\Omega}_{M}(l=0)\sim e^{-l^{}}{\Omega}_{M}(l^{})$, ${\Omega}_{M}(l^{})=\frac{{u}}{\alpha}$. Thus at this phase of this system, the explicit dependence of the Luttinger liquid parameter is absent. This gap of the system is of the order of bandwidth of the cavity QED system. (3) Now we consider the case, when ${\Delta_{1}}<<\|y_{\|\|}\|$. The system is in the deep massive phase. In this phase, one can write the RG equation as $\frac{dK(l)}{dl}=0$ and $\frac{d\Delta_{1}(l)}{dl}=2(1-2K){\Delta_{1}}(l)$. For this RG equation, we get ${\Delta_{1}}(l)={\Delta_{1}}(0)e^{2(1-2K)l}$ from this equation $e^{-l^{}}={\Delta(0)}^{1/2(1-2K)}$. The true gap of the system is ${\Omega}_{2}(l=0)\sim e^{-l^{}}{\Omega}_{2}(l^{})$, $\frac{\Omega_{2}(l=0)}{\Omega_{2}(l^{})}\sim{\Delta_{1}(0)}^{+1/2(1-2K)}$. This gap is the power law dependence of the bare $\Delta_{1}$. This gap varies with $K$, i.e., the gap of the system is now varying with the interaction of the system. It is very clear from the analytical expression that the gap gets smaller and tends to zero at $K=1/2$, this prediction is consistent with the physical scenario that at $K=1/2$ the system shows the phase transition. (4) Close to the transition point, Eq.(25) reduces to $\sqrt{{{\Delta_{1}}_{0}}^{2}-{(y_{\|\|})}^{2}}{l^{}}=\pi$. So the square root term goes to zero at the transition point and therefore ${y_{\|\|}}^{0}/{\sqrt{{{\Delta_{1}}_{0}}^{2}-{y_{\|\|}}^{2}}}\rightarrow\infty$ and the gap ${\Omega}(l=0)\sim{\Omega}({l^{}})e^{-\frac{\pi}{\sqrt{{{\Delta_{1}}_{0}}^{2}-{y_{\|\|}}^{2}}}}$. As one approach to the transition point, such as ${\Delta_{1}}_{0}\rightarrow\|{y_{\|\|}}\|$. The gap is exponentially small in the square root of the distance to the transition. It is therefore clear from our RG analysis that the system is in the massive phase for the two limits. For the first case when ${\Delta}>>{y_{\|\|}}$, the excitation gap of the system is of the order of the bandwidth of the system and the effect of photonic strong correlation is not explicit. The other limit of gapped state, i.e, when $\Delta<<\|y_{\|\|}\|$. The excitation gap of the system has a power law dependence which vary with $K$. It is very clear from the analytical expression of $J_{z}$ that one can tune it to zero by adjusting the Rabi frequencies oscillation of the system. In this situation, there is no z-component of intercavity exchange interaction. Therefore the three RG equations are reduce to two RG equations. $\displaystyle\frac{da}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}(2-1/K)a$ $\displaystyle\frac{dlnK}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}~{}{a^{2}}/{4K}~{},$ (27) If we do the following transformation $\tilde{K}=1/{2K}$ and $\tilde{a}=a/2$ and after that we follow another transformations $\tilde{K}=1+{y_{\|\|}}/2$ the above equations reduce to $\displaystyle\frac{dy_{\|\|}}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}-{\tilde{a}}^{2}$ $\displaystyle\frac{d\tilde{a}}{dl}~{}$ $\displaystyle=$ $\displaystyle~{}-~{}y_{\|\|}{\tilde{a}}~{},$ (28) The mathematical structure of these equations are the same as that of Eq. 22. Therefore the mathematical analysis for the different limits are the same but massive phase and deep massive phase for these situations are the photonic superfluid phase. The sine-Gordon coupling term is $\frac{a_{2}u}{2\pi{\alpha}}\int dxcos(2\theta(x)).$ where $a_{2}=\frac{a}{u}$. The $\theta(x)$ field locks into one of the minima of the cosine potential. Now we expand the potential for large $a_{2}$. We can write the effective Hamiltonian near to the minima $H=H_{0}+\frac{a_{2}u}{{\pi\alpha}}{\theta}^{2}(x).$ The total action of the Hamiltonian of the system can be written as $S=\frac{1}{2\pi K}\frac{1}{\beta\Omega}\sum_{k,{\omega}_{n}}[\frac{{{\omega}_{n}}^{2}}{u}+uk^{2}+\frac{2Ka_{2}u}{{\alpha}}]{\theta}^{}(k,\omega_{n}){\theta}(k,\omega_{n})$ (29) The excitations of the system which cost a finite energy even at $k=0$. In this situation the field, $\theta(x)$ is massive. The true gap of the system is ${\Omega}_{C}(l=0)\sim e^{-l^{}}{\Omega}_{C}(l^{})$, ${\Omega}_{C}(l^{})=(\frac{{u}}{\alpha})\sqrt{\alpha}$. Thus at this phase of this system, the explicit dependence of the Luttinger liquid parameter is absent. This gap of the system is much less than the excitation gap of the photon blocked state due to the presence of the extra factor $\sqrt{\alpha}$. In the deep massive phase, $\frac{\Omega_{C}(l=0)}{\Omega_{C}(l^{})}\sim{a_{2}(0)}^{+1/(2-1/K)}$. This gap is the power law dependence of the bare $a_{2}$. This gap varies with $K$, i.e., the gap of the system is now varying with the interaction of the system. It is very clear from the analytical expression that the gap gets smaller and tends to zero at $K=1/2$ and this prediction is consistent with the physical scenario that at $K=1/2$ the system shows the phase transition from photonic metallic state to photonic superfluid phase. Now we consider the effect of magnetic field ( ${\delta}_{1}$ term) on the quantum phases and phase boundaries of this cavity QED arrays. We do the following transformation to eliminate the magnetic field term from the Hamiltonian. We substitute $4\pi\sqrt{K}\phi\rightarrow 4\pi\sqrt{K}\phi+4\pi\delta x$, where $\delta=\sqrt{K}h$. This introduce the spatial oscillation in the $\Delta$ term. When the coupling $a$ is absent and the applied detuning field is larger than the photonic Mott gap, the system drives to the gapless photonic metallic state. This is nothing but the well known Porkovosky-Talapov model which shows the commensurate to incommensurate transition gia ; suj . When $\delta{a_{1}}>>1$ (where $a_{1}$ is the lattice spacing), the $cos(4\pi\sqrt{K}(\phi+\delta x))$ term is quickly oscillating and averages out zero. Thus the system reflects the competition between the $\Delta$ and $h$. As a results of it, the RG flow for $\Delta$ has to be cutoff when $4\pi\sqrt{K}\delta(l)a_{1}\sim 1$. To the lowest order in $\Delta,a$ and $\delta$, the RG flow equation is $\frac{d\delta}{dl}=\delta$ (30) As a result of this RG equation, the system shows two different response, either the flow ${\Delta}_{1}$ and $\delta$ or $a$ and $\delta$ flows of to the strong coupling phase. When $\Delta_{1}(l)$ reaches strong coupling before $\delta a_{1}$ become order one. The phase boundary is the same as we predict without magnetic field. The condition when $\Delta_{1}(l^{})\sim 1$ define a new length scale $l^{}$. It generates a self-consistent scenario that $4\pi\delta(l^{})a_{1}<<{{\Delta_{1}}(0)}^{1/(2-4K)}$. In the other limit when the RG flow of $a$ flows off to the strong coupling phase does not affected by the magnetic field term and thus the system is in the photonic superfluid phase. Conclusions: We have presented three sets of RG equations for the different physical situations for the Cavity QED system. We have predicted two different BKT transitions for the different physical situations. We have predicted the photonic superfluid phase, photonic metallic phase and photon blocked induced insulating phase. To the best of our knowledge, this explicit quantum phase analysis and their behavior based on the RG study for the microcavity array system is absent in the previous literature. Acknowledgments The author would like to acknowledge the series of lectures and also discussions of Prof. S. M. Girvin during the International Workshop/School on Dirac Materials and Chandrashekar Discussion Meeting, December’2012. The author would like to thank The Center for Condensed Matter Theory of the Physics Department of IISc for extended facility. Finally the author would like to thank Prof. Prabir Mukherjee for reading this manuscript critically. ## References * (1) G. S. Agarwal, Quantum Optics (Cambridge University Press, Cambridge 2013). * (2) S. M. Girvin, Circuit QED: Superconducting Qubits Coupled to Microwave Photons (Lecture Notes/Private Communication). * (3) Hartmann Michael J, Fernando G S, Brando L and Plenio Martin B 2006 Nature Phys 462 849; Hartmann Michael J, Fernando G S, Brando L and Plenio Martin B 2008, Laser and Photonics Rev. 2 527\. * (4) Hartmann Michael J, Fernando G S, Brando L and Plenio Martin B 2007, Phys. Rev. Lett 99 160501\. * (5) S. Sarkar, Physica B 407, 44 (2012). * (6) S. Horoche and J. M. Raimond in Exploring the Quantum Atoms, Cavities, and Photons, (Oxford University Press, 2006). * (7) E. T. Jaynes and F. W. Cummings, Proc IEEE 51, 89 (1963). * (8) A. C. Ji, X. C. Xie, and W. M. Liu, Phys. Rev.Lett. 99, 183602 (2007). * (9) T. Byrnes, N. Y. Kim, K. Kusudo, and Y. Yamamoto, Phys. Rev. B 78 075320 (2008). * (10) I. Carusotto $et~{}al.$, arXiv:0812.4195. * (11) M. J. Bhaseen $et~{}al.$, Phys. Rev. Lett. 102 135301 (2009). * (12) J. Zhao, A. W. Sandvik and K. Ueda, arXiv:0806.3603. * (13) P. Pippan, H. G. Evertz, and M. Hohenadler, arXiv: 0904.1350. * (14) M. Aichhorn $et~{}al.$, Phys. Rev. Lett. 100 216401 (2008). * (15) S. Schmidt, and G. Blatter, arXiv:0905.3344 . * (16) D. G. Angelakis, M. F. Santos and S. Bose, Phys. Rev. A 76 R031805, (2007). * (17) B. Kumar, J. Phys. A: Math. Theor 42, 245307 (2009); * (18) Sachdev Subir in ”Quantum Phase Transition” (Cambridge University Press, Cambridge, 1998). * (19) A. K. Pal and I. Bose, J. Phys. B: At. Mol. Opt 44, 045101 (2011). * (20) T. Giamarchi in Quantum Physics in One Dimension (Clarendon Press, Oxford, 2004). * (21) Itzykson and Drouffe in Statistical Field Theory, Volume 1 (Cambridge University Press, New York 1989). * (22) Here we present the Jordon-Wigner transformation of Eq. 15, $Q_{i}=e^{i\pi\sum_{n=1}^{i-1}{c_{i}}^{\dagger}c_{i}}$, ${S_{i}}^{-}=Q_{i}c_{i}$, ${S_{i}}^{+}={c_{i}}^{\dagger}{Q_{i}}^{\dagger}$. The Eq. 15 under this trnsformation become $\displaystyle H$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}({c_{i+1}}^{\dagger}c_{i}+h.c)+a({c_{i}}^{\dagger}{c_{i+1}}^{\dagger}+h.c)$ (31) $\displaystyle+\Delta\sum_{i}({c_{i}}^{\dagger}c_{i}-1/2)({c_{i+1}}^{\dagger}c_{i+1}-1/2)+h({c_{i}}^{\dagger}c_{i}-1/2)$ It is clear from the above expression that the anisotropic-exchange interaction induce creation or annihilation of photon pairs. * (23) K. M. Birnbaum $et~{}al.$, Nature 436, 87 (2005). * (24) F. V. James and J. Jerke, Can. J. Phys 85, 625 (2007). * (25) S. Sarkar, Phys. Rev. B 74, 052410 (2006). * (26) U. Schollwock, J. Richter, D. J. J. Farnell, and R. F. Bishop (Eds.), Lect. Notes Phys. 645, ”Quantum Magnetism” (Springer, Berlin, 2004). * (27) S. Lukyanov and A. Zamolodchikov, Nucl. Phys. B 493, 571 (1997). * (28) J. Cardy in Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996); Affleck I 1989 in Fields, Strings and and Critical Phenomena, ed Brezin E and Zinn-Justin J (North-Holland, Amstardam).	arxiv-papers	2013-09-30T09:52:49	2024-09-04T02:49:51.706002	{ "license": "Public Domain", "authors": "Sujit Sarkar", "submitter": "Sujit Sarkar", "url": "https://arxiv.org/abs/1309.7777" }
1309.7912	# An Image-Based Fluid Surface Pattern Model Mauro de Amorim\- [email protected] Ricardo Fabbri\- [email protected],br Lucia Maria dos Santos Pinto\- [email protected] Francisco Duarte Moura Neto\- [email protected] Polytechnic Institute at the Rio de Janeiro State University 28630-050 - Nova Friburgo, RJ, Brazil ###### Abstract This work aims at generating a model of the ocean surface and its motion from one or more video cameras. The idea is to model wave patterns from video as a first step towards a larger system of photogrammetric monitoring of marine conditions for use in offshore oil drilling platforms. The first part of the proposed approach consists in reducing the dimensionality of sensor data made up of the many pixels of each frame of the input video streams. This enables finding a concise number of most relevant parameters to model the temporal dataset, yielding an efficient data-driven model of the evolution of the observed surface. The second part proposes stochastic modeling to better capture the patterns embedded in the data. One can then draw samples from the final model, which are expected to simulate the behavior of previously observed flow, in order to determine conditions that match new observations. In this paper we focus on proposing and discussing the overall approach and on comparing two different techniques for dimensionality reduction in the first stage: principal component analysis and diffusion maps. Work is underway on the second stage of constructing better stochastic models of fluid surface motion as proposed here. Keywords: Inverse problems, fluid motion, nonrigid 3D reconstruction, dimensionality reduction, diffusion maps, pattern theory ## 1 Introduction The simulation of fluids is an imporant tool in computer graphics, e.g., for generating realistic animations of water flow. Many fluid simulations perform the evolution of liquid through time based on the Navier-Stokes equation or simpler wave and frequency models, among others. This work is an initial part of a larger effort to perform the inverse problem of generating graphics simulations, i.e., to automatically extract 3D models of fluid surfaces starting from real-world video data. The obtained model can then be used for simulating the motion of ocean surface patterns as _observed_ in the real world. Moreover, this simulation can be matched to video at new time instants in order to predict and track the actual conditions of the _observed_ fluid. In order to track the apparently complex dynamics of a large number of images of the ocean surface, our proposed system reduces the dimensionality of previsouly observed data and automatically learns a concise data-driven model. This enables the inference through synthesis of patterns that are intrisic to the observed phenomenon, while drastically reducing the search space with little to no loss. The remaining error in the low dimensional modeling is dealt with as part of a stochastic modeling stage. While detailed techniques for the plausible inference involving such patterns is ongoing work, we propose that the patterns in newly observed data are to be found by stochastic modeling, using the Bayesian paradigm, a method which can be described as “analysis by synthesis” (Mumford & Desolneux,, 2010). Finally, a stochastic model also enables generating plausible new images for a realistic image-based simulation. Figure 1 shows a diagram representing the steps of the proposed modeling approach dealt with in this manuscript. Two dimensionality reduction techniques were assessed: _(i)_ principal components analysis, which consists in performing orthogonal projections of data along linear directions of greater variance; and _(ii)_ diffusion maps, a recent non-linear technique which organizes data into graphs of local similarity and extracts global structure through a diffusion process (Lafon & Lee,, 2006). Figure 1: Diagram of the proposed approach for the modeling of fluid surface patterns aiming at applications of monitoring ocean conditions for offshore oil drilling processes, among others. ## 2 Dimensionality Reduction Techniques Assume a video is made up of $n$ video frames represented by a column vector $y_{i}$, $i=1,\dots,n$. Each video frame is an image with a total of $p$ pixels, so that each observation or data point is a $p$-dimensional vector $y_{i}\in\mathbbm{R}^{p}$. For instance, frames of a typical high-definition video $1920\times 1080$ are each comprised of more than 2 milion dimensions, an excessive number of parameters to capture the observable modes of variation of fluid surface patterns that are relevant for monitoring applications. We explore two different techniques for automatically reducing this number of parameters as a step towards generating a tractable yet meaningful model. ### 2.1 Principal Component Analysis The principal component analysis (PCA) is a linear dimentionality reduction technique. High-dimensional data, i.e., $p$-dimensional, gets reduced by PCA to a global linear submanifold of reduced dimension $q\ll p$, determined by the directions of greatest variability in the data. These directions are given by $q$ orthonormal vectors, the principal components, which are the eigenvectors corresponding to the largest eigenvalues of the covariance matrix of the data. Principal component analysis can be computed through the singular value decomposition (SVD) (Hastie & et al.,, 2013; Jolliffe,, 2002; Golub & Van Loan,, 1996) as follows. The $n$ observations $y_{i}\in\mathbbm{R}^{p}$ are mean-centered by subtracting their average $\bar{y}=\dfrac{1}{n}\sum^{n}_{j=1}y_{j}$. The new data vectors $x_{i}=y_{i}-\bar{y}$ form the columns of the $p\times n$ data matrix $X$ whose singluar value decomposition is given by $X=U\Sigma V^{\top},$ (1) with $U$ and $V$ being $p\times n$ and $n\times n$ orthonormal matrices, resp., and $\Sigma$ an $n\times n$ diagonal matrix, with diagonal elements $\sigma_{1}\geq\dots\geq\sigma_{n}\geq 0$ known as singular values. The first $q$ columns of $U$ form the $p\times q$ matrix $U_{q}$, the so-called first $q$ principal components, and $\alpha_{i}=U_{q}\top x_{i}$ are the coordinates of any given observation $x_{i}$ in the subspace of the principal components. The computation of PCA using SVD is equivalent to computing the eigenvalues and eigenvectors of the covariance matrix, which is given by $\dfrac{1}{n-1}XX^{\top}.$ (2) It is possible to show this fact by taking the SVD of the matrix $X$ as shown in (1), and building the product $XX^{\top}$, giving $XX^{\top}=(U\Sigma V^{\top})(V\Sigma U^{\top}).$ (3) Since $V$ is an orthogonal matrix, we have $XX^{\top}=U\Sigma^{2}U^{\top}.$ (4) From Equation (4) we clearly see the correspondence of the SVD and the covariance of the data samples. The singular values of $X$ are the square roots of the eigenvalues of the covariance matrix, and the singular vectors of $X$ are the eigenvectors of the covariance matrix. #### 2.1.1 Stochastic SVD In our application, the dataset is very large and in high dimensions, in which case computing the eigenvalues of the data matrix is a key challenge. Recent techniques enable the computation of large matrix decompositions in a robust manner without explicitly forming the entire data matrix in memory (Halko et al.,, 2010, 2011). The key main idea behind these methods is stochasticity. For the construction of the SVD using stochasticity, a method we refer in this manuscript as stochastic SVD is available (Halko et al.,, 2011) which can be broken in two stages. First, the construction of a subspace with reduced dimension in which to represent the input data in an approximate manner; a matrix is bult in which the columns form a reduced orthonormal basis for the data. Second, the projection of the observations onto that subspace and the subsequent computation of the SVD. Both stages are outlined in Algorithm 1. Algorithm 1 – Stochastic SVD Given an ${m\times n}$ matrix $A$, and integers $l$ and $q$, compute an approximate decomposition of $A\approx U\Sigma V^{\top}$, where $U$ and $V$ are orthonormal and $\Sigma$ is a non-negative diagonal matrix. 1:Create an ${n\times l}$ matrix $\Omega$, with independent entries and standard normal distribution. 2:Form $Y=A\Omega$. 3:Perform the $QR$ decomposition of $Y$, $Y=QR$, with $Q$ an $m\times l$ orthonormal matrix. 4:Form $B=Q^{\top}A$. 5:Compute an SVD of the reduced matrix $B=\tilde{U}\Sigma V^{\top}$. 6:Form the otrhonormal matrix $U=Q\tilde{U}$. The stochastic SVD provides a robust means to perform large-scale matrix decompositions, independently of the intrinisc structure of the data matrix. However, this approach yields an approximate solution whose quality must be assessed in the context of the original application. In this work, the purpose of the stochastic SVD is to obtain the principal components of a given dataset, i.e., to obtain the singular vectors to the left of an SVD of a data matrix. We know that the obtained vectors form a basis for a subspace, so that its precision can be assessed at each execution of the stochastic SVD. This assessment was done using a distance between subspaces as described in the next section. #### 2.1.2 Distance between subspaces Let $F$ and $G$ be subspaces of $\mathbbm{R}^{m}$, with $p=\textsf{Dim}(F)\geq\textsf{Dim}(G)\geq q\geq 1$. The princpal angles, $\theta_{1},\dots,\theta_{q}\in[0,\pi/2]$, between $F$ and $G$ are recursively defined as being real the numbers $\theta_{k}$ such that $\cos(\theta_{k})=\max_{u\in F,v\in G}\|u^{\top}v\|=\|u_{k}^{\top}v_{k}\|,$ subject to $u^{\top}u=v^{\top}v=1,\qquad u^{\top}u_{i}=v^{\top}v_{i}=0\ ,\qquad i=1,\dots,k-1.$ The vectors $\\{u_{1},\dots,u_{q}\\}$ e $\\{v_{1},\dots,v_{q}\\}$ are the so- called principal vectors between the subspaces $F$ e $G$. The greatest principal angle is related to the notion of distance between subspaces of the same dimension (Golub & Van Loan,, 1996). Thus, if $p=q$ then $dist(F,G)=\sqrt{1-\cos(\theta_{p})^{2}}$ $=\hskip 2.0pt\mathrm{sen}(\theta_{p})$. In a practical way, if the columns of $Q_{F}\in\mathbbm{R}^{m\times p}$ and those of $Q_{G}\in\mathbbm{R}^{m\times p}$ define orthonormal bases for $F$ and $G$, respectively, the cosine of the greatest principal angle is determined by computing an SVD of the matrix $Q_{F}^{\top}Q_{G}$ and taking the smallest of its singular values. It is important to note that $0\leq dist(F,G)\leq 1.$ The distance will be zero if $F=G$ and one if $F\cap G^{\perp}\neq\\{0\\}$, with $G^{\perp}$ denoting the orthogonal complement of $G$, i.e., the space of all orthogonal vectors to $G$. ### 2.2 Diffusion maps We have explored a more recent non-linear technique for dimensionality reduction and manifold learning, the so-called diffusion maps, which re- organizes the data according to a reduced set of parameters related to the approximate intrinsic geometry of the underlying phenomena (Lafon & Lee,, 2006; de la Porte et al.,, 2008). The reduced set of parametrs are computed from the eigenvalues and eigenvectors of a diffusion operator on data. It is robust to noise and outliers in data, and can be efficiently computed when its application is properly designed. Consider that the set of $n$ observations $y_{1},\dots,y_{n}\in\mathbbm{R}^{p}$ approximately sample one or more non- linear manifolds, each $y_{i}$ being a vector of all pixels of a video frame, for instance. We describe the diffusion maps algorithm in four steps. The first step consists in building an $n\times n$ matrix $W$ of pairwise (local) similarities between the observations. The similarities are defined by a kernel function $K:Y\times Y\rightarrow\mathbbm{R}$, satisfying $K(y_{i},y_{j})=K(y_{j},y_{i})$ and $K(y_{i},y_{j})\geq 0$. In the present work, we have used the heat kernel given as $W_{ij}=K(y_{i},y_{j})=e^{-\frac{\|\|y_{i}-y_{j}\|\|}{\epsilon}^{2}}.$ (5) By choosing the parameter $\epsilon$ one can adjust the size of the neighborhood with which to compute similarities, based on prior knowledge of the structure and density of data (de la Porte et al.,, 2008) and on sparsity considerations. In this work $\epsilon$ was taken as the largest squared Euclidean distance between all datapoints, although a significantly smaller $\epsilon$ could have been chosen, leading to a sparser matrix $W$ thus increasing the efficiency of the algorithm. The second stage consists in constructing the diffusion matrix $P$, which is a stochastic matrix, whose lines are normalized to 1. The matrix $P$ is obtained by $P=D^{-1}W,$ (6) where $D$ is a diagonal matrix whose entries are given by $D_{ii}=\sum_{j=1}^{n}W_{ij},\quad\text{for }i=1,\dots,n.$ (7) Each entry of $P$ provides the pairwise connectivity of the data in an underlying similarity graph. The graph can be seen as a Markov chain on the data points whose transition matrix is $P$; each entry $P_{ij}$ represents the probability of transitioning from data point $i$ to $j$ in one diffusion step. In other words, this is the probability of clustering together data points $i$ and $j$ in one step. When taking powers of the transition matrix $P$, one increases the number of steps taken to cluster nodes $i$ and $j$ to form a manifold. The third step consists in computing the spectral decomposition of the transition matrix $P$, thus obtaining the eigenvalues $\lambda_{i}$ and corresponding eigenvectors $\psi_{i}$. Since the matrix $P$ is stochastic, its greatest eigenvalue in absolute value, $\lambda_{0}$, equals $1$. When the transition matrix is positive definite, $P$ has a sequence of positive eigenvalues sorted in decreasing order $1=\lambda_{0}\geq\dots\geq\lambda_{n-1}>0.$ (8) The last step of diffusion maps is to actually perform the dimensionality reduction. This is done by discarding eigenvalues of smaller indices. In selecting the largest $q$ eigenvalues, we obtain a new feature vector $\widetilde{y_{j}}\in\mathbbm{R}^{q}$, using the diffusion map given by $\widetilde{y_{j}}=\begin{bmatrix}\lambda_{1}^{t}\psi_{j1}\\\ \vdots\\\ \lambda_{q}^{t}\psi_{jq}\\\ \end{bmatrix}\ ,$ (9) with, e.g., $\psi_{j1}$ being the $j-$th component of eigenvector $\psi_{1}$ and $t>0$ being a parameter corresponding to the power of the matrix $P$, whose effect is to cluster through the Markov chain in $t$ steps. ## 3 Image-Based Stochastic Modeling Our main goal is to devise a tractable stochastic model whose samples enable simulating a video of the observed behavior of ocean surface patterns. The same approach should be useful for image-based modeling and tracking of other continuous-time deformations. In addition to provide a framework for machine learning and inference, stochastic modeling is necessary to account for aspects of the phenomenon that can be difficult or impossible to model explicitly in an efficient manner. For the automatic construction of the model from video, towards an application of recognizing ocean patterns, we propose the use of Bayesian methods of probabilistic inference. The use of these methods require a “training” or “learning” stage from enough input data and then strategies for automatically fitting of the stochastic model to newly observed data, which we call the “testing” or “tracking” stage (Mumford & Desolneux,, 2010). The stochastic model will be built from the data-driven model given by either diffusion maps or PCA. Thus, consider that patterns of a signal are to be modeled, where $S(t)\in\mathbbm{R}^{q}$ is the video frame at instant $t$ in the diffusion map or PCA model space. In this work, we specifically reduce the frames to 3 dimensions, thus $q\in 3$, i.e., the motion patterns in the video are to be modulated by only 3 parameters. Using the Bayesian inference paradigm, one seeks to infer the state of the random variable $S(t)$ on a new time point given observable data $I(t)$ at other time points. In this work some examples of observable variables are: * • Number of sample points. * • The local curvature between sample points. * • The variation of sample time. In order to build the stochastic model, it is necessary to define a probability function $P(S(t),$ $I(t))$. The inference of $S(t)$ is carried out using the _a posteriori_ probability $P(S(t)\|I(t))$ through Bayes’ rule: $P(S(t)\|I(t))=\dfrac{P(S(t))P(I(t)\|S(t))}{P(I(t))},\text{ with }P(I(t))=\sum_{S(t)}P(S(t))P(I(t)\|S(t))$ (10) The above general approach can lead to three problems (Mumford & Desolneux,, 2010): * • The construction of the probability model, $P$; * • To find an algorithm to maximize the _a posteriori_ probability; * • To optimize the parameters $I$ of the model as to optimally fit the data. In order to validate the stochastic model, one must take samples to produce a stochastic simulation, and test if these reproduce the observed real-world flows of the sea surface. We have been actively working on adequate specific techniques for the above three problems for the overall approach proposed here. ## 4 Results and Discussion The experiments were preformed in the Scilab free software language and environment, together with the image processing toolbox SIP. The data consisted of a video of a beach front with a resolution of $360\times 640$ and $21\min 06s$ duration, illustrated in Figure 2.111HD Florida Beaches Sunset, Powerfloe Network, http://www.youtube.com/watch?v=0GBpGRXaruE, 02/10/2012. Used with permission. This video was chosen since it provides a recording without camera movement, as it simplifies the modeling of the problem without requiring a very large number of frames. We extracted four $15s$ video clips to be analyzed from the original video, forming the “training set” used to learn our model. For each clip all frames of the video were extracted and converted to grayscale, resulting in a set of $450$ images having $360\times 640=230\,400$ pixels or dimensions. These clips sample the temporal evolution of the behaviour of ocean surface patterns. The extracted frames of each clip go through dimensionality reduction using both stochastic PCA and diffusion maps in order to perform a comparative study of the power of each approach to represent the underlying patterns. Figure 2: Sample image of the video used in the experiments. The larger goal is to obtain a model of the 3D structure and motion of the ocean surface. We begin to tackle this by the construction of an efficient stochastic model of the image patterns in the video. ##### Reliability of the Stochastic PCA As we have used PCA with a stochastic SVD algorithm, the obtained subspace at each realization of the method has a certain variance. In order to assess this, we computed the precision of the obtained subspaces for multiple runs of stochastic PCA, using the distance betweeen subspaces described in Section 2.1.2. To the best of our knowledge, there is no information in the literature about the number of samples in stochastic SVD needed to obtain a set that generates a subspace that provides a good enough approximation to the principal components. For each set of images, the stochastic PCA was executed five times, followed by the pairwise computation of subspace distances in order to assess the variance of the result. Table 1 shows the average distance between the obtained subspaces, as well as the standard deviation for each video clip. Table 1: The average distance between the subspaces obtained through multiple runs of stochastic PCA, together with the standard deviation, for four different clips of the same original video. \| Average distance \| Standard deviation ---\|---\|--- Group 1 \| 0.1099 \| 0.003 Group 2 \| 0.1105 \| 0.039 Group 3 \| 0.1064 \| 0.056 Group 4 \| 0.1092 \| 0.005 Despite the average distances between the subspaces at each execution of stochastic PCA being close to $0.1$, which is around $10\%$ of the maximum distances between two observed subspaces, this result is acceptable for the applications up to this point. With the low standard deviation values, we have confirmed that the proposed algorithm gives consistent results in practice. ##### Diffusion Maps Vs. Stochastic PCA The results of applying dimensionality reduction using stochastic PCA and diffusion maps for the first videoclip are shown in Figure 3. (a) Stochastic PCA (b) Diffusion maps Figure 3: Projection of each frame of the first videoclip ($15\min$) in $\mathbbm{R}^{3}$ using 3(a) one instance of stochastic PCA and 3(b) diffusion map at scale $t=2$. From Figure 3 it is not possible to be conclusive about which technique has better potential as part of the proposed modeling pipeline. To further investigage this, we compare the decay of the eigenvalues used in the diffusion map with that of the eigenvalues used in the stochastic PCA, relative to the largest eigenvalues. The diffusion map eigenvalues present a much shaper decay than the eigenvalues used in the stochastic PCA, Figure 4. This indicates that diffusion maps can represent the observed flow with fewer parameters. Figure 4: The decay of the first $100$ eigenvalues used in the diffusion map compared to that of the eigenvalues used in the stochastic PCA, relative to the largest eigenvalues, for the first videoclip. ## 5 Conclusion and Future work The image-based fluid modeling approach proposed in this work is part of a larger effort with applications to the online monitoring of ocean conditions for aiding offshore processes of oil drilling, among others. With dimensionality reduction it is possible to represent the original data with a data-driven model with a small number of parameters. Using diffusion maps, these parameters reveal intrinsic structure of the original data, keeping the most relevant observable characteristics, while our experiments show that PCA was not as efficient in this sense. Building on such reduced model we proposed the main ideas of a stochastic model to account for remaining factors. Work is underway to produce techniques for detailed machine learning and inference based on this model. The final complete model should be able to capture the patterns of fluid surface motion in a way that, when statistically sampling from this model, we should obtain a sequence of images that can reproduce the observed fluid behavior through time. In the future this can then be used to help infer the state of the ocean surface in a new video sequence. We have also been considering the use of video streams from multiple views to improve quality and robustness. ## References * de la Porte et al., (2008) de la Porte, J., Herbst, B. M. Hereman, H. & van der Walt, S. J. An Introduction to Diffusion Maps. Proceedings of the 19th Symposium of the Pattern Recognition Association of South Africa; 26-28 de novembro de 2008; Prasa. Cape Town: F. Nicolls, University of Cape Town, South Africa, 2008. pp. 15-25. * Golub & Van Loan, (1996) Golub, G. H. & Van Loan, C. F. Matrix computations. 3 ed. Baltimore : Johns Hopkins University Press, 1996. * Halko et al., (2010) Halko, N.; Martinsson, P. G.; Shkolnisky, Y. & Tygert, M. An algorithm for the principal component analysis of large data sets. SIAM Journal on Scientific Computing, vol. 33, n. 5, pp. 2580-2594, 2011. * Halko et al., (2011) Halko, N.; Martinsson, P. G. & Tropp, J. A. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, vol. 53, n. 2, pp. 217-288, 2011. * Hastie & et al., (2013) Hastie, T.; Tibshirani, R. & Friedman, J. The elements of statistical learning: Data mining, inference, and prediction. Springer, 2013. * Jolliffe, (2002) Jolliffe, I. T. Principal component analysis. 2 ed. New York : Springer, 2002. * Lafon & Lee, (2006) Lafon, S. & Lee, A. Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parameterization. Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 28, n. 9, pp. 1393-1403, 2006. * Mumford & Desolneux, (2010) Mumford, D. & Desolneux, A. Pattern theory: the stochastic analysis of real-world signal. CRC Press, 2010.	arxiv-papers	2013-09-30T16:39:21	2024-09-04T02:49:51.718315	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Mauro de Amorim, Ricardo Fabbri, Lucia Maria dos Santos Pinto and\n Francisco Duarte Moura Neto", "submitter": "Ricardo Fabbri", "url": "https://arxiv.org/abs/1309.7912" }
1309.7928	DPF2013-45 September 30, 2013 Measurements of $B\rightarrow DK$ decays to constrain the CKM Unitarity Triangle angle $\gamma$ and related results at LHCb Daniel Craik 111On behalf of the LHCb Collaboration. Department of Physics University of Warwick, Coventry, UK > Constraints on the CKM angle $\gamma$ are presented from GLW, ADS, and GGSZ > analyses of $B^{\pm}\rightarrow DK^{\pm}$ at the LHCb experiment. The > branching fractions of $B^{0}\rightarrow\overline{D}^{0}K^{+}\pi^{-}$ and > $B_{s}^{0}\rightarrow\overline{D}^{0}K^{-}\pi^{+}$ are also reported, > measured relative to the related mode > $B^{0}\rightarrow\overline{D}^{0}\pi^{+}\pi^{-}$. > PRESENTED AT > > > > > DPF 2013 > The Meeting of the American Physical Society > Division of Particles and Fields > Santa Cruz, California, August 13–17, 2013 > ## 1 Measurements of $\gamma$ from $B^{\pm}\rightarrow DK^{\pm}$ The CKM angle $\gamma=\text{arg}(-V_{\text{ud}}V^{}_{\text{ub}}/V_{\text{cd}}V^{}_{\text{cb}})$ is currently the least well-constrained angle in the Unitarity Triangle. So far, the most-sensitive measurements of $\gamma$ from a single experiment have been performed by Belle [1] and BaBar [2]. These measurements yield values of $\left(68^{+15}_{-14}\right)^{\circ}$ and $\left(69^{+17}_{-16}\right)^{\circ}$, respectively. Tree-level processes such as $B^{\pm}\rightarrow DK^{\pm}$ provide a theoretically clean measurement of $\gamma$ with no contributions from new physics processes. This measurement can be compared with measurements from loop-mediated processes, which are sensitive to new physics, to provide a test of the Standard Model. The current limits on the CKM Unitarity Triangle due to tree-level and loop processes, as calculated by the CKMFitter group [3], are shown in Fig. 1. Figure 1: Constraints on the CKM Unitarity Triangle due to (left) tree-level processes and (right) loop-mediated processes. ### 1.1 GLW/ADS analysis of $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ The GLW method [4] uses $D$ decays to $C\\!P$ eigenstates such as $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$. Decays can proceed either via a $D^{0}$ or a $\overline{D}^{0}$ with a phase difference of $\delta_{B}+\gamma$. Suppression in the decay via $D^{0}$ with respect to the $\overline{D}^{0}$ decay limits interference to $\mathcal{O}(10\,\%)$ in $B^{\pm}\rightarrow DK^{\pm}$ and $\mathcal{O}(1\,\%)$ in $B^{\pm}\rightarrow D\pi^{\pm}$. The ADS method [5] uses $D$ decays to quasi-flavour-specific states such as $\pi^{+}K^{-}$ and $\pi^{-}K^{+}\pi^{+}\pi^{-}$. Here the suppression of one of the $B$ decays is partially balanced by the suppression of one of the $D$ decays, giving larger interference terms while also introducing an additional phase shift of $\delta_{D}$. Analyses have been performed on $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm}\rightarrow D\pi^{\pm}$ with the $D$ meson reconstructed from the final states $K^{+}K^{-}$, $\pi^{+}\pi^{-}$, $K^{+}\pi^{-}$, $\pi^{+}K^{-}$, $K^{-}\pi^{+}\pi^{+}\pi^{-}$ and $\pi^{-}K^{+}\pi^{+}\pi^{-}$ using LHCb data corresponding to $1\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass energy of 7 TeV [6, 7]. The invariant mass distributions of the two- and four- body suppressed ADS modes are shown in Fig. 2 and Fig. 3, respectively. Figure 2: Fits to the invariant mass distributions of the two-body suppressed ADS mode $\pi^{\mp}K^{\pm}$ in (top) $B^{\mp}\rightarrow DK^{\mp}$ and (bottom) $B^{\mp}\rightarrow D\pi^{\mp}$. The $B^{\mp}\rightarrow DK^{\mp}$ and $B^{\mp}\rightarrow D\pi^{\mp}$ components are shown in red and green, respectively. The shaded component indicates partially reconstructed backgrond, the dashed magenta line corresponds to partially reconstructed $\Lambda_{b}^{0}\rightarrow\Lambda_{c}^{+}h^{-}$ and the total shape also includes a combinatoric background. Figure 3: Fits to the invariant mass distributions of the four-body suppressed ADS mode $\pi^{\mp}K^{\pm}\pi^{\pm}\pi^{\mp}$ in (top) $B^{\mp}\rightarrow DK^{\mp}$ and (bottom) $B^{\mp}\rightarrow D\pi^{\mp}$. The $B^{\mp}\rightarrow DK^{\mp}$ and $B^{\mp}\rightarrow D\pi^{\mp}$ components are shown in red and green, respectively. The shaded component indicates partially reconstructed backgrond, the dashed magenta line corresponds to partially reconstructed $B_{s}^{0}\rightarrow DK^{-}\pi^{+}$ and the total shape also includes a combinatoric background. The observables measured are the ratio of $DK$ to $D\pi$ for each $D$ final state, $R_{K/\pi}^{f}=\frac{\Gamma\left(B^{-}\rightarrow D\left[\rightarrow f\right]K^{-}\right)+\Gamma\left(B^{+}\rightarrow D\left[\rightarrow\overline{f}\right]K^{+}\right)}{\Gamma\left(B^{-}\rightarrow D\left[\rightarrow f\right]\pi^{-}\right)+\Gamma\left(B^{+}\rightarrow D\left[\rightarrow\overline{f}\right]\pi^{+}\right)}\,,$ the charge asymmetry for each final state, $A_{h}^{f}=\frac{\Gamma\left(B^{-}\rightarrow D\left[\rightarrow f\right]h^{-}\right)-\Gamma\left(B^{+}\rightarrow D\left[\rightarrow\overline{f}\right]h^{+}\right)}{\Gamma\left(B^{-}\rightarrow D\left[\rightarrow f\right]h^{-}\right)+\Gamma\left(B^{+}\rightarrow D\left[\rightarrow\overline{f}\right]h^{+}\right)}\,,$ and the ratio of the suppressed to favoured modes for $D\rightarrow K\pi$ and $D\rightarrow K\pi\pi\pi$, $R_{h}^{\pm}=\frac{B^{\pm}\rightarrow D\left[f_{\text{sup}}\right]h^{\pm}}{B^{\pm}\rightarrow D\left[f\right]h^{\pm}}\,.$ The values obtained for each of these observables can be found in Refs. [6, 7]. These variables serve as inputs for the combined $\gamma$ measurements in Section 1.3 and Section 1.4. ### 1.2 GGSZ analysis of $B^{\pm}\rightarrow DK^{\pm}$ The GGSZ method [8] exploits the variation of the strong phase $\delta_{D}$ across the Dalitz plot in $D$ decays to three-body self-conjugate states such as $K_{S}^{0}\pi^{+}\pi^{-}$ and $K_{S}^{0}K^{+}K^{-}$. The Dalitz plot is divided into bins, as shown in Fig. 4, chosen to maximise statistical sensitivity. The populations of $B^{+}$ and $B^{-}$ decays in each bin are given by $N^{+}_{\pm i}=h_{B^{+}}\left[K_{\mp i}+(x^{2}_{+}+y^{2}_{+})K_{\pm i}+2\sqrt{K_{i}K_{-i}}(x_{+}c_{\pm i}\mp y_{+}s_{\pm i})\right]\,,$ $N^{-}_{\pm i}=h_{B^{-}}\left[K_{\pm i}+(x^{2}_{-}+y^{2}_{-})K_{\mp i}+2\sqrt{K_{i}K_{-i}}(x_{-}c_{\pm i}\pm y_{-}s_{\pm i})\right]\,,$ where $K_{\pm i}$ is the efficiency corrected yield in bin $\pm i$ due to $D^{0}$ flavour tagged events from BaBar [9, 10] and $c_{\pm i}$ and $s_{\pm i}$ are the cosine and sine of the strong phase $\delta_{D}$ in bin $\pm i$ from CLEO-c [11]. Figure 4: Binning schemes used for the Dalitz plots of (left) $D\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$ and (right) $D\rightarrow K_{S}^{0}K^{+}K^{-}$. Bins in the top-left half of the plots ($m^{2}_{K_{S}^{0}h^{-}}>m^{2}_{K_{S}^{0}h^{+}}$) are identified as $+i$ and bins in the bottom-right half are labeled $-i$. The remaining parameters are left free in the fit to the data: $h_{B^{\pm}}$ are normalisation factors for $B^{\pm}$, and $x_{\pm}=r_{B}\text{cos}(\delta_{B}\pm\gamma)$ and $y_{\pm}=r_{B}\text{sin}(\delta_{B}\pm\gamma)$ are the Cartesian parameters, which are sensitive to $\gamma$. Analyses have been performed on $B^{\pm}\rightarrow DK^{\pm}$ with the $D$ meson reconstructed in the final states $K_{S}^{0}\pi^{+}\pi^{-}$ and $K_{S}^{0}K^{+}K^{-}$ using LHCb data corresponding to $1\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass energy of 7 TeV [12] and $2\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass energy of 8 TeV [13]. The values obtained for the Cartesian parameters in the 8 TeV analysis are $x_{+}=\left(-8.7\pm 3.1\text{(stat.)}\pm 1.6\text{(syst.)}\pm 0.6\text{(ext.)}\right)\times 10^{-2}\,,$ $x_{-}=\left(\phantom{-}5.3\pm 3.2\text{(stat.)}\pm 0.9\text{(syst.)}\pm 0.9\text{(ext.)}\right)\times 10^{-2}\,,$ $y_{+}=\left(\phantom{-}0.1\pm 3.6\text{(stat.)}\pm 1.4\text{(syst.)}\pm 1.9\text{(ext.)}\right)\times 10^{-2}\,,$ $y_{-}=\left(\phantom{-}9.9\pm 3.6\text{(stat.)}\pm 2.2\text{(syst.)}\pm 1.6\text{(ext.)}\right)\times 10^{-2}\,,$ where the third uncertainty is due to the CLEO-c strong phase measurements used in the fit. Combining these values with the results from the 7 TeV analysis and fitting for $\gamma$, $r_{B}$ and $\delta_{B}$ yields values of $(57\pm 16)^{\circ}$, $(8.8^{+2.3}_{-2.4})\times 10^{-2}$ and $(124^{+15}_{-17})^{\circ}$, respectively, where the values for $\gamma$ and $\delta_{B}$ are modulo $180^{\circ}$. Two-dimensional projections of the confidence regions for these parameters are shown in Fig. 5. Figure 5: Two-dimensional projections of the confidence regions onto the (left) $(\gamma,r_{B})$ and (right) $(\gamma,\delta_{B})$ planes. Contours indicate the $1$, $2$ and $3\sigma$ boundaries and diamonds mark the central values. ### 1.3 Combination of results from $1\,\text{fb}^{-1}$ measurements The results in Section 1.1 and Section 1.2 are combined using a frequentist approach to obtain a more constraining measurement of $\gamma$ [14]. In addition to these results further measurements are included to improve the fit: measurements of the strong phases and coherence factors for $D\rightarrow K\pi$ and $D\rightarrow K\pi\pi\pi$ decays from CLEO-c[15], $C\\!P$ asymmetry measurements of the neutral $D$ mesons from the Heavy Flavour Averaging Group[16] and charm mixing parameters from LHCb[17]. A likelihood is constructed from the measured observables as ${\cal L}\left(\vec{\alpha}\right)={\displaystyle\prod_{i}}\xi_{i}\left(\vec{A}_{i}^{\text{obs}}\|\vec{\alpha}\right)\,,$ where the sum is over the different measurements, $\vec{\alpha}$ is the set of parameters and $\xi_{i}$ denotes the likelihood probability density functions (PDFs) of the observables $\vec{A}_{i}^{\text{obs}}$. For most observables a Gaussian PDF is assumed, however, where highly non-Gaussian behaviour is observed, the experimental likelihood is used. A combined $\gamma$ measurement has been performed including the results from Section 1.1 and a subset of the results from Section 1.2 corresponding to $1\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass energy of 7 TeV [12]. The best-fit values and confidence intervals (modulo $180^{\circ}$) of $\gamma$ are given in Table 1 and the $1-\text{CL}$ curves for $\gamma$ are shown in Fig. 6. combination \| $\gamma$ \| 68 % CL \| 95 % CL ---\|---\|---\|--- $DK$ \| $72.0^{\circ}$ \| $[56.4,86.7]^{\circ}$ \| $[42.6,99.6]^{\circ}$ $D\pi$ \| $18.9^{\circ}$ \| $[7.4,99.2]^{\circ}\cup[167.9,176.4]^{\circ}$ \| - $DK$ and $D\pi$ \| $72.6^{\circ}$ \| $[55.4,82.3]^{\circ}$ \| $[40.2,92.7]^{\circ}$ Table 1: Best-fit values and confidence intervals for $\gamma$ from the combination of $DK$ and $D\pi$ measurements. Figure 6: $1-\text{CL}$ curves for $\gamma$ from the combined $1\,\text{fb}^{-1}$ GLW/ADS and $1\,\text{fb}^{-1}$ GGSZ measurements using (left) only $DK$, (centre) only $D\pi$ and (right) both decay modes. ### 1.4 Combination including $3\,\text{fb}^{-1}$ GGSZ measurement Another combination [18] has been performed that incorporates all of the results reported in Section 1.2 but only those observables from Section 1.1 corresponding to $B^{\pm}\rightarrow DK^{\pm}$ decays. Mixing in the neutral $D$ mesons is also neglected in the equations used for the observables in this combination. The best-fit values and confidence intervals (all modulo $180^{\circ}$) for $\gamma$, $r_{B}$ and $\delta_{B}$ are given in Table 2. Figure 7 and Figure 8 show the $1-\text{CL}$ curve for $\gamma$, and the 2D projection of the likelihood in $\gamma$ and $r_{B}$, respectively. quantity \| value \| 68 % CL \| 95 % CL ---\|---\|---\|--- $\gamma$ \| $67.2^{\circ}$ \| $[55.1,79.1]^{\circ}$ \| $[43.9,89.5]^{\circ}$ $r_{B}$ \| $0.0923$ \| $[0.0843,0.1001]$ \| $[0.0762,0.1075]$ $\delta_{B}$ \| $114.3^{\circ}$ \| $[101.3,126.3]^{\circ}$ \| $[88.7,136.3]^{\circ}$ Table 2: Best-fit values and confidence intervals for $\gamma$, $r_{B}$ and $\delta_{B}$ from the combination of $DK$ measurements including GGSZ measurements from $3\,\text{fb}^{-1}$ of data. Figure 7: $1-\text{CL}$ curve for $\gamma$ from the combined $1\,\text{fb}^{-1}$ GLW/ADS and $3\,\text{fb}^{-1}$ GGSZ measurements. Figure 8: Two-dimensional projection of the confidence regions onto the $(\gamma,r_{B})$ plane. Contours show the $1$ and $2\sigma$ boundaries and markers indicate the central values. ## 2 Measurement of $B_{(s)}^{0}\rightarrow DK\pi$ branching fractions The decay mode $B^{0}\rightarrow DK^{+}\pi^{-}$ has potential for a significant future measurement of $\gamma$ [19, 20, 21]. Sensitivity to $\gamma$ comes from the interference of $b\rightarrow c$ and $b\rightarrow u$ amplitudes of a similar magnitude. $B_{s}^{0}\rightarrow DK^{-}\pi^{+}$ and the related mode $B_{s}^{0}\rightarrow D^{}K^{-}\pi^{+}$ form important backgrounds to this mode, therefore, an understanding of these modes is necessary. Branching fraction measurements of $B^{0}\rightarrow DK^{+}\pi^{-}$ and $B_{s}^{0}\rightarrow DK^{-}\pi^{+}$, relative to the normalisation mode $B^{0}\rightarrow D\pi^{+}\pi^{-}$, have been made using LHCb data corresponding to $1\,\text{fb}^{-1}$ of $pp$ collisions at a centre of mass energy of 7 TeV [22]. The invariant mass distributions of $D\pi\pi$ and $DK\pi$ candidates where the $D$ is reconstructed from $\overline{D}^{0}\rightarrow K^{+}\pi^{-}$ are shown in Fig. 9. The measured relative branching fractions are $\frac{{\cal B}\left(B^{0}\to\overline{D}^{0}K^{+}\pi^{-}\right)}{{\cal B}\left(B^{0}\to\overline{D}^{0}\pi^{+}\pi^{-}\right)}=0.106\pm 0.007\,\text{(stat.)}\pm 0.008\,\text{(syst.)}\,,$ $\frac{{\cal B}\left(B_{s}^{0}\to\overline{D}^{0}K^{-}\pi^{+}\right)}{{\cal B}\left(B^{0}\to\overline{D}^{0}\pi^{+}\pi^{-}\right)}=1.18\pm 0.05\,\text{(stat.)}\pm 0.12\,\text{(syst.)}\,.$ Figure 9: Fits to the $B^{0}_{(s)}$ candidate invariant mass distributions for the (a) $D\pi\pi$ and (b) $DK\pi$ samples. Data points are shown in black, the full fitted PDFs as solid blue lines and the components as detailed in the legends. These relative measurements yield absolute branching fractions of ${\cal B}\left(B^{0}\to\overline{D}^{0}K^{+}\pi^{-}\right)=(9.0\pm 0.6\,\text{(stat.)}\pm 0.7\,\text{(syst.)}\pm 0.9({\cal B}))\times 10^{-5}\,,$ ${\cal B}\left(B_{s}^{0}\to\overline{D}^{0}K^{-}\pi^{+}\right)=(1.00\pm 0.04\,\text{(stat.)}\pm 0.10\,\text{(syst.)}\pm 0.10({\cal B}))\times 10^{-3}\,,$ where the third uncertainty arises from the uncertainties on ${\cal B}(B^{0}\to\overline{D}^{0}\pi^{+}\pi^{-})$. This is the most precise measurement of ${\cal B}(B^{0}\to\overline{D}^{0}K^{+}\pi^{-})$ to date and the first measurement of ${\cal B}(B_{s}^{0}\to\overline{D}^{0}K^{-}\pi^{+})$. Although no quantitative analysis of the Dalitz plots has yet been attempted, the Dalitz plot distributions obtained (corrected for efficiency) are presented in Fig. 10. Figure 10: Efficiency corrected Dalitz plot distributions for (a) $B^{0}\rightarrow\overline{D}^{0}\pi^{+}\pi^{-}$, (b) $B^{0}\rightarrow\overline{D}^{0}K^{+}\pi^{-}$ and (c) $B_{s}^{0}\rightarrow\overline{D}^{0}K^{-}\pi^{+}$ candidates obtained from the signal weights. ## 3 Conclusions and prospects The $B^{\pm}\rightarrow DK^{\pm}$ decay mode offers an excellent opportunity to measure the CKM angle $\gamma$ from Standard Model processes. The combination in Section 1.4 gives the most sensitive measurement of $\gamma$ from a single experiment so far, yielding a value of $(67\pm 12)^{\circ}$. This measurement is expected to improve further with the completion of a GLW/ADS analysis on the remaining $2\,\text{fb}^{-1}$ of LHCb data currently available. In addition, other modes such as $B^{0}\rightarrow DK^{+}\pi^{-}$ offer great prospects for future $\gamma$ measurements. 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D 79 (2009) 051301 [arXiv:0810.2706 [hep-ph]]. * [21] T. Gershon and M. Williams, Phys. Rev. D 80 (2009) 092002 [arXiv:0909.1495 [hep-ph]]. * [22] R. Aaij et al. [LHCb Collaboration], Phys. Rev. D 87 (2013) 112009 [arXiv:1304.6317 [hep-ex]].	arxiv-papers	2013-09-30T17:29:46	2024-09-04T02:49:51.725944	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Daniel Craik", "submitter": "Daniel Craik", "url": "https://arxiv.org/abs/1309.7928" }
1309.7938	# The NIFFTE project J. Ruza Corresponding author. D.M Asnerb R.G Bakerc J. Bundgaardd E. Burgette M. Cunninghama J. Deavene D.L. Duked U. Greifed S. Grimesf M. Heffnera T. Hillg D. Isenhowerh J.L. Klayc V. Kleinrathe N. Kornilovf A.B. Laptevi W. Lovelandj T.N. Masseyf R. Meharchandi H. Quh S. Sangiorgioa B. Seilhana L. Snydera S. Staveb G. Tatishvilib R.T. Thorntonh F. Tovessoni D. Towellh R.S. Towellh S. Watsonh B. Wendte and L. Woodb aLawrence Livermore National Laboratory Livermore CA 94550 USA bPacific Northwest National Laboratory Richland WA 99354 USA cCalifornia Polytechnic State University San Luis Obispo CA 93407 USA dColorado School of Mines Golden CO 80401 USA eIdaho State University Pocatello ID 83209 USA fOhio University Athens OH 45701 USA gIdaho National Laboratory Idaho Falls ID 83415 USA hAbilene Christian University Abilene TX 79699 USA iLos Alamos National Laboratory Los Alamos NM 87545 USA jOregon State University Corvallis OR 97331 USA E-mail [email protected] ###### Abstract The Neutron Induced Fission Fragment Tracking Experiment (NIFFTE) is a double- sided Time Projection Chamber (TPC) with micromegas readout designed to measure the energy-dependent neutron-induced fission cross sections of the major and minor actinides with unprecedented accuracy. The NIFFTE project addresses the challenge of minimizing major sources of systematic uncertainties from previous fission chamber measurements such as: target and beam non-uniformities, misidentification of alpha and light charged particles as fission fragments, and uncertainties inherent to the reference standards used. In-beam tests of the NIFFTE TPC at the Los Alamos Neutron Science Center (LANSCE) started in 2010 and have continued in 2011, 2012 and 2013. An overview of the NIFFTE TPC status and performance at LANSCE will be presented. ###### keywords: Fission; TPC; Cross Section ## 1 Introduction Sensitivity studies indicate that high-accuracy measurements of fission cross sections are needed and the Neutron Induced Fission Fragment Tracking Experiment (NIFFTE) collaboration is developing a Time Projection Chamber (TPC) to meet this need. Neutron-induced fission cross sections have been measured with fission chambers (ICs) for decades, typically by placing a thin target of the fissile material of interest into a neutron beam and detecting the ionization left in a gas by the resulting fission products. To date, fission chambers have been limited to measuring the energy deposited in a gas by charged particles, and were therefore unable to distinguish between different particles of the same energy. Fragments are subject to energy loss and straggling within the target111A non negligible fraction of fission fragments might become indistinguishable from lighter charged particles within the fission chamber due to energy loss and straggling. resulting in a systematic error for the measured cross section. Based on well-established technology originally developed for high-energy physics experiments, the NIFFTE TPC is uniquely suited to achieve unprecedented accuracy. The detection principle of a TPC is rather simple: particles ionizing the gas traverse a uniform electric field. The electrons produced by the ionization are drifted by the electric field and collected at a segmented readout plane. A two-dimensional projection of the particle’s track is obtained by interpreting the amount of charge deposited in each segment. Combining this two dimensional projection with the drift time of the electrons provides a full, three-dimensional representation of the particle’s path through the detector. A schematic of a neutron-induced fission in the TPC is shown in figure 1. The fissionable sample to be studied is placed in the center of the detector at the midpoint of the cathode. The field cage covers the $\rm{54\,mm}$ drift distance at each side of the sample and it is placed inside a pressure vessel of $\rm{15\,cm}$ diameter that is filled up with a suitable gas mixture e.g. Argon($90\,\%$)-Methane($10\,\%$) at a nominal $\rm{550\,torr}$. At each end of the chamber charge is amplified and read out on a MICROMEGAS “pad” plane. The signals are digitized and processed by custom-made EtherDAQ card assemblies [1, 2] sampling at $50\,\rm{MHz}$ together with a software framework developed for the data acquisition, analysis, and simulation tasks [3]. An article describing the TPC hardware is in preparation. Figure 1: Schematic of a neutron-induced fission in the target of the NIFFTE TPC (left) and picture of the TPC installed in the LANSCE-WNR facility during the 2012 run cycle. ## 2 Beam experiments Building a TPC suitable for fission research is an ambitious task, involving complex systems and state-of-the-art technology. As new hardware and software capabilities are implemented, frequent evaluation is required. In-beam testing takes place at the 4FP90L flight path of Los Alamos Neutron Science Center (LANSCE) Weapons Neutron Research (WNR) facility [4] that makes use of an unmoderate tungsten spallation target to produce a high-energy neutron spectrum with energies ranging from $100\,\rm{keV}$ to $600\,\rm{MeV}$. The first TPC experiments at LANSCE-WNR took place in 2010. Data were collected first with 64 then 192 individual channels ($\rm{1\,\%}$ and $\rm{3\,\%}$ of the full TPC), with blank carbon samples and on carbon backings with $\rm{{}^{238}U}$ deposited onto them [5]. These early measurements validated the TPC technique, as particle tracks could be reconstructed for both fission fragments and alpha particles. During the 2011 LANSCE run cycle, one complete sextant (496 channels, $\rm{8\,\%}$ of the full TPC) was instrumented, and data were collected with samples containing $\rm{{}^{238}U}$, $\rm{{}^{235}U}$, and $\rm{{}^{239}Pu}$. During the most recent run cycle (2012), the TPC was instrumented with one complete pad plane (2976 channels, 1/2 of the full detector), and notably, for the first time in the TPC project, information about the neutron time-of-flight was available. ## 3 Performance Preliminary analysis of the data taken during the 2010 and 2011 LANSCE run cycles have been reported [6, 7] and the 2012 run cycle data is currently being analyzed. Data were collected on three samples: a split sample (semicircular deposits) containing $\rm{{}^{238}U}$ and $\rm{{}^{235}U}$; a split sample with $\rm{{}^{235}U}$ and $\rm{{}^{239}Pu}$; and a sample with $\rm{{}^{239}Pu}$ electro-plated onto metallized polyproplylene. In adition, off-beam data was collected with the use of $\rm{{}^{252}Cf}$ and $\rm{{}^{244}Cm}$ bottom sources to fine tune the acquisition software and the analysis codes. All results included in this paper correspond to the 2012 data taking period and should be considered preliminary. ### 3.1 Neutron Time of Flight Neutron-induced fission depends on the energy of the incident neutron. The neutron Time-Of-Flight (nTOF) between the spallation target at WNR and the fissionable target placed inside the TPC can be folded together with the overall WNR beam spacing structure -nominal $1.8\,\rm{\mu s}$ pulse spacing between neutron bunches -to reproduce the energy spectrum of the neutrons inducing fission events in the TPC. Figure 2 shows a reconstruction of the nTOF for the NIFFTE TPC during the 2012 run cycle. The peak around $28\,\rm{nsec}$ corresponds to photo-induced fission in the TPC target and, can be used to obtain an absolute calibration for both, the distance of the TPC relative to the spallation target and the energy of the incoming neutrons. The width of the photo-fission peak serves as a tool to determine the systematic error on the energy of the incident neutron. During 2012, the NIFFTE TPC was able to achieve $2.5-3.5\,\rm{nsec}$ FWHM for the photo-fission peak, which is competitive with the resolution obtained by previous fission chamber experiments. Figure 2: Neutron time of flight reconstruction for neutron-induced fission events in the TPC and zoom of the observed photo-fission peak. A split sample (semicircle deposits) containing $\rm{{}^{238}U}$ and $\rm{{}^{235}U}$ was used during this measurement. ### 3.2 Tracking and Particle Identification One of the major advantages of using a TPC to study neutron-induced fission cross sections is the ability to reconstruct and study the features of the ionizing tracks that the different particles generate on their way through the active volume of NIFFTE. One example of how particle identification (distinguishing alpha and light charged particles from fission fragments) might be done using the tracking information of the TPC can be shown by comparing the length of the particle’s track to its total energy deposited in the TPC (see figure 3). Fission fragments, have relatively short tracks and deposit large amount of energy in the fill gas, while light particles (alpha particles) have longer tracks and deposit less energy. Figure 3: Plot of length versus energy of track reconstructions in the TPC for a $\rm{{}^{252}Cf}$ buttom source. On the plot there are three regions that can be easily spotted, an almost vertical line at low energies with lengths from $\rm{1\,cm}$ up to $\rm{6\,cm}$ containing $\alpha$-particles and protons, a big elliptical zone centered around $\rm{80\,MeV}$ with tracks from $\rm{1.5\,cm}$ up to $\rm{3\,cm}$ gathering most of the fission fragment population and a small shadow to the elipse with track lengths around $\rm{1\,cm}$ and energies between $\rm{5}$ and $\rm{40\,MeV}$. The shadow region has been succesfully identified as fission fragments that come out of the target at high polar angles, suffering from straggling, and hit on the concentric copper ring of $\rm{2}$ by $\rm{4\,cm}$ diameter that holds the polypropylene backing foil where the $\rm{2\,mm}$ diameter $\rm{{}^{252}Cf}$ spot was deposited. Using precise tracking information, e.g. the differential ionization pattern of the particle within the gas, allows the search of the specific Bragg peaks created by the particles and the comparison of their ionization pattern with simulations and theoretical models to provide an ultimate tool for particle identification. Fission fragments deposit most of the energy at the begining of their short track through the TPC, while $\alpha$ particles and protons penetrate the gas further and have their Bragg peak towards the end of their path through the TPC (see figures 4 and 5). Figure 4: Ionization pattern of a fission fragment in the NIFFTE TPC. The Bragg peak is located at the begining of the track for these particles. Finally, tracking information also enables tracing of each individual particle back to a well-defined origin in the sample, which ultimately allows for a minimization on the systematics errors such as beam spot size, and sample uniformity. This makes it possible to study two or more actinides on the same backing, as described in [5], and to check sample uniformity using the decay activity of the actinide deposits. Figure 5: Ionization pattern of an observed $\alpha$-track in the NIFFTE TPC (black) and a theoretical prediction that includes the ionization nature of an $\alpha$-like particle together with the specific diffusion expected in the TPC for this particular track (blue). In red the residual of observed and modeled ionization pattern is shown. Figure 6: Track vertex reconstructions of $\alpha$-like events emerging from a $\rm{2\,mm}$ spot source of $\rm{{}^{252}Cf}$ placed in the center of the NIFFTE TPC. In the figure a picture of the sample used can be seen. Figure 6 shows the vertex reconstructions for spontaneous $\alpha$-like events from a $\rm{{}^{252}Cf}$ bottom source. The tracks seem to originate from the center of the cathode plane, coming out of a circle of approximately $\rm{2\,mm}$ diameter, which is consistent with the actual dimension of the $\rm{{}^{252}Cf}$ deposit. Note that the two different backing materials can easily be distinguished: one external ring of copper with inner and outer diameters of $\rm{2\,cm}$ and $\rm{4\,cm}$ respectively, and a $\rm{2\,cm}$ diameter polypropylene foil on top of which the $\rm{{}^{252}Cf}$ was deposited. This figure clearly shows the radiographic capabilities of NIFFTE to map sample uniformities. ## 4 Conclusion The NIFFTE TPC is being tested off-beam and in-beam at the neutron facility LANSCE-WNR. Preliminary results (with half detector instrumented) indicate that the TPC performance is meeting expectations. As the NIFFTE project moves from development into production phase, the collaboration is optmistic that the project is on-track to deliver neutron-induced fission cross sections with unprecedented accuracy. ###### Acknowledgements. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. DE-AC52-07NA27344. ## References * [1] M. Heffner, V. Riot, and L. Fabris _IEEE Transactions on Nuclear Science_ 60, _3_ (June 2013). * [2] L. Wood _et al._ , _Proceedings of the International Conference on Nuclear Data for Science and Technology_ , _Nuclear Data Sheets_ (2013). * [3] S. Stave _et al._ , _Proceedings of the International Conference on Nuclear Data for Science and Technology_ , _Nuclear Data Sheets_ (2013). * [4] P.W. Lisowski and K.F. Schoenberg, _Nuclear Instruments and Methods in Physics Search Section A_ 562, _910_ (2006). * [5] W. Loveland, L. Yao _et al._ , _Proceedings of the International Conference on Nuclear Data for Science and Technology_ , _Nuclear Data Sheets_ (2013). * [6] F. Tovesson, A.B. Laptev, R. Meharchand, _DOE-NE Technical Report LA-UR 11-05439_ , September 2011. * [7] R. Meharchand, T. Hill, L. Snyder, _Los Alamos National Laboratory Report LA-UR 12-25452_ , October 2012.	arxiv-papers	2013-09-30T17:39:47	2024-09-04T02:49:51.732312	{ "license": "Public Domain", "authors": "Ruz J., Asner D.M., Baker R.G., Bundgaard J., Burgett E., Cunningham\n M., Deaven J., Duke D.L., Greife U., Grimes S., Heffner M., Hill T.,\n Isenhower D., Klay J.L., Kleinrath V., Kornilov N., Laptev A.B., Loveland W.,\n Masseyf T.N., Meharchand R., Qu H., Sangiorgio S., Seilhan B., Snyder L.,\n Stave S., Tatishvili G., Thornton R.T., Tovesson F., Towell D., Towell R.S.,\n Watson S., Wendt B. and Wood L", "submitter": "Jaime Ruz Armendariz", "url": "https://arxiv.org/abs/1309.7938" }
1309.7961	# Prospects For Precision Measurements with Reactor Antineutrinos at Daya Bay Daya Bay Collaboration (September 27, 2013) ###### Abstract In 2012 the Daya Bay experiment made an unambiguous observation of reactor antineutrino disappearance over kilometer-long baselines and determined that the neutrino mixing angle $\theta_{13}$ is non-zero. The measurements of Daya Bay have provided the most precise determination of $\theta_{13}$ to date. This whitepaper outlines the prospects for precision studies of reactor antineutrinos at Daya Bay in the coming years. This includes precision measurements of sin22$\theta_{13}$ and $\Delta m^{2}_{ee}$ to $<$3%, high- statistics measurement of reactor flux and spectrum, and non-standard physics searches. ## I Introduction Since the discovery of a non-zero value for $\theta_{13}$ DYB1 , Daya Bay has provided the most precise measurement of sin22$\theta_{13}$ of $0.089\pm 0.010\pm 0.005$ with 139 days of 6 antineutrino detector (AD) data DYB2 in June 2012. Recently, with 217 days of 6-AD data, we have obtained the most precise measurement of sin22$\theta_{13}=0.090^{+0.008}_{-0.009}$ and the first measurement of $\Delta m^{2}_{ee}=(2.59^{+0.19}_{-0.20})\times 10^{-3}~{}eV^{2}$. Two more ADs were installed and have been in operation since October 19, 2012. We present the expected precision in sin22$\theta_{13}$ and $\Delta m^{2}_{ee}$ as a function of time in Sections II and III respectively. The prospects for precise measurement of the reactor antineutrino flux and spectrum are summarized in Section IV. ## II Precision Measurement of sin22$\theta_{13}$ Daya Bay’s currently published measurements of sin22$\theta_{13}$ are still statistics limited. With the data collected with 8 ADs as of October 2013 the statistical error will reach the current systematic uncertainty of 0.005 DYB2 . We have assessed improvements in the systematic uncertainty based on calibration data and increased precision that is statistical in nature (such as the uncertainties in estimating the 9Li/8He and fast neutron backgrounds). Based on the projected systematic uncertainty we estimate that the total uncertainty in sin22$\theta_{13}$ can be reduced from 0.008 to 0.006 in one year and 0.003 after 3 to 4 years as shown in Fig. 1. Figure 1: Projected total uncertainty in sin22$\theta_{13}$ for the Daya Bay experiment. The current systematic uncertainty of 0.005 is used at the beginning of the 8AD run. Figure 2: Significance with which CP violation can be observed, by NOvA+T2K (left) and NOvA+T2K+LBNE (right), as a function of the value of $\delta_{CP}$. Observation of CP violation is equivalent to measuring $\delta_{CP}\neq 0,\pi$. The significance is calculated by minimizing over normal and inverted hierarchies, as the hierarchy is assumed to be unknown. The impact of $\theta_{13}$ precision is shown. Daya Bay’s short-baseline antineutrino disappearance measurement is the most precise reactor measurement of sin22$\theta_{13}$ and will likely remain the most precise measurement of this fundamental parameter for the foreseeable future. Precise measurement of this quantity may shed light on symmetries between quarks and leptons at a fundamental level, and test unitarity of the neutrino mixing matrix. Comparison of future long baseline accelerator measurements of $\theta_{13}$ to Daya Bay will allow precision tests of the neutrino-standard model interpretation of neutrino oscillations with sensitivity to non-standard neutrino interactions and sterile neutrino scenarios. The expected improvement in sin22$\theta_{13}$ by Daya Bay will make $\theta_{13}$ the most precisely measured neutrino mixing angle and extend the CP reach (shown in Fig. 2) and $\theta_{23}$ octant determination of long baseline accelerator experiments. ## III Precision Measurement of $\Delta m^{2}_{ee}$ Our discovery of a large value for $\theta_{13}$ enables a measurement of the effective mass splitting $\Delta m^{2}_{ee}$ with a precision comparable to the MINOS measurement of $\Delta m^{2}_{\mu\mu}$. The disappearance of electron antineutrinos over km-long baselines observed at Daya Bay is a combination of antineutrino oscillations with mass splittings $\Delta m^{2}_{31}$ and $\Delta m^{2}_{32}$. Complementary to $\Delta m^{2}_{\mu\mu}$ determined by accelerator-based experiments, Daya Bay will provide a precise measurement of $\Delta m^{2}_{ee}$, as shown in Fig. 3, better than 3% after five years of data collection with the 8-AD configuration. In addition to the neutrino mixing angles and the Dirac CP-violating phase, the mass-squared differences are crucial for understanding the nature of neutrinos. First, independent determination of the three mass-squared differences will validate the sum rule of neutrino mixing, $\Delta m^{2}_{21}+\Delta m^{2}_{32}+\Delta m^{2}_{13}=0$. Any deviation from this sum rule could signal the existence of neutrinos beyond three generations. In addition, comparison of $\Delta m^{2}_{ee}$ and $\Delta m^{2}_{\mu\mu}$ will provide direct constraint to the neutrino mass hierarchy (sign of $\Delta m^{2}_{31}$). Figure 3: Expected Daya Bay uncertainty in $\Delta m^{2}_{ee}$ as a function of running time with 8 ADs. The current systematic uncertainty is used at the beginning of the run. The present error of the 6-AD data set is $0.2\times 10^{-3}~{}eV^{2}$. Since the fundamental parameters $\Delta m^{2}_{32}$ and $\Delta m^{2}_{31}$ are directly related to the effective mass splitting $\Delta m^{2}_{ee}$ in the L/E region for Daya Bay, the impact of the Daya Bay measurements in both sin22$\theta_{13}$ and $\Delta m^{2}_{ee}$ with three years of 8-AD data is illustrated in Fig. 4. Only statistical uncertainties are shown. The relative systematic uncertainty, which is relevant for the oscillation analysis, is still smaller than the expected statistical uncertainty of the far site. The overall normalization is not constrained in the fit, and the best-fit normalization is used in Fig. 4. Figure 4: Expected L/E distribution of Daya Bay with 8 ADs in three experimental halls, EH1, EH2, and EH3 after three years of running. Only statistical uncertainties are shown. ## IV Precision Measurement of the Reactor Antineutrino Flux and Spectrum Daya Bay collects reactor antineutrino data at a tremendous rate which enables a precision measurement of the reactor antineutrino spectra in the four near site detectors. For the data sets published to date based on six detectors, Daya Bay has collected $>$300k antineutrino events. Two additional detectors, one at the Ling Ao near site and one at the far site, have been operational since October 2012. The Daya Bay experimental configuration allows spectral and flux measurements as close as 360m from the reactor cores at the Daya Bay site and 480m at the Ling Ao site. At these detector locations the measured reactor antineutrino flux from the nearest reactor cores remains largely non- oscillated in the standard 3-neutrino oscillation framework. The contribution from the more distant reactors at $\sim$900m is approximately 20% (7.5%) of the total event rate at Daya Bay (Ling Ao). The oscillation effects of antineutrinos from the far reactor and resulting distortions in the measured antineutrino spectrum can be corrected for in any spectral analysis assuming a 3-neutrino framework. With this unique configuration of multiple, functionally identical antineutrino detectors at various baselines Daya Bay will be able to report the following measurements and physics analyses: ### IV.1 High-precision measurement of the reactor antineutrino spectrum Using data from the near and far sites Daya Bay will make a measurement of the reactor antineutrino spectrum with high precision. Amongst all running reactor experiments Daya Bay will collect the largest sample of antineutrino events and achieve a $<$1% statistical uncertainty in a 2-year run over a large range of energies at the near sites. ### IV.2 Test of the reactor antineutrino spectrum vs predictions and search for new interactions Using known reactor operation data such as thermal power output and fission fraction evolution from reactor core simulations Daya Bay can predict the expected non-oscillated energy spectrum of reactor antineutrinos emitted from each reactor. The Daya Bay collaboration works closely with the reactor company and relevant operation data are provided to the collaboration on a regular basis. A precise comparison of the predicted reactor antineutrino spectrum with the spectrum measured at Daya Bay will test our understanding and calculations of antineutrino emitted from reactors. This is particularly relevant in the context of recent discussions of our understanding of reactor flux calculations. Discrepancies in the spectral shape may point to (a) missing nuclear physics in the reactor spectrum predictions or (b) new physics beyond the 3-neutrino framework including non-standard interaction (NSI) effects, and sterile neutrinos. This spectral shape test is independent of the absolute flux normalization and the uncertainties in the predicted total antineutrino rate at Daya Bay. Due to the high statistics of the Daya Bay measurement, the statistical uncertainty in the 2011-2012 Daya Bay data set is already below the flux conversion uncertainty on the spectrum. Daya Bay’s measurement of the reactor antineutrino spectrum will be ultimately limited by our understanding of energy scale uncertainties and detector effects. Detector studies and simulations are ongoing to improve the energy response model of the Daya Bay detectors. ### IV.3 Absolute measurement of the reactor flux In addition to a measurement of the spectral shape, Daya Bay will measure the absolute reactor flux from the six Daya Bay and Ling Ao reactor cores. The absolute flux measurement tests our understanding of reactor flux models within the theoretical uncertainties of the predictions and the experimental uncertainties associated with the absolute detection efficiencies of the Daya Bay detectors. An absolute flux measurement in the antineutrino detectors at Daya Bay will provide unique data points at the baselines of the Daya Bay experiment (360-2,000 m) and will further our understanding if there is a deficit in the measured reactor neutrino flux at short baselines, also known as the “reactor anomaly”. An analysis of past reactor experiments compared with predictions has revealed a discrepancy of about 5.7% in the absolute antineutrino flux. While Daya Bay has demonstrated superb relative detector uncertainties, an absolute measurement will be systematics limited. A statistical precision of 0.1% will be achievable in the Daya Bay data set. Improvements in the analysis may eventually reduce absolute detection uncertainties to $\sim$1%. An absolute reactor flux measurement can test the theoretical flux prediction with the uncertainty currently estimated at about 2.7%. We can compare the Daya Bay’s measurements to previous reactor flux measurements, for example, the absolute Bugey-4 measurement with an uncertainty of 1.4%. Daya Bay’s measurement of the absolute flux and reactor antineutrino spectrum will provide important input to our understanding of the “reactor anomaly”. ### IV.4 Study of the time-evolution of the reactor antineutrino flux The large reactor antineutrino event rate measured and the unique combination of baselines and reactor cores at Daya Bay allows a detailed study of the time variation of the reactor antineutrino flux. This contains information on the operation of the reactors as well as the evolution and isotopic composition of the core’s fuel. Correlating the measured antineutrino flux with reactor operations is of interest to reactor monitoring, the safeguard community, and applied neutrino science. With six reactors and 4 near-site detectors Daya Bay will provide the largest data set on reactor flux variations as a function of time. ## References * (1) F. P. An et al., Phys. Rev. Lett. 108, 171803 (2012). * (2) F. P. An et al., Chin. Phys. C37, 011001 (2013).	arxiv-papers	2013-09-30T19:01:14	2024-09-04T02:49:51.738504	{ "license": "Public Domain", "authors": "The Daya Bay Collaboration", "submitter": "Kam Biu Luk", "url": "https://arxiv.org/abs/1309.7961" }
1310.0003	# Quantum criticality of geometric phase in coupled optical cavity arrays under linear quench Sujit Sarkar Poornaprajna Institute of Scientific Research, 4 Sadashivanagar, Bangalore 5600 80, India. e-mail: [email protected] phone: 0091-80-23611836 Fax: 0091-80-23619034 ###### Abstract The atoms trapped in microcavities and interacting through the exchange of virtual photons can be modeled as an anisotropic Heisenberg spin-1/2 lattice. We study the dynamics of the geometric phase of this system under the linear quenching process of laser field detuning which shows the XX criticality of the geometric phase in presence of single Rabi frequency oscillation. We also study the quantum criticality for different quenching rate in the presence of single or two Rabi frequencies oscillation in the system. PACS: 42.50.Pq, 03.65.Vf, 42.50.-p Keywords: Cavity Quantum Electrodynamics, Geometric Phase and Quantum Optics. Introduction: The recent experimental success in engineering strong interaction between the photons and atoms in high quality micro-cavities opens up the possibility to use light matter system as quantum simulators for many body physics [1-16]. The authors of Ref. (hart1 ),(hart2 ) and (sujop ) have shown that effective spin lattice can be generated with individual atom in the micro-cavities that are coupled to each other via exchange of virtual photons. The two states of spin polarization are represented by the two long lived atomic levels in the system. A Many body Hamiltonians can be created and probed in coupled cavity arrays. In our previous study, we have explained explicitly the basic physics of the formation of micro-optical cavity sujop . The atoms in the cavity are used for detection and also for generation of interaction between photons in the same cavity. As the distance between the adjacent cavities is considerably larger than the optical wave length of the resonant mode, individual cavities can be addressed. This artificial system can act as a quantum simulator. In this optical cavities system one can find different quantum phases of polariton ( a combined excitations of atom-photon interactions. ) by using the spin model that conserve the total number of excitations. This micro-cavity system shows the different quantum phases and quantum phase transitions. Quantum Phase Transition (QPT) associate with the fundamental changes that occurs in the macroscopic nature of the matter at zero temperature due to the variation of some external parameter. The quantum phase transitions are characterized by the drastic change in the ground state properties of the system driven by the quantum fluctuations ss . Motivation of this research paper: To the best of our knowledge, here we not only study the dynamics of geometric phase but also solve the nature of criticality explicitly under laser field detuning quenching process for different quenching rate which is absent in the previous literature of cavity QED [1-16]. Here we mention very briefly the essence of the geometric phase in the condensed matter. The geometric phases have been associated with a variety of condensed matter phenomena thou ; resta ; hatsugai ; bb since its inception berry . Besides, various theoretical investigations, the geometric phases have been experimentally tested in various cases, e.g. with photons p1 ; p2 ; p3 , with neutrons n1 ; n2 and with atoms a1 . The quantum state engineering of cavity QED is in the state of art due to the rapid experimental/technological progress of the subject. We hope that the theoretical scheme which we propose for the laser field detuning induce quenching process in the dynamics of geometric phase will be predicted by the experimental group based on photon. The generation of the geometric phase (GP) is a witness of a singular point in the energy spectrum that arises in all non-trivial geometric evolutions. In this respect, the connection of the geometric phase with quantum phase transition (QPT) has been explored very recently car ; zhu ; hamma ; sujgeo . The geometric phase can be used as a tool to probe QPT in many body systems. Since response times typically diverge in the vicinity of the critical point, sweeping through the phase transition with a finite velocity leads to a breakdown of adiabatic condition and generate interesting dynamical (non- equilibrium ) effects. In the case of thermal phase transitions, the Kibble- Zurek (KZ) mechanism kib ; zur explains the formation of defects via rapid cooling. This idea of defect formation in second order phase transition has been extended to zero temperature QPT zur1 ; dziar by studying the spin models under linear quench. We will use this concept in the present study. The micro-cavities of a photonic crystal are coupled through the exchange of photons. Each cavity consists of one atom with three levels in the energy spectrum, two of them are long lived and represent two spin states of the system and the other represent excited states Ref. hart1 ; hart2 ; sujop ). Externally applied laser and cavity modes couple to each atom of the cavity. It may induce the Raman transition between these two long lived energy levels. Under a suitable detuning between the laser and the cavity modes, virtual photons are created in the cavity which mediate interactions with another atom in a neighboring cavity. One can eliminate the excited states by choosing the appropriate detuning between the applied laser and cavity modes. Then one can achieve only two states per atom in the long lived state and the system can be described by a spin-1/2 Hamiltonian hart1 ; hart2 ; sujop . The Hamiltonian of our present study consists of three parts: $H~{}=~{}{H_{A}}~{}+~{}{H_{C}}~{}+~{}{H_{AC}}$ (1) Hamiltonians are the following ${H_{A}}~{}=~{}\sum_{j=1}^{N}{{\omega}_{e}}\|e_{j}><e_{j}\|~{}+~{}{\omega}_{ab}\|b_{j}><b_{j}\|$ (2) where $j$ is the cavity index. ${\omega}_{ab}$ and ${\omega}_{e}$ are the energies of the state $\|b>$ and the excited state respectively. The energy level of state $\|a>$ is set as zero. $\|a>$ and $\|b>$ are the two stable state of a atom in the cavity and $\|e>$ is the excited state of that atom in the same cavity. The following Hamiltonian describes the photons in the cavity, ${H_{C}}~{}=~{}{{\omega}_{C}}\sum_{j=1}^{N}{{c_{j}}}^{\dagger}{c_{j}}~{}+~{}{J_{C}}\sum_{j=1}^{N}({{c_{j}}}^{\dagger}{c_{j+1}}+h.c),$ (3) where ${c_{j}}^{\dagger}({c_{j}})$ is the photon creation(annihilation) operator for the photon field in the $j$’th cavity, ${\omega}_{C}$ is the energy of photons and $J_{C}$ is the tunneling rate of photons between neighboring cavities. The interaction between the atoms and photons and also by the driving lasers are described by ${H_{AC}}~{}=~{}\sum_{j=1}^{N}[(\frac{{\Omega}_{a}}{2}e^{-i{{\omega}_{a}}t}+{g_{a}}{a_{j}})\|e_{j}><a_{j}\|+h.c]+[a\leftrightarrow b].$ (4) Here ${g_{a}}$ and ${g_{b}}$ are the couplings of the cavity mode for the transition from the energy states $\|a>$ and $\|b>$ to the excited state. ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequencies of the lasers with frequencies ${\omega}_{a}$ and ${\omega}_{b}$ respectively. The authors of Ref. hart1 ; hart2 ; sujop have derived an effective spin model by considering the following physical processes: A virtual process regarding emission and absorption of photons between the two stable states of neighboring cavity yields the resulting effective Hamiltonian as ${H_{xy}}=\sum_{j=1}^{N}B{{\sigma}_{j}}^{z}~{}+~{}\sum_{j=1}^{N}(\frac{J_{1}}{2}{{\sigma}_{j}}^{\dagger}{{\sigma}_{j+1}}^{-}~{}+~{}\frac{J_{2}}{2}{{\sigma}_{j}}^{-}{{\sigma}_{j+1}}^{-}+h.c)$ (5) When $J_{2}$ is real then this Hamiltonian reduces to the XY model. Where ${{\sigma}_{j}}^{z}=\|b_{j}><b_{j}\|~{}-~{}\|a_{j}><a_{j}\|$, ${{\sigma}_{j}}^{+}=\|b_{j}><a_{j}\|$, ${{\sigma}_{j}}^{-}=\|a_{j}><b_{j}\|$ . $\displaystyle H_{xy}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N}(B{{\sigma}_{i}}^{z}~{}+~{}{J_{1}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}+{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y})$ (6) $\displaystyle+{J_{2}}({{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}-{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}))$ $\displaystyle=\sum_{i=1}^{N}B({{\sigma}_{i}}^{z}~{}+~{}{J_{x}}{{\sigma}_{i}}^{x}{{\sigma}_{i+1}}^{x}~{}+~{}{J_{y}}{{\sigma}_{i}}^{y}{{\sigma}_{i+1}}^{y}).$ With ${J_{x}}=(J_{1}+J_{2})$ and ${J_{y}}=(J_{1}-J_{2})$. We follow the references jame ; hart1 to present the analytical expression for the different physical parameters of the system. $B=\frac{\delta_{1}}{2}-\beta$ (7) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle\frac{1}{2}[\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}({\Delta}_{b}-\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}-$ (8) $\displaystyle\frac{{\|{\Omega_{b}}\|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-{\gamma_{b}}{g_{b}}^{2}-{\gamma_{1}}{g_{a}}^{2}+{\gamma_{1}}^{2}\frac{{g_{a}}^{4}}{{\Delta_{b}}}-(a\leftrightarrow b)]$ ${J_{1}}=\frac{\gamma_{2}}{4}(\frac{{\|{\Omega_{a}}\|}^{2}{g_{b}}^{2}}{{{\Delta}_{a}}^{2}}+\frac{{\|{\Omega_{b}}\|}^{2}{g_{a}}^{2}}{{{\Delta}_{b}}^{2}})$ (9) ${J_{2}}=\frac{\gamma_{2}}{2}(\frac{{\Omega_{a}}{\Omega_{b}}g_{a}g_{b}}{{\Delta}_{a}{\Delta_{b}}}).$ (10) Where $\gamma_{a,b}=\frac{1}{N}\sum_{k}\frac{1}{{\omega}_{a,b}-{\omega}_{k}}$ $\gamma_{1}=\frac{1}{N}\sum_{k}\frac{1}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$ and $\gamma_{2}=\frac{1}{N}\sum_{k}\frac{e^{ik}}{({\omega}_{a}+{\omega}_{b})/2-{\omega}_{k}}$ ${\delta_{1}}={\omega}_{ab}-({\omega}_{a}-{\omega}_{b})/2$, ${\Delta}_{a}={\omega}_{e}-{\omega}_{a}$. ${\Delta}_{b}={\omega}_{e}-{\omega}_{a}-({\omega}_{ab}-{\delta_{1}})$. ${{\delta}_{a}}^{k}={\omega}_{e}-{\omega}_{k}$, ${{\delta}_{b}}^{k}={\omega}_{e}-{\omega}_{k}-({\omega}_{ab}-{\delta_{1}})$, $g_{a}$ and $g_{b}$ are the couplings of respective transition to the cavity mode, ${\Omega}_{a}$ and ${\Omega}_{b}$ are the Rabi frequency of laser with frequency $\omega_{a}$ and $\omega_{b}$. Model Hamiltonian and Quantum Phases: We express our model Hamiltonian in a more explicit way as $H~{}=~{}\sum_{n}~{}[(1+\alpha)~{}S_{n}^{x}S_{n+1}^{x}~{}+~{}(1-\alpha)~{}S_{n}^{y}S_{n+1}^{y}+B\sum_{n}~{}S_{n}^{z}$ (11) where $S_{n}^{\alpha}$ are the spin-1/2 operators. We assume that the $XY$ anisotropy $0<\alpha\leq 1$ and magnetic field strength is $h\geq 0$. The parameters correspondence between the micro cavities and spin chain are the following: ${J_{1}=1}$ and ${J_{2}}=\alpha$. Figure 1: Color online, variation of geometric phase with time for different quenching time for $k=\pi/3$. The anisotropy parameter ($\alpha=1$, transverse Ising model) for the dash curves and inset for $\alpha=0$ (isotropic XX spin model). Figure 2: Color online, variation of the geometric phase with time for three different quenching time for $k=\pi/3$. The solid curve is for the $\alpha=0.2$ and dashed curve is for $\alpha=0.8$. Figure 3: Color online, variation of geometric phase with time for the quenching time ${\tau_{q}=1000}$ for different anisotropy in exchange couplings for $k=\pi/3$. The all curves are meeting in the point at $t={\tau_{q}}$. Geometric Phase and Criticality: Here, we calculate the geometric phase and its dynamics under the quenching of magnetic field. In this model, the geometric phase of the ground state is evaluated by applying a rotation of $\phi$ around the $z$-axis in a closed circuit to each spin car ; car1 . A new set of Hamiltonians $H_{\phi}$ is constructed from the Hamiltonian (H) as $H_{\phi}=U(\phi)~{}H~{}U^{\dagger}(\phi)$ (12) where $U(\phi)=\prod_{j=-M}^{+M}\exp(i\phi\sigma_{j}^{z}/2)$ and $\sigma_{j}^{z}$ is the $z$ component of the standard Pauli matrix at site $j$. Here $M$ is the integer which relates with the lattice site numbers by the following relation $2M+1=N$. In our atom-cavity system, the rotation of laser field around the z-axis is equivalent to the rotation of the quantum spin system around the z-axis. The family of Hamiltonians generated by varying $\phi$ has the same energy spectrum as the initial Hamiltonian and $H(\phi)$ is $\pi$-periodic in $\phi$. The ground state $\|g>$ of the system is expressed as $\|g>=\prod_{k>0}(\cos\frac{\theta_{k}}{2}\|0>_{k}\|0>_{-k}-i\exp(2i\phi)\sin\frac{\theta_{k}}{2}\|1>_{k}\|1>_{-k})$ (13) where $\|0>_{k}$ and $\|1>_{k}$ are the vacuum and single fermionic excitation of the $k$-th momentum mode respectively. The angle $\theta_{k}$ is given by $\cos\theta_{k}=\frac{\cos k-B}{\Lambda_{k}}$ (14) and $\displaystyle{\Lambda_{k}=\sqrt{(\cos k-B)^{2}+\alpha^{2}\sin^{2}k}}$ is the energy gap above the ground state. The ground state is a direct product of $N$ spins, each lying in the two-dimensional Hilbert space spanned by $\|0>_{k}\|0>_{-k}$ and $\|1>_{k}\|1>_{-k}$. For each value of $k$, the state in each of the two dimensional Hilbert space can be represented as a Bloch vector with coordinates $(2\phi,\theta_{k})$. In our previous study sujop ; bonodi we have shown explicitly the derivation of geometric phase for this problem. $\Gamma_{k}=\pi(1-\cos\theta_{k})$ (15) The total geometric phase of the state $\|g>$ is given by $\Gamma_{g}=\sum_{k}~{}\Gamma_{k}$ . For an adiabatic evolution, if the initial state is an eigenstate, the evolved state remains in the eigenstate. Now we derive the instantaneous geometric phases of this system due to a gradually decreasing magnetic field, i.e., the quenching field. $B(t<0)=-\frac{t}{\tau_{q}}$ (16) $B(t)$, driving the transition, is assumed to be linear with an adjustable time parameter $\tau_{q}$ ($1/{\tau_{q}}$ is the quenching rate). The system be initially at time $t(<0)<<\tau_{q}$ such that $B(t)>>1$, in our parameter space of cavity arrays system, ${\delta_{1}}(t)>>[\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}({\Delta}_{b}-\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}-\frac{{\|{\Omega_{b}}\|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-{\gamma_{b}}{g_{b}}^{2}-{\gamma_{1}}{g_{a}}^{2}+{\gamma_{1}}^{2}\frac{{g_{a}}^{4}}{{\Delta_{b}}}-(a\leftrightarrow b)]$. Here we consider the quenching of the magnetic field by considering quenching the detuning field ${\delta}_{1}$. The instantaneous ground state at any instant $t$ is given by $\|\psi_{0}(t)>=\prod_{k}(\cos\frac{\theta_{k}(t)}{2}\|0>_{k}\|0>_{-k}-i\exp(2i\phi)\sin\frac{\theta_{k}(t)}{2}\|1>_{k}\|1>_{-k})$ (17) We now use eqn. (14) and (13) to derive the geometric phase of the $k^{th}$ mode which yields $\Gamma_{k}(t)=\pi\left(1-\frac{\cos k+\frac{t}{\tau_{q}}}{\sqrt{(\cos k+\frac{t}{\tau_{q}})^{2}+\alpha^{2}\sin^{2}k}}\right)$ (18) The geometric phase for an isotropic system with $\alpha=0$ and transverse Ising model with $\alpha=1$ may now be easily obtained. $\displaystyle\rm{For}~{}\alpha=0,~{}~{}~{}~{}\Gamma_{k}(t)$ $\displaystyle=$ $\displaystyle 2\pi\Theta(\|t\|-{{\tau}_{q}})$ $\displaystyle\rm{and}~{}\rm{for}~{}\alpha=1,~{}~{}\Gamma_{k}(t)$ $\displaystyle=$ $\displaystyle\pi\left(1-\frac{\cos k+\frac{t}{\tau_{q}}}{\sqrt{1+\frac{t^{2}}{\tau_{q}^{2}}+2\frac{t}{\tau_{q}}\cos k}}\right)$ (19) For ${\alpha}=0$, one of the Rabi frequencies, either (${\Omega_{a}}$) or (${\Omega_{b}}$) is zero. For ${\alpha}=1$, where both of Rabi frequencies are non zero. Finally, at $t=0$, when the magnetic field is gradually turned off, the situation is a bit different. The configuration of the final state will depend on the number of kinks generated in the system due to phase transition at or near $t=-{\tau_{q}}$ and as such it will depend on the quench time $\tau_{q}$ zur . The number of kinks is the number of quasi-particles excited at $B=0$ ( in our parameter space of micro-cavities, ${{\delta}_{1}}/2=\beta$, and the quasiparticle excitation is the excitation of polariton) and is given by ${\cal{N}}={\sum_{k}}p_{k}$ where $p_{k}$ is the excitation probability (for the slow transition) and is given by the Landau Zener formula zener $p_{k}\approx\exp{(-2\pi\tau_{q}k^{2})}$. As different pairs of quasi- particles ($k,-k$) evolve independently for large values of $\tau_{q}$, it is likely that only one pair of quasi-particles with momenta $(k_{0},-k_{0})$ will be excited. Where $k_{0}(=\frac{\pi}{N})$ corresponds to the minimum value of the energy $\Lambda_{k}$. Thus the condition for adiabatic transition in a finite chain is given by $\tau_{q}>>\frac{N^{2}}{2\pi^{3}}$. Hence, well in the adiabatic regime, the final state at $t=0$ is given by $\displaystyle\|\psi_{final}>$ $\displaystyle=$ $\displaystyle\|1>_{k_{0}}\|0>_{-k_{0}}$ (20) $\displaystyle\prod_{k,k\neq\pm k_{0}}(\cos\frac{\theta_{k}}{2}\|0>_{k}\|0>_{-k}$ $\displaystyle-i\exp(2i\phi)\sin\frac{\theta_{k}}{2}\|1>_{k}\|1>_{-k})$ Figure 4: Color online, variation of the geometric phase with time for two different quenching time, $\tau_{q}=100,10,000$ for four different anisotropy parameters $\alpha=0.2,0.5,0.8,1.0$ for the blue, cyan, red, green respectively. Here $k=\pi/3$. Figure 5: Color online, this three dimensional figures shows the variation of first order derivative of geometric phase with time, with the anisotropy exchange parameter and time. Its shows the existence of XX criticality (${\alpha=0}$ ). This state is similar to the direct product of only $N-1$ spins oriented along $(2\phi,\theta_{k})$ where the state of the spin corresponding to momentum $k_{0}$ does not contribute to the geometric phase. The total geometric phase of this state is given by $\Gamma_{final}(t=0)=\sum_{k,k\neq\pm k_{0}}\pi(1-\cos\theta_{k})$. For isotropic, $\alpha=0$ and for quantum spin chain system, geometric phase shows a sharp drop and finally become zero at $t={\tau_{q}}$ (see the inset of the Fig. 1) for this case optical cavity array system has only single Rabi frequency oscillation. In our study, we set the energy scale $J_{1}=1$, that leads to the following relations in the Rabi- frequencies, laser frequencies and coupling strengths in the following way, $\frac{4}{\gamma_{2}}=(\frac{{\|{\Omega_{a}}\|}^{2}{g_{b}}^{2}}{{{\Delta}_{a}}^{2}}+\frac{{\|{\Omega_{b}}\|}^{2}{g_{a}}^{2}}{{{\Delta}_{b}}^{2}})$. We achieve the isotropic $H_{XX}$ Hamiltonian, either ${\Omega}_{a}$ or ${\Omega}_{b}$ is zero. For ${\Omega}_{a}=0$, the reduce analytical expression for the magnetic field and $J_{1}$ exchange are $B=\frac{\delta_{1}}{2}-\frac{1}{2}[\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}({\Delta}_{b}-\frac{{\|{\Omega_{b}}\|}^{2}}{4{\Delta}_{b}}-\frac{{\|{\Omega_{b}}\|}^{2}}{4({\Delta}_{a}-{\Delta}_{b})}-{\gamma_{b}}{g_{b}}^{2}-{\gamma_{1}}{g_{a}}^{2}+{\gamma_{1}}^{2}\frac{{g_{a}}^{4}}{{\Delta_{b}}}]$. and ${J_{1}}=\frac{\gamma_{2}}{4}(\frac{{\|{\Omega_{b}}\|}^{2}{g_{a}}^{2}}{{{\Delta}_{b}}^{2}})$ respectively. For ${\Omega_{b}}=0$, the analytical expression for $B$ and $J_{1}$ can be obtained from Eq. (7) and Eq. (8) by putting ${\Omega_{b}}=0$. For $\alpha=1$, the spin chain is in a transverse magnetic field. One can achieve this limit by adjusting the detuning field, Rabi frequencies and the coupling strengths. For this limit, the analytical expression between the Rabi frequency and coupling strength is ${\Delta}_{b}=\frac{\gamma_{2}\Omega_{a}\Omega_{b}g_{a}g_{b}}{2\Delta_{a}}$, two Rabi frequency oscialltion exist for this case. The analytical expression for effective magnetic filed and the $J_{1}$ coupling can be obtained by substituting the analytical expression of $\Delta_{b}$ in Eq (7) and Eq. (8). The dynamics of the geometric phase shows no sharp drop of the geometric phase through out the quenching period. The dynamics of the geometric phase under quenching is different for isotropic quantum spin chain system ($H_{XX}$). Inset of the Fig. 1 shows the dynamics of geometric phase for $H_{XX}$ model. We will also observe that the first order derivative of the geometric phase also behave differently for these two limit of our model Hamiltonian. Therefore it is very clear from this study that the presence of single Rabi frequency and two Rabi frequency oscillation in the system of cavity QED changes the dynamics of geometric phase differently. In Fig. 2, we present the study of the geometric phase for different values of anisotropy exchange interactions ${\alpha}=0.2,0.8$ and for three different quenching times ${\tau}_{q}=250,500,1000$, which can be achieved by manipulating two Rabi frequencies ${\Omega}_{a}$ and ${\Omega}_{b}$. It reveals from our study that the qualitative behavior of the geometric phase is same but for the lower value of exchange anisotropy decaying rate is higher than for the higher values of exchange anisotropy. The geometric phase for different anisotropic exchange parameters crosses at the point $t={\tau_{q}}$ for the same ${\tau}_{q}$. This common crossing point is nothing but the quantum critical point for the system at $B=1$ for the same $\tau_{q}$. This has shown explicitly in Fig. 5 of this study. Apart from that the geometric phase for lower value of $\alpha$ also crosses the geometric phases of other $\tau_{q}$ but at the time lower than $\tau_{q}$. Therefore it is clear from this study due to the smaller values of Rabi frequencies (${\Omega}_{a}$ or ${\Omega_{b}}$) decaying rate of geometric phase is sharper compare to the larger values of Rabi frequencies. The same physics can also be obtained by regulating ${\Delta}_{a}$ and ${\Delta}_{b}$. It is very clear from the analytical expression of Eq. 14 and Eq. 15 that the dynamics of geometric phase depends on the magnetic field (time dependent quenching field) and the exchange anisotropy, therefore for a fixed $\tau_{q}$ the common meeting point of the geometric phase occurs at the quenching time. In Fig.3, we study the effect of different exchange interactions for $\tau_{q}=1000$. We observe that the all geometric phases crosses at the point $t={\tau}_{q}$. It reveals from this study that for a fixed quenching rate, the different exchange anisotropy (the presence of different Rabi frequencies oscillation) has no effect to change the common crossing point of the geometric phase. In our study, $1/{\tau_{q}}$, is the quenching rate for smaller values of $\tau_{q}$ is larger and smaller for higher values of $\tau_{q}$. In Fig. 4 shows the dynamical evalution of geometric phase for higher and lower quenching rate. It appears from our study that for higher quenching rate $(\tau_{q}=100)$ all curves for different anisotropy. For lower quenching rate is smaller for this case there is no collapse of the dynamical behaviour of the geometric phase. In Fig. 4, we study the effect of slower and rapid quenching rate on the dynamical behaviour of geometric phase. It appears from our study that for higher quenching rate ($\tau_{q}=100$) all curves for different anisotropy merge into a single line as if the exchange anisotropy,i.e., the different values of two Rabi frequency oscillations has no effect on the dynamical behaviour of the geometric phase. The dynamical behaviour of the geometric phase for lower quenching rate ($\tau_{q}=10,000$) is the same as that of $\tau_{q}=1000$. The crossing point of the geometric phase at $B=1$ for a fixed $\tau_{q}$ is the same for all quenching rate. To study the criticality of the geometric phase, we study the variation of the geometric phase (${\Gamma}_{k}$) and its derivative with respect to the quench induced magnetic filed ($B$) i.e. $({\frac{d{\Gamma}_{k}}{dB}})$ with time. We find the non-analytic behavior of the derivative at $t={\tau}_{q}$. The analytical expression for the derivative is ${\frac{d{\Gamma}_{k}(t)}{dB}}=\frac{\pi{\alpha}^{2}sin^{2}(k)}{{({({cos(k)+t/{{\tau}_{q}})}^{2}+{\alpha}^{2}sin^{2}(k)})}^{3/2}}$ (21) Fig. 5, shows the total variation of ${\frac{d{\Gamma}_{k}(t)}{dB}}$. The non- analytical behavior for $\alpha=0$ at $t=-{\tau_{q}}$ helps us to predict XX criticality under the linear quenching process. The analysis with different values of $\tau_{q}$ shows that the appearance of XX criticality is independent of $\tau_{q}$, i.e., independent of fast and slow quenching rates. Therefore, it is clear from our study that the quantum criticality of the geometric phase only obtain in the presence of single Rabi frequency oscillation in the system. Conclusions: We have studied the quantum criticality of the geometric phase for optical cavity arrays. We have predicted XX criticality of the geometric phase, which is independent of the quenching rate. 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Soc.Lond.A 137(1932) 696.	arxiv-papers	2013-09-30T09:29:37	2024-09-04T02:49:51.749809	{ "license": "Public Domain", "authors": "Sujit Sarkar", "submitter": "Sujit Sarkar", "url": "https://arxiv.org/abs/1310.0003" }
1310.0056	# modernizing PHCpack through phcpy Jan Verschelde∗† * Corresponding author: [email protected]† University of Illinois at ChicagoCopyright © 2014 Jan Verschelde. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. http://creativecommons.org/licenses/by/3.0/ ###### Abstract PHCpack is a large software package for solving systems of polynomial equations. The executable phc is menu driven and file oriented. This paper describes the development of phcpy, a Python interface to PHCpack. Instead of navigating through menus, users of phcpy solve systems in the Python shell or via scripts. Persistent objects replace intermediate files. ## 1 Introduction Our mathematical problem is to solve a system of polynomial equations in several variables. The discrete part of the output data includes the number of solutions and degrees of positive dimensional solution sets. When the input is exact or if the coefficients of the polynomial can be given with any accuracy, the isolated solutions can be then approximated up to any accuracy and for each positive dimensional solution component, as many generic points as its degree can be computed. Version 1.0 of PHCpack was archived in [Ver99]. PHCpack incorporates two external software packages: MixedVol [GLW05] and QDlib [HLB01]. Although the original focus was to approximate all isolated complex solutions, PHCpack prototyped many of the early algorithms in numerical algebraic geometry [SVW03], [SVW05]. Recent updates are listed in [Ver10]. The Python interface to PHCpack got a first start when Kathy Piret met William Stein at the software for algebraic geometry workshop at the Institute for Mathematics and its Applications in Minneapolis in the Fall of 2006. The first version of this interface is described in [Pir08]. Sage [S+13] offers the interface phc.py, developed by William Stein, Marshall Hampton and Alex Jokela. Version 0.0.1 of phcpy originated at lecture 40 of the author in the graduate course MCS 507 in the Fall of 2012, as an illustration of Sphinx [Bra]. Version 0.1.0 was prepared for presentation at EuroSciPy 2013 (August 2013), version 0.1.4 corresponds to the first version of this paper. The current version of phcpy is 0.1.5. We first outline in the next section the application of numerical homotopy continuation methods to compute, over the complex numbers, all isolated solutions and all positive dimensional irreducible solution sets of a polynomial system. Then we describe how phcpy relates to other interfaces to PHCpack. The functionality of phcpy is then summarized briefly as the online Sphinx documentation is more extensive and still growing. The Python interface to PHCpack builds directly on the C interface to the Ada code. Related software packages that apply homotopy continuation methods to solve polynomial systems are (in alphabetical order): Bertini [BHSW], [BHSW08], HOMPACK90 [WSM+97] (the successor of HOMPACK [WBM87]), HOM4PS [GLL02], HOM4PS-2.0 [LLT08], NAG4M2 [Ley11], PHoM [GKK+04], and pss3.0.5 [Mal]. As polynomial homotopy continuation methods involve many algorithms from various fields of computing, every software has its unique strengths and the statement “ _no one package provides all such capabilities_ ” quoted from [BHSW08] remains true today. ## 2 Polynomial Homotopy Continuation Our mathematical problem is to solve a polynomial system $f({\bf x})={\bf 0}$ in several variables ${\bf x}=(x_{1},x_{2},\ldots,x_{n})$. A homotopy connects $f({\bf x})={\bf 0}$ to a system $g({\bf x})={\bf 0}$ with known solutions: $h({\bf x},t)=\gamma(1-t)g({\bf x})+tf({\bf x})={\bf 0},\quad\gamma\in{\mathbb{C}}.$ For almost all values for $\gamma$, the solutions of $h({\bf x},t)={\bf 0}$ are regular for all $t\in[0,1)$. Numerical continuation methods track solution paths defined by $h({\bf x},t)={\bf 0}$. For systems with natural parameters $\lambda$, we solve $f({\mbox{\boldmath$\lambda$}},{\bf x})={\bf 0}$ first for generic values of the parameters ${\mbox{\boldmath$\lambda$}}={\mbox{\boldmath$\lambda$}}_{0}$ and then use $h({\bf x},t)=(1-t)f({\mbox{\boldmath$\lambda$}}_{0},{\bf x})+tf({\mbox{\boldmath$\lambda$}}_{1},{\bf x})={\bf 0},$ to solve a specific instance $f({\mbox{\boldmath$\lambda$}}_{1},{\bf x})={\bf 0}$. The schematic in Figure 1 illustrates that singular solutions along the paths are avoided by a generic choice of the parameters ${\mbox{\boldmath$\lambda$}}_{0}$ at $t=0$. Figure 1: A generic choice for start parameters avoids singularities along the paths. Solving a system that has solution sets of positive dimension, e.g.: a curve or surface, means to compute its dimension and its degree. For a solution set of dimension $d$, we add to the system $d$ linear equations with random coefficients to reduce the problem to computing isolated solutions. If the dimension is not known in advance, then it can be computed in a cascade of homotopies [SV00]. Because the coefficients of the linear equation are random, the solutions of the system that satisfy the random linear equations will be isolated. If the solution set has multiplicity one, the solutions of the augmented system will be isolated points of multiplicity one. Moreover, the number of isolated solutions of the augmented system that lie on the $d$-dimensional solution set of the original system will be equal to the degree of the $d$-dimensional solution set. Thus a positive dimensional solution set of dimension $d$ is represented by a set of $d$ random linear equations and as many points in the intersection of the original system with those random linear equations as the degree of the $d$-dimensional solution set. In numerical algebraic geometry, this representation is called a witness set. For sparse polynomial systems with very few monomials appearing with nonzero coefficient (in an extreme case, we consider binomial systems that have exactly two monomials with nonzero coefficient in each equation), we can represent positive dimensional solution sets by monomial maps. For example, the two equations $x^{2}y-zx=0$, $x^{2}z-y^{2}x=0$ have as solutions three monomial maps: $(x=0,y=\lambda_{1},z=\lambda_{2})$, $(x=\lambda_{1},y=\lambda_{1}^{2},z=\lambda_{1}^{3})$, and $(x=\lambda_{1},y=0,z=0)$, for parameters $\lambda_{1}$ and $\lambda_{2}$. These monomial maps form the leading terms of Puiseux series developments for general algebraic sets. Surveys on homotopy continuation are [AG93], [AG97], [Li03], and [Wat86], [Wat89], [Wat02]. Book treatments are in [AG03], [Mor87], and [SW05]. ## 3 Interfaces to PHCpack and phc This paper is mainly concerned with software problems. There are at least three motivations to develop phcpy: * 1. PHCpack is a large Ada package, its executable phc operates via menus, with input and output to files. With phcpy we provide an interpreter interface to phc. * 2. The code in PHCpack lacks adequate _user_ documentation so that many of its features are not obviously accessible to users. The Python modules of phcpy refactor the functionality of PHCpack and beautiful documentation is generated by Sphinx [Bra]. * 3. As many new algorithms were first implemented with PHCpack, reproducibility [SBB13] of published computational results can be automated via regression tests with Python scripts. Because also other interfaces to PHCpack may accomplish the same goals outlined above, we first give an overview of the interfaces to PHCpack. The first interface to PHCpack was based on the OpenXM [MNO+11] protocol for the interaction of software components. The virtue of this protocol is that only an executable version of the software is required and one does not need to compile the code. The interfaces to PHCpack from Maple [LV04], MATLAB & Octave [GV08b], and Macaulay2 [GPV13] only require the executable phc. This type of interface works in three stages: (1) prepare an input file for phc; (2) call phc with some options, the input file, and the name of an output file; (3) parse the output file to extract the results. In principle, everything that can be done via the command-line menus of phc can thus also be performed via Maple procedures, MATLAB, Octave, or Macaulay2 scripts. Figure 2 shows the interfaces to PHCpack. Figure 2: Diagram of the interfaces to PHCpack and phc. The interfaces PHCpack.m2, PHCmaple, PHClab, depicted to the right of the antidiagonal line require only the executable version phc. The other interfaces PHClib, PHCmpi, and phcpy are based on the source code of PHCpack. The C interface to PHCpack, described in [LV06], offers the C programmer access to the path trackers of PHCpack. This interface was developed for use with the Message Passing Interface [SOH+98] and serves also as the basis for phcpy. In the C interface, the data structures for polynomials and solutions are not duplicated. Instead of data structure duplication, one can enter into the C interface routine a polynomial term after term. The interface then behaves like a state machine. Why would phcpy be any better than the other interfaces? Leaving aside the growing popularity of Python for scientific computing, the replacement of files by persistent objects enabled the implementation of a generator for the path trackers. After initialization of the homotopy (with target, start system, and one start solution), the user can call a “next” function to compute the next point at the solution path that originates at the start solution given at initialization. This “next” function (available for standard double, double double, quad double precision, and arbitrary multiprecision) allows a detailed investigation of the properties of a particular solution path. In addition, it gives the user a fine control over the order of execution. If desired, the tolerances and the step size can be adjusted as needed in an application that plots solution trajectories. Another (future) application of phcpy is a web interface, such as at https://kepler.math.uic.edu (beta version) presented by Xiangcheng Yu at the SIAM AG 2013 conference in the first week of August 2013. ## 4 Using phcpy The blackbox solver of PHCpack is its most widely used function. In phcpy, this blackbox solver is available in the function solve of the module solver. The solver takes on input a list of strings that contain valid representations of polynomials. On return is a list of strings, which contain the solutions of the system. >>> from phcpy.solver import solve>>> from phcpy.phcpy2c import py2c_set_seed>>> f = ["x*2y*2 + x + y;","xy + x + y + 1;"]>>> py2c_set_seed(21320)0>>> s = solve(f,silent=True)>>> len(s)4>>> print s[0]t : 1.00000000000000E+00 0.00000000000000E+00m : 1the solution for t :x : -1.00000000000000E+00 0.00000000000000E+00y : -1.61803398874989E+00 0.00000000000000E+00== err : 2.143E-101 = rco : 4.775E-02 = res : 2.220E-16 =With py2c_set_seed() we fix the seed of the random number generator for the coefficients of the start system in the homotopy, which makes for predictable runs. Otherwise, the solve() each time generates different coefficients in the homotopies and the order of the solutions on return may differ. For each solution, the triplet (err,rco,res) indicates the quality of the solution: * • err: the norm of the last update made by Newton’s method (forward error), * • rco: estimate for the inverse condition number of the Jacobian matrix, * • res: norm of the evaluated solution (backward error). With double double and quad double arithmetic we get more accurate solutions. To predict the number of isolated solutions with the mixed volume: >>> from phcpy.solver import mixed_volume>>> mixed_volume(f)4Version 0.1.5 of phcpy contains the following modules: * • solver: a blackbox solver, mixed-volume calculator, linear-product root count and start system, path trackers, deflation for isolated singular solutions. * • examples: a selection of interesting benchmark systems. Typing python examples.py at the command prompt calls the blackbox solver on all benchmark examples, thus providing an automatic regression test. * • families: some problems can be formulated for any number of variables. * • phcmaps: monomial maps as solutions of binomial systems. * • phcsols: conversion of PHCpack solution strings into Python dictionaries. * • phcsets: basic tools to manipulate positive dimensional solution sets. * • phcwulf: basic client/server setup to solve many systems. * • schubert: the Pieri homotopies solve particular polynomial systems arising in enumerative geometry. The number of exported functions, documented by Sphinx [Bra] runs in the several hundreds. The code of version 0.1.1 of phcpy was improved with the aid of Pylint [The], yielding a global rating of 9.73/10. ## 5 The Design of phcpy The design of phcpy is drawn in Figure 3. This design can be viewed as an application of a façade pattern (see Figure B.31 in [Bai08]). The façade pattern plays a strong role in converting legacy systems incrementally to more modern software and is appropriate as phcpy should be viewed as a modernization of PHCpack. The implementation of use_c2phc.adb applies the chain of responsibility pattern (see Figure B.12 in [Bai08]), calling handlers to specific packages in PHCpack. That we use the name phcpy and not PyPHC indicates that phcpy is more than just an interface. Figure 3: The design of phcpy depends on PHClib, a library of various collections of C functions, through one file phcpy2c.c (with documentation in the corresponding header phcpy2c.h) which encodes the Python bindings. PHClib interfaces to the Ada routines of PHCpack through one Ada procedure use_c2phc.adb. The collection of parallel programs (MPI2phc) using message passing (MPI) depends on PHClib. The code for phcpy builds directly on the C interface to PHCpack. The C interface was developed to use the Message Passing Interface (MPI) [SOH+98]. In joint work with Yusong Wang [VW02], [VW04a], [VW04b], Yan Zhuang [VZ06], Yun Guan [GV08a], and Anton Leykin [LV05], [LV09], [LVZ06], the main program was always a C program. The C interface described in [LV06] is centered around one gateway function use_c2phc. To the Ada programmer, this function has the specification function use_c2phc ( job : integer; a : C_intarrs.Pointer; b : C_intarrs.Pointer; c : C_dblarrs.Pointer ) return integer;The prototype of the corresponding C function is extern int _ada_use_c2phc ( int task, int a, int b, double c );With use_c2phc we obtain one uniform streamlined design of the interface: the C programmer calls one single Ada function _ada_use_c2phc. What use_c2phc executes depends on the job number. The (a,b,c) parameters are flexible enough to pass strings and still provide some form of type checking (which would not be possible had we wiped out all types with void). To make _ada_use_c2phc usable, we have written a number of C wrappers, responsible for parsing the arguments of the C functions to be passed to _ada_use_c2phc. The extension module and the shared object for the implementation of phcpy is a set of wrappers defined by phcpy2c.c and documented by phcpy2c.h. As a deliberate design decision of phcpy, all calls to functions in PHCpack pass through the C interface. By this design, the development of phcpy benefits the C and C++ programmers. ## 6 Obtaining, Installing, and Contributing PHCpack and phcpy are distributed under the GNU GPL license (version 2 or any later version). Recently a new repository PHCpack was added on github with the source code of version 2.3.85 of PHCpack, which contains version 0.1.5 of phcpy. Executable versions for Linux, Mac, and Windows are available via the homepage of the author. The code was developed on a Red Hat Enterprise Linux Workstation (Release 6.4) and a MacBook Pro laptop (Mac OS X 10.8.5) using the GNAT GPL 2013 compiler. Versions 2.6.6 and 2.7.3 of Python, respectively on Linux and Mac, were used to develop phcpy. Packaged binary distributions of phcpy for the platforms listed above are available via the homepage of the author. Although the blackbox solver of PHCpack has been in use since 1996, phcpy itself is still very much in beta stage. Suggestions for improvement and contributions to phcpy will be greatly appreciated. ## 7 Acknowledgments The author thanks Max Demenkov for his comments and questions at the poster session at EuroSciPy 2013. In particular the question on obtaining all solutions along a path led to the introduction of generator functions for the path trackers in version 0.1.4 of phcpy. This material is based upon work supported by the National Science Foundation under Grant No. 1115777. ## References * [AG93] E.L. Allgower and K. 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Wang. _Computing feedback laws for linear systems with a parallel Pieri homotopy_ , In Y. Yang, editor, Proceedings of the 2004 International Conference on Parallel Processing Workshops, 15-18 August 2004, Montreal, Quebec, Canada. High Performance Scientific and Engineering Computing, pages 222-229, IEEE Computer Society, 2004. * [VZ06] J. Verschelde and Y. Zhuang. _Parallel implementation of the polyhedral homotopy method_ , In T.M. Pinkston and F. Ozguner, editors, Proceedings of the 2006 International Conference on Parallel Processing Workshops, 14-18 Augustus 2006, Columbus, Ohio, High Performance Scientific and Engineering Computing, pages 481-488, IEEE Computer Society, 2006. * [Wat86] L.T. Watson. _Numerical linear algebra aspects of globally convergent homotopy methods_ , SIAM Rev., 28(4):529-545, 1986. * [Wat89] L.T. Watson. _Globally convergent homotopy methods: a tutorial_ , Appl. Math. Comput., 31(Spec. Issue):369-396, 1989. * [Wat02] L.T. Watson. _Probability-one homotopies in computational science_ , J. Comput. Appl. Math., 140(1&2):785-807, 2002. * [WBM87] L.T. Watson, S.C. Billups, and A.P. Morgan. _Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms_ , ACM Trans. Math. Softw., 13(3):281-310, 1987. * [WSM+97] L.T. Watson, M. Sosonkina, R.C. Melville, A.P. Morgan, and H.F. Walker. _Algorithm 777: HOMPACK90: A suite of Fortran 90 codes for globally convergent homotopy algorithms_ , ACM Trans. Math. Softw., 23(4):514-549, 1997.	arxiv-papers	2013-09-30T21:05:03	2024-09-04T02:49:51.759015	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jan Verschelde", "submitter": "Jan Verschelde", "url": "https://arxiv.org/abs/1310.0056" }
1310.0104	# Pure Subspaces, Generalizing the Concept of Pure Spinors Carlos Batista Departamento de Física Universidade Federal de Pernambuco 50670-901 Recife-PE, Brazil [email protected] ###### Abstract The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the choice of a spinorial connection is exploited in order to relate twistor equation to the integrability of maximally isotropic distributions. (Keywords: Isotropic Spaces, Pure Spinors, Integrability, Clifford Algebra, Twistors) ## 1 Introduction It is well-known that given a spinor $\hat{\varphi}$ one can construct a vector subspace $N_{\hat{\varphi}}$ spanned by the vectors that annihilate $\hat{\varphi}$ under the Clifford action, $v\in N_{\hat{\varphi}}$ whenever $v\cdot\hat{\varphi}=0$. This subspace is necessarily isotropic, which means that $\langle v,v\rangle=0$ for all $v\in N_{\hat{\varphi}}$. Particularly, if the dimension of $N_{\hat{\varphi}}$ is maximal $\hat{\varphi}$ is said to be a pure spinor. Thus, to every spinor it is associated an isotropic vector subspace. The idea of the present article is to go the other way around and associate to every isotropic subspace $I$ a spinorial subspace $\hat{L}_{I}$. As we shall see, these spinorial subspaces provide a natural generalization for the concept of pure spinors and, therefore, we shall say that $\hat{L}_{I}$ is the pure subspace associated to $I$. As a consequence, some classical results are shown to be just particular cases of broader theorems. Hence, this study can shed more light on the role played by the pure spinors on physics and mathematics. Spinors have been around for a long time, they were first discovered a century ago by Élie Cartan [1]. Since then, they led to great achievements in a multitude of areas of mathematics and physics. For instance, spinors are of fundamental importance in particle physics, since fermions are represented by spinorial fields (see [2] for an alternative treatment). In addition, spinors can often be used as helpful tools for calculating physical and geometrical quantities. As examples let us recall Witten’s elegant proof of energy positivity in general relativity [3] and the calculation of scattering amplitudes in quantum field theory [4]. From the geometrical point of view, a particularly important class of spinors is formed by the so-called pure spinors. Such spinors are also acquiring increasing significance in high- energy physics, as illustrates the pure spinor formalism in string theory [5]. In spite of such undeniable usefulness, the role played by spinors is certainly not completely understood yet. Hopefully, the work presented here will contribute for a further understanding of spinorial calculus and, specially, of the meaning of pure spinors. The isotropic subspaces are of great relevance in mathematical-physics, a fact that became clear after the works by R. Penrose wherein he introduced the spinorial calculus and the null tetrad formalism in 4-dimensional general relativity [6]. Such approaches, in which isotropic distributions play a prominent role, brought much progress to general relativity. Notably, Kinnersley was able to find analytically all solutions of Einstein’s vacuum equation in 4 dimensions for the case of the space-time admitting two independent integrable distributions of isotropic planes [7, 8, 9]. Particularly, all known 4-dimensional black-holes are contained in this class of solutions. Higher-dimensional manifolds have been a subject of great importance in theoretical physics, not to mention in mathematics. Following the successful track of 4-dimensional general relativity recent works have shown that isotropic structures do also play an important role in higher dimensions. In reference [10] it was proved a relation between integrable isotropic structures and the existence of a conformal Killing-Yano tensor, while in [11] these structures were used to classify the Weyl tensor as well as to partially generalize the Goldberg-Sachs theorem, see also [12]. The outline of this article is as follows. In Section 2 it is introduced the notation adopted throughout the article to deal with spinors. Section 3 defines the pure subspaces and present a theorem on algebraic identities satisfied by them. Then, Section 4 extends the formalism from vector spaces to fiber bundles over differential manifolds. In addition, this section reviews the issue of introducing a connection on the spinorial bundle and explore the freedom in its choice. Section 5 presents some differential results regarding the pure subspaces, connecting the integrability of an isotropic distribution with differential constraints satisfied by the elements of the associated pure subspace. Finally, in Section 6 it is proved a theorem relating twistors to the existence of integrable maximally isotropic distributions. All over this work it is being assumed that the vector spaces and the manifolds are even- dimensional and endowed with a non-degenerate metric of arbitrary signature. Unless otherwise stated, Einstein summation convention is used, meaning that repeated indices are summed. The results are all local. ## 2 Spinors and the Space $V\oplus V^{}$ Given a vector space $\mathcal{V}$ endowed with a non-degenerate inner product $\langle\,,\rangle$, the Clifford Algebra $Cl(\mathcal{V})$ is an algebra in this vector space such that: $v\,u\,+\,u\,v\,=\,2\,\langle v,u\rangle\,\;\;\;\forall\;v,u\,\in\;\mathcal{V}.$ (1) The space of spinors associated to $(\,\mathcal{V},\langle\,,\rangle\,)$ is a vectorial space $S$ where an irreducible and faithful representation of $Cl(\mathcal{V})$ acts. In even dimensions it is always possible to find a matrix representation for $Cl(\mathcal{V})$, if $\dim(\mathcal{V})=2n$ then the least-dimensional faithful representation of this algebra is provided by $2^{n}\times 2^{n}$ matrices. Therefore, in this case spinors are represented by the column vectors on which these matrices act, so that the space of spinors has dimension $2^{n}$. Now, let us deal with the vector space $V\oplus V^{}$, where $V$ is an $n$-dimensional vector space and $V^{}$ is its dual. The space $V\oplus V^{}$ is naturally endowed with a non-degenerate symmetric inner product $\langle\,,\rangle$ defined by: $\langle e+\theta,e^{\prime}+\theta^{\prime}\rangle\,\equiv\,\frac{1}{2}\left[\,\theta(e^{\prime})\,+\,\theta^{\prime}(e)\,\right]\;\;\;\;\;\;e,e^{\prime}\in V\;\textrm{ and }\;\theta,\theta^{\prime}\in V^{}\,.$ (2) Let us introduce the basis $\\{e_{1},e_{2},...,e_{n}\\}$ for $V$ and denote by $\\{\theta^{1},\theta^{2},...,\theta^{n}\\}$ the dual basis, $\theta^{i}(e_{j})=\delta_{j}^{i}$. So, thanks to (2) we have that the following relations hold: $\langle e_{i},e_{j}\rangle\,=\,0\,,\quad\;\quad\langle e_{i},\theta^{j}\rangle\,=\,\frac{1}{2}\,\delta_{i}^{j}\,,\quad\;\quad\langle\theta^{i},\theta^{j}\rangle\,=\,0\,.$ The interesting feature of the space $V\oplus V^{}$ endowed with this natural inner product is that the spinors can be constructed quite easily. Indeed, the algebra $Cl(V\oplus V^{})$ admits a representation in the space $\wedge V^{}$, the exterior algebra of $V^{}$. The action of $V\oplus V^{}$ vectors in this representation being defined by: $(e+\theta)\cdot\varphi\,\equiv\,e\lrcorner\varphi\,+\,\theta\wedge\varphi\quad\quad\;e\in V\;,\;\theta\in V^{}\;,\;\varphi\in\wedge V^{}\,,$ (3) where $e\lrcorner\varphi$ means the interior product of the vector $e$ on the form $\varphi$. The action of a scalar is defined in the trivial way, $\lambda\cdot\varphi=\lambda\varphi$. In order to see that this is indeed a representation of the Clifford algebra, note that by successive application of some vector $v=e+\theta$ on the spinor $\varphi$ we get $\displaystyle v\cdot\left[v\cdot\varphi\right]\,$ $\displaystyle=\,(e+\theta)\cdot\left[e\lrcorner\varphi\,+\,\theta\wedge\varphi\right]$ $\displaystyle=\,e\lrcorner e\lrcorner\varphi\,+\,e\lrcorner(\theta\wedge\varphi)\,+\,\theta\wedge(e\lrcorner\varphi)\,+\,\theta\wedge\theta\wedge\varphi$ $\displaystyle=\,\theta(e)\,\varphi\,=\,\langle v,v\rangle\,\varphi\,=\,(v\,v)\cdot\varphi\,,$ where in the last step it was used Eq. (1). Since the space $\wedge V^{}$ has $2^{n}$ dimensions we conclude that this is, indeed, the space of spinors, $S=\wedge V^{}$ [13]. To simplify the notation, it will be used the abbreviation $\theta^{12...k}\equiv\theta^{1}\wedge\theta^{2}\wedge...\wedge\theta^{k}$. Furthermore, in order to avoid any confusion it is worth distinguishing the elements of $Cl(V{\oplus}V^{})$ from the spinors of $S=\wedge V^{}$. With this intent from now on we shall always denote the spinors by a Greek letter with a hat on top. For instance, $\theta^{i}$ is an element of $V\oplus V^{}$ while $\hat{\theta}^{i}$ is a spinor. In the present formalism an inner product can be easily introduced on the space of spinors. Such product is non-degenerate and defined up to an arbitrary scale factor. This scale can be fixed by an arbitrary choice of basis for $S$, here let us choose the basis $\\{\hat{1},\hat{\theta}^{i},\hat{\theta}^{ij},...,\hat{\theta}^{12...n}\\}$. Then the inner product $(\,,):S\times S\rightarrow\mathbb{C}$ is defined to be such that: $<\,\hat{\varphi}^{\,t}\wedge\hat{\psi}\,>_{n}\,\,=\,\lambda\,\hat{\theta}^{12...n}\;\;\;\Longleftrightarrow\;\;\;(\hat{\varphi},\hat{\psi})\,\equiv\,\lambda\,.$ (4) Where $<\hat{\varphi}>_{k}$ means the component of degree $k$ of the form $\hat{\varphi}$, while $\hat{\varphi}^{\,t}$ means the reverse of $\hat{\varphi}$. For instance, if $\hat{\varphi}=\hat{1}+\hat{\theta}^{12...k}$ then $<\hat{\varphi}>_{k}=\hat{\theta}^{12...k}$ and $\hat{\varphi}^{\,t}=\hat{1}+\hat{\theta}^{k...21}$. It is not difficult convincing ourselves that $(v\cdot\hat{\varphi},\hat{\psi})=(\hat{\varphi},v\cdot\hat{\psi})$ for any $v\in V\oplus V^{}$. Such property implies that this inner product is invariant under the group $Spin_{+}(V\oplus V^{})$, the double cover of $SO_{+}(V\oplus V^{})$. More generally, if $\Omega\in Cl(V\oplus V^{})$ then it can be proved that $(\Omega\cdot\hat{\varphi},\hat{\psi})=(\hat{\varphi},\Omega^{t}\cdot\hat{\psi})$. If $v_{1},\,\ldots,v_{p}$ are vectors in $V\oplus V^{}$ then we shall define $v_{1}\wedge\ldots\wedge v_{p}$ to be the complete antisymmetric part of the Clifford product $v_{1}v_{2}\cdots v_{p}$. In particular, we have that $v_{1}\wedge v_{2}=\frac{1}{2!}(v_{1}v_{2}-v_{2}v_{1})$. Then, the so-called pseudo-scalar of the Clifford algebra $Cl(V\oplus V^{})$ is defined by: $\mathcal{I}\,=\,2^{n}\,e_{1}\wedge\theta^{1}\wedge e_{2}\wedge\theta^{2}\wedge\ldots\wedge e_{n}\wedge\theta^{n}\,.$ It is simple matter to prove that $\mathcal{I}\,\mathcal{I}=1$. So that such object can be used to split the spinor space as $S=S^{+}\oplus S^{-}$, where $\hat{\varphi}\,\in\,S^{\pm}$ if $\mathcal{I}\cdot\hat{\varphi}=\pm\hat{\varphi}$. The elements of $S^{+}$ are called the Weyl spinors of positive chirality, while those of $S^{-}$ are the Weyl spinors of negative chirality. It can be shown that $S^{+}$ is spanned by the spinors of even degree, while $S^{-}$ is spanned by the spinors of odd degree, so that $\dim(S^{\pm})=2^{n-1}$. More about the spinors of $Cl(V\oplus V^{})$ can be found in [13], where this formalism is applied to the space $TM\oplus TM^{}$, with $TM$ being the tangent bundle of a manifold $M$. In order for the reader to get acquainted with the language introduced so far, let us work out a simple example. Example 1 When $\dim(V)=2$ we have that $\\{e_{1},e_{2},\theta^{1},\theta^{2}\\}$ provides a basis for $V\oplus V^{}$. Thus, a basis for the Clifford algebra $Cl(V\oplus V^{})$ is formed by the following 16 elements: $\\{1,\,e_{i},\,\theta^{i},\,e_{1}\wedge e_{2},\,e_{i}\wedge\theta^{j},\,\theta^{1}\wedge\theta^{2},\,e_{i}\wedge\theta^{1}\wedge\theta^{2},\,e_{1}\wedge e_{2}\wedge\theta^{i},\,e_{1}\wedge e_{2}\wedge\theta^{1}\wedge\theta^{2}\\}\,.$ While a basis for the spinor space is given by $\\{\hat{1},\hat{\theta}^{1},\hat{\theta}^{2},\hat{\theta}^{12}\\}$. Since $\hat{1}$ and $\hat{\theta}^{12}$ are forms of even degree they are Weyl spinors of positive chirality, while $\hat{\theta}^{1}$ and $\hat{\theta}^{2}$ have negative chirality. Note, for instance, that the following relations hold: $e_{i}\cdot\hat{1}=0\,,\quad\;e_{i}\cdot\hat{\theta}^{j}\,=\,\delta^{j}_{i}\,\hat{1}\,,\quad\;e_{1}\cdot\hat{\theta}^{12}\,=\,\hat{\theta}^{2}\,,\quad\;\theta^{i}\cdot\hat{1}=\hat{\theta}^{i}\,,\quad\;\theta^{1}\cdot\hat{\theta}^{2}=\hat{\theta}^{12}\,,\quad\;\theta^{1}\cdot\hat{\theta}^{12}=0\,.$ Moreover, the non-zero inner products are given by: $(\hat{1},\hat{\theta}^{12})\,=\,-(\hat{\theta}^{12},\hat{1})\,=\,1\quad\textrm{and}\quad(\hat{\theta}^{1},\hat{\theta}^{2})\,=\,-(\hat{\theta}^{2},\hat{\theta}^{1})\,=\,1\,.$ In particular, note that this inner product is skew-symmetric. $\Box$ Although it may appear too restrictive working with vector spaces of the form $V{\oplus}V^{}$, this is not the case at all. Every even-dimensional vector space endowed with a non-degenerate metric can easily be cast in the form $V\oplus V^{}$ when complexified. For example, in the Minkowski space $\mathbb{R}^{1,3}$ a complex null tetrad basis can always be introduced. In the standard notation of General Relativity this null tetrad is denoted by $\\{l,m,n,\overline{m}\\}$, with $\langle l,n\rangle=1$ and $\langle m,\overline{m}\rangle=-1$, all other inner products between the basis vectors being zero. So it is possible to make the following associations: $l\leftrightarrow e_{1}$, $m\leftrightarrow e_{2}$, $n\leftrightarrow 2\theta^{1}$, $\overline{m}\leftrightarrow-2\theta^{2}$. Thus, $\\{l,m\\}$ can be seen as a basis for $V$ and $\\{n,\overline{m}\\}$ a basis for $V^{}$. So, from now on all calculations will be done on vector spaces of the form $V\oplus V^{}$ and over the complex field. But it must be clear that the results can be easily carried to all complexified even-dimensional spaces. When the metric on the even-dimensional space has split signature the results are also valid without complexification, since in this case the isotropic subspaces can have dimension equal to half of the dimension of the full vector space. The results on real even-dimensional vector spaces can be extracted from the complex case by choosing suitable reality conditions, in the spirit of [9, 12]. ## 3 Pure Spinorial Subspaces, Algebraic Results While dealing with spinors it is common, and often valuable, associating to every non-zero spinor $\hat{\varphi}$ a vector subspace $N_{\hat{\varphi}}\subset(V\oplus V^{})$ defined by: $N_{\hat{\varphi}}=\\{\,v\in(V\oplus V^{})\;\|\;v\cdot\hat{\varphi}=0\,\\}\,.$ These vector subspaces are called isotropic or totally null, because every vector belonging to $N_{\hat{\varphi}}$ has zero norm. This can be easily verified, if $v\in N_{\hat{\varphi}}$ then $\langle v,v\rangle\hat{\varphi}=(vv)\cdot\hat{\varphi}=v\cdot(v\cdot\hat{\varphi})=0$, so that $\langle v,v\rangle=0$. The maximum dimension that an isotropic subspace of $V\oplus V^{}$ can have is $n=\dim(V)$. Therefore, an isotropic subspace of dimension $n$ is called a maximally isotropic subspace. In the particular case of $N_{\hat{\varphi}}$ being maximally isotropic, the spinor $\hat{\varphi}$ is said to be a pure spinor. For example, in the notation of last section the spinor $\hat{1}$ is such that $N_{\hat{1}}=\textrm{Span}\\{e_{1},e_{2},...,e_{n}\\}$, so that $\hat{1}$ is a pure spinor. The pure spinors are very special objects in mathematics and have been studied since the beginning of XX century, more about them can be found in [14, 15, 16]. Pure spinors are also acquiring increasing relevance in physics, particularly in string theory [5]. However, it seems to have been overlooked that it is also possible to go the other way around and associate spinors to isotropic subspaces. More precisely, given an isotropic subspace $I$, one can define a spinorial subspace $\hat{L}_{I}$ spanned by all spinors annihilated by the Clifford action of $I$. The aim of the present article is to explore this new path. Suppose that $I\subset(V\oplus V^{})$ is an isotropic subspace. Then, let us define the subspace $\hat{L}_{I}\subset S$ as follows: $\hat{L}_{I}\,\equiv\,\left\\{\,\hat{\varphi}\in S\;\|\;v\cdot\hat{\varphi}=0\;\;\forall\;v\in I\,\right\\}\,.$ (5) It is trivial to see that $\hat{L}_{I}$ is, indeed, a vector subspace of the spinor space $S=\wedge V^{}$. In the particular case of $I$ being maximal, $\dim(I)=n$, it follows that $\hat{L}_{I}=\textrm{Span}\\{\hat{\psi}\\}$, where $\hat{\psi}$ is the pure spinor associated to $I$ and $\textrm{Span}\\{\hat{\psi}\\}$ is the one-dimensional subspace spanned by it. This section is devoted to enunciate and prove several algebraic results concerning the spaces $\hat{L}_{I}$, where $I$ is any totaly null subspace of $V\oplus V^{}$. Before proceeding let us see a simple example. Example 2 If $\dim(V)=3$ we have that $\\{e_{1},e_{2},e_{3},\theta^{1},\theta^{2},\theta^{3},\\}$ is a basis to $V\oplus V^{}$. The spinor space is generated by the basis $\\{\hat{1},\hat{\theta}^{1},\hat{\theta}^{2},\hat{\theta}^{3},\hat{\theta}^{12},\hat{\theta}^{13},\hat{\theta}^{23},\hat{\theta}^{123}\\}$. Then, let us define the following isotropic subspaces of $V\oplus V^{}$: $I_{1}\,=\,\textrm{Span}\\{e_{1}\\}\,,\quad I_{2}\,=\,\textrm{Span}\\{e_{1},\theta^{2}\\}\,,\quad I_{3}\,=\,\textrm{Span}\\{e_{1},\theta^{2},\theta^{3}\\}\,.$ Thus, the associated spinorial subspaces are respectively given by: $\hat{L}_{I_{1}}\,=\,\textrm{Span}\\{\hat{1},\hat{\theta}^{2},\hat{\theta}^{3},\hat{\theta}^{23}\\}\,,\quad\hat{L}_{I_{2}}\,=\,\textrm{Span}\\{\hat{\theta}^{2},\hat{\theta}^{23}\\}\,,\quad\hat{L}_{I_{3}}\,=\,\textrm{Span}\\{\hat{\theta}^{23}\\}\,.$ $\Box$ Now, let us enunciate, in the form of a theorem, several interesting algebraic results regarding the spinorial subspaces $\hat{L}_{I}$. ###### Theorem 1 Given two isotropic subspaces $I,I^{\prime}\subset(V\oplus V^{})$, with $V$ being a complexified vector space of dimension $n$, then it follows that: (1) $\dim(\hat{L}_{I})\,=\,2^{n-\dim(I)}$ (2) $\hat{L}_{I}=\hat{L}_{I^{\prime}}$ $\Longleftrightarrow$ $I=I^{\prime}$ (3) $(\hat{L}_{I}+\hat{L}_{I^{\prime}})\subset\hat{L}_{I\cap I^{\prime}}$ (4) If $I^{\prime\prime}=I+I^{\prime}$ is isotropic then $\hat{L}_{I+I^{\prime}}=\hat{L}_{I}\cap\hat{L}_{I^{\prime}}$ (5) If $u\in(V\oplus V^{})$ is such that $u\cdot\hat{\varphi}=0$ $\,\forall\,\;\hat{\varphi}\in\hat{L}_{I}$ then $u\in I$ (6) $I^{\prime}\subset I$ $\Longleftrightarrow$ $\hat{L}_{I}\subset\hat{L}_{I^{\prime}}$ (7) If $I\neq\\{0\\}$ then $(\hat{\varphi},\hat{\psi})=0$ $\,\forall\,\;\hat{\varphi},\hat{\psi}\in\hat{L}_{I}$ (8) $I^{\prime\prime}=I+I^{\prime}$ is isotropic $\Longleftrightarrow$ $\hat{L}_{I}\cap\hat{L}_{I^{\prime}}\neq\\{0\\}$ (9) If $I$ is not maximal then $\hat{L}_{I}=\hat{L}^{+}_{I}\oplus\hat{L}^{-}_{I}$ where $\hat{L}^{\pm}_{I}$ are spanned by Weyl spinors of $\pm$ chirality and $\dim(\hat{L}^{+}_{I})=\dim(\hat{L}^{-}_{I})$. Before proving the above results it is important to note that given an isotropic subspace $I\subset V\oplus V^{}$ of dimension $k$ then it is always possible to find a vector subspace $V^{\prime}=\textrm{Span}\\{e_{1},\ldots,e_{n}\\}$ such that $I$ is spanned by $\\{e_{1},e_{2},\ldots,e_{k}\\}$ and $V\oplus V^{}=V^{\prime}\oplus V^{\prime}$. Hence, the judicious and convenient choice $I=\textrm{Span}\\{e_{1},\ldots,e_{k}\\}$ represents no loss of generality. _Proof of Theorem 1:_ _(1)_ If $I=\textrm{Span}\\{e_{1},e_{2},\ldots,e_{k}\\}$ then it is not hard to conclude that $\hat{L}_{I}=\textrm{Span}\\{\,\hat{1},\,\hat{\theta}^{\alpha^{\prime}},\,\hat{\theta}^{\alpha^{\prime}\beta^{\prime}},\,\cdots,\,\hat{\theta}^{k+1\,k+2...n}\,\\}\,,$ where $\alpha^{\prime},\beta^{\prime}\,\in\\{k+1,k+2,...,n\\}$. Leading us to the following result: $\dim(\hat{L}_{I})=1+(n-k)+\frac{(n-k)(n-k-1)}{2!}+\cdots+1=\sum_{i=0}^{n-k}\,\dbinom{n-k}{i}=(1+1)^{n-k}.$ _(2)_ Suppose that $\hat{L}_{I}=\hat{L}_{I^{\prime}}$ but $I\neq I^{\prime}$, then there exists $u\in I$ such that $u$ does not belong to $I^{\prime}$ (or the converse). Now, if $\hat{\varphi}\in\hat{L}_{I}=\hat{L}_{I^{\prime}}$ is a non-zero spinor then $2\langle u,v^{\prime}\rangle\hat{\varphi}=(uv^{\prime}+v^{\prime}u)\cdot\hat{\varphi}=0$ $\,\forall\,\,v^{\prime}\in I^{\prime}$, so that $\langle u,v^{\prime}\rangle=0$ for all $v^{\prime}\in I^{\prime}$. This implies that $I^{\prime\prime}\equiv I^{\prime}+\textrm{Span}\\{u\\}$ is an isotropic subspace. Moreover, since $u\notin I^{\prime}$ it follows that $\dim(I^{\prime\prime})=\dim(I^{\prime})+1$, from which we conclude that $\dim(\hat{L}_{I^{\prime\prime}})<\dim(\hat{L}_{I^{\prime}})$. On the other hand, since $\hat{L}_{I}=\hat{L}_{I^{\prime}}$ then $u\cdot\hat{L}_{I^{\prime}}=0$, so if $\hat{\varphi}\in\hat{L}_{I^{\prime}}$ then $v^{\prime\prime}\cdot\hat{\varphi}=0$ for all $v^{\prime\prime}\in I^{\prime\prime}$. Thus, $\hat{L}_{I^{\prime}}\subset\hat{L}_{I^{\prime\prime}}$ which implies $\dim(\hat{L}_{I^{\prime\prime}})\geq\dim(\hat{L}_{I^{\prime}})$, contradicting the former inequality. Therefore, if $\hat{L}_{I}=\hat{L}_{I^{\prime}}$ then $I=I^{\prime}$. The converse is trivial. _(3)_ If $\hat{\psi}\in(\hat{L}_{I}+\hat{L}_{I^{\prime}})$ then $\hat{\psi}=\hat{\varphi}+\hat{\varphi}^{\prime}$, where $v\cdot\hat{\varphi}=0=v^{\prime}\cdot\hat{\varphi}^{\prime}$ for all $v\in I$ and $v^{\prime}\in I^{\prime}$. Thus, if $u\in I\cap I^{\prime}$ then $u\cdot\hat{\varphi}=0=u\cdot\hat{\varphi}^{\prime}$, which implies $u\cdot\hat{\psi}=0$. This means that $\hat{\psi}\in\hat{L}_{I\cap I^{\prime}}$, proving the wanted relation. _(4)_ Suppose that $I^{\prime\prime}=I+I^{\prime}$ is an isotropic subspace and let $v^{\prime\prime}\in I^{\prime\prime}$, then $v^{\prime\prime}=v+v^{\prime}$ where $v\cdot\hat{\varphi}=0=v^{\prime}\cdot\hat{\varphi}^{\prime}$ for all $\hat{\varphi}\in\hat{L}_{I}$ and $\hat{\varphi}^{\prime}\in\hat{L}_{I^{\prime}}$. Thus, if $\hat{\psi}\in\hat{L}_{I}\cap\hat{L}_{I^{\prime}}$ then $v^{\prime\prime}\cdot\hat{\psi}=0$. This means that $\hat{L}_{I}\cap\hat{L}_{I^{\prime}}\subset\hat{L}_{I^{\prime\prime}}$. Conversely, if $\hat{\psi}\in\hat{L}_{I^{\prime\prime}}$ then $(v+v^{\prime})\cdot\hat{\psi}=0$ for all $v\in I$ and $v^{\prime}\in I^{\prime}$. In particular, if we take $v^{\prime}=0$ then we get $v\cdot\hat{\psi}=0$ for all $v\in I$ $\Rightarrow$ $\hat{\psi}\in\hat{L}_{I}$. In an analogous fashion it is obtained that $\hat{\psi}\in\hat{L}_{I^{\prime}}$, so that $\hat{\psi}\in\hat{L}_{I}\cap\hat{L}_{I^{\prime}}$. Therefore, we get $\hat{L}_{I^{\prime\prime}}\subset\hat{L}_{I}\cap\hat{L}_{I^{\prime}}$, establishing the equality $\hat{L}_{I^{\prime\prime}}=\hat{L}_{I}\cap\hat{L}_{I^{\prime}}$. _(5)_ Suppose that $u\in(V\oplus V^{})$ is such that $u\cdot\hat{\varphi}=0$ for all $\hat{\varphi}\in\hat{L}_{I}$, then if $v\in I$ it follows that $2\langle u,v\rangle\hat{\varphi}=(uv+vu)\cdot\hat{\varphi}=0$. Hence, $\langle u,v\rangle=0$ for all $v$, so that $I^{\prime\prime}=I+\textrm{Span}\\{u\\}$ is an isotropic subspace of $V\oplus V^{}$. By hypothesis it follows that $\hat{L}_{I}\subset\hat{L}_{I^{\prime\prime}}$. On the other hand, by definition, we have $\dim(I^{\prime\prime})\geq\dim(I)$ which implies, by part _(1)_ of this theorem, that $\dim(\hat{L}_{I^{\prime\prime}})\leq\dim(\hat{L}_{I})$. Therefore, we conclude that $\hat{L}_{I}=\hat{L}_{I^{\prime\prime}}$. Now, using part _(2)_ of this theorem we get $I=I^{\prime\prime}\equiv I+\textrm{Span}\\{u\\}$, thus $u\in I$. _(6)_ If $I^{\prime}\subset I$ then $v^{\prime}\cdot\hat{\varphi}=0$ for all $v^{\prime}\in I^{\prime}$ and $\hat{\varphi}\in\hat{L}_{I}$, which is tantamount to $\hat{L}_{I}\subset\hat{L}_{I^{\prime}}$. Conversely, if $\hat{L}_{I}\subset\hat{L}_{I^{\prime}}$ then $v^{\prime}\cdot\hat{\varphi}=0$ for all $\hat{\varphi}\in\hat{L}_{I}$ and $v^{\prime}\in I^{\prime}$, which thanks to the item _(5)_ of this theorem, implies that $v^{\prime}\in I$, and so $I^{\prime}\subset I$. _(7)_ As explained above, if $\dim(I)=k>0$ then one can assume, without loss of generality, that $I=\textrm{Span}\\{e_{1},e_{2},\ldots,e_{k}\\}$. With this assumption $\hat{L}_{I}$ is generated by the basis $\\{\hat{1},\hat{\theta}^{\alpha^{\prime}},\hat{\theta}^{\alpha^{\prime}\beta^{\prime}},\ldots,\hat{\theta}^{k+1\,k+2...n}\\}$, where $\alpha^{\prime},\beta^{\prime}\,\in\\{k+1,k+2,...,n\\}$. The inner product of two elements of this basis, $(\hat{\varphi}_{1},\hat{\varphi}_{2})$, is always zero, since $<\hat{\varphi}_{1}^{\,t}\wedge\hat{\varphi}_{2}>_{n}=\pm<\hat{\varphi}_{1}\wedge\hat{\varphi}_{2}>_{n}=0$. With the last equality stemming from the fact that neither $\hat{\varphi}_{1}$ nor $\hat{\varphi}_{2}$ contains the term $\hat{\theta}^{1}$, so that it is impossible to get the spinor $\hat{\theta}^{12...n}$ after the wedge product. Then, using (4) we see that $(\hat{\varphi}_{1},\hat{\varphi}_{2})=0$. The rest of the statement follows from the bilinearity of this inner product. _(8)_ If $I^{\prime\prime}=I+I^{\prime}$ is isotropic then, from parts _(1)_ and _(4)_ of this theorem, it follows that $\dim(\hat{L}_{I}\cap\hat{L}_{I^{\prime}})=\dim(\hat{L}_{I^{\prime\prime}})\geq 1$, so $\hat{L}_{I}\cap\hat{L}_{I^{\prime}}\neq\\{0\\}$. Conversely, if $\hat{L}_{I}\cap\hat{L}_{I^{\prime}}\neq\\{0\\}$ then there exists $\hat{\varphi}\neq 0$ such that $v\cdot\hat{\varphi}=0=v^{\prime}\cdot\hat{\varphi}$ for all $v\in I$ and $v^{\prime}\in I^{\prime}$. Thus, $2\langle v,v^{\prime}\rangle\hat{\varphi}=(vv^{\prime}+v^{\prime}v)\cdot\hat{\varphi}=0$, so that $\langle v,v^{\prime}\rangle=0$ for all $v\in I$ and $v^{\prime}\in I^{\prime}$, implying that $I^{\prime\prime}=I+I^{\prime}$ is isotropic. _(9)_ This result is a simple consequence of the fact that if $I$ is non- maximal, $k\neq n$, then it can always be chosen to be $I=\textrm{Span}\\{e_{1},e_{2},\ldots,e_{k}\\}$, so that $\hat{L}_{I}=\textrm{Span}\\{\hat{1},\hat{\theta}^{\alpha^{\prime}},\hat{\theta}^{\alpha^{\prime}\beta^{\prime}},\ldots,\hat{\theta}^{k+1\,k+2...n}\\}$, where $\alpha^{\prime},\beta^{\prime}\,\in\\{k+1,k+2,...,n\\}$. In this basis for $\hat{L}_{I}$ there are $2^{n-k-1}$ spinors of even grade and $2^{n-k-1}$ spinors of odd grade. $\Box$ A corollary that easily follows from items _(1)_ , _(4)_ and _(8)_ is that if two isotropic subspaces $I$ and $I^{\prime}$ are such that the dimension of $\hat{L}_{I}\cap\hat{L}_{I^{\prime}}$ is not even then $\dim(\hat{L}_{I}\cap\hat{L}_{I^{\prime}})=1$ and $(I+I^{\prime})$ is a maximally isotropic subspace. Furthermore, part _(3)_ of Theorem 1 states that $(\hat{L}_{I}+\hat{L}_{I^{\prime}})$ is contained in the space $\hat{L}_{I\cap I^{\prime}}$. Then, a natural question to be posed is whether this result can be made stronger. For instance, is it true that both spaces are equal? The answer is no, generally there are elements of $\hat{L}_{I\cap I^{\prime}}$ that are not contained in $(\hat{L}_{I}+\hat{L}_{I^{\prime}})$. More precisely, it can be proved that if $(\hat{L}_{I}+\hat{L}_{I^{\prime}})=\hat{L}_{I\cap I^{\prime}}$ then one of the special cases must happen: (i) $I\subset I^{\prime}$, (ii) $I^{\prime}\subset I$ or (iii) $\dim(I)=\dim(I^{\prime})=\dim(I\cap I^{\prime})+1$ with $(I+I^{\prime})$ non-isotropic. It is not hard to see that the pseudo-scalar $\mathcal{I}\in Cl(V\oplus V^{})$ anti-commutes with every vector $u\in V\oplus V^{}$, $\mathcal{I}u=-u\mathcal{I}$. Thus, if $I\subset V\oplus V^{}$ is some isotropic subspace and $\hat{\varphi}\in\hat{L}_{I}$ then $v\cdot(\mathcal{I}\cdot\hat{\varphi})=-\mathcal{I}\cdot(v\cdot\hat{\varphi})=0$ for all $v\in I$. Therefore, the spinor $(\mathcal{I}\cdot\hat{\varphi})$ must belong to $\hat{L}_{I}$. In particular, if $\hat{\varphi}\in\hat{L}_{I}$ then the Weyl spinors of positive and negative chirality $\hat{\varphi}^{\pm}=\frac{1}{2}(\hat{\varphi}\pm\mathcal{I}\cdot\hat{\varphi})$ belong to $\hat{L}_{I}$ as well. This, in turn, implies that $\hat{L}_{I}$ can be split as the direct sum of a subspace of positive chirality and a subspace of negative chirality, which agrees with item _(9)_ of Theorem 1. Particularly, such reasoning entails the well-known result that every pure spinor must be a Weyl spinor. It is also worth noting that part _(7)_ of this theorem guarantees that every pure spinor $\hat{\psi}$ must be orthogonal to itself, $(\hat{\psi},\hat{\psi})=0$. Another interesting problem that can be posed is the following: Given a proper subspace of the spinor space, $\Upsilon\subset S$, then how can one know whether $\Upsilon$ it is related to some isotropic subspace $I\subset V\oplus V^{}$ ? More precisely, given $\Upsilon$ how to know if there exists some isotropic subspace $I$ such that $\Upsilon=\hat{L}_{I}$? The above theorem provides several clues for the answer: If one of the conditions (i) $\dim(\Upsilon)\neq 2^{s}$ for some $s\in\\{0,1,...,n-1\\}$, (ii) $(\hat{\varphi}_{1},\hat{\varphi}_{2})\neq 0$ for some $\hat{\varphi}_{1},\hat{\varphi}_{2}\,\in\Upsilon$, (iii) $\dim(\Upsilon)=1$ and $\hat{\varphi}\in\Upsilon$ is not a Weyl spinor or (iv) $\dim(\Upsilon)\neq 1$ and $\Upsilon$ does not admit a basis such that half of its elements are Weyl spinors of positive chirality while the other half are Weyl spinors of negative chirality happen then the spinorial subspace $\Upsilon$ is not related to any isotropic subspace. But the converse is not true. For instance, if $\Upsilon$ is the space spanned by some $\hat{\varphi}^{+}\in S^{+}$ then in general $\Upsilon$ is not related to an isotropic subspace, since it is well-known that although every pure spinor is a Weyl spinor not all Weyl spinors are pure111If $n\leq 3$ then all Weyl spinors are pure. But if $n>3$ there are more Weyl spinors than pure spinors.. Then, it is convenient to make the following definition: Definition: _A proper spinorial subspace $\Upsilon\subset S$ is called pure if there exists some isotropic subspace $I\subset V\oplus V^{}$, such that $\Upsilon=\hat{L}_{I}$. In the latter case we shall say that $\Upsilon$ is the pure subspace associated to $I$._ Since the constraint that a spinor must obey in order for it to be pure is quadratic in the spinor [14, 16], then, probably, the algebraic conditions that a proper spinorial subspace $\Upsilon\subset S$ might obey in order for it to be a pure subspace might consist of quadratic equations in the spinors of $\Upsilon$. As a last comment in this section, note that the space $\hat{L}_{I}$ can be defined in a different way from the one presented in Eq. (5). When $I$ is maximal then $\hat{L}_{I}$ is the pure spinor line related to $I$. When $I$ is not maximal there are several ways to complete $I$ in order to create maximally isotropic subspaces. Now, let $\\{I_{a}\\}$ be the set of all isotropic subspaces of dimension $n$ such that $I\subset I_{a}$. Associated to each maximally isotropic subspace $I_{a}$ there is a pure spinor line spanned by $\hat{\varphi}_{a}$. Then, the space $\hat{L}_{I}$ is just the one spanned by the pure spinors $\\{\hat{\varphi}_{a}\\}$. Remark:_It is not difficult to note that if $\hat{\varphi}_{1},\hat{\varphi}_{2}\in\hat{L}_{I}$ and $(e+\theta)\in I$ then $(e+2\theta)\cdot(\hat{\varphi}_{1}\wedge\hat{\varphi}_{2})=0$. In particular, this implies that if $I^{\prime}=\textrm{Span}\\{e_{1},e_{2},\ldots,e_{k}\\}$ then $(\hat{\varphi}^{\prime}_{1}\wedge\hat{\varphi}^{\prime}_{2})\in\hat{L}_{I^{\prime}}$ for all $\hat{\varphi}^{\prime}_{1},\hat{\varphi}^{\prime}_{2}\in\hat{L}_{I^{\prime}}$. But this is not true in the case of $I^{\prime}$ being an arbitrary isotropic subspace. So, probably there exists some bilinear operation $\diamond:S\times S\rightarrow S$ such that given a general isotropic subspace $I$ then $(\hat{\varphi}_{1}\diamond\hat{\varphi}_{2})$ belongs to $\hat{L}_{I}$ for all $\hat{\varphi}_{1},\hat{\varphi}_{2}\in\hat{L}_{I}$ and such that $\diamond$ reduces to $\wedge$ when $I$ is equal to $I^{\prime}$. _ ## 4 Manifolds and Spinorial Connections From now on we are going to work on a $2n$-dimensional manifold $(M,g)$ endowed with a non-degenerate metric $g$. Furthermore, the tangent bundle $TM$ is assumed to be endowed with a torsion-free and metric-compatible derivative, the Levi-Civita connection. Since in the present article we are concerned only with local results it follows that we are allowed to identify the complexified tangent spaces $\mathbb{C}\otimes T_{p}M$, at any point $p\in M$, with a vector space of the form $V\oplus V^{}$, so that the results of the previous sections can be used. More precisely, if $\\{e_{i},\theta^{j}\\}$ is a frame in a patch of $M$, with $i,j\in\\{1,2,\ldots,n\\}$, such that $g(e_{i},e_{j})\,=\,0\,,\quad\;g(e_{i},\theta^{j})\,=\,\frac{1}{2}\,\delta^{j}_{i}\,,\quad\;g(\theta^{i},\theta^{j})\,=\,0\,,$ then we shall say that $V$ is the vector bundle spanned by $\\{e_{i}\\}$ while $V^{}$ is the dual vector bundle spanned by $\\{\theta^{i}\\}$, with $\theta^{j}(e_{i})\equiv\delta^{j}_{i}$. So that an equation analogous to (2) holds: $g(e+\theta,e^{\prime}+\theta^{\prime})\,\equiv\,\frac{1}{2}\left[\,\theta(e^{\prime})\,+\,\theta^{\prime}(e)\,\right]\;\;\;\;\;\;e,e^{\prime}\in\Gamma(V)\;\textrm{ and }\;\theta,\theta^{\prime}\in\Gamma(V^{})\,.$ Where $\Gamma(V)$ is the space of sections of the bundle $V$. So, in this approach a fiber of the spinorial bundle $S$ will be identified with $\wedge V^{}$. It is worth remarking that the identifications $\mathbb{C}\otimes T_{p}M\sim V\|_{p}\oplus V^{}\|_{p}$ and $S_{p}\sim\wedge V^{}\|_{p}$ are not canonical at all, they depend on an arbitrary choice of frame. Generally, the frame $\\{e_{i},\theta^{j}\\}$ cannot be globally defined (non-parallelizable manifolds), so that these objects are well-defined just in a local neighborhood of the arbitrary point $p\in M$. Therefore, the space $V$ might, actually, be understood as a local trivialization of some bundle, which is sufficient for our purposes. Before proceeding, let us establish some notation. It is useful to denote the frame $\\{e_{i},\theta^{j}\\}$ by $\\{e_{a}\\}$, where the indices $a,b,\ldots$ run from $1$ to $2n$ and $e_{i+n}\equiv\theta^{i}$. The components of the metric in this frame are denoted by the symmetric matrix $g_{ab}\equiv g(e_{a},e_{b})$, while $g^{ab}$ stands for its inverse, $g^{ab}g_{bc}=\delta^{a}_{c}$. Finally, let $\omega_{ab}^{\phantom{ab}c}$ be the connection coefficients in this frame: $\nabla_{a}\,e_{b}\,\equiv\,\nabla_{e_{a}}\,e_{b}\,=\,\omega_{ab}^{\phantom{ab}c}\,e_{c}\,.$ It is customary to raise or lower the indices by means of $g^{ab}$ and $g_{ab}$ respectively. For instance, $\omega_{abd}=\omega_{ab}^{\phantom{ab}c}g_{cd}$ and $\omega_{a}^{\phantom{a}dc}=\omega_{ab}^{\phantom{ab}c}g^{bd}$. Since the components of the matrix $g_{ab}$ are constant and the connection is metric compatible, it follows that $\omega_{abc}=-\omega_{acb}$. Now, let us introduce a connection $\hat{\nabla}_{a}$ on the spinorial bundle $S$. In order for this connection to be an extension of the Levi-Civita connection one might impose for it to satisfy the Leibniz rule with respect to the Clifford action: $\hat{\nabla}_{a}\,(v\cdot\hat{\varphi})\,=\,(\nabla_{a}\,v)\cdot\hat{\varphi}\,+\,v\cdot\,\hat{\nabla}_{a}\,\hat{\varphi}\quad\;\;\forall\;\;v\in\Gamma(TM)\;,\,\,\hat{\varphi}\in\Gamma(S)\,.$ (6) Then, let $\\{\hat{\psi}_{\alpha}\\}$ be a local frame of the spinorial bundle $S$, with $\alpha,\beta,\ldots$ running from $1$ to $2^{n}$, such that the Clifford action of the frame $\\{e_{a}\\}$ on the spinors $\hat{\psi}_{\alpha}$ is constant in the relevant patch of $M$. This means that the following relation holds: $e_{a}\cdot\hat{\psi}_{\alpha}\,\equiv\,(e_{a})^{\beta}_{\phantom{\beta}\alpha}\,\hat{\psi}_{\beta}\,,\quad\textrm{with the matrices }\;(e_{a})^{\beta}_{\phantom{\beta}\alpha}\;\textrm{ being constant}.$ (7) In physics the matrices $(e_{a})^{\beta}_{\phantom{\beta}\alpha}$ are the so- called Dirac matrices. For instance, we can assume that the frame $\\{\hat{\psi}_{\alpha}\\}$ is given by $\\{\hat{1},\hat{\theta^{i}},\hat{\theta}^{ij},\ldots,\hat{\theta}^{1\ldots n}\\}$. Then, without loss of generality, define the action of the connection on this frame of spinors to be: $\hat{\nabla}_{a}\,\hat{\psi}_{\alpha}\,\equiv\,\Omega_{a}\cdot\hat{\psi}_{\alpha}\,=\,(\Omega_{a})^{\beta}_{\phantom{\beta}\alpha}\,\hat{\psi}_{\beta}\,.$ (8) Now, we might look for the general expression of $\Omega_{a}$ that satisfies Eq. (6). Computing $\hat{\nabla}_{a}(e_{b}\cdot\hat{\psi}_{\alpha})$ in one hand by means of (7) and on the other hand by means of (6) and then equating both calculations lead us to the following result: $(e_{b})^{\beta}_{\phantom{\beta}\alpha}\,(\Omega_{a})^{\sigma}_{\phantom{\sigma}\beta}\,\hat{\psi}_{\sigma}\,\,=\,\,\omega_{ab}^{\phantom{ab}c}\,(e_{c})^{\sigma}_{\phantom{\sigma}\alpha}\,\hat{\psi}_{\sigma}\,+\,(\Omega_{a})^{\beta}_{\phantom{\beta}\alpha}\,(e_{b})^{\sigma}_{\phantom{\sigma}\beta}\,\hat{\psi}_{\sigma}\,.$ In terms of Clifford algebra this equation is tantamount to: $\omega_{ab}^{\phantom{ab}c}\,e_{c}\,=\,\Omega_{a}\,e_{b}\,-\,e_{b}\,\Omega_{a}\,.$ (9) Such relation makes clear that $\Omega_{a}$ is defined up to the sum of a term on the centre of the Clifford algebra. But when the dimension is even, as assumed throughout this paper, the centre of the Clifford algebra is spanned by the identity operator. Then, the general solution to Eq. (9) is: $\Omega_{a}\,=\,-\,\frac{1}{4}\omega_{a}^{\phantom{a}bc}\,(e_{b}\wedge e_{c})\,+\,A_{a}\,=\,-\,\frac{1}{4}\omega_{a}^{\phantom{a}bc}\,e_{b}\,e_{c}\,+\,A_{a}\,.$ (10) Where $A_{a}$ are the components of an arbitrary $1$-form $A\in\Gamma(TM^{})$. Thus, if $v=v^{b}e_{b}$ and $\hat{\varphi}=\varphi^{\beta}\hat{\psi}_{\beta}$ are general vector and spinor fields respectively, then Eqs. (9) and (8) imply that: $\nabla_{a}\,v\,=\,(\partial_{a}v^{b})e_{b}+v^{b}(\Omega_{a}e_{b}-e_{b}\Omega_{a})\quad;\quad\hat{\nabla}_{a}\,\hat{\varphi}\,=\,(\partial_{a}\varphi^{\beta})\hat{\psi}_{\beta}+\varphi^{\beta}\,\Omega_{a}\cdot\hat{\psi}_{\beta}\,\,.$ (11) A nice review about connections on the spinorial bundle is available in [18], see also [19]. Note that since the spinorial space is defined to be the space where an irreducible and minimal representation of the Clifford algebra acts, it follows that spinors are defined up to a conformal scale. For instance, if the frame $\\{\hat{\psi}_{\alpha}\\}$ obeys to Eq. (7) then the frame formed by the spinors $\hat{\psi}^{\prime}_{\alpha}\equiv e^{-\lambda}\hat{\psi}_{\alpha}$, for some function $\lambda$, also obeys to the same equation. Likewise, it is worth recalling that the inner product (4) was also defined up to a global multiplicative factor, so that one can easily redefine this inner product, $(\,,)\rightarrow\,\prec\,,\succ$, in such a way that $(\hat{\psi}_{\alpha},\hat{\psi}_{\beta})=\prec\hat{\psi}^{\prime}_{\alpha},\hat{\psi}^{\prime}_{\beta}\succ$. Thus one can see the transformation $\hat{\varphi}\mapsto\hat{\varphi}^{\prime}=e^{-\lambda}\hat{\varphi}$ as an intrinsic symmetry of the spinorial formalism. However, in order for this symmetry to be compatible with the spinorial connection we must assume that the connection introduced in (10) and (11) change in a way that $A_{a}\mapsto A^{\prime}_{a}\,=\,A_{a}+\partial_{a}\lambda$. More precisely, we shall define the connection $\hat{\nabla}^{\prime}_{a}$ to be such that $\hat{\nabla}^{\prime}_{a}\,\hat{\psi}_{\alpha}\,=\,\Omega^{\prime}_{a}\cdot\hat{\psi}_{\alpha}\,=\,(\Omega_{a}+\partial_{a}\lambda)\cdot\hat{\psi}_{\alpha}\,=\,\hat{\nabla}_{a}\,\hat{\psi}_{\alpha}+(\partial_{a}\lambda)\,\hat{\psi}_{\alpha}\,.$ With this definition one arrives at the following relation: $\hat{\varphi}^{\prime}\,=\,e^{-\lambda}\,\hat{\varphi}\;\textrm{ and }A^{\prime}_{a}\,=\,A_{a}+\partial_{a}\lambda\;\;\;\Rightarrow\quad\hat{\nabla}^{\prime}_{a}\,\hat{\varphi}^{\prime}\,=\,e^{-\lambda}\hat{\nabla}_{a}\,\hat{\varphi}\,=\,\left(\hat{\nabla}_{a}\,\hat{\varphi}\right)^{\prime}\,.$ Physically, the above equation says that $\hat{\nabla}_{a}$ is the covariant derivative associated with the scaling symmetry of the spinors. The curvature of the spinorial bundle is defined by the following action: $\hat{\mathfrak{R}}_{ab}\,\hat{\psi}\,=\,\left(\hat{\nabla}_{a}\hat{\nabla}_{b}-\hat{\nabla}_{b}\hat{\nabla}_{a}-\hat{\nabla}_{[e_{a},e_{b}]}\right)\,\hat{\psi}\,.$ Where $[e_{a},e_{b}]$ means the Lie bracket of the vector fields $e_{a}$ and $e_{b}$. Then, using (8) and (10), one can explicitly prove that this curvature is given by: $\hat{\mathfrak{R}}_{ab}\,\hat{\psi}\,=\,-\frac{1}{4}\,R_{ab}^{\phantom{ab}cd}\,(e_{c}e_{d})\cdot\hat{\psi}\,+\,F_{ab}\,\hat{\psi}\;\,\,.$ (12) Where, in the above equation, $R_{ab}^{\phantom{ab}cd}=g^{cf}g^{dh}R_{abfh}$ is the Riemann tensor of the Riemannian manifold $(M,g)$, while $F_{ab}$ are the components of the exterior derivative of the $1$-form $A$, $F_{ab}=(dA)_{ab}$. In particular, note that the spinorial curvature is invariant under the scaling transformation, $A_{a}\mapsto A^{\prime}_{a}\,=\,A_{a}+\partial_{a}\lambda$. Two other operators that are worth mentioning are the Dirac and the twistor operators, defined respectively by: $\hat{\boldsymbol{D}}\,=\,g^{ab}e_{a}\cdot\hat{\nabla}_{b}\quad\;;\;\quad\hat{\boldsymbol{T}}_{a}\,=\,\hat{\nabla}_{a}\,-\,\frac{1}{2n}\,e_{a}\cdot\hat{\boldsymbol{D}}\,.$ The twistor operator is characterized by the property $g^{ab}e_{a}\cdot\hat{\boldsymbol{T}}_{b}=0$. We shall say that a spinor $\hat{\psi}$ is a twistor if it is annihilated by the action of the twistor operator, $\hat{\boldsymbol{T}}_{a}\hat{\psi}=0$ $\forall\,a$. As an aside, note that the square of the Dirac operator is given by: $\hat{\boldsymbol{D}}^{2}\,\hat{\psi}\,=\,\hat{\Box}\,\hat{\psi}\,+\,\frac{1}{2}F^{ab}\,(e_{a}e_{b})\cdot\,\hat{\psi}\,+\,\frac{1}{4}\,R\,\hat{\psi}\,.$ Where $\hat{\Box}$ is the “spinorial Laplacian”, $\hat{\Box}=\hat{\nabla}^{a}\hat{\nabla}_{a}-\hat{\nabla}_{\nabla^{a}e_{a}}$, and $R=R_{ab}^{\phantom{ab}ab}$ is the Ricci scalar. So far, in order to define a connection on the spinorial bundle we have just imposed that this connection obeys to the Leibniz rule with respect to the Clifford action, see (6). As a consequence, we have found that such connection is unique up to an additive $1$-form $A$. It turns out that such freedom can be fixed once we require the spinorial connection to obey the Leibniz rule with respect to the inner product on the spinorial bundle, in addition to Eq. (6). More precisely, if one requires that $\nabla_{a}\,(\hat{\varphi},\hat{\psi})\,=\,(\hat{\nabla}_{a}\hat{\varphi},\hat{\psi})\,+\,(\hat{\varphi},\hat{\nabla}_{a}\hat{\psi})$ (13) then the choice of $A_{a}$ in Eq. (10) is unique. For example, if we choose the spinorial frame $\\{\hat{\psi}_{\alpha}\\}$ to be $\\{\hat{1},\hat{\theta^{i}},\hat{\theta}^{ij},\ldots,\hat{\theta}^{1\ldots n}\\}$ then Eq. (13) holds if, and only if, we set $A_{a}=0$. For instance, this was the choice of connection made by R. Penrose when he introduced the spinorial formalism in 4-dimensional general relativity [17], in index notation Eq. (13) means that the symplectic form $\epsilon_{AB}$ is covariantly constant. For sake of generality, in the following sections it will not be assumed that Eq. (13) holds, so that the freedom in the choice of $A_{a}$ can be exploited. ## 5 Pure Subspaces and Integrability Once we have introduced a connection on the spinorial bundle, one can look for theorems on the integrability of isotropic distributions and its relation with the pure subspaces. For instance, let $I$ be some maximally isotropic distribution of vector fields over $(M,g)$ and $\hat{\psi}$ the associated pure spinor, meaning that $\hat{\psi}$ is annihilated by $I$. Then, it is well-known that the distribution $I$ is integrable if, and only if, $\hat{\nabla}_{X}\hat{\psi}\propto\hat{\psi}$ for all $X\in I$. Using the formalism introduced in Section 3, it turns out that this result can be seen as a particular case of a more general result stated here in the form of the following theorem. ###### Theorem 2 An isotropic distribution of vector fields $I$ is integrable if, and only if, $X\cdot\hat{\nabla}_{Y}\hat{\varphi}=Y\cdot\hat{\nabla}_{X}\hat{\varphi}$ for all $X,Y\in I$ and $\hat{\varphi}\in\hat{L}_{I}$. _Proof of Theorem 2:_ Let $I=I_{k}=\textrm{Span}\\{e_{1},e_{2},...,e_{k}\\}$, with $k$ fixed, and $\hat{\varphi}\in\hat{L}_{I_{k}}$. Then, $e_{\beta}\cdot\hat{\varphi}=0$, where $\beta\in\\{1,2,...,k\\}$. Hence, $\displaystyle 0=$ $\displaystyle\hat{\nabla}_{\alpha}(e_{\beta}\cdot\hat{\varphi})=(\nabla_{\alpha}e_{\beta})\cdot\hat{\varphi}+e_{\beta}\cdot(\hat{\nabla}_{\alpha}\hat{\varphi})\;\;\Rightarrow$ $\displaystyle[e_{\alpha},e_{\beta}]\cdot\hat{\varphi}=e_{\alpha}\cdot(\hat{\nabla}_{\beta}\hat{\varphi})-e_{\beta}\cdot(\hat{\nabla}_{\alpha}\hat{\varphi})\,.$ (14) Now, if $I$ is integrable then $[e_{\alpha},e_{\beta}]=f_{\alpha\beta}^{\gamma}e_{\gamma}$, where $\alpha,\beta,\gamma\in\\{1,2,...,k\\}$, so that $[e_{\alpha},e_{\beta}]\cdot\hat{\varphi}=0$. Conversely, by item _(5)_ of Theorem 1, if $[e_{\alpha},e_{\beta}]\cdot\hat{\varphi}=0$ for all $\hat{\varphi}\in\hat{L}_{I_{k}}$ then $[e_{\alpha},e_{\beta}]\in I$, so $I$ is integrable. Therefore, (14) implies that $I_{k}$ is integrable $\Leftrightarrow$ $e_{\alpha}\cdot(\nabla_{\beta}\hat{\varphi})=e_{\beta}\cdot(\nabla_{\alpha}\hat{\varphi})$, for all $\alpha,\beta\in\\{1,2,...,k\\}$ and for all $\hat{\varphi}\in\hat{L}_{I_{k}}$, proving Theorem 2. $\Box$ Now, since $I_{k}$ is an isotropic distribution it follows that $g(e_{\alpha},e_{\beta})=0$, which in terms of Clifford algebra means that $e_{\alpha}e_{\beta}=-e_{\beta}e_{\alpha}$. Then, supposing that $I_{k}$ is integrable and using the last theorem we have, $\displaystyle(e_{\alpha}e_{\beta})\cdot\hat{\nabla}_{\gamma}\hat{\varphi}\,$ $\displaystyle=\,-(e_{\beta}e_{\alpha})\cdot\hat{\nabla}_{\gamma}\hat{\varphi}\,=\,-(e_{\beta}e_{\gamma})\cdot\hat{\nabla}_{\alpha}\hat{\varphi}\,=\,(e_{\gamma}e_{\beta})\cdot\hat{\nabla}_{\alpha}\hat{\varphi}$ $\displaystyle=\,(e_{\gamma}e_{\alpha})\cdot\hat{\nabla}_{\beta}\hat{\varphi}\,=\,-(e_{\alpha}e_{\gamma})\cdot\hat{\nabla}_{\beta}\hat{\varphi}\,=\,-(e_{\alpha}e_{\beta})\cdot\hat{\nabla}_{\gamma}\hat{\varphi}\,,$ with $\hat{\varphi}\in\hat{L}_{I_{k}}$ and $\alpha,\beta,\gamma\in\\{1,2,...,k\\}$. This means that $v\cdot(e_{\beta}\cdot\hat{\nabla}_{\gamma}\hat{\varphi})=0$ for all $v\in I_{k}$. Hence, by definition, we conclude that $(e_{\beta}\cdot\hat{\nabla}_{\gamma}\hat{\varphi})$ belongs to $\hat{L}_{I_{k}}$, proving the following corollary. ###### Corollary 1 If $I$ is an integrable isotropic distribution then $[X\cdot(\hat{\nabla}_{Y}\hat{\varphi})]$ belongs to $\hat{L}_{I}$ for all $X,Y\in I$ and $\hat{\varphi}\in\hat{L}_{I}$. In the case of $I_{k}$ being maximal we have $I_{k}=I_{n}=\textrm{Span}\\{e_{1},e_{2},...,e_{n}\\}$, so that $\hat{L}_{I_{k}}$ is generated by the pure spinor $\hat{1}$. So, the above corollary implies that if $I_{n}$ is integrable then $e_{i}\cdot(\hat{\nabla}_{j}\hat{1})\propto\hat{1}$ for all $i,j\in\\{1,2,...,n\\}$, which lead us to the relation $\hat{\nabla}_{j}\hat{1}=\lambda_{j}\hat{1}+\kappa_{ji}\hat{\theta}^{i}$. But the covariant derivative of a chiral spinor cannot change its chirality, so that we must have $\kappa_{ji}=0$. Hence, if $I$ is a maximally isotropic distribution that is integrable then the parallel transport of its pure spinor in a direction tangent to $I$ does not change the direction of the spinor. By Theorem 2, it is clear that the converse of this result is also valid. This is a well-known result that in the language of the pure subspaces assumes the following form. ###### Corollary 2 A maximally isotropic distribution $I$ is integrable if, and only if, $\hat{\nabla}_{X}\,\hat{\varphi}\propto\hat{\varphi}$ for all $X\in I$ and $\hat{\varphi}\in\hat{L}_{I}$. Now, let $I_{1}=\textrm{Span}\\{e_{1}\\}$ be some one-dimensional distribution generated by the null vector field $e_{1}$, and $\hat{L}_{I_{1}}$ its associated pure subspace. So, if $\hat{\varphi}\in\hat{L}_{I_{1}}$ then $(\nabla_{1}e_{1})\cdot\hat{\varphi}=-e_{1}\cdot(\hat{\nabla}_{1}\hat{\varphi})$. Such relation, along with item _(5)_ of Theorem 1, implies that the vector field $e_{1}$ is geodesic if, and only if, $(\hat{\nabla}_{1}\hat{\varphi})$ belong to $\hat{L}_{I_{1}}$ for all $\hat{\varphi}\in\hat{L}_{I_{1}}$. This simple result is just a particular case of a broader theorem concerning the totally geodesic character of isotropic foliations. Before stating and proving such theorem, let us recall some properties of totally geodesic submanifolds. If $M^{\prime}$ is a submanifold of $(M,g)$ then a point $p\in M^{\prime}\subset M$ is said to be geodesic when every geodesic of $M$ that is tangent to $M^{\prime}$ at the point $p$ remains in $M^{\prime}$ forever. The submanifold $M^{\prime}$ is said to be totally geodesic if all its points are geodesic. As an example, note that a geodesic curve in $M$ is always a one- dimensional totally geodesic submanifold of $M$. In order to proceed it is important to explicitly show which restrictions are imposed to the connection coefficients by the existence of a totally geodesic submanifold. Let $\\{x^{1},x^{2},...,x^{2n}\\}$ be local coordinates for $M$ in the neighborhood of $p\in M^{\prime}\subset M$ such that $\\{x^{1},x^{2},...,x^{k}\\}$ are local coordinates for $M^{\prime}$ in this neighborhood. Thus, $\\{\partial_{1},\partial_{2},...,\partial_{k}\\}$ spans the tangent spaces of $M^{\prime}$ near $p$. Now, let $x^{\mu}(t)$ be a geodesic of $M$ such that $x(0)=p$ and $\frac{dx}{dt}(0)=\partial_{1}\|_{p}$. Then, using the geodesic equation it is trivial to see that near $p$ $\displaystyle x^{\mu}(t)=$ $\displaystyle x^{\mu}(0)+t\,\delta_{1}^{\phantom{1}\mu}-\frac{1}{2}t^{2}\,\Gamma_{11}^{\phantom{11}\mu}+O(t^{3})\;\;\Rightarrow$ $\displaystyle\frac{dx^{\mu}}{dt}(t)=\delta_{1}^{\phantom{1}\mu}-t\,\Gamma_{11}^{\phantom{11}\mu}+O(t^{2})\,.$ (15) Where $\Gamma_{\mu\nu}^{\phantom{\mu\nu}\rho}$ is the Christoffel symbol of the metric $g$ evaluated at $p$. Now, if $p$ is a geodesic point then $x^{\mu}(t)$ must be a point in $M^{\prime}$ and $\frac{dx^{\mu}}{dt}(t)$ must be tangent to $M^{\prime}$ for all $t$. In this case Eq. (15) implies that $\Gamma_{11}^{\phantom{11}\mu}\|_{p}=0$ if $\mu>k$. In general, if $p$ is a geodesic point then $\Gamma_{\alpha\alpha}^{\phantom{11}\mu}\|_{p}=0$ (no sum in $\alpha$) for all $\alpha\in\\{1,2,...,k\\}$ and $\mu\in\\{k+1,k+2,...,2n\\}$. It is not hard to note that this is also a sufficient condition. By means of this result along with the Frobenius theorem, we are led to the following conclusion: _The leaves of an integrable distribution $I^{\prime}$ are totally geodesic submanifolds if, and only if, $\nabla_{X}Y$ is tangent to $I^{\prime}$ for all $X,Y$ tangent to $I^{\prime}$_. There are some other equivalent ways to characterize an integrable distribution of totally geodesic leaves. Let $\\{E_{1},E_{2},...,E_{2n}\\}$ be vector fields that form a frame in $(M,g)$ such that $\\{E_{1},E_{2},...,E_{k}\\}$ span the leaves of an integrable distribution. Then, these leaves are totally geodesic submanifolds if, and only if, $\nabla_{X}(E_{1}\wedge E_{2}\wedge\cdots\wedge E_{k})\propto(E_{1}\wedge E_{2}\wedge\cdots\wedge E_{k})\quad\;\forall\;X\,\in\textrm{Span}\\{E_{1},E_{2},...,E_{k}\\}\,.$ Analogously, if $\\{E^{1},E^{2},...,E^{2n}\\}$ is the dual frame of 1-forms, $E^{a}(E_{b})=\delta_{b}^{\phantom{b}a}$, then the leaves of the integrable distribution are totally geodesic if, and only if, $\nabla_{X}(E^{k+1}\wedge\cdots\wedge E^{2n})\propto(E^{k+1}\wedge\cdots\wedge E^{2n})\quad\;\forall\;X\,\in\textrm{Span}\\{E_{1},E_{2},...,E_{k}\\}$ Finally, it is worth remarking that an embedded Euclidean manifold $M^{\prime}$ is totally geodesic if, and only if, its second fundamental form vanishes. Now, we are ready to prove the following theorem: ###### Theorem 3 An isotropic distribution of vector fields $I$ is integrable and its leaves are totally geodesic submanifolds if, and only if, $(\nabla_{X}\hat{\varphi})$ belongs to $\hat{L}_{I}$ for all $X\in I$ and $\hat{\varphi}\in\hat{L}_{I}$. _Proof of Theorem 3:_ If $I$ is integrable and generates totally geodesic submanifolds then, by what was seen above, it follows that $(\nabla_{X}Y)\in I$ for all $X,Y\in I$, so that $0=(\nabla_{X}Y)\cdot\hat{\varphi}=-Y\cdot(\hat{\nabla}_{X}\hat{\varphi})\;\;\;\forall\;X,Y\in I\;\,\textrm{and}\;\,\hat{\varphi}\in\hat{L}_{I}.$ Since $Y$ is any vector field of $I$ then, by definition, $(\hat{\nabla}_{X}\hat{\varphi})$ belongs to $\hat{L}_{I}$. Conversely, if $(\hat{\nabla}_{X}\hat{\varphi})\in\hat{L}_{I}$ for all $X\in I$ and $\hat{\varphi}\in\hat{L}_{I}$ then it follows that $Y\cdot(\nabla_{X}\hat{\varphi})=0=X\cdot(\hat{\nabla}_{Y}\hat{\varphi})$. So, by Theorem 2 this implies that the distribution $I$ is integrable. Moreover, since $0=Y\cdot(\nabla_{X}\hat{\varphi})=-(\nabla_{X}Y)\cdot\hat{\varphi}$ for all $\hat{\varphi}\in\hat{L}_{I}$ then by item _(5)_ of Theorem 1 we find that $(\nabla_{X}Y)$ must belong to $I$, implying that the leaves of this isotropic distribution are totally geodesic submanifolds. $\Box$ Combining this theorem along with Corollary 2 we find that if a maximally isotropic distribution is integrable then its leaves are totally geodesic, a known result that was proved in [10]. Particularly, if the signature of the manifold is Lorentzian and $I$ is a maximally isotropic distribution then $\dim(I\cap\overline{I})=1$, with $\overline{I}$ denoting the complex conjugate of the distribution $I$ [20]. Thus, if $I$ is integrable then $(I\cap\overline{I})$ generates a null geodesic congruence. In four dimensions, $2n=4$, if the Ricci tensor vanishes then this geodesic congruence is shear-free and the Weyl tensor is algebraically special [9]. Note that all manipulations of the present section assumed just that the spinorial connection $\hat{\nabla}_{a}$ obeys to the Leibniz rule with respect to the Clifford action, meaning that Eq. (6) holds. Thus, in the above results the $1$-form $A$ of Eq. (10) is arbitrary. In particular, one can use this freedom and Corollary 2 to prove that: If $\hat{\varphi}$ is a pure spinor that generates an integrable maximally isotropic distribution then, it is always possible to choose $A_{a}$ in (10) to be such that $\hat{\nabla}_{X}\,\hat{\varphi}=0$ for all $X$ tangent to this distribution. Moreover, by means of Eq. (12), one can verify that this required $1$-form can be a pure gauge, $A_{a}=\partial_{a}\lambda$ for some function $\lambda$, if, and only if, $\hat{\mathfrak{R}}_{XY}\,\hat{\varphi}=0$ for all $X,Y$ tangent to the distribution. ## 6 Twistor Equation and Integrability of Maximally Isotropic Distributions It is well-known that in four dimensions a pure spinor obeying to the twistor equation generates an integrable distribution of isotropic planes. The aim of the present section is to investigate whether an analogous property holds in higher dimensions. Namely, the following questions are going to be answered: Does a pure spinor obeying to the twistor equation necessarily generate an integrable maximally isotropic distribution? What about the converse, does a pure spinor generating an integrable distribution obeys to the twistor equation when we judiciously use the freedom in the choice of the spinorial connection? If $\hat{\varphi}\neq 0$ is an arbitrary pure spinor then one can always make a convenient choice of frame such that $\hat{\varphi}=\hat{1}$. In order to facilitate the calculations, let us assume that this choice was made, meaning that $e_{i}\cdot\hat{\varphi}=0$ and $\theta^{i}\cdot\hat{\varphi}=\hat{\theta}^{i}$. Where, as previously defined, the indices $i,j,k,l$ run from $1$ to $n$ while the indices $a,b$ belong to $\\{1,2,\ldots,2n\\}$, with $2n$ being the dimension of the manifold. Then, one can easily prove the following relations: $\displaystyle(e_{i}\theta^{k}\theta^{l})\cdot\hat{\varphi}=(\delta^{k}_{i}\,\hat{\theta}^{l}-\delta^{l}_{i}\,\hat{\theta}^{k})\quad\;;$ $\displaystyle\;\quad(e_{j}e_{i}\theta^{k}\theta^{l})\cdot\hat{\varphi}=(\delta^{k}_{i}\,\delta^{l}_{j}-\delta^{l}_{i}\,\delta^{k}_{j})\,\hat{1}$ $\displaystyle\hat{\nabla}_{a}\hat{\varphi}=(A_{a}\,+\,\frac{1}{2}\,$ $\displaystyle\omega_{aj}^{\phantom{aj}j})\,\hat{1}+\omega_{aij}\,\hat{\theta}^{ji}\,.$ Where Eqs. (8) and (10) were used. By means of the above relations one can prove, after some algebra, that the action of the twistor operator in $\hat{\varphi}$ is given by: $\displaystyle\hat{\boldsymbol{T}}_{j}\,\hat{\varphi}\,=\,$ $\displaystyle\frac{1}{n}\left[(n-1)(A_{j}+\frac{1}{2}\,\omega_{jk}^{\phantom{jk}k})\,+\,\omega^{i}_{\phantom{i}ij}\right]\hat{1}\,+$ $\displaystyle+\,$ $\displaystyle\frac{1}{n}\left[(n-1)\omega_{jkl}\,+\,2\,\omega_{[lk]j}\right]\,\hat{\theta}^{lk}$ (16) $\displaystyle\hat{\boldsymbol{T}}_{j+n}\,\hat{\varphi}\,=$ $\displaystyle\,\frac{1}{2}\left(A^{j}-\frac{1}{2}\omega^{ji}_{\phantom{ji}i}\right)\,\hat{1}\,+\,\frac{1}{n}\,\omega_{[ikl]}\,\hat{\theta}^{jikl}\,+$ $\displaystyle+\,$ $\displaystyle\left[\,\frac{1}{2}\,\omega^{j}_{\phantom{j}kl}\,+\,\frac{1}{n}\,\delta^{j}_{[k}A_{l]}\,+\,\frac{1}{2n}\,\delta^{j}_{[k}\omega_{l]i}^{\phantom{l]i}i}\,+\,\frac{1}{n}\,\omega^{i}_{\phantom{i}i[k}\delta^{j}_{l]}\,\right]\,\hat{\theta}^{lk}\,.$ (17) Where in the above equations $V_{[a_{1}a_{2}\ldots a_{p}]}$ means the anti- symmetric part of $V_{a_{1}a_{2}\ldots a_{p}}$. For instance, $V_{[ab]}=\frac{1}{2!}(V_{ab}-V_{ba})$. If $\hat{\varphi}$ is a twistor then the right hand side of both equations above must vanish for all $j$, let us analyse these two conditions separately. First note that if the dimension is different from six, $n\neq 3$, then the right hand side of (16) vanishes if, and only if, $A_{j}\,=\,\frac{1}{n-1}\,\omega^{i}_{\phantom{i}ji}-\frac{1}{2}\omega_{jk}^{\phantom{jk}k}\quad\;\;\textrm{and}\quad\;\;\omega_{ijk}\,=\,0\,.$ (18) While in six dimensions $\hat{\boldsymbol{T}}_{j}\hat{\varphi}=0$ if, and only if, $A_{j}\,=\,\frac{1}{3-1}\,\omega^{i}_{\phantom{i}ji}-\frac{1}{2}\omega_{jk}^{\phantom{jk}k}\quad\;\;\textrm{and}\quad\;\;\omega_{ijk}\,=\,\omega_{[ijk]}\,.$ Since the maximally isotropic distribution associated to the pure spinor $\hat{\varphi}=\hat{1}$ is spanned by $\\{e_{1},e_{2},\ldots,e_{n}\\}$ and this distribution is integrable if, and only if, $\omega_{ijk}=0$ then the above conditions lead us to the following theorem: ###### Theorem 4 If a pure spinor $\hat{\varphi}$ generates an inetgrable maximally isotropic distribution then one can always choose the 1-form $A_{a}$ so that $\hat{\boldsymbol{T}}_{X}\hat{\varphi}=0$ for all vector fields $X$ tangent to such distribution. Conversely, if the dimension is different from six and $\hat{\boldsymbol{T}}_{X}\hat{\varphi}=0$ for all $X$ tangent to the maximally isotropic distribution generated by $\hat{\varphi}$ then this distribution is integrable. Now, assume that we choose the $1$-form $A$ so that the components $A_{j}$ are just as in Eq. (18). Then, inserting this choice into (17) leads us to the following equation: $\displaystyle\hat{\boldsymbol{T}}_{j+n}\,\hat{\varphi}\,=$ $\displaystyle\,\frac{1}{2}\left(A^{j}-\frac{1}{2}\omega^{ji}_{\phantom{ji}i}\right)\,\hat{1}\,+\,\frac{1}{n}\,\omega_{[ikl]}\,\hat{\theta}^{jikl}\,+$ $\displaystyle+$ $\displaystyle\,\left[\,\frac{1}{2}\,\omega^{j}_{\phantom{j}kl}\,+\,\frac{1}{n-1}\,\omega^{i}_{\phantom{i}i[k}\delta^{j}_{l]}\,\right]\,\hat{\theta}^{lk}\,.$ (19) So, in order for the equation $\hat{\boldsymbol{T}}_{j+n}\hat{\varphi}=0$ to hold the three terms on the right hand side of the above equation must vanish. The first term can always be made to vanish by a suitable choice of the $1$-form $A$, namely we must set $2A_{j+n}\,=\,A^{j}\,=\,\frac{1}{2}\omega^{ji}_{\phantom{ji}i}\,.$ Due to the anti-symmetry in its indices, the spinor $\hat{\theta}^{jikl}$ is zero in four and six dimensions. Therefore, in these cases the second term on the right hand side of Eq. (19) is automatically zero. While if the dimension is greater or equal to eight this term will vanish if, and only if, $\omega_{[ijk]}=0$. But it is worth recalling that if we assume that $\hat{\boldsymbol{T}}_{j}\hat{\varphi}=0$ then this condition is already satisfied. Finally, the third term on the right hand side of Eq. (19) vanish if, and only if, $\omega^{i}_{\phantom{i}jk}\,=\,0\;\;\;\forall\;i\neq j,k\,\quad\textrm{and}\quad\,\omega^{i}_{\phantom{i}ik}\,=\,\omega^{j}_{\phantom{j}jk}\;(\textrm{no sum in }i,j)\;\;\;\forall\;i\neq k\neq j\,.$ (20) Where it is worth reemphasizing that in the second condition above no sum is assumed in the repeated indices $i$ and $j$. Since in four dimensions $i,j,k\in\\{1,2\\}$ it follows that the conditions of (20) are identically satisfied in this case. While in higher dimensions these constraints are non- trivial. In order to give some sort of insight on the meaning of the constraints displayed in (20), let us register that the integrability of the maximally isotropic distribution $\textrm{Span}\\{e_{i}\\}$ along with (20) is tantamount to the following restrictions: $\displaystyle g(\nabla_{X}e_{i}-\nabla_{Y}e_{j},e_{k})\,=\,0\quad\;\forall\;\,X,Y\in\,\Gamma(TM)\textrm{ such that}$ (21) $\displaystyle\;g(X,e_{i})\,=\,g(Y,e_{j})\;\textrm{ and }\;g(X,e_{k})\,=\,g(Y,e_{k})\,=\,0\,.$ The table below summarizes the joint analysis of Eqs. (16) and (17), displaying the necessary and sufficient conditions for the pure spinor $\hat{\varphi}$ to be a twistor. Since the condition $\omega_{ijk}=0$ is equivalent to the integrability of the maximally isotropic distribution generated by $\hat{\varphi}=\hat{1}$, such table leads us to Theorem 5, presented on the sequence. Dimension \| $A_{j}$ \| $A_{j+n}$ \| Constraints over $\omega_{abc}$ ---\|---\|---\|--- 4 \| $\frac{1}{2-1}\omega^{i}_{\phantom{i}ji}-\frac{1}{2}\omega_{jk}^{\phantom{jk}k}$ \| $\frac{1}{4}\omega^{ji}_{\phantom{ji}i}$ \| $\omega_{ijk}=0$ 6 \| $\frac{1}{3-1}\omega^{i}_{\phantom{i}ji}-\frac{1}{2}\omega_{jk}^{\phantom{jk}k}$ \| $\frac{1}{4}\omega^{ji}_{\phantom{ji}i}$ \| $\omega_{ijk}=\omega_{[ijk]}$ , Eq. (20) $2n\geq 8$ \| $\frac{1}{n-1}\omega^{i}_{\phantom{i}ji}-\frac{1}{2}\omega_{jk}^{\phantom{jk}k}$ \| $\frac{1}{4}\omega^{ji}_{\phantom{ji}i}$ \| $\omega_{ijk}=0$ , Eq. (20) Table 1: Depending on the dimension, the pure spinor $\hat{\varphi}=\hat{1}$ is a twistor if, and only if, the components of the $1$-form $A$ and the connection coefficients $\omega_{abc}$ are as displayed in this table. Recall that the constraints in the last row of the last column can be replaced by the single Eq. (21). ###### Theorem 5 If $\hat{\varphi}\neq 0$ is a pure spinor then the following results hold depending on the dimension of the manifold: (1) $2n=4\rightarrow$ If $\hat{\varphi}$ is a twistor then the maximally isotropic distribution generated by $\hat{\varphi}$ is integrable. Conversely, if $\hat{\varphi}$ generates an integrable distribution then one can always choose $A_{a}$ so that $\hat{\varphi}$ obeys to the twistor equation with respect to the connection of Eq. (10). (2) $2n=6\rightarrow$ The fact that $\hat{\varphi}$ obeys to the twistor equation does not imply that the distribution generated by $\hat{\varphi}$ is integrable. Conversely, the integrability of the distribution generated by $\hat{\varphi}$ does not guarantee that one can arrange $A_{a}$ in order to make $\hat{\varphi}$ obey to the twistor equation. (3) $2n\geq 8\rightarrow$ If $\hat{\varphi}$ is a twistor then the maximally isotropic distribution generated by $\hat{\varphi}$ is integrable. On the other hand, the integrability of the distribution generated by $\hat{\varphi}$ does not imply that one can find $A_{a}$ such that $\hat{\varphi}$ obeys to the twistor equation. Hopefully, the results presented in the last two sections will be valuable to some branches differential geometry. Specially, since the null directions play an important role on the study of holonomy [21, 22] it follows that some applications on this subject might appear. Although the study of Euclidean restricted holonomy is well established [23], for other signatures some questions remain open. In particular, recently some progress has been accomplished in the Lorentzian case [22]. It is also worth mentioning the interplay between holonomy groups and the twistor equation [24]. Concerning physics, the formalism introduced in the present article might be useful for string theory and supergravity [25] as well as in the study of geometrical properties of Black-Holes [26]. ## Acknowledgments I want to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil) and CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brazil) for the financial support. I also wish to thank the anonymous referee for helpful comments. ## References * [1] E. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. France 41 (1913), 53. * [2] W. Graf, Differential forms as spinors, Ann. Inst. Henri Poincaré XXIX (1978), 85; I. Benn and R. Tucker, Fermions without spinors, Commun. Math. Phys. 89 (1983), 341. * [3] E. Witten, A new proof of the positive energy theorem, Commun. Math. Phys. 80 (1981), 381. * [4] R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005), 181602. arXiv:hep-th/0501052 * [5] N. Berkovits and D. Marchioro, Relating the Green-Schwarz and pure spinor formalisms for the superstring, J. High Energy Phys. 01(2005). arXiv:hep-th/0412198 * [6] R. Penrose, A spinor approach to general relativity, Ann. Phys. 10 (1960), 171; E. Newman and R. Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962), 566. * [7] W. Kinnersley, Type D vacuum metrics, J. Math. Phys. 10 (1969), 1195. * [8] J. F. Plebański and S. Hacyan, Null geodesic surfaces and Goldberg-Sachs theorem in complex Riemannian spaces, J. Math. Phys. 16 (1975), 2403. * [9] C. Batista, A generalization of the Goldberg-Sachs theorem and its consequences, Gen. Relativ. Gravit. 45 (2013), 1411. arXiv:1205.4666 * [10] L. Mason and A. Taghavi-Chabert, Killing-Yano tensors and multi-hermitian structures, J. Geom. Phys. 60 (2010), 907. arXiv:0805.3756 * [11] A. Taghavi-Chabert, The complex Goldberg-Sachs theorem in higher dimensions, J. Geom. Phys. 62 (2012), 981. arXiv:1107.2283 * [12] C. Batista, On the Weyl tensor classification in all dimensions and its relation with integrability properties, J. Math. Phys. 54 (2013), 042502. arXiv:1301.2016 * [13] M. Gualtieri, Generalized complex geometry, PhD Thesis, University of Oxford (2003). * [14] E. Cartan, The theory of spinors, Dover (1966). * [15] C. Chevalley, The algebraic theory of spinors, Columbia University Press (1954) * [16] P. Charlton, The geometry of pure spinors, with applications, PhD Thesis, Newcastle University (1997). * [17] R. Penrose and W. Rindler, Spinors and space-time 1 and 2, Cambridge University Press (1986). * [18] A. Trautman, Connections and the Dirac operator on spinor bundles, J. Geom. Phys. 58 (2008), 238. * [19] I. Benn and R. Tucker, An introduction to spinors and geometry with applications in Physics, Adam Hilger (1987). * [20] W. Kopczynski and A. Trautman, Simple spinors and real structures, J. Math. Phys. 33 (1992), 550. * [21] H. Baum, K. Lärz and T. Leistner, On the full holonomy group of special Lorentzian manifolds (2012). arXiv:1204.5657 * [22] T. Leistner, On the classification of Lorentzian holonomy groups, J. Differential Geom. 76 (2007), 423. arXiv:math/0305139 ; A. S. Galaev, Metrics that realize all Lorentzian holonomy algebras, Int. J. Geom. Methods Mod. Phys. 3 (2006), 1025. arXiv:math/0502575 * [23] M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279. * [24] A. Lischewski, Towards a classification of pseudo-Riemannian geometries admitting twistor spinors (2013). arXiv:1303.7246 * [25] J. M. Figueroa-O’Farrill, Breaking the M-waves, Class. Quant. Grav. 17 (2000), 2925. arXiv:hep-th/9904124 * [26] B. Carneiro da Cunha and A. R. de Queiroz, Killing Horizons and Spinors (2013). arXiv:1312.4911	arxiv-papers	2013-10-01T00:19:21	2024-09-04T02:49:51.768056	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Carlos Batista", "submitter": "Carlos A. Batista da S. Filho", "url": "https://arxiv.org/abs/1310.0104" }
1310.0212	[mult] # MODELING MICROWAVE/ELECTRON-CLOUD INTERACTION M. Mattes [email protected] E. Sorolla [email protected] EPFL-STI-IEL- LEMA Station 11 1015 Lausanne Switzerland F. Zimmermann [email protected] CERN Accelerators and Beam Physics Group 1211 Geneva 23 Switzerland ###### Abstract Starting from the separate codes BI-RME and ECLOUD or PyECLOUD, we are developing a novel joint simulation tool, which models the combined effect of a charged particle beam and of microwaves on an electron cloud. Possible applications include the degradation of microwave transmission in tele- communication satellites by electron clouds; the microwave-transmission tecchniques being used in particle accelerators for the purpose of electron- cloud diagnostics; the microwave emission by the electron cloud itself in the presence of a magnetic field; and the possible suppression of electron-cloud formation in an accelerator by injecting microwaves of suitable amplitude and frequency. A few early simulation results are presented. ## Section 1 Motivation Electron multiplication on surfaces exposed to either an oscillating electromagnetic field or to a pulsed electric field gives rise to the phenomenon of multipacting, which can significantly degrade the performance of radiofrequency devices operating in high-vacuum conditions. These include, for example, components for satellite tele-communication like waveguide filters but also accelerating cavities for particle accelerators. In both examples, the electron multiplication can result in a quasi-stationary “electron cloud” inside the devices perturbing their performance. Though studies of electron multipacting in microwave devices as well as of beam-induced multipactor in particle accelerators have both been ongoing separately for several decades already [1, 2, 3, 4], little is known about the simultaneous interaction of an electron cloud with both microwaves and a particle beam [5, 6]. This lack of knowledge has so far prevented the proper interpretation of microwave transmission measurements which have been applied in several accelerators afflicted by an electron cloud. A related phenomenon is encountered in satellite-based telecommunication systems, which nowadays suffer from the problem of an electron-cloud build up interacting with the microwave field used to transmit information. ## Section 2 Model In the associated oral contribution at ECLOUD12 we have outlined the model applied to describe the interaction between electrons and microwaves. Differently from [7], the modelling is based on time-domain Green functions describing the radiation of a single electron represented as a point charge. Using the modal expansion representation of the involved Green functions, we are able to take into account the exact boundary conditions of the devices, e.g. waveguides or cavities. ## Section 3 Example Results Figure 1 shows the example of the LHC beam pipe representing the transverse electric (red arrows) and magnetic field (blue arrows) of the first TE and TM mode, respectively. The modes are computed using the BI-RME algorithm [8]. Fig. 1: First TE (top) and TM (bottom) mode of LHC beam pipe. The red arrows represent the transverse electric field, the blue ones the transverse magnetic field. In Fig. 2 the example of a single electron under multipactor regime between two parallel plates separated by 0.2 mm is depicted. The electric field, driving the electron, oscillates at 5 GHz. The upper figure shows the position of the electron, the lower one the part of the radiated electric field density related to the time-derivative of the vector potential ${A}$. The spectrum includes all harmonics that are odd multiples of the fundamental one and are generated by the moving electron. The model developed is useful for microwave diagnosis of devices prone to microwave breakdown due to multipaction. The upper picture of Fig. 3 shows the evolution of the electron density for the case of the parallel plate waveguide of Fig. 2 applying the model of [9]. The blue curve represents the electron density, the red one the associated radiated electric field contribution from the time-derivative of the vector potential ${A}$. There is a clear correlation between the two curves. The lower part of Fig. 3 is an example from the LHC beam pipe. The space charge build up was simulated with PyECLOUD [10] (blue curve). Based on the electron trajectories the radiated electromagnetic field has been computed (red curve). Again, a correlation of the two curves can be observed. Fig. 2: Multipacting electron (top) and the associated radiated electric field spectrum (bottom) inside a parallel-plate waveguide of height 0.2mm. Simulation parameters: RF frequency: 5 GHz; RF voltage: 70 V, initial velocity of electrons: 2 eV. Fig. 3: Example of an electron cloud build-up inside the parallel plate waveguide of Fig. 2 (top) and in the LHC beam pipe. The red curve repreesents the electron density (top) respectively the space charge (bottom). The blue lines are the associated time derivatives of the vector potential${A}$, thus the electric field. ## Section 4 ACKNOWLEDGMENT This project has been financed by the Swiss National Science Foundation (SNF) under Grant No. 200021_129661 “Modelling microwave-electron interaction in the LHC beam pipe”. ## References * [1] K. Ohmi, Beam and Photo-electron Interactions in Positron Storage Rings, Phys. Rev. Lett. 75, 1526-1529 (1995) * [2] M.A. Furman and G.R. Lambertson, The Electron-Cloud Instability in the Arcs of the PEP-II Positron Ring, presented at the International Workshop on Multibunch Instabilities in Future Electron and Positron Accelerators (MBI97), KEK, Tsukuba, Japan, 15-18 July 1997, KEK Proceedings 97-17 (1997) p. 170 * [3] F. Zimmermann, A Simulation Study of Electron-Cloud Instability and Beam-Induced Multipacting in the LHC, CERN LHC Project Report 95 (1997) * [4] C. Vicente, A. Cervello, M. Mattes, D. Wolk, D. Raboso, B. Mottet, H.L. Hartnagel, and J.R. Mosig, Multipactor and corona discharge prediction in rectangular waveguide based microwave components, in Proc. International Workshop on Microwave Filters, CNES, Toulouse, France, 2004 * [5] F.J. Decker, F. Caspers and F. Zimmermann, Impact of microwaves on the electron cloud and incoherent effects, in Proc. Mini-Workshop on Electron Cloud Simulations for Proton and Positron Beams (ECLOUD’02), pages 277-282, CERN, Geneva, Switzerland, 2002 * [6] F. Caspers and F. Zimmermann, Interactions of microwaves and electron clouds, in 2009 IEEE Particle Accelerator Conference, PAC’09, program code WE1PBI02, Vancouver, B.C., Canada, 2009 * [7] Edén Sorolla et al, An analytical model to evaluate the radiated power spectrum of a multipactor discharge in a parallel-plate region, IEEE Transactions on Electron Devices, 55(8):2252-2258, August 2008 * [8] G. Conciauro, R. Sorrentino and M. Guglielmi, Advanced modal analysis, John Wiley & Sons Inc., New York, NY, USA, 1999d. * [9] Edén Sorolla and Michael Mattes, Multipactor saturation in parallel-plate waveguides, Physics of Plasmas, Vol. 19, p. 072304, 2012 * [10] G. Iadarola and G. Rumolo, PyECLOUD and build up simulations at CERN, in Proc.of ECLOUD12, La Biodola, Isola d’Elba, Italy, 5-9 June 2012.	arxiv-papers	2013-10-01T09:43:15	2024-09-04T02:49:51.781465	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M. Mattes (EPFL, Lausanne), E. Sorolla (EPFL, Lausanne), F. Zimmermann\n (CERN)", "submitter": "Scientific Information Service CERN", "url": "https://arxiv.org/abs/1310.0212" }
1310.0239	# Exact reconstruction formulas for a Radon transform over cones Markus Haltmeier ###### Abstract Inversion of Radon transforms is the mathematical foundation of many modern tomographic imaging modalities. In this paper we study a conical Radon transform, which is important for computed tomography taking Compton scattering into account. The conical Radon transform we study integrates a function in $\mathbb{R}^{d}$ over all conical surfaces having vertices on a hyperplane and symmetry axis orthogonal to this plane. As the main result we derive exact reconstruction formulas of the filtered back-projection type for inverting this transform. Keywords. Radon transform, conical projections, computed tomography, inversion formula, filtered back-projection. AMS subject classifications. 44A12, 45Q05, 92C55. ## 1 Introduction Suppose that $f\colon\mathbb{R}^{d}\to\mathbb{R}$, with $d\geq 2$, is a smooth function supported in the half space $\mathbb{R}^{d-1}\times\left(0,\infty\right)$, and let $p$ be some real number. We study the problem of reconstructing the function $f$ from the integrals $\left(\mathcal{R}^{(p)}f\right)\left(\mathbf{u},\theta\right)\coloneqq\int_{0}^{\infty}\frac{1}{s^{p}}\int_{S^{d-2}}f\left(\mathbf{u}+s\sin\left(\theta\right)\mathbf{n},s\cos\left(\theta\right)\right)\left(s\sin\left(\theta\right)\right)^{d-2}\mathrm{d}\mathbf{n}\mathrm{d}s$ (1.1) for $\mathbf{u}\in\mathbb{R}^{d-1}$ and $\theta\in\left(0,\pi/2\right)$. (Here $S^{d-2}$ is the unit sphere in $\mathbb{R}^{d-1}$ and $\mathrm{d}\mathbf{n}$ the surface measure on $S^{d-2}$). We call the function $\mathcal{R}^{(p)}f\colon\mathbb{R}^{d-1}\times(0,\pi/2)\to\mathbb{R}$ the conical Radon transform of $f$. As illustrated in Figure 1.1, $\bigl{(}\mathcal{R}^{(p)}f\bigr{)}\left(\mathbf{u},\theta\right)$ is the integral of $f$ over the one sided conical surface $C\left(\mathbf{u},\theta\right)$ having vertex $\left(\mathbf{u},0\right)$ on the plane $\mathbb{R}^{d-1}\times\left(0,\infty\right)$, symmetry axis $\mathbf{e}_{d}\coloneqq\left(0,\dots,0,1\right)$, and half opening angle $\theta\in\left(0,\pi/2\right)$. The product $\left(s\sin\left(\theta\right)\right)^{d-2}\mathrm{d}\mathbf{n}\mathrm{d}s$ is the standard surface measure on $C\left(\mathbf{u},\theta\right)$, and $1/s^{p}$ is an additional radial weight that can be adapted to a particular application at hand. For $\theta\in(0,\pi/2)$, the function $\bigl{(}\mathcal{R}^{(p)}f\bigr{)}\left(\,\cdot\,,\theta\right)$ may be considered as a conical projection of $f$ onto $\mathbb{R}^{d-1}\times\left\\{0\right\\}$. Figure 1.1: The conical Radon transform integrates a function with support in the upper half space over one sided conical surfaces $C\left(\mathbf{u},\theta\right)$ centered at $(\mathbf{u},0)\in\mathbb{R}^{d-1}\times\left(0,\infty\right)$ having symmetry axis $\mathbf{e}_{d}=\left(0,\dots,0,1\right)$ and half-angle $\theta\in\left(0,\pi/2\right)$. Any point on $C\left(\mathbf{u},\theta\right)$ can be written in the form $\left(\mathbf{u}+s\sin\left(\theta\right)\mathbf{n},s\cos\left(\theta\right)\right)$ with $\mathbf{n}\in S^{d-2}$ and $s>0$. The $d-1$ dimensional surface measure on $C\left(\mathbf{u},\theta\right)$ is given by $\left(s\sin\left(\theta\right)\right)^{d-2}\mathrm{d}\mathbf{n}\mathrm{d}s$, with $\mathrm{d}\mathbf{n}$ denoting the standard surface measure on $S^{d-2}$. Inversion of the conical Radon transform in three spatial dimension is important for computed tomography taking Compton scattered photons into account [BZG98, CB94, Par00]. In [CB94, NTG05] Fourier reconstruction formulas have been derived for the cases $p\in\left\\{0,2\right\\}$. For two spatial dimensions, $\mathcal{R}^{(p)}$ has been studied with $p\in\left\\{0,2\right\\}$ in [BZG97, TN11], where reconstruction formulas of the back-projection type have been derived. In dimensions $d>3$, the conical Radon transform has, to the best of our knowledge, not been studied so far. In this paper we study $\mathcal{R}^{(p)}$ for any $d\geq 2$ and any $p\in\mathbb{R}$. We derive explicit reconstruction formulas of the back- projection type (see Theorem 1.1) as well as a Fourier slice identity (see Theorem 1.3) similar to the one of the classical Radon transform. ### 1.1 Statement of the main results Before we present our main results we introduce some notation. By $C^{\infty}_{c}(\mathbb{R}^{d-1}\times\left(0,\infty\right))$ we denote the space of all functions defined on $\mathbb{R}^{d}$, that are $C^{\infty}$ and have compact support in $\mathbb{R}^{d-1}\times\left(0,\infty\right)$. Likewise $C^{\infty}(\mathbb{R}^{d-1}\times\left(0,\pi/2\right))$ denotes the space of all infinitely smooth functions defined on $\mathbb{R}^{d-1}\times\left(0,\pi/2\right)$. As can easily be seen, the conical Radon transform defined by (1.1) is well defined as an operator $\mathcal{R}^{(p)}\colon C_{c}^{\infty}\bigl{(}\mathbb{R}^{d-1}\times(0,\infty)\bigr{)}\to C^{\infty}\bigl{(}\mathbb{R}^{d-1}\times(0,\pi/2)\bigr{)}$. Points in $\mathbb{R}^{d}$ will be written in the form $\left(\mathbf{x},y\right)$ with $\mathbf{x}\in\mathbb{R}^{d-1}$ and $y\in\mathbb{R}$. The Fourier transform of a function $f\colon\mathbb{R}^{d-1}\times\mathbb{R}\to\mathbb{C}$ with respect to the first component is denoted by $\left(\mathcal{F}f\right)\left(\mathbf{k},y\right)\coloneqq\int_{\mathbb{R}^{d-1}}e^{-i\mathbf{k}\mathbf{x}}f\left(\mathbf{x},y\right)\mathrm{d}\mathbf{x}$ for $\left(\mathbf{k},y\right)\in\mathbb{R}^{d-1}\times\mathbb{R}$. The Hankel transform of order $\left(d-3\right)/2$ in the second argument is denoted by $\left(\mathcal{H}_{(d-3)/2}f\right)\left(\mathbf{x},\lambda\right)\coloneqq\int_{0}^{\infty}J_{(d-3)/2}\left(y\lambda\right)f\left(\mathbf{x},y\right)y\mathrm{d}y$ for $\left(\mathbf{x},\lambda\right)\in\mathbb{R}^{d-1}\times\left(0,\infty\right)$, where $J_{(d-3)/2}$ is the Bessel function of the first kind of order $\left(d-3\right)/2$. Note that for $d=2$, we have $J_{-1/2}\left(y\right)=\sqrt{2/\pi y}\cos\left(y\right)$ and hence $\mathcal{H}_{-1/2}$ is closely related to the cosine transform. Similarly, we denote by $\mathcal{F}g$ the Fourier transform of a function $g\colon\mathbb{R}^{d-1}\times\left(0,\pi/2\right)\to\mathbb{C}$ with respect to the first argument. Finally, we denote by $\mathcal{I}^{(1-d)}g\colon\mathbb{R}^{d-1}\times\left(0,\pi/2\right)\to\mathbb{C}$ the Riesz potential of $g$, defined by $\left(\mathcal{F}\mathcal{I}^{(1-d)}g\right)\left(\mathbf{k},\theta\right)\coloneqq\left\|\mathbf{k}\right\|^{d-1}\left(\mathcal{F}g\right)\left(\mathbf{k},\theta\right)\quad\text{ for }\left(\mathbf{k},\theta\right)\in\mathbb{R}^{d-1}\times\left(0,\frac{\pi}{2}\right)\,.$ (1.2) The Riesz potential is well defined if $\left(\mathcal{F}g\right)\left(\,\cdot\,,\theta\right)\in L^{1}\bigl{(}\mathbb{R}^{d-1}\bigr{)}$ for every $\theta\in\left(0,\pi/2\right)$, which will always be the case in our considerations. #### Explicit reconstruction formulas The central results of this paper are the following explicit reconstruction formulas for inverting the conical Radon transform. ###### Theorem 1.1 (Reconstruction formulas for the conical Radon transform). For every $p\in\mathbb{R}$, every $f\in C^{\infty}_{c}\bigl{(}\mathbb{R}^{d-1}\times\left(0,\infty\right)\bigr{)}$ and every $\left(\mathbf{x},y\right)\in\mathbb{R}^{d-1}\times\left(0,\infty\right)$, we have $\displaystyle f\left(\mathbf{x},y\right)$ $\displaystyle=\frac{y^{p}}{\left(2\pi\right)^{d-1}}\int_{0}^{\pi/2}\frac{1}{\cos\left(\theta\right)^{1+p}}\int_{S^{d-2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(p)}f\right)\left(\mathbf{x}+y\tan\left(\theta\right)\mathbf{n},\theta\right)\mathrm{d}\mathbf{n}\mathrm{d}\theta\,,$ (1.3) $\displaystyle f\left(\mathbf{x},y\right)$ $\displaystyle=\frac{1}{\left(2\pi\right)^{d-1}}\int_{\mathbb{R}^{d-1}}\frac{\left(\left\|\mathbf{u}-\mathbf{x}\right\|^{2}+y^{2}\right)^{\frac{p-1}{2}}}{\left\|\mathbf{u}-\mathbf{x}\right\|^{d-2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(p)}f\right)\left(\mathbf{u},\arctan\left(\frac{\left\|\mathbf{u}-\mathbf{x}\right\|}{y}\right)\right)\,\mathrm{d}\mathbf{u}\,.$ (1.4) Here $\mathcal{I}^{(1-d)}$ is the Riesz potential defined by (1.2). ###### Proof. See Sections 2.2 and 2.3. ∎ The reconstruction formulas (1.3), (1.4) are of the filtered back-projection type: The Riesz potential can be interpreted as a filtration step in the first argument and the integrations actually sum over all conical surfaces that pass through the reconstruction point $\left(\mathbf{x},y\right)\in\mathbb{R}^{d-1}\times\left(0,\infty\right)$. In analogy to the classical Radon transform the integration process may therefore be called _conical back-projection_. Note that (1.3), (1.4) only differ up to a different parametrization of the set of all conical surfaces passing through the reconstruction point. For practical applications, the two and three dimensional situations are the most relevant ones. In these cases the formulas of Theorem 1.1 read as follows. ###### Corollary 1.2 (Reconstruction formulas for $d=2,3$). 1. (a) Suppose $d=2$. Then, for every $f\in C^{\infty}_{c}(\mathbb{R}\times\left(0,\infty\right))$ and every $\left(x,y\right)\in\mathbb{R}\times\left(0,\infty\right)$, $\displaystyle f\left(x,y\right)$ $\displaystyle=\frac{y^{p}}{2\pi}\int_{0}^{\pi/2}\frac{\bigl{(}\partial_{u}\mathrm{H}_{u}\mathcal{R}^{(p)}f\bigr{)}\left(x+\mathbf{n}y\tan\left(\theta\right),\theta\right)+\bigl{(}\partial_{u}\mathrm{H}_{u}\mathcal{R}^{(p)}f\bigr{)}\left(x-\mathbf{n}y\tan\left(\theta\right),\theta\right)}{\cos\left(\theta\right)^{1+p}}\,\mathrm{d}\theta\,,$ $\displaystyle f\left(x,y\right)$ $\displaystyle=\frac{1}{2\pi}\int_{\mathbb{R}}\left(\left\|u-x\right\|^{2}+y^{2}\right)^{\frac{p-1}{2}}\left(\partial_{u}\mathrm{H}_{u}\mathcal{R}^{(p)}f\right)\left(x,\arctan\left(\frac{\left\|u-x\right\|}{y}\right)\right)\,\mathrm{d}u\,.$ Here $\partial_{u}$ and $\mathrm{H}_{u}$ denote the derivative and the Hilbert transform in first argument. 2. (b) Suppose $d=3$. Then, for every $f\in C^{\infty}_{c}\bigl{(}\mathbb{R}^{2}\times\left(0,\infty\right)\bigr{)}$ and every $\left(\mathbf{x},y\right)\in\mathbb{R}^{2}\times\left(0,\infty\right)$, $\displaystyle f\left(\mathbf{x},y\right)$ $\displaystyle=-\frac{y^{p}}{4\pi^{2}}\int_{0}^{\pi/2}\frac{1}{\cos\left(\theta\right)^{1+p}}\int_{S^{1}}\left(\Delta_{\mathbf{u}}\mathcal{R}^{(p)}f\right)\left(\mathbf{x}+y\tan\left(\theta\right)\mathbf{n},\theta\right)\mathrm{d}\mathbf{n}\mathrm{d}\theta$ $\displaystyle f\left(\mathbf{x},y\right)$ $\displaystyle=-\frac{1}{4\pi^{2}}\int_{\mathbb{R}^{2}}\frac{\left(\left\|\mathbf{u}-\mathbf{x}\right\|^{2}+y^{2}\right)^{\frac{p-1}{2}}}{\left\|\mathbf{u}-\mathbf{x}\right\|}\left(\Delta_{\mathbf{u}}\mathcal{R}^{(p)}f\right)\left(\mathbf{u},\arctan\left(\frac{\left\|\mathbf{u}-\mathbf{x}\right\|}{y}\right)\right)\,\mathrm{d}\mathbf{u}\,.$ Here $\Delta_{\mathbf{u}}$ denotes the Laplacian in the first component. ###### Proof. For $d=2$ we have the Fourier representation $\bigl{(}\mathcal{F}\mathrm{H}_{u}f\bigr{)}\left(k\right)=-i\operatorname{sign}\left(k\right)\bigl{(}\mathcal{F}f\bigr{)}\left(k\right)$ and $\bigl{(}\mathcal{F}\partial_{u}f\bigr{)}\left(k\right)=ik\bigl{(}\mathcal{F}f\bigr{)}\left(k\right)$ of the Hilbert transform and the one dimensional derivative, respectively. This shows $\mathcal{I}^{-1}=\partial_{u}\mathrm{H}_{u}$. Hence Item (a) follows from Theorem 1.1. Similarly, for $d=3$, we have $\mathcal{I}^{-2}=-\Delta_{\mathbf{u}}$ and hence Item (b) again follows from Theorem 1.1. ∎ For $d=2$, formulas equivalent to the ones of Theorem 1.2 (a) have been first derived in [BZG97, TN11]. The three dimensional reconstruction formulas of Theorem 1.2 (b) (as well as the higher dimensional generalizations of Theorem 1.1) are new. One notes, that in three spatial dimensions the reconstruction formulas are particularly simple and further local: The reconstruction of $f$ at some reconstruction point $\left(\mathbf{x},y\right)$ only requires the integrals over cones passing through an arbitrarily small neighbourhood of $\left(\mathbf{x},y\right)$. Since for any odd $d$, the Riesz potential satisfies $\mathcal{I}^{(1-d)}=\left(-1\right)^{(d-1)/2}\Delta_{\mathbf{u}}^{(d-1)/2}$, the reconstruction formulas (1.3), (1.4) are in fact local for every odd space dimensions. Contrary, in even space dimension (1.3), (1.4) are non-local: Recovering a function at a single point requires knowledge of the integrals over all conical surfaces. This behaviour is similar to the one of the classical Radon transform, where also the inversion is local in odd and non- local in even dimensions (see, for example, [Nat01, p. 20]). #### A Fourier slice identity Theorem 1.1 will be established using the following Theorem 1.3, which an analogon of the well known Fourier slice identity [Nat01, Chapter 1, Theorem 1.1] satisfied by the classical Radon transform. ###### Theorem 1.3 (Fourier slice identity for the conical Radon transform). For every $p\in\mathbb{R}$, every $f\in C_{c}^{\infty}\bigl{(}\mathbb{R}^{d-1}\times(0,\infty)\bigr{)}$ and every $\left(\mathbf{k},\theta\right)\in\mathbb{R}\times(0,\pi/2)$, we have $\left(\mathcal{H}_{\frac{d-3}{2}}\mathcal{F}y^{\frac{d-3}{2}-p}f\right)\left(\mathbf{k},\left\|\mathbf{k}\right\|\tan\left(\theta\right)\right)=\left(2\pi\right)^{\frac{1-d}{2}}\frac{\cos\left(\theta\right)^{1-p}}{\tan\left(\theta\right)^{\frac{d-1}{2}}}\left\|\mathbf{k}\right\|^{\frac{d-3}{2}}\left(\mathcal{F}\mathcal{R}^{(p)}f\right)\left(\mathbf{k},\theta\right)\,.$ (1.5) Here $y^{(d-3)/2-p}f$ is the function $\left(\mathbf{x},y\right)\mapsto y^{(d-3)/2-p}f\left(\mathbf{x},y\right)$, $\mathcal{F}$ the Fourier transform in the first argument, and $\mathcal{H}_{(d-3)/2}$ the Hankel transform of order $(d-3)/2$ in the second argument. ###### Proof. See Section 2.1. ∎ The Fourier slice identity is of course of interest on its own. The argument $(\mathbf{k},\left\|\mathbf{k}\right\|\tan\theta)$, for $\mathbf{k}\in\mathbb{R}^{d-1}$ and $\alpha\in(0,\pi/2)$, appearing on the left hand side of (1.5), fills in the whole upper half-space, which is required to invert the Fourier-Hankel transform using well known explicit and stable inversion formulas. Hence the function $f$ can be reconstructed based on (1.5) by means of a $d-1$-dimensional Fourier transform, followed by an interpolation, and finally performing an inverse $d$-dimensional Fourier- Hankel transform. ### 1.2 Outline The remainder of the paper is mainly devoted to the proofs of Theorems 1.1 and 1.3 that we will establish in the following Section 2. We will first derive the Fourier slice identity of Theorem 1.3, which will then be used to proof the reconstruction formulas of Theorem 1.1. The paper ends with a discussion in Section 3. ## 2 Proofs of the main results In this section we derive Theorems 1.1 and 1.3. The following elementary Lemma shows that it suffices to derive these results for the special case $p=0$. ###### Lemma 2.1 (Relation between $\mathcal{R}^{(p)}$ and $\mathcal{R}^{(0)}$). For every $p\in\mathbb{R}$, every $f\in C_{c}^{\infty}\bigl{(}\mathbb{R}^{d-1}\times(0,\infty)\bigr{)}$ and every $\left(\mathbf{k},\theta\right)\in\mathbb{R}^{d-1}\times(0,\pi/2)$, we have $\left(\mathcal{R}^{(p)}f\right)\left(\mathbf{u},\theta\right)=\left(\cos\left(\theta\right)^{p}\mathcal{R}^{(0)}y^{-p}f\right)\left(\mathbf{u},\theta\right)\,.$ (2.1) Here $y^{-p}$ stands for the operator that multiplies a function $f\left(\mathbf{x},y\right)$ by $y^{-p}$ and likewise $\cos\left(\theta\right)^{p}$ stands for the operator that multiplies $g\left(\mathbf{u},\theta\right)$ by $\cos\left(\theta\right)^{p}$. ###### Proof. The definition of $\mathcal{R}^{(p)}$ and the substitution $s=y/\cos\left(\theta\right)$ yield $\left(\mathcal{R}^{(p)}f\right)\left(\mathbf{u},\theta\right)=\int_{0}^{\infty}\frac{1}{s^{p}}\int_{S^{d-2}}f\left(\mathbf{u}+s\sin\left(\theta\right)\mathbf{n},s\cos\left(\theta\right)\right)\left(s\sin\left(\theta\right)\right)^{d-2}\mathrm{d}\mathbf{n}\mathrm{d}s\\\ =\cos\left(\theta\right)^{p-1}\int_{0}^{\infty}\int_{S^{d-2}}y^{-p}f\left(\mathbf{u}+y\tan\left(\theta\right)\mathbf{n},y\right)\left(y\tan\left(\theta\right)\right)^{d-2}\mathrm{d}\mathbf{n}\mathrm{d}y\,.$ Comparing the last expression for $\mathcal{R}^{(p)}f$ with the corresponding expression for $\mathcal{R}^{(0)}f$ obviously shows (2.1). ∎ ### 2.1 Proof of Theorem 1.3 (the Fourier slice identity) We start by showing (1.5) for the special case $p=0$. The general case will then be a consequence of Lemma 2.1. The definition of the conical Radon transform, the definition of the Fourier transform and some basic manipulations yield $\displaystyle\left(\mathcal{F}\mathcal{R}^{(0)}f\right)\left(\mathbf{k},\theta\right)$ $\displaystyle=\int_{\mathbb{R}^{d-1}}e^{-i\mathbf{k}\mathbf{u}}\left(\mathcal{R}^{(0)}f\right)\left(\mathbf{u},\theta\right)\mathrm{d}\mathbf{u}$ $\displaystyle=\int_{0}^{\infty}\int_{S^{d-2}}\int_{\mathbb{R}^{d-1}}e^{-i\mathbf{k}\mathbf{u}}f\left(\mathbf{u}+s\sin\left(\theta\right)\mathbf{n},s\cos\left(\theta\right)\right)\left(s\sin\left(\theta\right)\right)^{d-2}\,\mathrm{d}\mathbf{u}\mathrm{d}\mathbf{n}\mathrm{d}s$ $\displaystyle=\int_{0}^{\infty}\left(s\sin\left(\theta\right)\right)^{d-2}\int_{S^{d-2}}e^{i\mathbf{k}s\sin\left(\theta\right)\mathbf{n}}\int_{\mathbb{R}^{d-1}}e^{-i\mathbf{k}\mathbf{u}}f\left(\mathbf{x},s\cos\left(\theta\right)\right)\,\mathrm{d}\mathbf{u}\mathrm{d}\mathbf{n}\mathrm{d}s$ $\displaystyle=\int_{0}^{\infty}\left(s\sin\left(\theta\right)\right)^{d-2}\left(\mathcal{F}f\right)\left(\mathbf{k},s\cos\left(\theta\right)\right)\left[\int_{S^{d-2}}e^{i\mathbf{k}s\sin\left(\theta\right)\mathbf{n}}\mathrm{d}\mathbf{n}\right]\mathrm{d}s\,.$ Now we use the identity (see, for example, [Nat01, page 198]), $\int_{S^{d-2}}e^{-i\mathbf{k}r\mathbf{n}}\mathrm{d}\mathbf{n}=\left(2\pi\right)^{\frac{d-1}{2}}\left\|\mathbf{k}\right\|^{\frac{3-d}{2}}r^{\frac{3-d}{2}}J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|r\right)\quad\text{ for all }\left(\mathbf{k},r\right)\in\mathbb{R}^{d-1}\times\left(0,\infty\right)\,.$ (2.2) Application of (2.2) with $r=s\sin\left(\theta\right)$ followed by the substitution $s=y/\cos\left(\theta\right)$ yields $\displaystyle\left(\mathcal{F}\mathcal{R}^{(0)}f\right)\left(\mathbf{k},\theta\right)$ $\displaystyle=\left(2\pi\right)^{\frac{d-1}{2}}\int_{0}^{\infty}\left(s\sin\left(\theta\right)\right)^{d-2}\left(\mathcal{F}f\right)\left(\mathbf{k},s\cos\left(\theta\right)\right)\left\|\mathbf{k}\right\|^{\frac{3-d}{2}}\left(s\sin\left(\theta\right)\right)^{\frac{3-d}{2}}J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|s\sin\left(\theta\right)\right)\mathrm{d}s$ $\displaystyle=\left(2\pi\right)^{\frac{d-1}{2}}\int_{0}^{\infty}\left(s\sin\left(\theta\right)\right)^{\frac{d-1}{2}}\left(\mathcal{F}f\right)\left(\mathbf{k},s\cos\left(\theta\right)\right)\left\|\mathbf{k}\right\|^{\frac{3-d}{2}}J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|s\sin\left(\theta\right)\right)\mathrm{d}s$ $\displaystyle=\left(2\pi\right)^{\frac{d-1}{2}}\int_{0}^{\infty}\left(y\tan\left(\theta\right)\right)^{\frac{d-1}{2}}\left(\mathcal{F}f\right)\left(\mathbf{k},y\right)\left\|\mathbf{k}\right\|^{\frac{3-d}{2}}J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|y\tan\left(\theta\right)\right)\frac{\mathrm{d}y}{\cos\left(\theta\right)}$ $\displaystyle=\left(2\pi\right)^{\frac{d-1}{2}}\frac{\tan\left(\theta\right)^{\frac{d-1}{2}}}{\cos\left(\theta\right)}\left\|\mathbf{k}\right\|^{\frac{3-d}{2}}\int_{0}^{\infty}y^{\frac{d-3}{2}}\left(\mathcal{F}f\right)\left(\mathbf{k},y\right)J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|y\tan\left(\theta\right)\right)y\mathrm{d}y\,.$ The last displayed equation is recognised as the Hankel transform of order $\left(d-3\right)/2$ of $\mathcal{F}f$ in the second argument. We conclude, that $\left(\mathcal{F}\mathcal{R}^{(0)}f\right)\left(\mathbf{k},\theta\right)=\left(2\pi\right)^{\frac{d-1}{2}}\frac{\tan\left(\theta\right)^{\frac{d-1}{2}}}{\cos\left(\theta\right)}\left\|\mathbf{k}\right\|^{\frac{3-d}{2}}\left(\mathcal{H}_{\frac{d-3}{2}}\mathcal{F}y^{\frac{d-3}{2}}f\right)\left(\mathbf{k},\left\|\mathbf{k}\right\|\tan\left(\theta\right)\right)\,.$ (2.3) This shows (1.5) for the special case $p=0$. For general $p\in\mathbb{R}$ we use the relation $\mathcal{R}^{(0)}y^{-p}f=\cos\left(\theta\right)^{-p}\mathcal{R}^{(p)}f$ from Lemma 2.1. Together with (2.3) this yields $\left(\mathcal{H}_{\frac{d-3}{2}}\mathcal{F}y^{\frac{d-3}{2}-p}f\right)\left(\mathbf{k},\left\|\mathbf{k}\right\|\tan\left(\theta\right)\right)=\left(\mathcal{H}_{\frac{d-3}{2}}\mathcal{F}y^{\frac{d-3}{2}}y^{-p}f\right)\left(\mathbf{k},\left\|\mathbf{k}\right\|\tan\left(\theta\right)\right)\\\ =\left(2\pi\right)^{\frac{1-d}{2}}\frac{\cos\left(\theta\right)}{\tan\left(\theta\right)^{\frac{d-1}{2}}}\left\|\mathbf{k}\right\|^{\frac{d-3}{2}}\left(\mathcal{F}\mathcal{R}^{(0)}y^{-p}f\right)\left(\mathbf{k},\theta\right)=\left(2\pi\right)^{\frac{1-d}{2}}\frac{\cos\left(\theta\right)^{1-p}}{\tan\left(\theta\right)^{\frac{d-1}{2}}}\left\|\mathbf{k}\right\|^{\frac{d-3}{2}}\left(\mathcal{F}\mathcal{R}^{(p)}f\right)\left(\mathbf{k},\theta\right)\,.$ This is (1.5) for the case of general $p\in\mathbb{R}$ and concludes the proof of Theorem 1.3. ### 2.2 Proof of reconstruction formula (1.3) We start with the proof of (1.3) for $p=0$. Application of the inversion formulas for the Fourier and the Hankel transform followed by the substitution $\lambda=\left\|\mathbf{k}\right\|\tan\left(\theta\right)$ shows $\displaystyle y^{\frac{d-3}{2}}f\left(\mathbf{x},y\right)$ $\displaystyle=\frac{1}{\left(2\pi\right)^{d-1}}\int_{\mathbb{R}^{d-1}}\int_{0}^{\infty}\left(\mathcal{H}_{\frac{d-3}{2}}\mathcal{F}y^{\frac{d-3}{2}}f\right)\left(\mathbf{k},\lambda\right)J_{\frac{d-3}{2}}\left(\lambda y\right)e^{i\mathbf{k}\mathbf{x}}\,\lambda\mathrm{d}\lambda\,\mathrm{d}\mathbf{k}$ $\displaystyle=\frac{1}{\left(2\pi\right)^{d-1}}\int_{\mathbb{R}^{d-1}}\int_{0}^{\pi/2}\left(\mathcal{H}_{\frac{d-3}{2}}\mathcal{F}y^{\frac{d-3}{2}}f\right)\left(\mathbf{k},\tan\left(\theta\right)y\right)J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|\tan\left(\theta\right)y\right)e^{i\mathbf{k}\mathbf{x}}\frac{\left\|\mathbf{k}\right\|^{2}\tan\left(\theta\right)}{\cos\left(\theta\right)^{2}}\,\mathrm{d}\theta\mathrm{d}\mathbf{k}\,.$ Application of the Fourier slice identity (Theorem 1.3) with $p=0$ and interchanging the order of integration then yields $y^{\frac{d-3}{2}}f\left(\mathbf{x},y\right)=\left(2\pi\right)^{\frac{3\left(1-d\right)}{2}}\int_{0}^{\pi/2}\frac{\left(\tan\left(\theta\right)\right)^{\frac{3-d}{2}}}{\cos\left(\theta\right)}\int_{\mathbb{R}^{d-1}}\left\|\mathbf{k}\right\|^{\frac{d+1}{2}}\left(\mathcal{F}\mathcal{R}^{(0)}f\right)\left(\mathbf{k},\theta\right)J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|\tan\left(\theta\right)y\right)e^{i\mathbf{k}\mathbf{x}}\mathrm{d}\mathbf{k}\,\mathrm{d}\theta\,.$ (2.4) By (2.2), we have $J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|\tan\left(\theta\right)y\right)=\left(2\pi\right)^{\frac{1-d}{2}}\left\|\mathbf{k}\right\|^{\frac{d-3}{2}}\left(\tan\left(\theta\right)y\right)^{\frac{d-3}{2}}\int_{S^{d-2}}e^{-i\mathbf{k}\tan\left(\theta\right)y\mathbf{n}}\mathrm{d}\mathbf{n}\,.$ Therefore, $\int_{\mathbb{R}^{d-1}}\left\|\mathbf{k}\right\|^{\frac{d+1}{2}}\left(\mathcal{F}\mathcal{R}^{(0)}f\right)\left(\mathbf{k},\theta\right)J_{\frac{d-3}{2}}\left(\left\|\mathbf{k}\right\|\tan\left(\theta\right)y\right)e^{i\mathbf{k}\mathbf{x}}\mathrm{d}\mathbf{k}\\\ \begin{aligned} &=\left(2\pi\right)^{\frac{1-d}{2}}\int_{S^{d-2}}\left(\tan\left(\theta\right)y\right)^{\frac{d-3}{2}}\left[\int_{\mathbb{R}^{d-1}}\left\|\mathbf{k}\right\|^{d-1}\left(\mathcal{F}\mathcal{R}^{(0)}f\right)\left(\mathbf{k},\theta\right)e^{i\mathbf{k}\left(\mathbf{x}-\tan\left(\theta\right)y\mathbf{n}\right)}\mathrm{d}\mathbf{k}\right]\\\ &=\left(2\pi\right)^{\frac{d-1}{2}}\int_{S^{d-2}}\left(\tan\left(\theta\right)y\right)^{\frac{d-3}{2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(0)}f\right)\left(\mathbf{x}-\tan\left(\theta\right)y\mathbf{n},\theta\right)\mathrm{d}\mathbf{n}\,.\end{aligned}$ Together with (2.4) this further implies $y^{\frac{d-3}{2}}f\left(\mathbf{x},y\right)=\frac{1}{\left(2\pi\right)^{d-1}}\int_{0}^{\pi/2}\frac{\left(\tan\left(\theta\right)\right)^{\frac{3-d}{2}}}{\cos\left(\theta\right)}\int_{S^{d-2}}\left(\tan\left(\theta\right)y\right)^{\frac{d-3}{2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(0)}f\right)\left(\mathbf{x}-\tan\left(\theta\right)y\mathbf{n},\theta\right)\mathrm{d}\mathbf{n}\,\mathrm{d}\theta\\\ =\frac{y^{\frac{d-3}{2}}}{\left(2\pi\right)^{d-1}}\int_{0}^{\pi/2}\frac{1}{\cos\left(\theta\right)}\int_{S^{d-2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(0)}f\right)\left(\mathbf{x}-\tan\left(\theta\right)y\mathbf{n},\theta\right)\mathrm{d}\mathbf{n}\,\mathrm{d}\theta\,.$ This shows formula (1.3) for the special case $p=0$. To show (1.3) in the general case $p\in\mathbb{R}$, we again use the relation $\cos\left(\theta\right)^{-p}\mathcal{R}^{(p)}f=\mathcal{R}^{(0)}y^{-p}f$ from Lemma 2.1. Hence application of the reconstruction formula for the special case $p=0$ to $\mathcal{R}^{(0)}y^{-p}f$ yields $y^{-p}f\left(\mathbf{x},y\right)=\left(2\pi\right)^{1-d}\int_{0}^{\pi/2}\frac{1}{\cos\left(\theta\right)^{1+p}}\int_{S^{d-2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(p)}f\right)\left(\mathbf{x}-\tan\left(\theta\right)y\mathbf{n},\theta\right)\mathrm{d}\mathbf{n}\,\mathrm{d}\theta\,.$ This shows (1.3) in the general case $\mu\in\mathbb{R}$. ### 2.3 Proof of reconstruction formula (1.4) Finally we derive (1.4) as an easy consequence of (1.3). To that end we first substitute $\theta=\arctan\left(r/y\right)$ with $r\in\left(0,\infty\right)$. Then $\mathrm{d}\theta=y^{-1}\cos\left(\theta\right)^{2}\mathrm{d}r$ and $\cos\left(\theta\right)=1/\sqrt{1+r^{2}/y^{2}}$. Consequently, (1.3) implies $f\left(\mathbf{x},y\right)=\frac{y^{p-1}}{\left(2\pi\right)^{d-1}}\int_{0}^{\infty}\int_{S^{d-2}}\left(1+r^{2}/y^{2}\right)^{\frac{p-1}{2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(0)}f\right)\left(\mathbf{x}-r\mathbf{n},\arctan\left(\frac{r}{y}\right)\right)\mathrm{d}\mathbf{n}\,\mathrm{d}r\\\ =\frac{1}{\left(2\pi\right)^{d-1}}\int_{0}^{\infty}\int_{S^{d-2}}\left(r^{2}+y^{2}\right)^{\frac{p-1}{2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(0)}f\right)\left(\mathbf{x}+r\mathbf{n},\arctan\left(\frac{r}{y}\right)\right)\mathrm{d}\mathbf{n}\,\mathrm{d}r\,.$ Now we substitute $\mathbf{x}+r\mathbf{n}=\mathbf{u}$ (polar coordinates in the plane $\mathbb{R}^{d-1}$ around the center $\mathbf{x}$). Then $\mathrm{d}\mathbf{u}=r^{d-2}\mathrm{d}\mathbf{n}\mathrm{d}r$ and $r=\left\|\mathbf{u}-\mathbf{x}\right\|$. Consequently, $f\left(\mathbf{x},y\right)=\frac{1}{\left(2\pi\right)^{d-1}}\int_{\mathbb{R}^{d-1}}\frac{\left(\left\|\mathbf{u}-\mathbf{x}\right\|^{2}+y^{2}\right)^{\frac{p-1}{2}}}{\left\|\mathbf{u}-\mathbf{x}\right\|^{d-2}}\left(\mathcal{I}^{(1-d)}\mathcal{R}^{(0)}f\right)\left(\mathbf{x},\arctan\left(\frac{\left\|\mathbf{u}-\mathbf{x}\right\|}{y}\right)\right)\;\mathrm{d}\mathbf{x}\,.$ This is the reconstruction formula (1.4). ## 3 Discussion In this paper we derived explicit reconstruction formulas for the conical Radon transform, which integrates a function in $d$ spatial variables over all cones with vertices on a hyperplane and symmetry axis orthogonal to this plane. The derived formulas are of the back-projection type and are theoretically exact. Further, they are local for odd $d$, and non-local for even $d$. Among others, inversion of the conical Radon transform is relevant for emission tomography using Compton cameras as proposed in [EFTN77, Sin83, TNE74]. Such a device measures the direction as well as the scattering angle of an incoming photon at the front of the camera. The location of the photon emission can therefore be traced back to the surface of a cone. Recovering the density of the photon source therefore yields to the inversion of the conical Radon transform in a natural manner. Radon transforms are the theoretical foundation of many medical imaging and remote sensing application. Certainly the most well known instance is the classical Radon transform, which integrates a function over hyperplanes. Among others, inversion of the classical Radon transform is important for classical transmission computed tomography and has been studied in many textbooks (see, for example, [Hel99, Nat01]). Closed form reconstruction formulas are known for a long time and have first been derived already in 1917 by J. Radon [Rad17]. Another Radon transform that has been studied in detail more recently is the spherical Radon transform. This transform integrates a function over spherical surfaces (for some restricted centers of integration) and is, among others, important for photo- and thermoacoustic tomography [KK08]. Closed form reconstruction formulas for planar and spherical center sets have been found in [And88, Kun07, Faw85, FHR07, FPR04]. The conical Radon transform, on the other hand, is much less studied. In particular, closed form reconstruction formulas have only been known for the case $d=2$, see [BZG97, TN11]. In this paper we derived such reconstruction formulas for arbitrary dimension $d\geq 2$. For computed tomography with Compton cameras [BZG98, CB94], the three dimensional case is of course the most relevant one. In this case, our reconstruction formulas have a particularly simple structure and consist of an application of the Laplacian followed by a conical back-projection. The numerical implementation seems quite straight forward following the ones of the classical or the spherical Radon transform (see, for example, [FHR07, Nat01]). Numerical studies, however, will be subject of future research. ## References * [And88] L. E. Andersson. On the determination of a function from spherical averages. SIAM J. Math. Anal., 19(1):214–232, 1988. * [BZG97] R. Basko, G. L. Zeng, and G. T. Gullberg. Analytical reconstruction formula for one-dimensional compton camera. IEEE Trans. Nucl. Sci., 44(3):1342–1346, 1997. * [BZG98] R. Basko, G. L. Zeng, and G. T. Gullberg. Application of spherical harmonics to image reconstruction for the compton camera. Physics in Medicine and Biology, 43(4):887, 1998. * [CB94] M. J. Cree and P. J. Bones. Towards direct reconstruction from a gamma camera based on compton scattering. IEEE Trans. Med. Imag., 13(2):398–407, 1994. * [EFTN77] D. B. Everett, J. S. Fleming, R. W. Todd, and J. M. Nightingale. Gamma-radiation imaging system based on the compton effect. Proc. IEEE, 124(11):995–1000, 1977. * [Faw85] J. A. Fawcett. Inversion of $n$-dimensional spherical averages. SIAM J. Appl. Math., 45(2):336–341, 1985. * [FHR07] D. Finch, M. Haltmeier, and Rakesh. Inversion of spherical means and the wave equation in even dimensions. SIAM J. Appl. Math., 68(2):392–412, 2007. * [FPR04] D. Finch, S. Patch, and Rakesh. Determining a function from its mean values over a family of spheres. SIAM J. Math. Anal., 35(5):1213–1240, 2004. * [Hel99] S. Helgason. The Radon Transform, volume 5 of Progress in Mathematics. Birkhäuser, Boston, second edition, 1999. * [KK08] P. Kuchment and L. A. Kunyansky. Mathematics of thermoacoustic and photoacoustic tomography. European J. Appl. Math., 19:191–224, 2008. * [Kun07] L. A. Kunyansky. Explicit inversion formulae for the spherical mean Radon transform. Inverse Probl., 23(1):373–383, 2007. * [Nat01] F. Natterer. The Mathematics of Computerized Tomography, volume 32 of Classics in Applied Mathematics. SIAM, Philadelphia, 2001. * [NTG05] M. K. Nguyen, T. T. Truong, and P. Grangeat. Radon transforms on a class of cones with fixed axis direction. J. Phys. A, 38(37):8003–8015, 2005. * [Par00] L. C. Parra. Reconstruction of cone-beam projections from compton scattered data. IEEE Trans. Nucl. Sci., 47(4):1543–1550, 2000. * [Rad17] J. Radon. Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verh. Kön. Sächs. Ges. Wiss. Leipzig Math. Phys. Kl., 69:262–277, 1917. * [Sin83] M. Singh. An electronically collimated gamma camera for single photon emission computed tomography. Part I: Theoretical considerations and design criteria. Med. Phys., 10(421):1983, 1983. * [TN11] T. T. Truong and M. K. Nguyen. On new V-line Radon transforms in $\mathbb{R}^{2}$ and their inversion. J. Phys. A, 44(7):075206, 13, 2011. * [TNE74] R. W. Todd, J. M. Nightingale, and D. B. Everett. A proposed gamma camera. Nature, 251:132–134, 1974.	arxiv-papers	2013-10-01T11:06:44	2024-09-04T02:49:51.787451	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Markus Haltmeier", "submitter": "Markus Haltmeier", "url": "https://arxiv.org/abs/1310.0239" }
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k1,40:19112968,17998583:0 ) (1,40:3729360,18785015:15383608,446948,9168 x1,40:4749086,18785015 k1,39:5023846,18785015:274760 x1,39:5569758,18785015 k1,39:5844517,18785015:274759 x1,39:6645356,18785015 k1,39:6920116,18785015:274760 x1,39:7975227,18785015 x1,39:9031002,18785015 k1,39:9305761,18785015:274759 x1,39:10761960,18785015 k1,39:11036720,18785015:274760 x1,39:12201934,18785015 k1,39:12476693,18785015:274759 x1,39:13277532,18785015 k1,39:13552292,18785015:274760 x1,39:15153971,18785015 k1,39:15428730,18785015:274759 x1,39:17012065,18785015 k1,39:17547984,18785015:535919 x1,39:19112968,18785015 k1,39:19112968,18785015:0 ) (1,40:3729360,19571447:15383608,446948,141880 x1,40:4020336,19571447 x1,40:4338192,19571447 x1,40:4956848,19571447 x1,40:5357263,19571447 x1,40:6922893,19571447 k1,39:7182172,19571447:259279 x1,39:7800828,19571447 x1,39:8115404,19571447 x1,39:8396556,19571447 k1,39:8655835,19571447:259279 x1,39:9893147,19571447 k1,39:10152425,19571447:259278 x1,39:11426441,19571447 k1,39:11685720,19571447:259279 x1,39:12013400,19571447 x1,39:12331256,19571447 x1,39:12830640,19571447 k1,39:13089919,19571447:259279 x1,39:13380895,19571447 k1,39:13640174,19571447:259279 x1,39:14914181,19571447 k1,39:15173459,19571447:259278 x1,39:16629003,19571447 k1,39:16888282,19571447:259279 x1,39:17434194,19571447 k1,39:17693473,19571447:259279 x1,39:19112968,19571447 k1,39:19112968,19571447:0 ) (1,40:3729360,20357879:15383608,446948,141880 x1,40:4348016,20357879 x1,40:4638992,20357879 x1,40:4857224,20357879 x1,40:6065039,20357879 k1,39:6330027,20357879:264988 x1,39:7495232,20357879 x1,39:9296791,20357879 k1,39:9561778,20357879:264987 x1,39:9889458,20357879 x1,39:11150370,20357879 k1,39:11395129,20357879:244759 x1,39:12013785,20357879 x1,39:13669856,20357879 k1,39:13934844,20357879:264988 x1,39:14881180,20357879 k1,39:15125939,20357879:244759 x1,39:15854681,20357879 k1,39:16099439,20357879:244758 x1,39:16900278,20357879 x1,39:18338117,20357879 k1,39:18603105,20357879:264988 x1,39:19112968,20357879 k1,39:19112968,20357879:0 ) (1,40:3729360,21144311:15383608,446948,141880 x1,40:4057040,21144311 x1,39:5363814,21144311 k1,39:5501096,21144311:137282 x1,39:6083704,21144311 x1,39:6649928,21144311 x1,39:7923926,21144311 k1,39:8061208,21144311:137282 x1,39:8571071,21144311 x1,39:9116983,21144311 x1,39:10136054,21144311 x1,39:11155780,21144311 k1,39:11293062,21144311:137282 x1,39:11802925,21144311 k1,39:11940206,21144311:137281 x1,39:12558862,21144311 k1,39:12696144,21144311:137282 x1,39:13787959,21144311 k1,39:13925241,21144311:137282 x1,39:14435104,21144311 k1,39:14572386,21144311:137282 x1,39:15810344,21144311 k1,39:15947626,21144311:137282 x1,39:16857258,21144311 x1,39:17657442,21144311 x1,39:18130601,21144311 x1,39:19112968,21144311 k1,39:19112968,21144311:0 ) (1,40:3729360,21930743:15383608,446948,9168 x1,40:4530199,21930743 k1,39:4706579,21930743:176380 x1,39:6344306,21930743 x1,39:8673424,21930743 x1,39:10129623,21930743 k1,39:10306003,21930743:176380 x1,39:10815866,21930743 x1,39:11289025,21930743 x1,39:12089209,21930743 x1,39:13108935,21930743 k1,39:13285315,21930743:176380 x1,39:14231651,21930743 k1,39:14408030,21930743:176379 x1,39:14735710,21930743 x1,39:15192485,21930743 x1,39:15374668,21930743 x1,39:16466483,21930743 k1,39:16642863,21930743:176380 x1,39:17443702,21930743 k1,39:17620082,21930743:176380 x1,39:18567056,21930743 x1,39:18894736,21930743 x1,39:19112968,21930743 k1,40:19112968,21930743:0 ) (1,40:3729360,22717175:15383608,446948,141880 x1,40:4202519,22717175 x1,40:5222245,22717175 k1,39:5362079,22717175:139834 x1,39:6235671,22717175 x1,39:8001190,22717175 k1,39:8196349,22717175:195159 x1,39:9215429,22717175 k1,39:9355263,22717175:139834 x1,39:10810807,22717175 k1,39:10950642,22717175:139835 x1,39:11496554,22717175 k1,39:11636388,22717175:139834 x1,39:13055883,22717175 k1,39:13195718,22717175:139835 x1,39:14142054,22717175 x1,39:14651917,22717175 x1,39:15926571,22717175 k1,39:16066405,22717175:139834 x1,39:16503516,22717175 k1,39:16643351,22717175:139835 x1,39:17589687,22717175 x1,39:18463271,22717175 k1,39:18603105,22717175:139834 x1,39:19112968,22717175 k1,39:19112968,22717175:0 ) (1,40:3729360,23503607:15383608,446948,141880 x1,40:4348016,23503607 k1,39:4496627,23503607:148611 x1,39:5042531,23503607 k1,39:5191142,23503607:148611 x1,39:6537885,23503607 k1,39:6686496,23503607:148611 x1,39:7232400,23503607 k1,39:7381010,23503607:148610 x1,39:8291298,23503607 x1,39:8728409,23503607 x1,39:9529248,23503607 k1,39:9677859,23503607:148611 x1,39:10187722,23503607 k1,39:10336333,23503607:148611 x1,39:11719780,23503607 k1,39:11868391,23503607:148611 x1,39:12414303,23503607 k1,39:12562914,23503607:148611 x1,39:13363753,23503607 k1,39:13512364,23503607:148611 x1,39:14568140,23503607 k1,39:14716735,23503607:148595 x1,39:16172934,23503607 x1,39:17010477,23503607 k1,39:17159088,23503607:148611 x1,39:18378703,23503607 k1,39:18530360,23503607:151657 x1,39:19112968,23503607 k1,39:19112968,23503607:0 ) (1,40:3729360,24290039:15383608,446948,141880 x1,40:4712382,24290039 k1,39:4898634,24290039:186252 x1,39:5699473,24290039 k1,39:5885725,24290039:186252 x1,39:6213405,24290039 x1,39:6670180,24290039 x1,39:6852363,24290039 x1,39:8271858,24290039 k1,39:8458110,24290039:186252 x1,39:9405084,24290039 x1,39:9732764,24290039 x1,39:10205923,24290039 x1,39:11225649,24290039 k1,39:11411901,24290039:186252 x1,39:12285493,24290039 x1,39:13887172,24290039 k1,39:14073425,24290039:186253 x1,39:14983057,24290039 k1,39:15169309,24290039:186252 x1,39:15787965,24290039 k1,39:15974217,24290039:186252 x1,39:17175481,24290039 k1,39:17361733,24290039:186252 x1,39:17871596,24290039 k1,39:18057848,24290039:186252 x1,39:18894736,24290039 x1,39:19112968,24290039 k1,40:19112968,24290039:0 ) (1,40:3729360,25076471:15383608,446948,141880 x1,40:4712382,25076471 k1,39:4870161,25076471:157779 x1,39:5671000,25076471 k1,39:5828779,25076471:157779 x1,39:6447435,25076471 x1,39:7211579,25076471 x1,39:9140282,25076471 k1,39:9298061,25076471:157779 x1,39:9843973,25076471 k1,39:10001752,25076471:157779 x1,39:11348495,25076471 k1,39:11506275,25076471:157780 x1,39:12816978,25076471 k1,39:12974757,25076471:157779 x1,39:14139971,25076471 k1,39:14297750,25076471:157779 x1,39:14515982,25076471 x1,39:14806958,25076471 x1,39:15124814,25076471 x1,39:15958429,25076471 k1,39:16116208,25076471:157779 x1,39:16626071,25076471 k1,39:16783850,25076471:157779 x1,39:19112968,25076471 k1,40:19112968,25076471:0 ) (1,40:3729360,25862903:15383608,446948,141880 x1,40:5403791,25862903 x1,40:7186997,25862903 g1,39:7350837,25862903 x1,39:8297173,25862903 x1,39:9462396,25862903 x1,39:10535868,25862903 k1,40:19112968,25862903:8577100 g1,40:19112968,25862903 ) (1,42:3729360,26725001:15383608,446948,145636 h1,41:3729360,26725001:655361,0,0 x1,41:5950360,26725001 k1,41:6207253,26725001:256893 x1,41:7008092,26725001 k1,41:7264984,26725001:256892 x1,41:7883640,26725001 x1,41:8340424,26725001 k1,41:8597317,26725001:256893 x1,41:9544291,26725001 x1,41:9871971,26725001 x1,41:10345130,26725001 x1,41:11364856,26725001 k1,41:11621749,26725001:256893 x1,41:12859061,26725001 k1,41:13115953,26725001:256892 x1,41:15868441,26725001 k1,41:16125240,26725001:256799 x1,41:16635103,26725001 k1,41:16891996,26725001:256893 x1,41:17838970,26725001 x1,41:18166650,26725001 x1,41:19112968,26725001 k1,41:19112968,26725001:0 ) (1,42:3729360,27511433:15383608,446948,141880 x1,42:4530199,27511433 k1,41:4707327,27511433:177128 x1,41:5617615,27511433 x1,41:7837958,27511433 k1,41:8015086,27511433:177128 x1,41:8524294,27511433 x1,41:9034157,27511433 x1,41:9325133,27511433 x1,41:10344859,27511433 k1,41:10521988,27511433:177129 x1,41:10812964,27511433 x1,41:11140644,27511433 x1,41:11624298,27511433 x1,41:12497890,27511433 x1,41:13972432,27511433 k1,41:14152882,27511433:180450 x1,41:14443858,27511433 k1,41:14620986,27511433:177128 x1,41:14948666,27511433 x1,41:15514234,27511433 x1,41:16679457,27511433 k1,41:16856585,27511433:177128 x1,41:18276080,27511433 x1,41:19112968,27511433 k1,41:19112968,27511433:0 ) (1,42:3729360,28297865:15383608,446948,141880 x1,42:4348016,28297865 x1,42:5440487,28297865 k1,41:5587413,28297865:146926 x1,41:6460997,28297865 k1,41:6607923,28297865:146926 x1,41:7845235,28297865 k1,41:7992162,28297865:146927 x1,41:8319842,28297865 x1,41:8610818,28297865 x1,41:8922114,28297865 x1,41:9385449,28297865 x1,41:10659465,28297865 k1,41:10806391,28297865:146926 x1,41:11570535,28297865 k1,41:11717461,28297865:146926 x1,41:12518300,28297865 k1,41:12665226,28297865:146926 x1,41:13721002,28297865 k1,41:13867912,28297865:146910 x1,41:15324111,28297865 x1,41:16161654,28297865 k1,41:16308580,28297865:146926 x1,41:17364355,28297865 k1,41:17511281,28297865:146926 x1,41:19112968,28297865 k1,41:19112968,28297865:0 ) (1,42:3729360,29084297:15383608,446948,141880 x1,42:5112151,29084297 x1,41:5423447,29084297 x1,41:5959527,29084297 k1,41:6211703,29084297:252176 x1,41:7012542,29084297 k1,41:7264718,29084297:252176 x1,41:7773926,29084297 x1,41:9448357,29084297 k1,41:9700534,29084297:252177 x1,41:10173693,29084297 x1,41:10484989,29084297 x1,41:10948324,29084297 k1,41:11200500,29084297:252176 x1,41:11746412,29084297 k1,41:11998588,29084297:252176 x1,41:12981628,29084297 x1,41:13491491,29084297 x1,41:14546602,29084297 x1,41:15602377,29084297 k1,41:15854553,29084297:252176 x1,41:16764841,29084297 x1,41:17856657,29084297 x1,41:19112968,29084297 k1,41:19112968,29084297:0 ) (1,42:3729360,29870729:15383608,446948,141880 x1,42:4129784,29870729 x1,41:4405039,29870729 k1,41:4679532,29870729:274493 x1,41:5298188,29870729 x1,41:5771347,29870729 x1,41:7300281,29870729 k1,41:7574773,29870729:274492 x1,41:8375612,29870729 k1,41:8650105,29870729:274493 x1,41:8941081,29870729 x1,41:9258937,29870729 x1,41:10969408,29870729 k1,41:11243901,29870729:274493 x1,41:12299012,29870729 x1,41:13354787,29870729 k1,41:13629279,29870729:274492 x1,41:15085478,29870729 k1,41:15359971,29870729:274493 x1,41:16306307,29870729 x1,41:17253281,29870729 k1,41:17527773,29870729:274492 x1,41:18037636,29870729 k1,41:18312129,29870729:274493 x1,41:19112968,29870729 k1,41:19112968,29870729:0 ) (1,42:3729360,30657161:15383608,446948,141880 x1,42:4948975,30657161 k1,41:5135197,30657161:186222 x1,41:5972741,30657161 k1,41:6154469,30657161:181728 x1,41:6482149,30657161 x1,41:8144131,30657161 k1,41:8325876,30657161:181745 x1,41:9272850,30657161 x1,41:9600530,30657161 x1,41:10073689,30657161 x1,41:11348343,30657161 k1,41:11530089,30657161:181746 x1,41:12148745,30657161 x1,41:12463321,30657161 x1,41:12744473,30657161 k1,41:12926218,30657161:181745 x1,41:14163530,30657161 k1,41:14345276,30657161:181746 x1,41:15619292,30657161 k1,41:15801038,30657161:181746 x1,41:17293278,30657161 k1,41:17475023,30657161:181745 x1,41:18130383,30657161 k1,41:18312129,30657161:181746 x1,41:19112968,30657161 k1,41:19112968,30657161:0 ) (1,42:3729360,31443593:15383608,446948,141880 x1,42:4676334,31443593 x1,42:5004014,31443593 x1,42:6278012,31443593 k1,41:6485830,31443593:207818 x1,41:6995038,31443593 x1,41:7504901,31443593 x1,41:7795877,31443593 x1,41:8815603,31443593 k1,41:9023421,31443593:207818 x1,41:9314397,31443593 x1,41:9642077,31443593 x1,41:10125731,31443593 x1,41:10999323,31443593 x1,41:12310025,31443593 k1,41:12517843,31443593:207818 x1,41:13464179,31443593 k1,41:13671996,31443593:207817 x1,41:14327356,31443593 k1,41:14535174,31443593:207818 x1,41:15336013,31443593 k1,41:15543831,31443593:207818 x1,41:15834807,31443593 x1,41:16152663,31443593 x1,41:17863134,31443593 k1,41:18070952,31443593:207818 x1,41:18689608,31443593 x1,41:18907840,31443593 x1,41:19112968,31443593 k1,42:19112968,31443593:0 ) (1,42:3729360,32230025:15383608,446948,141880 x1,42:4275272,32230025 x1,42:6203975,32230025 k1,41:6348218,32230025:144243 x1,41:6894130,32230025 k1,41:7038373,32230025:144243 x1,41:7839212,32230025 k1,41:7983455,32230025:144243 x1,41:9039231,32230025 k1,41:9183455,32230025:144224 x1,41:10639654,32230025 x1,41:11477197,32230025 k1,41:11621440,32230025:144243 x1,41:12677215,32230025 k1,41:12821458,32230025:144243 x1,41:14240953,32230025 x1,41:15077841,32230025 k1,41:15222084,32230025:144243 x1,41:16096313,32230025 x1,41:16278496,32230025 x1,41:16606176,32230025 x1,41:16893879,32230025 x1,41:17913605,32230025 k1,41:18057848,32230025:144243 x1,41:18894736,32230025 x1,41:19112968,32230025 k1,42:19112968,32230025:0 ) (1,42:3729360,33016457:15383608,446948,141880 x1,42:5003359,33016457 x1,42:6441853,33016457 k1,41:6776029,33016457:334176 x1,41:7795109,33016457 k1,41:8002621,33016457:207512 x1,41:8293597,33016457 x1,41:8611453,33016457 x1,41:10321924,33016457 k1,41:10529437,33016457:207513 x1,41:11475773,33016457 x1,41:12422747,33016457 k1,41:12630259,33016457:207512 x1,41:13503843,33016457 k1,41:13711355,33016457:207512 x1,41:14948667,33016457 k1,41:15156179,33016457:207512 x1,41:16284698,33016457 x1,41:17740242,33016457 k1,41:17947754,33016457:207512 x1,41:19112968,33016457 k1,41:19112968,33016457:0 ) (1,42:3729360,33802889:15383608,446948,141880 x1,42:4348016,33802889 x1,42:4638992,33802889 x1,42:5331038,33802889 x1,42:5513221,33802889 x1,42:6277356,33802889 k1,41:6525701,33802889:248345 x1,41:7071605,33802889 x1,41:7435971,33802889 x1,41:8964897,33802889 k1,41:9213242,33802889:248345 x1,41:9759154,33802889 k1,41:10007498,33802889:248344 x1,41:10808337,33802889 k1,41:11056682,33802889:248345 x1,41:11566545,33802889 x1,41:12294632,33802889 x1,41:12986678,33802889 x1,41:13678724,33802889 k1,41:13927069,33802889:248345 x1,41:15528748,33802889 x1,41:16530131,33802889 k1,41:16799601,33802889:269470 x1,41:17309464,33802889 k1,41:17557809,33802889:248345 x1,41:17885489,33802889 x1,41:18166641,33802889 x1,41:19112968,33802889 k1,41:19112968,33802889:0 ) (1,42:3729360,34589321:15383608,446948,141880 x1,42:4712382,34589321 k1,41:4890550,34589321:178168 x1,41:5691389,34589321 k1,41:5869557,34589321:178168 x1,41:6925333,34589321 k1,41:7103487,34589321:178154 x1,41:8559686,34589321 x1,41:9397229,34589321 k1,41:9575397,34589321:178168 x1,41:10631172,34589321 k1,41:10809340,34589321:178168 x1,41:11137020,34589321 x1,41:11702588,34589321 x1,41:12867811,34589321 k1,41:13045979,34589321:178168 x1,41:14465474,34589321 x1,41:15302362,34589321 k1,41:15480530,34589321:178168 x1,41:16099186,34589321 x1,41:17191657,34589321 k1,41:17369825,34589321:178168 x1,41:17806936,34589321 k1,41:17985104,34589321:178168 x1,41:18603760,34589321 x1,41:18894736,34589321 x1,41:19112968,34589321 k1,42:19112968,34589321:0 ) (1,42:3729360,35375753:15383608,446948,141880 x1,42:5148855,35375753 k1,41:5315108,35375753:166253 x1,41:5824971,35375753 k1,41:5991224,35375753:166253 x1,41:6901512,35375753 x1,41:8248256,35375753 k1,41:8414509,35375753:166253 x1,41:9215348,35375753 k1,41:9381601,35375753:166253 x1,41:10436712,35375753 x1,41:11492487,35375753 k1,41:11658740,35375753:166253 x1,41:13114939,35375753 k1,41:13281192,35375753:166253 x1,41:15065054,35375753 x1,41:15392734,35375753 x1,41:16653646,35375753 k1,41:16819899,35375753:166253 x1,41:18275443,35375753 x1,41:19112968,35375753 k1,41:19112968,35375753:0 ) (1,42:3729360,36162185:15383608,446948,141880 x1,42:4239223,36162185 x1,42:4967310,36162185 x1,42:5659356,36162185 x1,42:6351402,36162185 x1,42:6472650,36162185 k1,42:19112968,36162185:12640318 g1,42:19112968,36162185 ) (1,42:3729360,38017577:15383608,541848,14544 k1,42:4878874,38017577:1149514 (1,42:4878874,38017577:0,0,0 g1,42:4878874,38017577 ) x1,42:10209263,38017577 x1,42:11230837,38017577 g1,42:11427445,38017577 x1,42:13742684,38017577 x1,42:14497666,38017577 x1,42:16063444,38017577 g1,42:16260052,38017577 x1,42:17963455,38017577 k1,42:19112968,38017577:1149513 ) (1,42:3729360,38935081:15383608,541848,139977 k1,42:8803160,38935081:5073800 x1,42:12517436,38935081 x1,42:14039169,38935081 g1,42:14039169,38935081 k1,42:19112968,38935081:5073799 ) (1,44:3729360,40160188:15383608,446948,141880 h1,43:3729360,40160188:655361,0,0 x1,43:5403801,40160188 k1,43:5689842,40160188:286041 x1,43:6745617,40160188 x1,43:7218776,40160188 x1,43:8019615,40160188 k1,43:8305656,40160188:286041 x1,43:8596632,40160188 x1,43:8914488,40160188 x1,43:9533144,40160188 x1,43:9933559,40160188 x1,43:11062078,40160188 x1,43:11717420,40160188 k1,43:12003461,40160188:286041 x1,43:13314163,40160188 x1,43:14224451,40160188 k1,43:14510493,40160188:286042 x1,43:14728725,40160188 (1,43:14728725,40160188:327680,446948,9168 x1,43:15056405,40160188 ) x1,43:15220245,40160188 k1,43:15506286,40160188:286041 (1,43:15506286,40160188:327680,446948,9168 x1,43:15833966,40160188 ) x1,43:15997806,40160188 k1,43:16283847,40160188:286041 (1,43:16283847,40160188:327680,437780,4912 x1,43:16611527,40160188 ) x1,43:16829759,40160188 k1,43:17115800,40160188:286041 x1,43:18317064,40160188 k1,43:18603105,40160188:286041 x1,43:19112968,40160188 k1,43:19112968,40160188:0 ) (1,44:3729360,40946620:15383608,446948,141880 x1,44:5149511,40946620 k1,43:5300528,40946620:151017 x1,43:6101367,40946620 k1,43:6252384,40946620:151017 x1,43:6907744,40946620 x1,43:8036263,40946620 x1,43:8764350,40946620 k1,43:8915367,40946620:151017 x1,43:9461279,40946620 k1,43:9612296,40946620:151017 x1,43:10413135,40946620 k1,43:10564152,40946620:151017 x1,43:11983647,40946620 k1,43:12134663,40946620:151016 x1,43:12644526,40946620 k1,43:12795543,40946620:151017 x1,43:13596382,40946620 k1,43:13747399,40946620:151017 x1,43:14839215,40946620 x1,43:16039823,40946620 k1,43:16190840,40946620:151017 x1,43:16736752,40946620 k1,43:16887769,40946620:151017 x1,43:17688608,40946620 k1,43:17839625,40946620:151017 x1,43:18894736,40946620 x1,43:19112968,40946620 k1,44:19112968,40946620:0 ) (1,44:3729360,41733052:15383608,446948,7531 x1,44:4785135,41733052 g1,43:4948975,41733052 x1,43:6405174,41733052 g1,43:6569014,41733052 x1,43:7551381,41733052 k1,44:19112968,41733052:11561587 g1,44:19112968,41733052 ) (1,46:5040082,43270772:14072886,446948,141880 (1,45:5040082,43270772:0,291271,0 g1,45:5040082,43270772 g1,45:3729360,43270772 g1,45:3401680,43270772 (1,45:3401680,43270772:1310722,291271,0 k1,45:4712402,43270772:1310722 (1,45:4712402,43270772:0,291271,0 k1,45:4384721,43270772:-327681 x1,45:4712402,43270772 ) ) g1,45:5040082,43270772 ) x1,45:5367762,43270772 x1,45:6846906,43270772 x1,45:7720498,43270772 x1,45:8048178,43270772 x1,45:9309090,43270772 k1,45:9711830,43270772:402740 x1,45:10840349,43270772 k1,45:11243089,43270772:402740 x1,45:12334904,43270772 x1,45:13426720,43270772 x1,45:14992350,43270772 k1,45:15395089,43270772:402739 x1,45:15977697,43270772 x1,45:16815240,43270772 k1,45:17217980,43270772:402740 x1,45:18164316,43270772 k1,45:18567056,43270772:402740 x1,45:19112968,43270772 k1,45:19112968,43270772:0 ) (1,46:5040082,44057204:14072886,446948,141880 x1,46:6386833,44057204 k1,45:6546629,44057204:159796 x1,45:6874309,44057204 x1,45:7962851,44057204 k1,45:8122647,44057204:159796 x1,45:9724983,44057204 x1,45:10107711,44057204 k1,45:10268316,44057204:160605 x1,45:10778179,44057204 k1,45:10937975,44057204:159796 x1,45:12539662,44057204 k1,45:12699458,44057204:159796 x1,45:12990434,44057204 k1,45:13150231,44057204:159797 x1,45:14751918,44057204 k1,45:14911714,44057204:159796 x1,45:16112968,44057204 k1,45:16272764,44057204:159796 x1,45:16818676,44057204 k1,45:16978472,44057204:159796 x1,45:17306152,44057204 x1,45:19112968,44057204 k1,45:19112968,44057204:0 ) (1,46:5040082,44843636:14072886,446948,141880 x1,46:5477193,44843636 k1,45:5674453,44843636:197260 x1,45:6293109,44843636 x1,45:6911765,44843636 x1,45:8440699,44843636 k1,45:8637960,44843636:197261 x1,45:8928936,44843636 x1,45:9246792,44843636 x1,45:10684630,44843636 k1,45:10890245,44843636:205615 x1,45:11436157,44843636 k1,45:11633417,44843636:197260 x1,45:12143280,44843636 k1,45:12340540,44843636:197260 x1,45:13942227,44843636 k1,45:14139488,44843636:197261 x1,45:14649351,44843636 x1,45:14977031,44843636 x1,45:15159214,44843636 x1,45:15980373,44843636 x1,45:16781212,44843636 k1,45:16978472,44843636:197260 x1,45:17306152,44843636 x1,45:19112968,44843636 k1,45:19112968,44843636:0 ) (1,46:5040082,45630068:14072886,446948,92400 x1,46:5840266,45630068 g1,45:6004106,45630068 x1,45:6295082,45630068 x1,45:6612938,45630068 x1,45:8069119,45630068 k1,46:19112968,45630068:11043849 g1,46:19112968,45630068 ) (1,36:3729360,47050772:15383608,435814,113504 (1,36:3729360,47050772:943720,367985,0 k1,36:4345400,47050772:616040 (1,36:4345400,47050772:327680,367985,0 $1,36:4345400,47050772(1,36:4345400,46865742:283988,182955,0x1,36:4596620,46865742)$1,36:4629388,47050772 k1,36:4673080,47050772:43692 ) ) (1,36:4673080,47050772:0,435814,0 r1,47:4673080,47050772:0,435814,0 ) x1,36:5168005,47050772 x1,36:5824939,47050772 k1,36:6103556,47050772:278617 x1,36:6831787,47050772 x1,36:8171333,47050772 k1,36:8449951,47050772:278618 x1,36:8974239,47050772 k1,36:9252856,47050772:278617 x1,36:9893528,47050772 k1,36:10172145,47050772:278617 x1,36:10842180,47050772 k1,36:11120649,47050772:278469 x1,36:11731964,47050772 x1,36:12926274,47050772 k1,36:13204891,47050772:278617 x1,36:13874920,47050772 x1,36:14835407,47050772 k1,36:15114025,47050772:278618 x1,36:15405526,47050772 x1,36:16184099,47050772 x1,36:16679024,47050772 x1,36:17494806,47050772 k1,36:17773423,47050772:278617 x1,36:18996578,47050772 x1,36:19112968,47050772 k1,36:19112968,47050772:0 ) (1,36:3729360,47673364:15383608,357558,113504 x1,36:5695440,47673364 k1,36:5861950,47673364:166510 x1,36:6910519,47673364 x1,36:7026909,47673364 x1,36:8992989,47673364 k1,36:9159498,47673364:166509 x1,36:9916567,47673364 k1,36:10075990,47673364:159423 x1,36:11124559,47673364 x1,36:11240949,47673364 x1,36:13381615,47673364 k1,36:13548124,47673364:166509 x1,36:14218159,47673364 k1,36:14377553,47673364:159394 x1,36:14988868,47673364 x1,36:15804126,47673364 x1,36:16852689,47673364 k1,36:17012112,47673364:159423 x1,36:17448842,47673364 k1,36:17608264,47673364:159422 x1,36:18190747,47673364 x1,36:18598114,47673364 x1,36:19112968,47673364 k1,36:19112968,47673364:0 ) (1,36:3729360,48295956:15383608,357558,186779 x1,36:4777405,48295956 x1,36:6117476,48295956 x1,36:8928177,48295956 g1,36:9059249,48295956 x1,36:9816318,48295956 g1,36:9947390,48295956 x1,36:10588062,48295956 g1,36:10719134,48295956 x1,36:12728709,48295956 g1,36:12859781,48295956 x1,36:13471621,48295956 x1,36:13995909,48295956 x1,36:14228690,48295956 x1,36:14636057,48295956 x1,36:15451839,48295956 g1,36:15582911,48295956 x1,36:16311142,48295956 x1,36:17388544,48295956 r1,47:17388544,48295956:0,186779,186779 k1,36:19112968,48295956:1724424 g1,36:19112968,48295956 ) ] g1,97:19112968,48295956 ) k1,97:19579138,48295956:466170 r1,97:19579138,48295956:0,33750705,0 k1,97:20045308,48295956:466170 (1,97:20045308,48295956:15383608,33750705,0 [1,97:20045308,48295956:15383608,33750705,0 (1,47:21356030,15200611:14072886,446948,141880 (1,46:21356030,15200611:0,291271,0 g1,46:21356030,15200611 g1,46:20045308,15200611 g1,46:19717628,15200611 (1,46:19717628,15200611:1310722,291271,0 k1,46:21028350,15200611:1310722 (1,46:21028350,15200611:0,291271,0 k1,46:20700669,15200611:-327681 x1,46:21028350,15200611 ) ) g1,46:21356030,15200611 ) x1,46:21683710,15200611 x1,46:23162854,15200611 x1,46:24036446,15200611 x1,46:24364126,15200611 x1,46:25625038,15200611 k1,46:25840109,15200611:215071 x1,46:26567549,15200611 x1,46:28132541,15200611 x1,46:28450397,15200611 k1,46:28665468,15200611:215071 x1,46:29830028,15200611 x1,46:30703602,15200611 k1,46:30918672,15200611:215070 x1,46:32010487,15200611 x1,46:33102303,15200611 x1,46:34667933,15200611 k1,46:34883004,15200611:215071 x1,46:35428916,15200611 k1,46:35428916,15200611:0 ) (1,47:21356030,15987043:14072886,446948,141880 x1,47:22520581,15987043 x1,47:22811557,15987043 x1,46:24403411,15987043 k1,46:24594148,15987043:190737 x1,46:24921828,15987043 x1,46:26182740,15987043 k1,46:26373476,15987043:190736 x1,46:27993498,15987043 k1,46:28190958,15987043:197460 x1,46:28773566,15987043 x1,46:29611109,15987043 k1,46:29801846,15987043:190737 x1,46:30092822,15987043 k1,46:30283559,15987043:190737 x1,46:31667014,15987043 x1,46:33342075,15987043 x1,46:34345417,15987043 x1,46:34466665,15987043 k1,46:34664125,15987043:197460 x1,46:35210684,15987043 x1,46:35428916,15987043 k1,47:35428916,15987043:0 ) (1,47:21356030,16773475:14072886,446948,141880 x1,47:22666077,16773475 g1,46:22829917,16773475 x1,46:23776253,16773475 g1,46:23940093,16773475 x1,46:24267773,16773475 x1,46:25356315,16773475 g1,46:25520155,16773475 x1,46:27068099,16773475 g1,46:27231939,16773475 x1,46:28178275,16773475 g1,46:28342115,16773475 x1,46:28633091,16773475 g1,46:28796931,16773475 x1,46:29961491,16773475 x1,46:31199449,16773475 g1,46:31363289,16773475 x1,46:31981945,16773475 x1,46:32272921,16773475 x1,46:33765151,16773475 k1,47:35428916,16773475:1663765 g1,47:35428916,16773475 ) (1,48:21356030,18045338:14072886,446948,141880 (1,47:21356030,18045338:0,291271,0 g1,47:21356030,18045338 g1,47:20045308,18045338 g1,47:19717628,18045338 (1,47:19717628,18045338:1310722,291271,0 k1,47:21028350,18045338:1310722 (1,47:21028350,18045338:0,291271,0 k1,47:20700669,18045338:-327681 x1,47:21028350,18045338 ) ) g1,47:21356030,18045338 ) x1,47:21683710,18045338 x1,47:23162854,18045338 x1,47:24036446,18045338 x1,47:24364126,18045338 x1,47:25788878,18045338 k1,47:26229447,18045338:440569 x1,47:27503454,18045338 x1,47:28377046,18045338 x1,47:29432821,18045338 k1,47:29818046,18045338:385225 x1,47:31237541,18045338 k1,47:31622765,18045338:385224 x1,47:32641828,18045338 k1,47:33027053,18045338:385225 x1,47:34118868,18045338 x1,47:35210684,18045338 x1,47:35428916,18045338 k1,48:35428916,18045338:0 ) (1,48:21356030,18831770:14072886,446948,141880 x1,48:22921660,18831770 k1,47:23242309,18831770:320649 x1,47:23788221,18831770 k1,47:24108869,18831770:320648 x1,47:25273420,18831770 x1,47:25564396,18831770 x1,47:27156250,18831770 k1,47:27476899,18831770:320649 x1,47:27804579,18831770 x1,47:29065491,18831770 k1,47:29386140,18831770:320649 x1,47:31006162,18831770 k1,47:31366012,18831770:359850 x1,47:31948620,18831770 x1,47:32786163,18831770 k1,47:33106811,18831770:320648 x1,47:33725467,18831770 k1,47:34046116,18831770:320649 x1,47:34592028,18831770 x1,47:35428916,18831770 k1,47:35428916,18831770:0 ) (1,48:21356030,19618202:14072886,446948,102232 x1,48:22775525,19618202 g1,47:22939365,19618202 x1,47:23958428,19618202 g1,47:24122268,19618202 x1,47:25287491,19618202 x1,47:25469674,19618202 x1,47:26197769,19618202 g1,47:26361609,19618202 x1,47:26579841,19618202 (1,47:26579841,19618202:327680,437780,4912 x1,47:26907521,19618202 ) x1,47:27307936,19618202 g1,47:27471776,19618202 x1,47:28418112,19618202 k1,48:35428916,19618202:7010804 g1,48:35428916,19618202 ) (1,49:21356030,20890065:14072886,446948,141880 (1,48:21356030,20890065:0,291271,0 g1,48:21356030,20890065 g1,48:20045308,20890065 g1,48:19717628,20890065 (1,48:19717628,20890065:1310722,291271,0 k1,48:21028350,20890065:1310722 (1,48:21028350,20890065:0,291271,0 k1,48:20700669,20890065:-327681 x1,48:21028350,20890065 ) ) g1,48:21356030,20890065 ) x1,48:23649772,20890065 k1,48:23852359,20890065:202587 x1,48:24834726,20890065 k1,48:25037313,20890065:202587 x1,48:28786574,20890065 k1,48:28989161,20890065:202587 x1,48:29535073,20890065 k1,48:29737659,20890065:202586 x1,48:30757385,20890065 x1,48:32379383,20890065 k1,48:32581970,20890065:202587 x1,48:32909650,20890065 x1,48:34716466,20890065 k1,48:34919053,20890065:202587 x1,48:35428916,20890065 k1,48:35428916,20890065:0 ) (1,49:21356030,21676497:14072886,446948,141880 x1,49:22812212,21676497 k1,48:23030526,21676497:218314 x1,48:23321502,21676497 x1,48:24913356,21676497 k1,48:25131670,21676497:218314 x1,48:25641533,21676497 k1,48:25859847,21676497:218314 x1,48:27498229,21676497 k1,48:27716543,21676497:218314 x1,48:28662879,21676497 k1,48:28881193,21676497:218314 x1,48:30336737,21676497 x1,48:31174262,21676497 k1,48:31392576,21676497:218314 x1,48:33267543,21676497 k1,48:33499475,21676497:231932 x1,48:34919626,21676497 k1,48:35137940,21676497:218314 x1,48:35428916,21676497 k1,48:35428916,21676497:0 ) (1,49:21356030,22462929:14072886,446948,141880 x1,49:22739485,22462929 x1,49:24414546,22462929 x1,48:25417888,22462929 x1,48:25539136,22462929 k1,48:25792203,22462929:253067 x1,48:27648809,22462929 k1,48:27884031,22462929:235222 x1,48:28830367,22462929 k1,48:29065589,22462929:235222 x1,48:29393269,22462929 x1,48:30481811,22462929 k1,48:30717034,22462929:235223 x1,48:32264978,22462929 k1,48:32500200,22462929:235222 x1,48:32791176,22462929 k1,48:33026398,22462929:235222 x1,48:35428916,22462929 k1,48:35428916,22462929:0 ) (1,49:21356030,23249361:14072886,446948,141880 x1,49:23575709,23249361 x1,48:23713340,23249361 k1,48:23934774,23249361:221434 x1,48:24225750,23249361 k1,48:24435665,23249361:209915 x1,48:26037344,23249361 x1,48:27082632,23249361 k1,48:27292547,23249361:209915 x1,48:28421066,23249361 x1,48:28923721,23249361 k1,48:29133636,23249361:209915 x1,48:30079972,23249361 k1,48:30289888,23249361:209916 x1,48:30908544,23249361 k1,48:31118459,23249361:209915 x1,48:32138187,23249361 x1,48:32720139,23249361 x1,48:33726761,23249361 k1,48:33936676,23249361:209915 x1,48:35428916,23249361 k1,48:35428916,23249361:0 ) (1,49:21356030,24035793:14072886,446948,7531 x1,49:24050187,24035793 x1,48:24177987,24035793 k1,49:35428916,24035793:11250929 g1,49:35428916,24035793 ) (1,51:20045308,25268798:15383608,446948,141880 x1,51:21064388,25268798 k1,50:21241641,25268798:177253 x1,50:22151911,25268798 k1,50:22329163,25268798:177252 x1,50:23420978,25268798 x1,50:24877177,25268798 k1,50:25054430,25268798:177253 x1,50:25491541,25268798 k1,50:25668793,25268798:177252 x1,50:26652453,25268798 k1,50:26829706,25268798:177253 x1,50:27448362,25268798 x1,50:28285905,25268798 k1,50:28463158,25268798:177253 x1,50:28790838,25268798 x1,50:29081814,25268798 x1,50:29393110,25268798 x1,50:29856445,25268798 x1,50:31294301,25268798 k1,50:31474906,25268798:180605 x1,50:32421242,25268798 k1,50:32598495,25268798:177253 x1,50:33618212,25268798 k1,50:33795464,25268798:177252 x1,50:34633007,25268798 k1,50:34810260,25268798:177253 x1,50:35428916,25268798 k1,50:35428916,25268798:0 ) (1,51:20045308,26055230:15383608,446948,141880 x1,51:20810099,26055230 x1,51:22557267,26055230 k1,50:22779956,26055230:222689 x1,50:23325868,26055230 x1,50:24563171,26055230 k1,50:24785860,26055230:222689 x1,50:25295723,26055230 k1,50:25518412,26055230:222689 x1,50:26465386,26055230 k1,50:26688075,26055230:222689 x1,50:27306731,26055230 x1,50:28143619,26055230 x1,50:28271419,26055230 k1,50:28651126,26055230:379707 x1,50:29051541,26055230 k1,50:29274230,26055230:222689 x1,50:30183862,26055230 k1,50:30406551,26055230:222689 x1,50:31061911,26055230 x1,50:31862750,26055230 x1,50:33210138,26055230 k1,50:33432827,26055230:222689 x1,50:34051483,26055230 k1,50:34274172,26055230:222689 x1,50:34601852,26055230 x1,50:35428916,26055230 k1,50:35428916,26055230:0 ) (1,51:20045308,26841662:15383608,446948,141880 x1,51:20918892,26841662 x1,51:21810827,26841662 k1,50:22116148,26841662:305321 x1,50:23462891,26841662 k1,50:23739917,26841662:277026 x1,50:24831732,26841662 x1,50:25923548,26841662 x1,50:27489178,26841662 k1,50:27766203,26841662:277025 x1,50:28312115,26841662 k1,50:28589141,26841662:277026 x1,50:28916821,26841662 x1,50:30395965,26841662 x1,50:31269557,26841662 x1,50:31597237,26841662 x1,50:32858149,26841662 k1,50:33135174,26841662:277025 x1,50:34591373,26841662 x1,50:35428916,26841662 k1,50:35428916,26841662:0 ) (1,51:20045308,27628094:15383608,446948,141880 x1,51:21282603,27628094 k1,50:21532275,27628094:249672 x1,50:22078187,27628094 x1,50:22405867,27628094 x1,50:23524562,27628094 k1,50:23774233,27628094:249671 x1,50:24392889,27628094 x1,50:24902752,27628094 x1,50:26395637,27628094 k1,50:26645309,27628094:249672 x1,50:27155172,27628094 x1,50:27701084,27628094 x1,50:28720155,27628094 x1,50:29739881,27628094 k1,50:29989553,27628094:249672 x1,50:30644913,27628094 k1,50:30894584,27628094:249671 x1,50:31695423,27628094 k1,50:31945095,27628094:249672 x1,50:32563751,27628094 x1,50:33582822,27628094 k1,50:33832493,27628094:249671 x1,50:34378405,27628094 k1,50:34628077,27628094:249672 x1,50:35428916,27628094 k1,50:35428916,27628094:0 ) (1,51:20045308,28414526:15383608,446948,141880 x1,51:20336284,28414526 x1,51:21338316,28414526 x1,51:21702682,28414526 x1,51:22013978,28414526 x1,51:22295130,28414526 k1,50:22453541,28414526:158411 x1,50:25292531,28414526 k1,50:25450942,28414526:158411 x1,50:25996854,28414526 k1,50:26155266,28414526:158412 x1,50:26956105,28414526 k1,50:27114516,28414526:158411 x1,50:28570715,28414526 k1,50:28729126,28414526:158411 x1,50:30330813,28414526 k1,50:30489224,28414526:158411 x1,50:31398856,28414526 x1,50:32236399,28414526 k1,50:32394811,28414526:158412 x1,50:33013467,28414526 k1,50:33171878,28414526:158411 x1,50:33827238,28414526 x1,50:35428916,28414526 k1,50:35428916,28414526:0 ) (1,51:20045308,29200958:15383608,446948,141880 x1,51:20555171,29200958 g1,50:20719011,29200958 x1,50:21337667,29200958 x1,50:21655523,29200958 g1,50:21819363,29200958 x1,50:23166114,29200958 g1,50:23329954,29200958 x1,50:23803122,29200958 x1,50:24415225,29200958 x1,50:24536473,29200958 k1,51:35428916,29200958:10892443 g1,51:35428916,29200958 ) (1,52:20045308,30699541:15383608,541848,14544 k1,52:20833478,30699541:788170 (1,52:20833478,30699541:0,0,0 g1,52:20833478,30699541 ) x1,52:25334185,30699541 g1,52:25530793,30699541 x1,52:27715483,30699541 g1,52:27912091,30699541 x1,52:34640747,30699541 g1,52:34640747,30699541 k1,52:35428916,30699541:788169 ) (1,65:20045308,31780884:15383608,446948,141880 h1,63:20045308,31780884:655361,0,0 x1,63:21173837,31780884 k1,63:21453041,31780884:279204 x1,63:22144432,31780884 x1,63:22751296,31780884 k1,63:23030616,31780884:279320 x1,63:23358296,31780884 x1,63:23632888,31780884 x1,63:24033303,31780884 x1,63:24834142,31780884 k1,63:25113463,31780884:279321 x1,63:25659375,31780884 k1,63:25938695,31780884:279320 x1,63:26557351,31780884 x1,63:27176007,31780884 x1,63:28195078,31780884 k1,63:28474399,31780884:279321 x1,63:29602918,31780884 k1,63:29882238,31780884:279320 x1,63:31047444,31780884 k1,63:31326764,31780884:279320 x1,63:32418579,31780884 k1,63:32697900,31780884:279321 x1,63:33316556,31780884 x1,63:33631132,31780884 x1,63:33912284,31780884 k1,63:34191604,31780884:279320 x1,63:35428916,31780884 k1,63:35428916,31780884:0 ) (1,65:20045308,32567316:15383608,446948,141880 x1,65:20846147,32567316 x1,63:21622090,32567316 k1,63:21886153,32567316:264063 x1,63:22504809,32567316 x1,63:22822665,32567316 x1,63:23103817,32567316 x1,63:24159592,32567316 k1,63:24403611,32567316:244019 x1,63:24694587,32567316 k1,63:24938606,32567316:244019 x1,63:26212613,32567316 k1,63:26456632,32567316:244019 x1,63:27912176,32567316 k1,63:28156195,32567316:244019 x1,63:28702107,32567316 k1,63:28946126,32567316:244019 x1,63:30365621,32567316 k1,63:30609640,32567316:244019 x1,63:31555976,32567316 k1,63:31799995,32567316:244019 x1,63:32600834,32567316 x1,63:34020329,32567316 k1,63:34264348,32567316:244019 x1,63:35210684,32567316 x1,63:35428916,32567316 k1,65:35428916,32567316:0 ) (1,65:20045308,33353748:15383608,446948,141880 x1,65:20555171,33353748 x1,65:21993665,33353748 k1,63:22192125,33353748:198460 x1,63:23211205,33353748 k1,64:23360944,33353748:149739 x1,64:24307280,33353748 x1,64:24889878,33353748 x1,64:25217558,33353748 x1,64:25532134,33353748 x1,64:26551860,33353748 k1,64:26701598,33353748:149738 x1,64:27211461,33353748 k1,64:27361200,33353748:149739 x1,64:28162039,33353748 k1,64:28311778,33353748:149739 x1,64:28639458,33353748 x1,64:30118602,33353748 x1,64:30992194,33353748 x1,64:31319874,33353748 x1,64:32580786,33353748 k1,64:32730524,33353748:149738 x1,64:33859043,33353748 k1,64:34008782,33353748:149739 x1,64:35428916,33353748 k1,64:35428916,33353748:0 ) (1,65:20045308,34140180:15383608,446948,141880 x1,65:21319315,34140180 k1,64:21552521,34140180:233206 x1,64:22681031,34140180 k1,64:22914236,34140180:233205 x1,64:23860572,34140180 k1,64:24093778,34140180:233206 x1,64:25732160,34140180 k1,64:25965365,34140180:233205 x1,64:27384860,34140180 x1,64:28367882,34140180 k1,64:28601088,34140180:233206 x1,64:29074247,34140180 x1,64:29365223,34140180 x1,64:30930853,34140180 k1,64:31164058,34140180:233205 x1,64:31782714,34140180 x1,64:32097290,34140180 x1,64:32378442,34140180 k1,64:32611648,34140180:233206 x1,64:33848960,34140180 k1,64:34082165,34140180:233205 x1,64:35210684,34140180 x1,64:35428916,34140180 k1,65:35428916,34140180:0 ) (1,65:20045308,34926612:15383608,446948,141880 x1,65:21501507,34926612 k1,64:21691401,34926612:189894 x1,64:22965408,34926612 k1,64:23155302,34926612:189894 x1,64:23956141,34926612 k1,64:24146035,34926612:189894 x1,64:25201146,34926612 x1,64:26074720,34926612 k1,64:26264614,34926612:189894 x1,64:27429829,34926612 k1,64:27619722,34926612:189893 x1,64:28456610,34926612 x1,64:28966473,34926612 x1,64:30022240,34926612 k1,64:30212134,34926612:189894 x1,64:30758046,34926612 k1,64:30947940,34926612:189894 x1,64:31748779,34926612 k1,64:31938673,34926612:189894 x1,64:33066537,34926612 x1,64:33673401,34926612 k1,64:33863295,34926612:189894 x1,64:34628086,34926612 x1,64:35210684,34926612 x1,64:35428916,34926612 k1,65:35428916,34926612:0 ) (1,65:20045308,35713044:15383608,446948,141880 x1,65:20372988,35713044 x1,65:20687564,35713044 x1,65:21962218,35713044 k1,64:22188108,35713044:225890 x1,64:23024996,35713044 x1,64:24626674,35713044 k1,64:24852564,35713044:225890 x1,64:25507924,35713044 k1,64:25733814,35713044:225890 x1,64:26534653,35713044 k1,64:26760542,35713044:225889 x1,64:27815653,35713044 x1,64:28871428,35713044 k1,64:29097318,35713044:225890 x1,64:30553517,35713044 k1,64:30779407,35713044:225890 x1,64:31726381,35713044 x1,64:32054061,35713044 x1,64:32527220,35713044 x1,64:33546946,35713044 k1,64:33772836,35713044:225890 x1,64:35428916,35713044 k1,64:35428916,35713044:0 ) (1,65:20045308,36499476:15383608,446948,141880 x1,65:21064388,36499476 k1,64:21284076,36499476:219688 x1,64:22339851,36499476 x1,64:23613858,36499476 k1,64:23833546,36499476:219688 x1,64:24124522,36499476 x1,64:24439098,36499476 x1,64:24720250,36499476 x1,64:25848114,36499476 k1,64:26067802,36499476:219688 x1,64:27196321,36499476 k1,64:27416010,36499476:219689 x1,64:28836144,36499476 k1,64:29055832,36499476:219688 x1,64:30329184,36499476 k1,64:30548872,36499476:219688 x1,64:31677391,36499476 k1,64:31897079,36499476:219688 x1,64:32661214,36499476 x1,64:33279870,36499476 x1,64:34553868,36499476 k1,64:34773556,36499476:219688 x1,64:35428916,36499476 k1,64:35428916,36499476:0 ) (1,65:20045308,37285908:15383608,446948,141880 x1,65:20846147,37285908 x1,65:21683690,37285908 k1,64:21930657,37285908:246967 x1,64:22221633,37285908 k1,64:22468600,37285908:246967 x1,64:23778647,37285908 x1,64:26181173,37285908 k1,64:26428140,37285908:246967 x1,64:26719116,37285908 x1,64:27033692,37285908 x1,64:27314844,37285908 x1,64:28442708,37285908 k1,64:28689676,37285908:246968 x1,64:29235588,37285908 k1,64:29482555,37285908:246967 x1,64:30283394,37285908 k1,64:30530361,37285908:246967 x1,64:31695576,37285908 k1,64:31942543,37285908:246967 x1,64:32779431,37285908 x1,64:33289294,37285908 x1,64:34345061,37285908 k1,64:34592028,37285908:246967 x1,64:35428916,37285908 k1,64:35428916,37285908:0 ) (1,65:20045308,38072340:15383608,446948,141880 x1,65:21464803,38072340 x1,65:22447825,38072340 g1,64:22611665,38072340 x1,64:23084824,38072340 x1,64:23375800,38072340 x1,64:24850342,38072340 g1,64:25014182,38072340 x1,64:25960518,38072340 g1,64:26124358,38072340 x1,64:27252877,38072340 x1,64:28708421,38072340 g1,64:28872261,38072340 x1,64:30037475,38072340 g1,64:30201315,38072340 x1,64:31293130,38072340 x1,64:32384946,38072340 x1,64:34114416,38072340 k1,65:35428916,38072340:1314500 g1,65:35428916,38072340 ) (1,77:20045308,38858772:15383608,446948,141880 h1,75:20045308,38858772:655361,0,0 x1,75:22156859,38858772 k1,75:22456624,38858772:299765 x1,75:23767344,38858772 k1,75:24067109,38858772:299765 x1,75:25485949,38858772 x1,75:26796651,38858772 k1,75:27096415,38858772:299764 x1,75:27314647,38858772 (1,75:27314647,38858772:327680,446948,9168 x1,75:27642327,38858772 ) x1,75:27806167,38858772 k1,75:28105932,38858772:299765 (1,75:28105932,38858772:327680,446948,14416 x1,75:28433612,38858772 ) x1,75:28597452,38858772 k1,75:28897217,38858772:299765 (1,75:28897217,38858772:655360,446948,9168 x1,75:29552577,38858772 ) x1,75:29716417,38858772 k1,75:30016182,38858772:299765 (1,75:30016182,38858772:655360,446948,0 x1,75:30671542,38858772 ) x1,75:30889774,38858772 k1,75:31189538,38858772:299764 x1,75:32063122,38858772 k1,75:32362887,38858772:299765 x1,75:33600199,38858772 k1,75:33899964,38858772:299765 x1,75:35428916,38858772 k1,75:35428916,38858772:0 ) (1,77:20045308,39645204:15383608,446948,141880 x1,77:20663964,39645204 x1,77:22228947,39645204 k1,75:22458302,39645204:229355 x1,75:23259141,39645204 k1,75:23488495,39645204:229354 x1,75:24580310,39645204 x1,75:25672126,39645204 x1,75:27237756,39645204 k1,75:27467111,39645204:229355 x1,75:28413447,39645204 k1,75:28642802,39645204:229355 x1,75:29443641,39645204 k1,75:29672996,39645204:229355 x1,75:30801515,39645204 x1,75:32257714,39645204 k1,75:32487068,39645204:229354 x1,75:33615587,39645204 k1,75:33844942,39645204:229355 x1,75:35428916,39645204 k1,75:35428916,39645204:0 ) (1,77:20045308,40431636:15383608,446948,141880 x1,77:20627916,40431636 x1,77:21465459,40431636 k1,76:21915010,40431636:449551 x1,76:22388169,40431636 x1,76:23407240,40431636 k1,76:23856792,40431636:449552 x1,76:24402704,40431636 k1,76:24852255,40431636:449551 x1,76:25653094,40431636 k1,76:26102646,40431636:449552 x1,76:26430326,40431636 x1,76:28092308,40431636 k1,76:28541859,40431636:449551 x1,76:30034099,40431636 k1,76:30483650,40431636:449551 x1,76:32892077,40431636 k1,76:33341629,40431636:449552 x1,76:33559861,40431636 (1,76:33559861,40431636:655360,446948,0 x1,76:34215221,40431636 ) x1,76:34433453,40431636 k1,76:34883004,40431636:449551 x1,76:35428916,40431636 k1,77:35428916,40431636:0 ) (1,77:20045308,41218068:15383608,446948,141880 x1,77:22109674,41218068 k1,76:22274547,41218068:164873 x1,76:22492779,41218068 (1,76:22492779,41218068:655360,446948,9168 x1,76:23148139,41218068 ) x1,76:23530211,41218068 k1,76:23736471,41218068:206260 x1,76:24901686,41218068 k1,76:25066560,41218068:164874 x1,76:26485400,41218068 x1,76:27959942,41218068 k1,76:28125073,41218068:165131 x1,76:29726760,41218068 k1,76:29891634,41218068:164874 x1,76:30328745,41218068 k1,76:30493618,41218068:164873 x1,76:32022570,41218068 k1,76:32187443,41218068:164873 x1,76:32951587,41218068 k1,76:33116461,41218068:164874 x1,76:33917300,41218068 k1,76:34082173,41218068:164873 x1,76:35428916,41218068 k1,76:35428916,41218068:0 ) (1,77:20045308,42004500:15383608,446948,141880 x1,77:20773395,42004500 k1,76:20964828,42004500:191433 x1,76:21510740,42004500 k1,76:21702172,42004500:191432 x1,76:22649146,42004500 x1,76:22976826,42004500 x1,76:23449985,42004500 x1,76:24469711,42004500 k1,76:24661144,42004500:191433 x1,76:25279800,42004500 x1,76:26298871,42004500 x1,76:26589847,42004500 x1,76:27536166,42004500 k1,76:27727599,42004500:191433 x1,76:28346255,42004500 x1,76:29656957,42004500 k1,76:29848389,42004500:191432 x1,76:31486109,42004500 k1,76:31677542,42004500:191433 x1,76:31968518,42004500 k1,76:32159950,42004500:191432 x1,76:33433957,42004500 k1,76:33625390,42004500:191433 x1,76:33953070,42004500 x1,76:34227662,42004500 x1,76:34628077,42004500 x1,76:35428916,42004500 k1,76:35428916,42004500:0 ) (1,77:20045308,42790932:15383608,446948,141880 x1,77:20591220,42790932 k1,76:20774184,42790932:182964 x1,76:21575023,42790932 x1,76:22994518,42790932 k1,76:23177482,42790932:182964 x1,76:24123818,42790932 x1,76:24633681,42790932 x1,76:26072175,42790932 k1,76:26259920,42790932:187745 x1,76:27425143,42790932 k1,76:27608106,42790932:182963 x1,76:28554442,42790932 x1,76:30265552,42790932 k1,76:30448516,42790932:182964 x1,76:31577035,42790932 x1,76:31894891,42790932 k1,76:32077855,42790932:182964 x1,76:32623767,42790932 k1,76:32806731,42790932:182964 x1,76:33607570,42790932 k1,76:33790534,42790932:182964 x1,76:35428916,42790932 k1,76:35428916,42790932:0 ) (1,77:20045308,43577364:15383608,446948,141880 x1,77:20663964,43577364 x1,77:21683035,43577364 x1,77:21974011,43577364 x1,77:22920330,43577364 k1,76:23092308,43577364:171978 x1,76:24038644,43577364 k1,76:24210622,43577364:171978 x1,76:24720485,43577364 x1,76:25031781,43577364 x1,76:25567861,43577364 k1,76:25739839,43577364:171978 x1,76:26686175,43577364 x1,76:27050551,43577364 x1,76:28578839,43577364 k1,76:28750817,43577364:171978 x1,76:29733839,43577364 k1,76:29905818,43577364:171979 x1,76:30706657,43577364 k1,76:30878635,43577364:171978 x1,76:32043858,43577364 x1,76:32771945,43577364 k1,76:32943923,43577364:171978 x1,76:33599283,43577364 k1,76:33771261,43577364:171978 x1,76:34281124,43577364 k1,76:34453102,43577364:171978 x1,76:34671334,43577364 x1,76:35428916,43577364 k1,76:35428916,43577364:0 ) (1,77:20045308,44363796:15383608,446948,141880 x1,77:20882196,44363796 x1,77:21865218,44363796 k1,76:22021256,44363796:156038 x1,76:22603208,44363796 x1,76:23221864,44363796 x1,76:24714094,44363796 k1,76:24870133,44363796:156039 x1,76:25670972,44363796 k1,76:25827010,44363796:156038 x1,76:26117986,44363796 x1,76:26432562,44363796 x1,76:26713714,44363796 x1,76:27841578,44363796 k1,76:27997616,44363796:156038 x1,76:28943952,44363796 x1,76:29890926,44363796 k1,76:30046965,44363796:156039 x1,76:30592877,44363796 k1,76:30748915,44363796:156038 x1,76:31549754,44363796 k1,76:31705793,44363796:156039 x1,76:32760904,44363796 x1,76:33816679,44363796 k1,76:33972717,44363796:156038 x1,76:35428916,44363796 k1,76:35428916,44363796:0 ) (1,77:20045308,45150228:15383608,446948,9168 x1,77:21137123,45150228 x1,77:22556619,45150228 g1,76:22720459,45150228 x1,76:23230322,45150228 g1,76:23394162,45150228 (1,76:23394162,45150228:2091112,446948,9168 x1,76:25073050,45150228 x1,76:25485274,45150228 ) x1,76:25649114,45150228 k1,77:35428916,45150228:9779802 g1,77:35428916,45150228 ) (1,97:20045308,45936660:15383608,446948,141880 h1,78:20045308,45936660:655361,0,0 x1,78:21101093,45936660 x1,78:21376348,45936660 k1,78:21595214,45936660:218866 x1,78:22723733,45936660 k1,78:22942600,45936660:218867 x1,78:23852232,45936660 x1,78:24652416,45936660 x1,78:25125575,45936660 x1,78:26107942,45936660 k1,78:26326808,45936660:218866 x1,78:27127647,45936660 k1,78:27346514,45936660:218867 x1,78:28329554,45936660 x1,78:28839417,45936660 x1,78:29894528,45936660 x1,78:31205231,45936660 k1,78:31424097,45936660:218866 x1,78:33025784,45936660 k1,78:33244651,45936660:218867 x1,78:34409211,45936660 k1,78:34628077,45936660:218866 x1,78:35428916,45936660 k1,78:35428916,45936660:0 ) (1,97:20045308,46723092:15383608,446948,141880 x1,97:21501507,46723092 k1,78:21764082,46723092:262575 x1,78:22273945,46723092 k1,78:22536520,46723092:262575 (1,78:22536520,46723092:2091112,446948,9168 x1,78:24215408,46723092 x1,78:24627632,46723092 ) x1,78:24791472,46723092 k1,78:25078730,46723092:287258 x1,78:26025066,46723092 k1,78:26287641,46723092:262575 x1,78:26797504,46723092 k1,78:27060079,46723092:262575 x1,78:27533238,46723092 x1,78:28043101,46723092 x1,78:28499885,46723092 k1,78:28762460,46723092:262575 x1,78:29344412,46723092 x1,78:29963068,46723092 x1,78:30945435,46723092 k1,78:31208010,46723092:262575 x1,78:31498986,46723092 x1,78:31816842,46723092 x1,78:32835913,46723092 x1,78:33163593,46723092 x1,78:33636752,46723092 x1,78:34656478,46723092 k1,78:34919053,46723092:262575 x1,78:35428916,46723092 k1,78:35428916,46723092:0 ) (1,97:20045308,47509524:15383608,446948,141880 x1,97:21392059,47509524 k1,78:21635985,47509524:243926 x1,78:22145193,47509524 x1,78:22655056,47509524 x1,78:22946032,47509524 x1,78:23965758,47509524 k1,78:24209683,47509524:243925 x1,78:24500659,47509524 x1,78:24828339,47509524 x1,78:25311993,47509524 x1,78:26185585,47509524 x1,78:27915055,47509524 k1,78:28179001,47509524:263946 x1,78:28469977,47509524 k1,78:28713903,47509524:243926 x1,78:29332559,47509524 x1,78:29789343,47509524 k1,78:30033268,47509524:243925 x1,78:30980242,47509524 x1,78:31307922,47509524 x1,78:31781081,47509524 x1,78:32800807,47509524 k1,78:33044733,47509524:243926 x1,78:33918325,47509524 x1,78:35428916,47509524 k1,97:35428916,47509524:0 ) (1,97:20045308,48295956:15383608,446948,145636 x1,97:22797796,48295956 k1,78:23067078,48295956:269282 x1,78:23285310,48295956 (1,78:23285310,48295956:655360,446948,0 x1,78:23940670,48295956 ) x1,78:24104510,48295956 k1,78:24373792,48295956:269282 (1,78:24373792,48295956:655360,446948,9168 x1,78:25029152,48295956 ) x1,78:25411224,48295956 k1,78:25706865,48295956:295641 x1,78:26580449,48295956 k1,78:26849731,48295956:269282 x1,78:28087043,48295956 k1,78:28356325,48295956:269282 x1,78:28684005,48295956 x1,78:28974981,48295956 x1,78:29286277,48295956 x1,78:29749612,48295956 x1,78:31187468,48295956 k1,78:31483109,48295956:295641 x1,78:33084796,48295956 k1,78:33354078,48295956:269282 x1,78:34190966,48295956 x1,78:35428916,48295956 k1,78:35428916,48295956:0 ) ] g1,97:35428916,48295956 ) ) ] h1,97:3729360,50262036:31699556,0,0 ] ] !58819 1 !10 2 [1,103:1,50262038:35428915,50262037,0 [1,103:3729360,50262038:31699556,49329697,0 [1,103:3729360,1718773:31699556,786432,0 h1,103:3729360,1718773:31699556,0,0 ] [1,103:3729360,48295958:31699556,44938785,0 [1,62:3729360,17426433:31699556,14069260,1 (1,61:3729360,8327917:31699556,4970744,4643064 g1,61:3729360,8327917 h1,54:3729360,8327917:0,0,0 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$1,58:35971920,12735049g1,58:35971920,12735049g1,58:36299600,12735049)g1,58:36299600,12735049)]$1,58:36299600,8327917 g1,61:36299600,8327917 g1,61:36299600,8327917 ) (1,61:3729360,14138825:31699556,446948,141880 h1,61:3729360,14138825:0,0,0 x1,61:5440487,14138825 k1,61:5678851,14138825:238364 x1,61:6188714,14138825 k1,61:6540923,14138825:352209 k1,61:6540923,14138825:0 x1,61:9124979,14138825 k1,61:9363344,14138825:238365 x1,61:10491863,14138825 k1,61:10730227,14138825:238364 x1,61:12150361,14138825 k1,61:12388725,14138825:238364 x1,61:14026452,14138825 k1,61:14264817,14138825:238365 x1,61:16085383,14138825 k1,61:16323747,14138825:238364 x1,61:17070195,14138825 k1,61:17496928,14138825:426733 x1,61:17824608,14138825 x1,61:19085520,14138825 k1,61:19323885,14138825:238365 x1,61:21325996,14138825 x1,61:21463627,14138825 k1,61:21720622,14138825:256995 x1,61:22484766,14138825 k1,61:22723130,14138825:238364 x1,61:23014106,14138825 k1,61:23252470,14138825:238364 x1,61:24453724,14138825 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g1,61:35428916,17284553 ) ] (1,103:3729360,25521564:31699556,6784410,0 (1,103:3729360,25521564:15383608,6784410,0 [1,97:3729360,25521564:15383608,6784410,0 (1,97:3729360,19392514:15383608,446948,141880 x1,97:4530199,19392514 k1,78:4755335,19392514:225136 x1,78:5520126,19392514 x1,78:6102724,19392514 x1,78:6430404,19392514 x1,78:6744980,19392514 x1,78:7764706,19392514 k1,78:7989842,19392514:225136 x1,78:8936178,19392514 k1,78:9161314,19392514:225136 x1,78:9779970,19392514 x1,78:10289178,19392514 x1,78:10932746,19392514 k1,78:11157882,19392514:225136 x1,78:11703794,19392514 k1,78:11928930,19392514:225136 x1,78:12547586,19392514 x1,78:14294762,19392514 k1,78:14519898,19392514:225136 x1,78:15430186,19392514 x1,78:17650529,19392514 k1,78:17875665,19392514:225136 x1,78:18384873,19392514 x1,78:18894736,19392514 x1,78:19112968,19392514 k1,97:19112968,19392514:0 ) (1,97:3729360,20178946:15383608,446948,141880 x1,97:4020336,20178946 x1,97:5040062,20178946 k1,78:5239547,20178946:199485 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x1,78:11434270,23324674 k1,78:11672797,23324674:238527 x1,78:12436941,23324674 k1,78:12675468,23324674:238527 x1,78:13294124,23324674 x1,78:13611980,23324674 k1,78:13850508,23324674:238528 x1,78:14687396,23324674 x1,78:15633732,23324674 k1,78:15872259,23324674:238527 x1,78:16418163,23324674 x1,78:17838314,23324674 x1,78:19112968,23324674 k1,78:19112968,23324674:0 ) (1,97:3729360,24111106:15383608,446948,141880 x1,97:5185559,24111106 k1,78:5360012,24111106:174453 x1,78:6160851,24111106 k1,78:6335304,24111106:174453 x1,78:7318344,24111106 x1,78:8155887,24111106 k1,78:8330340,24111106:174453 x1,78:9095131,24111106 x1,78:9677729,24111106 x1,78:10005409,24111106 x1,78:10319985,24111106 x1,78:11758479,24111106 k1,78:11993479,24111106:235000 x1,78:12393903,24111106 x1,78:13310750,24111106 k1,78:13485203,24111106:174453 x1,78:14905337,24111106 k1,78:15079790,24111106:174453 x1,78:16208309,24111106 x1,78:17664508,24111106 k1,78:17838961,24111106:174453 x1,78:19112968,24111106 k1,78:19112968,24111106:0 ) (1,97:3729360,24897538:15383608,446948,145636 x1,97:4057040,24897538 x1,78:5719022,24897538 k1,78:5902598,24897538:183576 x1,78:6849572,24897538 x1,78:7177252,24897538 x1,78:7650411,24897538 x1,78:8925065,24897538 k1,78:9108642,24897538:183577 x1,78:10273856,24897538 k1,78:10457432,24897538:183576 x1,78:10967295,24897538 x1,78:11804838,24897538 k1,78:11988414,24897538:183576 x1,78:13262421,24897538 k1,78:13445997,24897538:183576 x1,78:16198485,24897538 k1,78:16382042,24897538:183557 x1,78:17800882,24897538 k1,78:17984458,24897538:183576 x1,78:19112968,24897538 k1,78:19112968,24897538:0 ) ] g1,103:19112968,25521564 ) k1,103:19579138,25521564:466170 r1,103:19579138,25521564:0,6784410,0 k1,103:20045308,25521564:466170 (1,103:20045308,25521564:15383608,6784410,0 [1,103:20045308,25521564:15383608,6784410,0 (1,97:20045308,19392514:15383608,446948,141880 x1,97:21210522,19392514 g1,78:21374362,19392514 x1,78:22466177,19392514 x1,78:23557993,19392514 x1,78:25123623,19392514 g1,78:25287463,19392514 x1,78:26342575,19392514 g1,78:26506415,19392514 x1,78:27544494,19392514 g1,78:27708334,19392514 x1,78:28036014,19392514 h1,78:28036014,19392514:0,0,0 x1,78:28418086,19392514 g1,78:28621246,19392514 g1,86:28621246,19392514 k1,97:35428916,19392514:6807670 g1,97:35428916,19392514 ) (1,97:20045308,21086375:15383608,541848,14544 k1,97:22604911,21086375:2559603 (1,97:22604911,21086375:0,0,0 g1,97:22604911,21086375 ) x1,97:24440421,21086375 x1,97:27690712,21086375 g1,97:27887320,21086375 x1,97:32869313,21086375 k1,97:35428916,21086375:2559603 ) (1,97:20045308,22003879:15383608,541848,14544 k1,97:24372784,22003879:4327476 x1,97:31101440,22003879 g1,97:31101440,22003879 k1,97:35428916,22003879:4327476 ) (1,103:20045308,23162268:15383608,446948,141880 h1,101:20045308,23162268:655361,0,0 x1,101:22266308,23162268 k1,101:22475407,23162268:209099 x1,101:22766383,23162268 k1,101:22975482,23162268:209099 x1,101:24358937,23162268 x1,101:26033998,23162268 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x1,101:33389520,23948700 x1,101:34080911,23948700 k1,101:34214532,23948700:133621 x1,101:34833188,23948700 x1,101:35147764,23948700 x1,101:35428916,23948700 k1,101:35428916,23948700:0 ) (1,103:20045308,24735132:15383608,446948,141880 x1,103:21282620,24735132 k1,101:21517657,24735132:235037 x1,101:22318496,24735132 x1,101:23883470,24735132 k1,101:24118506,24735132:235036 x1,101:24882650,24735132 k1,101:25117687,24735132:235037 x1,101:25408663,24735132 k1,101:25643699,24735132:235036 x1,101:25971379,24735132 x1,101:26245971,24735132 x1,101:26646386,24735132 x1,101:27447225,24735132 k1,101:27682262,24735132:235037 x1,101:28228174,24735132 k1,101:28463210,24735132:235036 x1,101:29264049,24735132 x1,101:30683544,24735132 k1,101:30918581,24735132:235037 x1,101:31864917,24735132 k1,101:32099953,24735132:235036 x1,101:33519448,24735132 k1,101:33754485,24735132:235037 x1,101:34700821,24735132 x1,101:35210684,24735132 x1,101:35428916,24735132 k1,103:35428916,24735132:0 ) (1,103:20045308,25521564:15383608,446948,141880 x1,103:21483802,25521564 k1,101:21709769,25521564:225967 x1,101:22219632,25521564 k1,101:22433173,25521564:213541 x1,101:23051829,25521564 x1,101:23998147,25521564 k1,101:24211689,25521564:213542 x1,101:24976480,25521564 x1,101:26032247,25521564 k1,101:26245789,25521564:213542 x1,101:26791701,25521564 k1,101:27005243,25521564:213542 x1,101:28789105,25521564 x1,101:29116785,25521564 x1,101:30377697,25521564 k1,101:30591238,25521564:213541 x1,101:31101101,25521564 x1,101:31829188,25521564 x1,101:32521234,25521564 x1,101:32703417,25521564 x1,101:33613687,25521564 k1,101:33827229,25521564:213542 x1,101:35428916,25521564 k1,101:35428916,25521564:0 ) ] g1,103:35428916,25521564 ) ) [1,72:3729360,48295957:31699556,21463673,1 (1,71:3729360,45925853:31699556,19093569,0 k1,71:3831000,45925853:101640 h1,68:3831000,45925853:0,0,0 (1,68:3831000,45925853:31496277,19093569,0 (1,68:3831000,45925853:31496747,19093616,0 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[1,135:3729360,50262038:31699556,49329697,0 [1,135:3729360,1718773:31699556,786432,0 h1,135:3729360,1718773:31699556,0,0 ] [1,135:3729360,48295958:31699556,44938785,0 [1,86:3729360,15774426:31699556,12417253,1 (1,85:3729360,9074778:31699556,5717605,5389924 k1,85:3767675,9074778:38315 h1,80:3767675,9074778:0,0,0 $1,84:3767675,9074778[1,84:3767675,9074778:31622926,5717605,5389924(1,84:3767675,14228770:31622926,10871597,235932g1,82:3767675,14228770(1,82:3767675,14228770:15811463,10871597,235932r1,82:3767675,14228770:0,786432,235932g1,82:4095355,14228770g1,82:4095355,14228770$1,82:4095355,14228770 (1,82:4095355,14228770:15156103,10871597,0 (1,82:4095355,14228770:15155886,10871624,0 (1,82:4095355,14228770:15155886,10871624,0 (1,82:4095355,14228770:0,28878193,0 (1,82:4095355,14228770:40258437,28878193,0 ) k1,82:4095355,14228770:-40258437 ) ) ) ) $1,82:19251458,14228770g1,82:19251458,14228770g1,82:19579138,14228770)g1,82:19579138,14228770(1,84:19579138,14228770:15811463,10871597,235932g1,83:19906818,14228770g1,83:19906818,14228770$1,84:19906818,14228770 (1,83:19906818,14228770:15156103,10723010,0 (1,83:19906818,14228770:15155886,10723037,0 (1,83:19906818,14228770:15155886,10723037,0 (1,83:19906818,14228770:0,28483502,0 (1,83:19906818,14228770:40258437,28483502,0 ) k1,83:19906818,14228770:-40258437 ) ) ) ) $1,84:35062921,14228770g1,84:35062921,14228770g1,84:35390601,14228770)g1,84:35390601,14228770)]$1,84:35390601,9074778 g1,85:35390601,9074778 k1,85:35428916,9074778:38315 ) (1,85:3729360,15632546:31699556,446948,141880 k1,85:3758925,15632546:29565 (1,85:3758925,15632546:31640426,446948,141880 x1,85:5470052,15632546 g1,85:5633892,15632546 x1,85:6143755,15632546 g1,85:6346915,15632546 g1,85:6346915,15632546 x1,85:9477555,15632546 x1,85:10738467,15632546 g1,85:10902307,15632546 x1,85:11557667,15632546 x1,85:12523659,15632546 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x1,102:19112968,24031963 k1,102:19112968,24031963:0 ) (1,103:3729360,24818395:15383608,446948,7531 x1,103:4530199,24818395 g1,102:4694039,24818395 x1,102:6059133,24818395 k1,103:19112968,24818395:13053835 g1,103:19112968,24818395 ) (1,106:3729360,25773141:15383608,446948,141880 h1,104:3729360,25773141:655361,0,0 x1,104:5185569,25773141 k1,104:5362674,25773141:177105 x1,104:5653650,25773141 x1,104:5971506,25773141 x1,104:6772345,25773141 x1,104:7573184,25773141 k1,104:7750289,25773141:177105 x1,104:8296201,25773141 k1,104:8473306,25773141:177105 x1,104:8764282,25773141 k1,104:8941386,25773141:177104 x1,104:9269066,25773141 x1,104:10748210,25773141 x1,104:11621802,25773141 x1,104:11949482,25773141 x1,104:13210394,25773141 k1,104:13387499,25773141:177105 x1,104:14042859,25773141 x1,104:15008851,25773141 k1,104:15185956,25773141:177105 x1,104:16350516,25773141 x1,104:17588474,25773141 k1,104:17765579,25773141:177105 x1,104:18129945,25773141 x1,104:18894736,25773141 x1,104:19112968,25773141 k1,106:19112968,25773141:0 ) (1,106:3729360,26559573:15383608,446948,141880 x1,106:4602934,26559573 x1,106:5440477,26559573 k1,104:5665445,26559573:224968 x1,104:7012188,26559573 k1,104:7237156,26559573:224968 x1,104:8037340,26559573 x1,104:9311339,26559573 k1,104:9536307,26559573:224968 x1,104:9973418,26559573 k1,104:10198386,26559573:224968 x1,104:11035274,26559573 x1,104:12709697,26559573 k1,104:12934665,26559573:224968 x1,104:13444528,26559573 k1,104:13669496,26559573:224968 x1,104:14707575,26559573 k1,104:14932543,26559573:224968 x1,104:15260223,26559573 h1,104:15260223,26559573:0,0,0 x1,104:15424063,26559573 k1,104:15664313,26559573:240250 x1,104:17266000,26559573 k1,104:17490968,26559573:224968 x1,104:18401256,26559573 x1,104:19112968,26559573 k1,104:19112968,26559573:0 ) (1,106:3729360,27346005:15383608,446948,141880 x1,106:4530199,27346005 k1,104:4749191,27346005:218992 x1,104:5913751,27346005 x1,104:6787325,27346005 k1,104:7006316,27346005:218991 x1,104:7661676,27346005 x1,104:8627668,27346005 k1,104:8846660,27346005:218992 x1,104:9593108,27346005 k1,104:9961723,27346005:368615 x1,104:10689163,27346005 x1,104:12254155,27346005 x1,104:12572011,27346005 k1,104:12791002,27346005:218991 x1,104:13555146,27346005 k1,104:13774138,27346005:218992 x1,104:14974746,27346005 k1,104:15193738,27346005:218992 x1,104:15739650,27346005 k1,104:15958641,27346005:218991 x1,104:16759480,27346005 k1,104:16978472,27346005:218992 x1,104:17306152,27346005 x1,104:19112968,27346005 k1,104:19112968,27346005:0 ) (1,106:3729360,28132437:15383608,446948,141880 x1,106:5185559,28132437 k1,104:5417699,28132437:232140 x1,104:6218538,28132437 k1,104:6450679,28132437:232141 x1,104:7815773,28132437 k1,104:8223834,28132437:408061 x1,104:9389049,28132437 k1,104:9621190,28132437:232141 x1,104:10313246,28132437 x1,104:11313974,28132437 k1,104:11563189,28132437:249215 x1,104:12073052,28132437 x1,104:13710779,28132437 k1,104:13942919,28132437:232140 x1,104:15108133,28132437 k1,104:15340274,28132437:232141 x1,104:16378353,28132437 k1,104:16610493,28132437:232140 x1,104:16938173,28132437 h1,104:16938173,28132437:0,0,0 x1,104:17102013,28132437 k1,104:17351229,28132437:249216 x1,104:18389308,28132437 k1,104:18621448,28132437:232140 x1,104:18949128,28132437 h1,104:18949128,28132437:0,0,0 x1,104:19112968,28132437 k1,104:19112968,28132437:0 ) (1,106:3729360,28918869:15383608,446948,141880 x1,106:4767439,28918869 k1,104:5024352,28918869:256913 x1,104:5352032,28918869 h1,104:5352032,28918869:0,0,0 x1,104:5515872,28918869 k1,104:5796051,28918869:280179 x1,104:6834130,28918869 k1,104:7091043,28918869:256913 x1,104:7418723,28918869 h1,104:7418723,28918869:0,0,0 x1,104:7582563,28918869 k1,104:7862743,28918869:280180 x1,104:8809079,28918869 k1,104:9065991,28918869:256912 x1,104:10104070,28918869 k1,104:10360983,28918869:256913 x1,104:10688663,28918869 h1,104:10688663,28918869:0,0,0 k1,104:10945575,28918869:256912 x1,104:12000695,28918869 k1,104:12257608,28918869:256913 x1,104:14242014,28918869 k1,104:14522194,28918869:280180 x1,104:15177536,28918869 k1,104:15434448,28918869:256912 x1,104:16053104,28918869 x1,104:16780544,28918869 x1,104:18346192,28918869 k1,104:18603105,28918869:256913 x1,104:19112968,28918869 k1,104:19112968,28918869:0 ) (1,106:3729360,29705301:15383608,446948,92400 x1,106:4530199,29705301 k1,104:4773591,29705301:243392 x1,104:6120334,29705301 k1,104:6363725,29705301:243391 x1,104:7455540,29705301 k1,104:7698932,29705301:243392 x1,104:8590859,29705301 k1,104:9032674,29705301:441815 x1,104:9397050,29705301 x1,104:9933138,29705301 k1,104:10176529,29705301:243391 x1,104:11123503,29705301 k1,104:11366895,29705301:243392 x1,104:12458710,29705301 k1,104:12702102,29705301:243392 x1,104:13594029,29705301 k1,104:13857308,29705301:263279 x1,104:14148284,29705301 k1,104:14391675,29705301:243391 x1,104:14719355,29705301 x1,104:15980267,29705301 k1,104:16223659,29705301:243392 x1,104:17424913,29705301 k1,104:17668304,29705301:243391 x1,104:18214216,29705301 k1,104:18457608,29705301:243392 x1,104:19112968,29705301 k1,104:19112968,29705301:0 ) (1,106:3729360,30491733:15383608,446948,141880 x1,106:4057040,30491733 x1,104:6027696,30491733 k1,104:6226996,30491733:199300 x1,104:7392210,30491733 k1,104:7584419,30491733:192209 x1,104:8203075,30491733 k1,104:8395283,30491733:192208 x1,104:8686259,30491733 x1,104:9000835,30491733 x1,104:9281987,30491733 x1,104:10409851,30491733 k1,104:10602060,30491733:192209 x1,104:10929740,30491733 x1,104:12190652,30491733 k1,104:12382860,30491733:192208 x1,104:13001516,30491733 x1,104:14238819,30491733 k1,104:14431028,30491733:192209 x1,104:14976940,30491733 k1,104:15169148,30491733:192208 x1,104:16316028,30491733 k1,104:16508237,30491733:192209 x1,104:17491268,30491733 k1,104:17683448,30491733:192180 x1,104:18629784,30491733 k1,104:18821992,30491733:192208 x1,104:19112968,30491733 k1,104:19112968,30491733:0 ) (1,106:3729360,31278165:15383608,446948,141880 x1,106:4057040,31278165 x1,104:5317952,31278165 k1,104:5534142,31278165:216190 x1,104:6698702,31278165 x1,104:7536245,31278165 k1,104:7752435,31278165:216190 x1,104:8298347,31278165 k1,104:8514538,31278165:216191 x1,104:9169898,31278165 k1,104:9386088,31278165:216190 x1,104:10132536,31278165 k1,104:10361813,31278165:229277 x1,104:10834981,31278165 x1,104:11374332,31278165 k1,104:11590522,31278165:216190 x1,104:12245882,31278165 x1,104:13337698,31278165 x1,104:14110370,31278165 k1,104:14470581,31278165:360211 x1,104:15489661,31278165 k1,104:15705851,31278165:216190 x1,104:17125346,31278165 k1,104:17341536,31278165:216190 x1,104:17960192,31278165 x1,104:18469400,31278165 x1,104:19112968,31278165 k1,104:19112968,31278165:0 ) (1,106:3729360,32064597:15383608,446948,92400 x1,106:4202528,32064597 x1,104:4741879,32064597 k1,104:5005311,32064597:263432 x1,104:5824511,32064597 k1,104:6087944,32064597:263433 x1,104:7325256,32064597 x1,104:7404559,32064597 k1,104:7667992,32064597:263433 x1,104:8687072,32064597 k1,105:8950504,32064597:263432 x1,105:9605864,32064597 x1,105:10006279,32064597 x1,105:10952615,32064597 x1,105:11607957,32064597 k1,105:11871389,32064597:263432 x1,105:12817725,32064597 k1,105:13081158,32064597:263433 x1,105:13408838,32064597 x1,105:13689990,32064597 x1,105:14272588,32064597 x1,105:15036723,32064597 k1,105:15300156,32064597:263433 x1,105:15918812,32064597 x1,105:16391971,32064597 x1,105:17447746,32064597 k1,105:17711178,32064597:263432 x1,105:19112968,32064597 k1,105:19112968,32064597:0 ) (1,106:3729360,32851029:15383608,447828,141880 x1,106:5403792,32851029 k1,105:5565306,32851029:161514 x1,105:6292746,32851029 x1,105:7566762,32851029 x1,105:8585825,32851029 k1,105:8747339,32851029:161514 x1,105:9547523,32851029 k1,105:9709037,32851029:161514 x1,105:10182196,32851029 x1,105:11783874,32851029 k1,105:11945387,32851029:161513 x1,105:12600747,32851029 k1,105:12762261,32851029:161514 $1,105:12762261,32851029x1,105:13183694,32851029(1,105:13183694,32949332:339591,318577,0x1,105:13490517,32949332)$1,105:13523285,32851029 k1,105:13684799,32851029:161514 x1,105:14631135,32851029 k1,105:14792649,32851029:161514 $1,105:14792649,32851029x1,105:15214082,32851029(1,105:15214082,32949332:311374,197518,0x1,105:15476228,32949332)$1,105:15525456,32851029 x1,105:15689296,32851029 k1,105:15851275,32851029:161979 x1,105:16360483,32851029 x1,105:17525043,32851029 x1,105:17889409,32851029 x1,105:18200705,32851029 x1,105:18991720,32851029 x1,105:19112968,32851029 k1,105:19112968,32851029:0 ) ] g1,135:19112968,32851029 ) k1,135:19579138,32851029:466170 r1,135:19579138,32851029:0,15765882,0 k1,135:20045308,32851029:466170 (1,135:20045308,32851029:15383608,15765882,0 [1,135:20045308,32851029:15383608,15765882,0 (1,106:20045308,17740507:15383608,446948,141880 x1,106:20845492,17740507 k1,105:21026522,17740507:181030 x1,105:22592160,17740507 x1,105:23720679,17740507 k1,105:23901710,17740507:181031 x1,105:24665854,17740507 k1,105:24846884,17740507:181030 x1,105:25502226,17740507 k1,105:25683257,17740507:181031 x1,105:26484096,17740507 k1,105:26665126,17740507:181030 x1,105:26992806,17740507 x1,105:28799622,17740507 k1,105:28980653,17740507:181031 x1,105:29490516,17740507 k1,105:29671546,17740507:181030 x1,105:30472385,17740507 k1,105:30653415,17740507:181030 x1,105:32018509,17740507 k1,105:32273241,17740507:254732 x1,105:33292321,17740507 k1,105:33473351,17740507:181030 x1,105:33801031,17740507 x1,105:34082183,17740507 x1,105:34664781,17740507 x1,105:35428916,17740507 k1,105:35428916,17740507:0 ) (1,106:20045308,18526939:15383608,446948,141880 x1,106:20663964,18526939 x1,106:21137123,18526939 x1,106:22192898,18526939 k1,105:22388426,18526939:195528 x1,105:23371448,18526939 k1,105:23566977,18526939:195529 x1,105:24440569,18526939 x1,105:25152281,18526939 k1,105:25347809,18526939:195528 x1,105:26804008,18526939 k1,105:26999537,18526939:195529 x1,105:27800376,18526939 k1,105:27995904,18526939:195528 x1,105:29360998,18526939 k1,105:29659223,18526939:298225 x1,105:30678303,18526939 k1,105:30873832,18526939:195529 x1,105:31201512,18526939 x1,105:31482664,18526939 x1,105:32065262,18526939 x1,105:32829397,18526939 k1,105:33024925,18526939:195528 $1,105:33024925,18526939x1,105:33600340,18526939k1,105:33841062,18526939:240722x1,105:34350788,18526939k1,105:34591509,18526939:240721x1,105:35428916,18526939$1,105:35428916,18526939 k1,106:35428916,18526939:0 ) (1,106:20045308,19313371:15383608,446948,141880 x1,106:21500852,19313371 x1,106:22338377,19313371 k1,105:22582119,19313371:243742 x1,105:23892175,19313371 k1,105:24135836,19313371:243661 x1,105:25537626,19313371 k1,105:25801342,19313371:263716 x1,105:26674926,19313371 x1,105:26965902,19313371 x1,105:28275949,19313371 k1,105:28519690,19313371:243741 x1,105:29793697,19313371 k1,105:30037439,19313371:243742 x1,105:30838278,19313371 k1,105:31082019,19313371:243741 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x1,105:22265103,20886235 k1,105:22410051,20886235:144948 x1,105:23793498,20886235 k1,105:23938447,20886235:144949 x1,105:25176415,20886235 k1,105:25321364,20886235:144949 x1,105:27141292,20886235 k1,105:27286241,20886235:144949 x1,105:27613921,20886235 x1,105:28874833,20886235 k1,105:29019781,20886235:144948 x1,105:29838981,20886235 k1,105:29987708,20886235:148727 x1,105:30934044,20886235 k1,105:31078993,20886235:144949 x1,105:32134760,20886235 k1,105:32279709,20886235:144949 x1,105:33153301,20886235 x1,105:33610085,20886235 k1,105:33755033,20886235:144948 x1,105:34264896,20886235 k1,105:34409845,20886235:144949 x1,105:35210684,20886235 x1,105:35428916,20886235 k1,106:35428916,20886235:0 ) (1,106:20045308,21672667:15383608,446948,141880 x1,106:20737354,21672667 x1,106:21865873,21672667 g1,105:22029713,21672667 x1,105:22793857,21672667 g1,105:22957697,21672667 x1,105:24122247,21672667 g1,105:24286087,21672667 x1,105:24613767,21672667 x1,105:26584423,21672667 k1,106:35428916,21672667:8844493 g1,106:35428916,21672667 ) (1,107:20045308,23129425:15383608,538305,63696 (1,107:20045308,23129425:0,0,0 g1,107:20045308,23129425 ) x1,107:23453661,23129425 g1,107:23650269,23129425 x1,107:24829917,23129425 g1,107:25026525,23129425 x1,107:27429028,23129425 g1,107:27625636,23129425 x1,107:29504393,23129425 x1,107:30255439,23129425 g1,107:30452047,23129425 x1,107:31937588,23129425 k1,107:35428916,23129425:3491328 g1,107:35428916,23129425 ) (1,109:20045308,24200277:15383608,446948,141880 h1,108:20045308,24200277:655361,0,0 x1,108:21738748,24200277 k1,108:21942406,24200277:203658 x1,108:22270086,24200277 h1,108:22270086,24200277:0,0,0 x1,108:22433926,24200277 k1,108:22647539,24200277:213613 x1,108:23684953,24200277 k1,108:23898566,24200277:213613 x1,108:24808854,24200277 x1,108:25520566,24200277 k1,108:25724224,24200277:203658 x1,108:26525063,24200277 k1,108:26728721,24200277:203658 x1,108:27056401,24200277 x1,108:28535545,24200277 x1,108:29409137,24200277 x1,108:29736817,24200277 x1,108:30997729,24200277 k1,108:31201388,24200277:203659 x1,108:31856748,24200277 x1,108:32822740,24200277 k1,108:33026398,24200277:203658 x1,108:34190958,24200277 x1,108:35428916,24200277 k1,108:35428916,24200277:0 ) (1,109:20045308,24986709:15383608,446948,141880 x1,109:21173172,24986709 k1,108:21424876,24986709:251704 x1,108:22225715,24986709 k1,108:22477418,24986709:251703 x1,108:23132778,24986709 x1,108:23533193,24986709 x1,108:24479529,24986709 x1,108:25134871,24986709 k1,108:25386575,24986709:251704 x1,108:26005231,24986709 x1,108:26478390,24986709 x1,108:27534165,24986709 k1,108:27785868,24986709:251703 x1,108:28932730,24986709 k1,108:29399481,24986709:466751 x1,108:30927769,24986709 k1,108:31179472,24986709:251703 x1,108:31616583,24986709 k1,108:31868287,24986709:251704 x1,108:32159263,24986709 k1,108:32410966,24986709:251703 x1,108:33211805,24986709 x1,108:33939900,24986709 k1,108:34191604,24986709:251704 x1,108:35428916,24986709 k1,108:35428916,24986709:0 ) (1,109:20045308,25773141:15383608,446948,141880 x1,109:21646995,25773141 k1,108:21901319,25773141:254324 x1,108:23321453,25773141 k1,108:23575777,25773141:254324 x1,108:24231137,25773141 k1,108:24485461,25773141:254324 x1,108:24995324,25773141 k1,108:25249648,25773141:254324 x1,108:25977088,25773141 x1,108:27542080,25773141 x1,108:27859936,25773141 k1,108:28114260,25773141:254324 x1,108:28769620,25773141 k1,108:29023943,25773141:254323 x1,108:30480142,25773141 k1,108:30734466,25773141:254324 x1,108:31062146,25773141 k1,108:31316470,25773141:254324 x1,108:32408294,25773141 k1,108:32662618,25773141:254324 x1,108:33317978,25773141 x1,108:34373753,25773141 k1,108:34628077,25773141:254324 x1,108:35428916,25773141 k1,108:35428916,25773141:0 ) (1,109:20045308,26559573:15383608,446948,7531 x1,109:20372988,26559573 x1,108:21633900,26559573 k1,108:21810001,26559573:176101 x1,108:23175095,26559573 k1,108:23415038,26559573:239943 x1,108:24580253,26559573 k1,108:24756354,26559573:176101 x1,108:25921560,26559573 k1,108:26097662,26559573:176102 x1,108:26534773,26559573 k1,108:26710874,26559573:176101 x1,108:27184033,26559573 x1,108:28094311,26559573 x1,108:29682885,26559573 k1,108:29858986,26559573:176101 x1,108:30368849,26559573 k1,108:30544950,26559573:176101 x1,108:31345789,26559573 k1,108:31521890,26559573:176101 x1,108:32577001,26559573 x1,108:33632776,26559573 k1,108:33808877,26559573:176101 x1,108:35428916,26559573 k1,108:35428916,26559573:0 ) (1,109:20045308,27346005:15383608,446948,141880 x1,109:20591220,27346005 k1,108:20810940,27346005:219720 x1,108:21429596,27346005 x1,108:21939459,27346005 x1,108:23123025,27346005 k1,108:23356715,27346005:233690 x1,108:24666762,27346005 k1,108:24886482,27346005:219720 x1,108:25323593,27346005 k1,108:25543313,27346005:219720 x1,108:25834289,27346005 k1,108:26054009,27346005:219720 x1,108:26563872,27346005 x1,108:27529864,27346005 k1,108:27749584,27346005:219720 x1,108:28550423,27346005 x1,108:29242469,27346005 x1,108:30370988,27346005 k1,108:30590708,27346005:219720 x1,108:32301835,27346005 x1,108:33138723,27346005 x1,108:33593539,27346005 k1,108:33827229,27346005:233690 x1,108:35428916,27346005 k1,108:35428916,27346005:0 ) (1,109:20045308,28132437:15383608,446948,141880 x1,109:20663964,28132437 x1,109:22046756,28132437 k1,108:22346558,28132437:299802 x1,108:22856421,28132437 k1,108:23156223,28132437:299802 x1,108:23774879,28132437 k1,108:24074681,28132437:299802 x1,108:26113496,28132437 k1,108:26413298,28132437:299802 x1,108:27359634,28132437 x1,108:28597584,28132437 k1,108:28897385,28132437:299801 x1,108:29407248,28132437 k1,108:29707050,28132437:299802 x1,108:30434490,28132437 x1,108:31999482,28132437 x1,108:32317338,28132437 k1,108:32617140,28132437:299802 x1,108:33272500,28132437 x1,108:34328275,28132437 k1,108:34628077,28132437:299802 x1,108:35428916,28132437 k1,108:35428916,28132437:0 ) (1,109:20045308,28918869:15383608,446948,7531 x1,109:20372988,28918869 x1,108:21633900,28918869 k1,108:21866778,28918869:232878 x1,108:23231872,28918869 k1,108:23642145,28918869:410273 x1,108:24807360,28918869 k1,108:25040237,28918869:232877 x1,108:25550100,28918869 x1,108:26096012,28918869 x1,108:26896196,28918869 x1,108:27915922,28918869 k1,108:28148800,28918869:232878 x1,108:28694712,28918869 k1,108:28927589,28918869:232877 x1,108:29728428,28918869 k1,108:29961306,28918869:232878 x1,108:30616666,28918869 x1,108:31017081,28918869 x1,108:31963417,28918869 x1,108:32618759,28918869 k1,108:32851637,28918869:232878 x1,108:33834659,28918869 k1,108:34067536,28918869:232877 x1,108:34504647,28918869 k1,108:34737525,28918869:232878 x1,108:35210684,28918869 x1,108:35428916,28918869 k1,109:35428916,28918869:0 ) (1,109:20045308,29705301:15383608,446948,141880 x1,109:20955586,29705301 x1,109:22544160,29705301 k1,108:22713881,29705301:169721 x1,108:23223744,29705301 k1,108:23393464,29705301:169720 x1,108:24194303,29705301 k1,108:24364024,29705301:169721 x1,108:24982680,29705301 x1,108:25929016,29705301 x1,108:27457304,29705301 k1,108:27627024,29705301:169720 x1,108:28172936,29705301 k1,108:28342657,29705301:169721 x1,108:29143496,29705301 k1,108:29313216,29705301:169720 x1,108:29968576,29705301 x1,108:30368991,29705301 x1,108:31315327,29705301 x1,108:31970669,29705301 k1,108:32140390,29705301:169721 x1,108:33268909,29705301 k1,108:33438629,29705301:169720 x1,108:34603835,29705301 k1,108:34773556,29705301:169721 x1,108:35428916,29705301 k1,108:35428916,29705301:0 ) (1,109:20045308,30491733:15383608,446948,9168 x1,109:20846147,30491733 k1,108:21076690,30491733:230543 x1,108:22532889,30491733 k1,108:22763431,30491733:230542 x1,108:23309343,30491733 k1,108:23539886,30491733:230543 x1,108:25142210,30491733 x1,108:25469890,30491733 x1,108:26730802,30491733 k1,108:26961344,30491733:230542 x1,108:27543952,30491733 x1,108:28199294,30491733 x1,108:28672453,30491733 x1,108:29692179,30491733 k1,108:29922722,30491733:230543 x1,108:30468634,30491733 k1,108:30699177,30491733:230543 x1,108:31500016,30491733 k1,108:31730558,30491733:230542 x1,108:33095652,30491733 k1,108:33498920,30491733:403268 x1,108:34555342,30491733 x1,108:34737525,30491733 x1,108:35428916,30491733 k1,108:35428916,30491733:0 ) (1,109:20045308,31278165:15383608,446948,102232 x1,109:20700668,31278165 x1,109:21246572,31278165 x1,109:21792484,31278165 x1,109:22067076,31278165 x1,109:23341730,31278165 k1,108:23511777,31278165:170047 x1,108:24130433,31278165 x1,108:24445009,31278165 x1,108:24726161,31278165 k1,108:24896208,31278165:170047 x1,108:26133520,31278165 k1,108:26303567,31278165:170047 x1,108:27723062,31278165 k1,108:27893109,31278165:170047 x1,108:28366268,31278165 k1,108:28536315,31278165:170047 x1,108:29337154,31278165 k1,108:29507200,31278165:170046 x1,108:30600328,31278165 k1,108:30770375,31278165:170047 x1,108:30988607,31278165 (1,108:30988607,31278165:655360,446948,9168 x1,108:31643967,31278165 ) x1,108:31862199,31278165 k1,108:32032246,31278165:170047 x1,108:32978582,31278165 k1,108:33148629,31278165:170047 x1,108:34421981,31278165 k1,108:34592028,31278165:170047 x1,108:35210684,31278165 x1,108:35428916,31278165 k1,109:35428916,31278165:0 ) (1,109:20045308,32064597:15383608,446948,141880 x1,109:20336284,32064597 x1,109:21755779,32064597 k1,108:21948294,32064597:192515 x1,108:22530902,32064597 x1,108:23368445,32064597 k1,108:23560961,32064597:192516 x1,108:23851937,32064597 k1,108:24044452,32064597:192515 x1,108:24991426,32064597 x1,108:25792265,32064597 k1,108:25984780,32064597:192515 x1,108:26785619,32064597 x1,108:27695897,32064597 k1,108:27888413,32064597:192516 x1,108:29526795,32064597 k1,108:29719310,32064597:192515 x1,108:30229173,32064597 k1,108:30421688,32064597:192515 x1,108:31040344,32064597 x1,108:32605319,32064597 k1,108:32797835,32064597:192516 x1,108:33598674,32064597 k1,108:33791189,32064597:192515 x1,108:35428916,32064597 k1,109:35428916,32064597:0 ) 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(1,90:4095355,46098218:40258437,31377900,0 ) k1,90:4095355,46098218:-40258437 ) ) ) ) $1,90:19251458,46098218g1,90:19251458,46098218g1,90:19579138,46098218)g1,90:19579138,46098218(1,92:19579138,46098218:15811463,11936469,235932g1,91:19906818,46098218g1,91:19906818,46098218$1,92:19906818,46098218 (1,91:19906818,46098218:15156103,11936469,0 (1,91:19906818,46098218:15155886,11936498,0 (1,91:19906818,46098218:15155886,11936498,0 (1,91:19906818,46098218:0,31706808,0 (1,91:19906818,46098218:40258437,31706808,0 ) k1,91:19906818,46098218:-40258437 ) ) ) ) $1,92:35062921,46098218g1,92:35062921,46098218g1,92:35390601,46098218)g1,92:35390601,46098218)]$1,92:35390601,40411790 g1,95:35390601,40411790 k1,95:35428916,40411790:38315 ) (1,95:3729360,47501994:31699556,446948,141880 h1,95:3729360,47501994:0,0,0 x1,95:5440487,47501994 k1,95:5619895,47501994:179408 x1,95:6129758,47501994 k1,95:6364054,47501994:234296 k1,95:6364054,47501994:0 x1,95:7619709,47501994 k1,95:7803009,47501994:183300 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(1,95:3729360,48288426:31699556,446948,7531 x1,95:5731471,48288426 x1,95:5859271,48288426 k1,95:35428916,48288426:29569645 g1,95:35428916,48288426 ) ] ] h1,135:3729360,50262038:31699556,0,0 ] ] !34569 3 !10 4 [1,176:1,50262036:35428915,50262035,0 [1,176:3729360,50262036:31699556,49329695,0 [1,176:3729360,1718773:31699556,786432,0 h1,176:3729360,1718773:31699556,0,0 ] [1,176:3729360,48295956:31699556,44938783,0 (1,176:3729360,48295956:31699556,44938783,0 (1,176:3729360,48295956:15383608,44938783,0 [1,135:3729360,48295956:15383608,44938783,0 [1,116:3729360,16368051:15383608,13010878,0 (1,115:3729360,12450643:15383608,9093470,0 k1,115:3843113,12450643:113753 h1,113:3843113,12450643:0,0,0 (1,113:3843113,12450643:15156103,9093470,0 (1,113:3843113,12450643:15155822,9093493,0 (1,113:3843113,12450643:15155822,9093493,0 (1,113:3843113,12450643:0,31575245,0 (1,113:3843113,12450643:52625408,31575245,0 ) k1,113:3843113,12450643:-52625408 ) ) ) ) g1,115:18999216,12450643 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k1,115:16345429,15465299:242105 x1,115:17292403,15465299 x1,115:17620083,15465299 x1,115:18093242,15465299 x1,115:19112968,15465299 k1,115:19112968,15465299:0 ) (1,115:3729360,16251731:15383608,446948,116320 x1,115:5585319,16251731 g1,115:5749159,16251731 x1,115:7023166,16251731 g1,115:7187006,16251731 x1,115:9759273,16251731 k1,115:19112968,16251731:9353695 g1,115:19112968,16251731 ) ] [1,124:3729360,31138191:15383608,13980197,0 (1,123:3729360,28768087:15383608,11610093,0 k1,123:3843113,28768087:113753 h1,121:3843113,28768087:0,0,0 (1,121:3843113,28768087:15156103,11610093,0 (1,121:3843113,28768087:15156223,11610122,0 (1,121:3843113,28768087:15156223,11610122,0 (1,121:3843113,28768087:0,30234482,0 (1,121:3843113,28768087:39469056,30234482,0 ) k1,121:3843113,28768087:-39469056 ) ) ) ) g1,123:18999216,28768087 k1,123:19112968,28768087:113752 ) (1,123:3729360,30209879:15383608,446948,141880 h1,123:3729360,30209879:0,0,0 x1,123:5440487,30209879 k1,123:5677874,30209879:237387 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g1,123:14000774,30996311 x1,123:15856733,30996311 g1,123:16020573,30996311 x1,123:16390189,30996311 g1,123:16554029,30996311 x1,123:17700909,30996311 g1,123:17864749,30996311 x1,123:19011620,30996311 k1,123:19112968,30996311:101348 g1,123:19112968,30996311 ) ] [1,131:3729360,46326365:15383608,14398230,0 (1,130:3729360,43956261:15383608,12028126,0 k1,130:3843113,43956261:113753 h1,128:3843113,43956261:0,0,0 (1,128:3843113,43956261:15156103,12028126,0 (1,128:3843113,43956261:15156223,12028155,0 (1,128:3843113,43956261:15156223,12028155,0 (1,128:3843113,43956261:0,31323103,0 (1,128:3843113,43956261:39469056,31323103,0 ) k1,128:3843113,43956261:-39469056 ) ) ) ) g1,130:18999216,43956261 k1,130:19112968,43956261:113752 ) (1,130:3729360,45398053:15383608,446948,141880 h1,130:3729360,45398053:0,0,0 x1,130:5440487,45398053 k1,130:5677874,45398053:237387 x1,130:6187737,45398053 k1,130:6537992,45398053:350255 k1,130:6537992,45398053:0 x1,130:7011160,45398053 x1,130:8576124,45398053 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x1,130:16881709,46184485 g1,130:17045549,46184485 x1,130:18192420,46184485 k1,130:19112968,46184485:920548 g1,130:19112968,46184485 ) ] v1,135:3729360,48295956:0,0,0 ] g1,176:19112968,48295956 ) k1,176:19579138,48295956:466170 r1,176:19579138,48295956:0,44938783,0 k1,176:20045308,48295956:466170 (1,176:20045308,48295956:15383608,44938783,0 [1,176:20045308,48295956:15383608,44938783,0 (1,135:20045308,4012533:15383608,446948,141880 h1,134:20045308,4012533:655361,0,0 x1,134:21738748,4012533 k1,134:21892626,4012533:153878 x1,134:22220306,4012533 h1,134:22220306,4012533:0,0,0 x1,134:22384146,4012533 k1,134:22540017,4012533:155871 x1,134:23941815,4012533 k1,134:24097686,4012533:155871 x1,134:25007974,4012533 x1,134:25719686,4012533 k1,134:25873564,4012533:153878 x1,134:26674403,4012533 k1,134:26828281,4012533:153878 x1,134:27155961,4012533 x1,134:28635105,4012533 x1,134:29508697,4012533 x1,134:29836377,4012533 x1,134:31097289,4012533 k1,134:31251168,4012533:153879 x1,134:31906528,4012533 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x1,135:20700668,5585397 k1,134:20905477,5585397:204809 x1,134:21415340,5585397 k1,134:21620150,5585397:204810 x1,134:22347590,5585397 x1,134:23912582,5585397 x1,134:24230438,5585397 k1,134:24435247,5585397:204809 x1,134:24981159,5585397 k1,134:25185968,5585397:204809 x1,134:25986807,5585397 k1,134:26191616,5585397:204809 x1,134:26992455,5585397 x1,134:27720550,5585397 k1,134:27925360,5585397:204810 x1,134:29162672,5585397 k1,134:29367481,5585397:204809 x1,134:30022841,5585397 k1,134:30227650,5585397:204809 x1,134:31683849,5585397 k1,134:31888658,5585397:204809 x1,134:32216338,5585397 k1,134:32421148,5585397:204810 x1,134:33512972,5585397 k1,134:33717781,5585397:204809 x1,134:34373141,5585397 x1,134:35428916,5585397 k1,134:35428916,5585397:0 ) (1,135:20045308,6371829:15383608,446948,141880 x1,135:20846147,6371829 k1,134:21043058,6371829:196911 x1,134:21370738,6371829 x1,134:22631650,6371829 k1,134:22828560,6371829:196910 x1,134:24193654,6371829 k1,134:24398833,6371829:205179 x1,134:26000520,6371829 k1,134:26197430,6371829:196910 x1,134:26634541,6371829 k1,134:26831452,6371829:196911 x1,134:27450108,6371829 x1,134:28247674,6371829 k1,134:28444585,6371829:196911 x1,134:28917744,6371829 x1,134:29828022,6371829 x1,134:31416596,6371829 k1,134:31613507,6371829:196911 x1,134:32123370,6371829 k1,134:32320280,6371829:196910 x1,134:33121119,6371829 k1,134:33318030,6371829:196911 x1,134:34373141,6371829 x1,134:35428916,6371829 k1,134:35428916,6371829:0 ) (1,135:20045308,7158261:15383608,446948,141880 x1,135:21665347,7158261 k1,134:21976051,7158261:310704 x1,134:22521963,7158261 k1,134:22721651,7158261:199688 x1,134:23340307,7158261 x1,134:23850170,7158261 x1,134:25033736,7158261 k1,134:25242386,7158261:208650 x1,134:26552433,7158261 k1,134:26752121,7158261:199688 x1,134:27189232,7158261 k1,134:27388920,7158261:199688 x1,134:27679896,7158261 k1,134:27879585,7158261:199689 x1,134:29808926,7158261 k1,134:30008614,7158261:199688 x1,134:31245926,7158261 k1,134:31445614,7158261:199688 x1,134:31918773,7158261 k1,134:32118461,7158261:199688 x1,134:32409437,7158261 k1,134:32609125,7158261:199688 x1,134:34283556,7158261 k1,134:34483244,7158261:199688 x1,134:35210684,7158261 x1,134:35428916,7158261 k1,135:35428916,7158261:0 ) (1,135:20045308,7944693:15383608,446948,141880 x1,135:21610300,7944693 x1,135:21928156,7944693 x1,135:22049404,7944693 k1,134:22273750,7944693:224346 x1,134:24166405,7944693 k1,134:24378649,7944693:212244 x1,134:24669625,7944693 x1,134:24980921,7944693 x1,134:25589753,7944693 k1,134:25801998,7944693:212245 x1,134:26457358,7944693 k1,134:26669603,7944693:212245 x1,134:27470442,7944693 k1,134:27682687,7944693:212245 x1,134:28702406,7944693 k1,134:28914651,7944693:212245 x1,134:29242331,7944693 x1,134:30667083,7944693 k1,134:30891428,7944693:224345 x1,134:32493115,7944693 k1,134:32705360,7944693:212245 x1,134:33324016,7944693 x1,134:34706808,7944693 k1,134:34919053,7944693:212245 x1,134:35428916,7944693 k1,134:35428916,7944693:0 ) (1,135:20045308,8731125:15383608,446948,141880 x1,135:20918900,8731125 x1,134:21375684,8731125 k1,134:21566376,8731125:190692 x1,134:22076239,8731125 k1,134:22266932,8731125:190693 x1,134:23067771,8731125 x1,134:23759817,8731125 x1,134:24888336,8731125 k1,134:25079028,8731125:190692 x1,134:26025364,8731125 k1,134:26216057,8731125:190693 x1,134:27235128,8731125 x1,134:27841992,8731125 k1,134:28032684,8731125:190692 x1,134:29197898,8731125 k1,134:29388590,8731125:190692 x1,134:30189429,8731125 k1,134:30380122,8731125:190693 x1,134:31690824,8731125 k1,134:31881516,8731125:190692 x1,134:33118828,8731125 k1,134:33309521,8731125:190693 x1,134:34437385,8731125 k1,134:34628077,8731125:190692 x1,134:35428916,8731125 k1,134:35428916,8731125:0 ) (1,135:20045308,9517557:15383608,446948,9168 x1,135:20991644,9517557 k1,134:21167068,9517557:175424 x1,134:21712980,9517557 k1,134:21888404,9517557:175424 x1,134:22689243,9517557 k1,134:22864667,9517557:175424 x1,134:23192347,9517557 x1,134:24453259,9517557 k1,134:24628682,9517557:175423 x1,134:25993776,9517557 k1,134:26231688,9517557:237912 x1,134:27396903,9517557 k1,134:27572327,9517557:175424 x1,134:28846326,9517557 x1,134:29865397,9517557 k1,134:30040821,9517557:175424 x1,134:30550684,9517557 k1,134:30726107,9517557:175423 x1,134:31526946,9517557 k1,134:31702370,9517557:175424 x1,134:32030050,9517557 x1,134:32311202,9517557 x1,134:32893800,9517557 x1,134:33657935,9517557 k1,134:33833359,9517557:175424 x1,134:34816381,9517557 k1,134:34991805,9517557:175424 x1,134:35428916,9517557 k1,134:35428916,9517557:0 ) (1,135:20045308,10303989:15383608,446948,141880 x1,135:20882851,10303989 g1,134:21046691,10303989 x1,134:21702051,10303989 x1,134:22320707,10303989 x1,134:24122930,10303989 g1,134:24286770,10303989 x1,134:24614450,10303989 x1,134:25111209,10303989 g1,134:25275049,10303989 x1,134:25712160,10303989 g1,134:25876000,10303989 x1,134:26568046,10303989 x1,134:27362332,10303989 g1,134:27526172,10303989 x1,134:28036035,10303989 g1,134:28199875,10303989 x1,134:28818531,10303989 g1,134:28982371,10303989 x1,134:29273347,10303989 g1,134:29437187,10303989 x1,134:31221049,10303989 x1,134:31548729,10303989 x1,134:32809641,10303989 g1,134:32973481,10303989 x1,134:33264457,10303989 x1,134:34612512,10303989 k1,135:35428916,10303989:816404 g1,135:35428916,10303989 ) (1,137:20045308,11107141:15383608,446948,141880 h1,136:20045308,11107141:655361,0,0 x1,136:21719749,11107141 k1,136:21900644,11107141:180895 x1,136:23137956,11107141 k1,136:23318852,11107141:180896 x1,136:23974212,11107141 x1,136:24940204,11107141 k1,136:25121099,11107141:180895 x1,136:25630962,11107141 k1,136:25811857,11107141:180895 x1,136:26612696,11107141 k1,136:26793591,11107141:180895 x1,136:29451044,11107141 k1,136:29631940,11107141:180896 x1,136:30578276,11107141 k1,136:30759171,11107141:180895 x1,136:31086851,11107141 x1,136:32714736,11107141 k1,136:32895631,11107141:180895 x1,136:34297421,11107141 k1,136:34482580,11107141:185159 x1,136:35428916,11107141 k1,136:35428916,11107141:0 ) (1,137:20045308,11893573:15383608,446948,141880 x1,137:20846147,11893573 k1,136:21089434,11893573:243287 x1,136:22217953,11893573 k1,136:22461241,11893573:243288 x1,136:23626447,11893573 k1,136:23869734,11893573:243287 x1,136:24415646,11893573 k1,136:24658934,11893573:243288 x1,136:26005677,11893573 k1,136:26248964,11893573:243287 x1,136:27486914,11893573 k1,136:27730202,11893573:243288 x1,136:29186401,11893573 k1,136:29429688,11893573:243287 x1,136:30230527,11893573 k1,136:30473814,11893573:243287 x1,136:31838908,11893573 k1,136:32102057,11893573:263149 x1,136:32539168,11893573 k1,136:32782456,11893573:243288 x1,136:34603021,11893573 k1,136:34846308,11893573:243287 x1,136:35428916,11893573 k1,136:35428916,11893573:0 ) (1,137:20045308,12680005:15383608,446948,141880 x1,137:20372988,12680005 x1,136:21633900,12680005 k1,136:21855215,12680005:221315 x1,136:23857326,12680005 k1,136:24078641,12680005:221315 x1,136:24588504,12680005 k1,136:24809819,12680005:221315 x1,136:25847898,12680005 k1,136:26069213,12680005:221315 x1,136:26396893,12680005 h1,136:26396893,12680005:0,0,0 x1,136:26560733,12680005 k1,136:26936317,12680005:375584 x1,136:27955397,12680005 k1,136:28176712,12680005:221315 x1,136:30287598,12680005 k1,136:30508913,12680005:221315 x1,136:31965112,12680005 k1,136:32186427,12680005:221315 x1,136:34079737,12680005 k1,136:34301052,12680005:221315 x1,136:35428916,12680005 k1,136:35428916,12680005:0 ) (1,137:20045308,13466437:15383608,446948,141880 x1,137:20846147,13466437 k1,136:21075143,13466437:228996 x1,136:22494638,13466437 k1,136:22723634,13466437:228996 x1,136:23633266,13466437 k1,136:23862262,13466437:228996 x1,136:24480918,13466437 k1,136:24709914,13466437:228996 x1,136:27658343,13466437 k1,136:27887340,13466437:228997 x1,136:29889433,13466437 k1,136:30118429,13466437:228996 x1,136:31392436,13466437 k1,136:31621432,13466437:228996 x1,136:32422271,13466437 k1,136:32651267,13466437:228996 x1,136:33779786,13466437 k1,136:34008782,13466437:228996 x1,136:35428916,13466437 k1,136:35428916,13466437:0 ) (1,137:20045308,14252869:15383608,446948,141880 x1,137:21465459,14252869 g1,136:21629299,14252869 x1,136:22430138,14252869 g1,136:22593978,14252869 x1,136:25870087,14252869 g1,136:26033927,14252869 x1,136:28036020,14252869 k1,137:35428916,14252869:7392896 g1,137:35428916,14252869 ) (1,139:24597709,16040269:10831207,891184,613385 (1,139:24597709,16040269:6278806,891184,613385 (1,139:24597709,16040269:1115534,491520,163840 (1,139:24597709,16040269:254863,491520,163840 x1,139:24852572,16040269 ) x1,139:25191403,16040269 (1,139:25191403,16138572:266977,197518,0 x1,139:25425612,16138572 ) (1,139:25458380,16040269:254863,491520,163840 x1,139:25713243,16040269 ) ) g1,139:25895283,16040269 x1,139:26405009,16040269 g1,139:26587049,16040269 x1,139:26926349,16040269 (1,139:26962757,16040269:3731713,891184,613385 (1,139:26962757,16040269:3731713,891184,613385 h1,139:26962757,16040269:78643,0,0 [1,139:27041400,16040269:3574427,891184,613385 (1,139:27041400,15596913:3574427,447828,187504 x1,139:28105633,15596913 (1,139:28105633,15695216:329956,197518,0 x1,139:28402821,15695216 ) g1,139:28581221,15596913 x1,139:29090947,15596913 g1,139:29236579,15596913 x1,139:30300812,15596913 (1,139:30300812,15695216:315015,197518,89201 x1,139:30566599,15695216 ) ) (1,139:27041400,16489813:3574427,491520,163840 k1,139:27715706,16489813:674306 x1,139:28011377,16489813 (1,139:28011377,16588116:281419,197518,0 x1,139:28260028,16588116 ) g1,139:28402020,16489813 (1,139:28402020,16489813:915002,491520,163840 (1,139:28402020,16489813:254863,491520,163840 x1,139:28656883,16489813 ) x1,139:29027571,16489813 (1,139:29062159,16489813:254863,491520,163840 x1,139:29317022,16489813 ) ) g1,139:29426246,16489813 x1,139:29894646,16489813 k1,139:30615827,16489813:674305 ) ] h1,139:30615827,16040269:78643,0,0 ) ) x1,139:30876515,16040269 ) (1,139:34664772,16040269:764144,446948,116320 (1,139:34664772,16040269:764144,446948,116320 x1,139:35428916,16040269 ) ) ) (1,141:20045308,17899509:15383608,491520,163840 x1,141:20555171,17899509 k1,140:20812642,17899509:257471 x1,140:22414329,17899509 k1,140:22671800,17899509:257471 $1,140:22671800,17899509(1,140:22671800,17899509:915002,491520,163840(1,140:22671800,17899509:254863,491520,163840x1,140:22926663,17899509)x1,140:23297351,17899509(1,140:23331939,17899509:254863,491520,163840x1,140:23586802,17899509))$1,140:23586802,17899509 k1,140:23844273,17899509:257471 x1,140:24281384,17899509 k1,140:24538855,17899509:257471 x1,140:25339694,17899509 k1,140:25597165,17899509:257471 x1,140:25888141,17899509 x1,140:26202717,17899509 x1,140:27611733,17899509 k1,140:27869204,17899509:257471 x1,140:28961019,17899509 k1,140:29218490,17899509:257471 x1,140:31566624,17899509 k1,140:31847502,17899509:280878 $1,140:31847502,17899509x1,140:32315902,17899509$1,140:32362778,17899509 k1,140:32620249,17899509:257471 x1,140:33057360,17899509 k1,140:33314831,17899509:257471 x1,140:34115670,17899509 k1,140:34373141,17899509:257471 x1,140:35428916,17899509 k1,140:35428916,17899509:0 ) (1,141:20045308,18685941:15383608,446948,127431 x1,141:23957106,18685941 k1,140:24161585,18685941:204479 $1,140:24161585,18685941x1,140:24500885,18685941$1,140:24537293,18685941 k1,140:24733645,18685941:196352 x1,140:25170756,18685941 k1,140:25367108,18685941:196352 x1,140:26167947,18685941 k1,140:26364299,18685941:196352 x1,140:27783794,18685941 k1,140:27980146,18685941:196352 x1,140:30018297,18685941 k1,140:30214648,18685941:196351 x1,140:30432880,18685941 x1,140:31736374,18685941 x1,140:31874005,18685941 k1,140:32078485,18685941:204480 x1,140:33024821,18685941 k1,140:33221172,18685941:196351 $1,140:33221172,18685941x1,140:33516843,18685941(1,140:33516843,18784244:281419,197518,0x1,140:33765494,18784244)$1,140:33798262,18685941 k1,140:33994614,18685941:196352 x1,140:34431725,18685941 k1,140:34628077,18685941:196352 x1,140:35428916,18685941 k1,140:35428916,18685941:0 ) (1,141:20045308,19472373:15383608,446948,141880 x1,141:22265617,19472373 k1,140:22474274,19472373:208657 x1,140:24585160,19472373 k1,140:24793818,19472373:208658 x1,140:26559337,19472373 k1,140:26896948,19472373:337611 x1,140:29226712,19472373 x1,140:29538008,19472373 x1,140:30329023,19472373 x1,140:30450271,19472373 k1,140:30670133,19472373:219862 x1,140:31470972,19472373 k1,140:31679629,19472373:208657 x1,140:35428916,19472373 k1,140:35428916,19472373:0 ) (1,141:20045308,20258805:15383608,446948,141880 x1,141:21938618,20258805 k1,140:22066325,20258805:127707 x1,140:22975957,20258805 k1,140:23103665,20258805:127708 x1,140:23722321,20258805 k1,140:23850028,20258805:127707 x1,140:26107066,20258805 k1,140:26234773,20258805:127707 x1,140:27508780,20258805 k1,140:27636488,20258805:127708 x1,140:27927464,20258805 k1,140:28055171,20258805:127707 x1,140:30822710,20258805 k1,140:30950417,20258805:127707 x1,140:32552104,20258805 k1,140:32679812,20258805:127708 x1,140:33116923,20258805 k1,140:33244630,20258805:127707 x1,140:35428916,20258805 k1,140:35428916,20258805:0 ) (1,141:20045308,21045237:15383608,446948,141880 x1,141:20555171,21045237 k1,140:20748160,21045237:192989 x1,140:22568070,21045237 k1,140:22761060,21045237:192990 x1,140:23561899,21045237 k1,140:23754888,21045237:192989 x1,140:26266199,21045237 k1,140:26459189,21045237:192990 x1,140:27587708,21045237 k1,140:27780697,21045237:192989 x1,140:29200831,21045237 k1,140:29393820,21045237:192989 x1,140:30304108,21045237 x1,140:31088572,21045237 k1,140:31281562,21045237:192990 x1,140:31791425,21045237 k1,140:31984414,21045237:192989 x1,140:32785253,21045237 k1,140:32978243,21045237:192990 x1,140:34216201,21045237 k1,140:34409190,21045237:192989 x1,140:35428916,21045237 k1,140:35428916,21045237:0 ) (1,141:20045308,21831669:15383608,446948,141880 x1,141:20555171,21831669 k1,140:20772332,21831669:217161 x1,140:21810411,21831669 k1,140:22027571,21831669:217160 x1,140:22355251,21831669 h1,140:22355251,21831669:0,0,0 x1,140:22519091,21831669 k1,140:22882213,21831669:363122 x1,140:23901293,21831669 k1,140:24118454,21831669:217161 x1,140:25574653,21831669 k1,140:25791813,21831669:217160 x1,140:27685123,21831669 k1,140:27902284,21831669:217161 x1,140:28648732,21831669 k1,140:29011854,21831669:363122 x1,140:29339534,21831669 x1,140:30600446,21831669 k1,140:30817606,21831669:217160 x1,140:32819717,21831669 k1,140:33036878,21831669:217161 x1,140:33327854,21831669 x1,140:35428916,21831669 k1,140:35428916,21831669:0 ) (1,141:20045308,22618101:15383608,446948,141880 x1,141:21319315,22618101 g1,140:21483155,22618101 x1,140:22283994,22618101 g1,140:22447834,22618101 x1,140:23576353,22618101 g1,140:23740193,22618101 x1,140:25160327,22618101 g1,140:25324167,22618101 x1,140:25761278,22618101 g1,140:25925118,22618101 x1,140:26835406,22618101 x1,140:27619870,22618101 g1,140:27783710,22618101 x1,140:28293573,22618101 g1,140:28457413,22618101 x1,140:29495492,22618101 g1,140:29659332,22618101 x1,140:29987012,22618101 h1,140:29987012,22618101:0,0,0 x1,140:30150852,22618101 k1,141:35428916,22618101:5278064 g1,141:35428916,22618101 ) (1,145:20045308,24025390:15383608,538305,162393 (1,145:20045308,24025390:0,0,0 g1,145:20045308,24025390 ) x1,145:22010591,24025390 x1,145:23059667,24025390 k1,145:23249909,24025390:190242 x1,145:24735450,24025390 k1,145:24925692,24025390:190242 x1,145:26105340,24025390 k1,145:26295582,24025390:190242 x1,145:26907418,24025390 x1,145:27825168,24025390 x1,145:28655624,24025390 x1,145:29442035,24025390 k1,145:29632277,24025390:190242 x1,145:30550027,24025390 x1,145:31569230,24025390 x1,145:32574282,24025390 k1,145:32764524,24025390:190242 x1,145:33506911,24025390 x1,145:34074702,24025390 x1,145:35167037,24025390 x1,145:35428916,24025390 k1,145:35428916,24025390:0 ) (1,145:20045308,24942894:15383608,516288,6681 x1,145:20831740,24942894 x1,145:22055412,24942894 k1,145:35428916,24942894:13373504 g1,145:35428916,24942894 ) (1,148:20045308,25994765:15383608,446948,141880 h1,146:20045308,25994765:655361,0,0 x1,146:21319325,25994765 x1,146:21557876,25994765 k1,146:21800197,25994765:242321 x1,146:22273356,25994765 x1,146:23183634,25994765 x1,146:23971369,25994765 k1,146:24213691,25994765:242322 x1,146:25014530,25994765 k1,146:25256851,25994765:242321 x1,146:26348667,25994765 x1,146:27549275,25994765 k1,146:27791597,25994765:242322 x1,146:28337509,25994765 k1,146:28579830,25994765:242321 x1,146:29380669,25994765 k1,146:29622991,25994765:242322 x1,146:31078535,25994765 x1,146:31916060,25994765 k1,146:32158381,25994765:242321 x1,146:33396331,25994765 k1,146:33638653,25994765:242322 x1,146:34148516,25994765 k1,146:34390837,25994765:242321 x1,146:35428916,25994765 k1,146:35428916,25994765:0 ) (1,148:20045308,26781197:15383608,446948,141880 x1,148:20372988,26781197 h1,146:20372988,26781197:0,0,0 k1,146:20566596,26781197:193608 x1,146:21076459,26781197 k1,146:21270067,26781197:193608 x1,146:21561043,26781197 k1,146:21754651,26781197:193608 x1,146:22082331,26781197 x1,146:22363483,26781197 x1,146:22946081,26781197 x1,146:23710216,26781197 k1,146:23903824,26781197:193608 x1,146:25359368,26781197 x1,146:26196893,26781197 k1,146:26390502,26781197:193609 x1,146:26900365,26781197 x1,146:27628452,26781197 x1,146:28320498,26781197 x1,146:29012544,26781197 k1,146:29206152,26781197:193608 x1,146:29716015,26781197 x1,146:31281007,26781197 k1,146:31474615,26781197:193608 x1,146:32129975,26781197 k1,146:32323583,26781197:193608 x1,146:33124422,26781197 k1,146:33318030,26781197:193608 x1,146:34373141,26781197 x1,146:35428916,26781197 k1,146:35428916,26781197:0 ) (1,148:20045308,27567629:15383608,446948,141880 x1,148:21665347,27567629 k1,146:22057510,27567629:392163 x1,146:22603422,27567629 k1,146:22830263,27567629:226841 x1,146:23868342,27567629 k1,146:24095183,27567629:226841 x1,146:24422863,27567629 h1,146:24422863,27567629:0,0,0 x1,146:24586703,27567629 k1,146:24829294,27567629:242591 x1,146:25630133,27567629 k1,146:25856974,27567629:226841 x1,146:27094286,27567629 k1,146:27321127,27567629:226841 x1,146:27976487,27567629 x1,146:28942479,27567629 k1,146:29169320,27567629:226841 x1,146:29679183,27567629 k1,146:29906024,27567629:226841 x1,146:31252767,27567629 k1,146:31479608,27567629:226841 x1,146:32717558,27567629 k1,146:32944399,27567629:226841 x1,146:33381510,27567629 k1,146:33608351,27567629:226841 x1,146:34628077,27567629 x1,146:35428916,27567629 k1,146:35428916,27567629:0 ) (1,148:20045308,28354061:15383608,446948,141880 x1,148:20791756,28354061 k1,146:20990548,28354061:198792 x1,146:21318228,28354061 x1,146:22579140,28354061 k1,146:22729876,28354061:150736 x1,146:23895099,28354061 x1,146:24731987,28354061 x1,146:24859787,28354061 k1,146:25058580,28354061:198793 x1,146:25823380,28354061 k1,147:25974116,28354061:150736 x1,147:27102635,28354061 x1,147:27721291,28354061 x1,147:28777066,28354061 k1,147:28927802,28354061:150736 x1,147:29874776,28354061 k1,147:30025512,28354061:150736 x1,147:30717568,28354061 x1,147:31554456,28354061 k1,147:31705192,28354061:150736 x1,147:32870406,28354061 k1,147:33021142,28354061:150736 x1,147:33821981,28354061 k1,147:33972717,28354061:150736 x1,147:35428916,28354061 k1,147:35428916,28354061:0 ) (1,148:20045308,29140493:15383608,446948,141880 x1,148:22156850,29140493 x1,148:22484530,29140493 x1,147:23745442,29140493 k1,147:23924273,29140493:178831 x1,147:25089496,29140493 x1,147:25926384,29140493 k1,147:26105214,29140493:178830 x1,147:26723870,29140493 x1,147:27451310,29140493 x1,147:27924469,29140493 x1,147:28944195,29140493 k1,147:29123026,29140493:178831 x1,147:30033314,29140493 x1,147:30817778,29140493 k1,147:30996608,29140493:178830 x1,147:31506471,29140493 k1,147:31685302,29140493:178831 x1,147:32723381,29140493 k1,147:32902211,29140493:178830 x1,147:33229891,29140493 h1,147:33229891,29140493:0,0,0 x1,147:33393731,29140493 k1,147:33576309,29140493:182578 x1,147:34340453,29140493 k1,147:34519284,29140493:178831 x1,147:35428916,29140493 k1,147:35428916,29140493:0 ) (1,148:20045308,29926925:15383608,446948,141880 x1,148:20991644,29926925 x1,148:22411139,29926925 k1,147:22651470,29926925:240331 x1,147:23634492,29926925 k1,147:23874823,29926925:240331 x1,147:24675662,29926925 k1,147:24915992,29926925:240330 x1,147:25571352,29926925 x1,147:26299439,29926925 k1,147:26539770,29926925:240331 x1,147:27085682,29926925 k1,147:27326013,29926925:240331 x1,147:28126852,29926925 k1,147:28367183,29926925:240331 x1,147:29240775,29926925 k1,147:29481029,29926925:240254 x1,147:30718979,29926925 k1,147:30959310,29926925:240331 x1,147:31577966,29926925 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$1,149:27854502,33875805x1,149:28182183,33875805g1,149:28327815,33875805x1,149:28837541,33875805g1,149:28983173,33875805x1,149:29638535,33875805(1,149:29638535,33637980:555238,295640,0x1,149:30161005,33637980)$1,149:30193773,33875805 g1,149:30357613,33875805 x1,149:30867476,33875805 $1,149:30867476,33875805(1,149:30867476,33637980:703604,295640,54614x1,149:31538312,33637980)$1,149:31571080,33875805 x1,149:31734920,33875805 k1,150:35428916,33875805:3693996 g1,150:35428916,33875805 ) (1,167:20045308,35283093:15383608,538305,162393 (1,167:20045308,35283093:0,0,0 g1,167:20045308,35283093 ) x1,167:23235020,35283093 k1,167:23465387,35283093:230367 x1,167:27484791,35283093 k1,167:27715158,35283093:230367 x1,167:28632908,35283093 k1,167:28863275,35283093:230367 x1,167:29824282,35283093 k1,167:30054650,35283093:230368 x1,167:30753780,35283093 x1,167:31270451,35283093 k1,167:31500818,35283093:230367 x1,167:31894034,35283093 x1,167:33013912,35283093 x1,167:33395339,35283093 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k1,173:32343046,38842052:250911 x1,173:33107190,38842052 k1,173:33358102,38842052:250912 x1,173:34449917,38842052 k1,173:34700829,38842052:250912 x1,173:35428916,38842052 k1,173:35428916,38842052:0 ) (1,174:20045308,39628484:15383608,446948,141880 x1,174:21028348,39628484 k1,173:21190274,39628484:161926 x1,173:21627385,39628484 k1,173:21789311,39628484:161926 x1,173:22699599,39628484 x1,173:23484063,39628484 k1,173:23645989,39628484:161926 x1,173:24701109,39628484 k1,173:24863035,39628484:161926 x1,173:26518450,39628484 k1,173:26720972,39628484:202522 x1,173:27994979,39628484 k1,173:28156904,39628484:161925 x1,173:28957743,39628484 k1,173:29119669,39628484:161926 x1,173:30211485,39628484 x1,173:31412093,39628484 k1,173:31574019,39628484:161926 x1,173:32119931,39628484 k1,173:32281857,39628484:161926 x1,173:32572833,39628484 k1,173:32734759,39628484:161926 x1,173:34300397,39628484 x1,173:35428916,39628484 k1,173:35428916,39628484:0 ) (1,174:20045308,40414916:15383608,446948,141880 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!10 5 [1,178:1,50262038:35428915,50262037,0 [1,178:3729360,50262038:31699556,49329697,0 [1,178:3729360,1718773:31699556,786432,0 h1,178:3729360,1718773:31699556,0,0 ] [1,178:3729360,48295958:31699556,44938785,0 [1,157:3729360,27582839:31699556,24225666,1 (1,155:3729360,23639871:31699556,20282698,0 k1,155:3949378,23639871:220018 h1,153:3949378,23639871:0,0,0 (1,153:3949378,23639871:31259521,20282698,0 (1,153:3949378,23639871:31259629,20282749,0 (1,153:3949378,23639871:31259629,20282749,0 (1,153:3949378,23639871:0,26121608,0 (1,153:3949378,23639871:40258437,26121608,0 ) k1,153:3949378,23639871:-40258437 ) ) ) ) g1,155:35208899,23639871 k1,155:35428916,23639871:220017 ) (1,155:3729360,25081663:31699556,446948,141880 h1,155:3729360,25081663:0,0,0 x1,155:5440487,25081663 k1,155:5700347,25081663:259860 x1,155:6210210,25081663 k1,155:6605409,25081663:395199 k1,155:6605409,25081663:0 x1,155:7842712,25081663 k1,155:8102572,25081663:259860 x1,155:8830659,25081663 k1,155:9090518,25081663:259859 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x1,155:22984440,27440959 g1,155:23148280,27440959 x1,155:23694192,27440959 g1,155:23858032,27440959 x1,155:24149008,27440959 g1,155:24312848,27440959 x1,155:26533182,27440959 g1,155:26697022,27440959 x1,155:29317126,27440959 g1,155:29480966,27440959 x1,155:29918077,27440959 g1,155:30081917,27440959 x1,155:31137684,27440959 g1,155:31301524,27440959 x1,155:33867881,27440959 k1,155:35428916,27440959:1561035 g1,155:35428916,27440959 ) ] [1,165:3729360,44190585:31699556,15821313,1 (1,164:3729360,41820481:31699556,13451209,0 k1,164:3949378,41820481:220018 h1,162:3949378,41820481:0,0,0 (1,162:3949378,41820481:31259521,13451209,0 (1,162:3949378,41820481:31259629,13451242,0 (1,162:3949378,41820481:31259629,13451242,0 (1,162:3949378,41820481:0,17323494,0 (1,162:3949378,41820481:40258437,17323494,0 ) k1,162:3949378,41820481:-40258437 ) ) ) ) g1,164:35208899,41820481 k1,164:35428916,41820481:220017 ) (1,164:3729360,43262273:31699556,446948,141880 h1,164:3729360,43262273:0,0,0 x1,164:5440487,43262273 k1,164:5628832,43262273:188345 x1,164:6138695,43262273 k1,164:6390864,43262273:252169 k1,164:6390864,43262273:0 x1,164:6864032,43262273 x1,164:8428996,43262273 k1,164:8617341,43262273:188345 x1,164:10072885,43262273 x1,164:10910410,43262273 k1,164:11098754,43262273:188344 x1,164:12336704,43262273 k1,164:12525049,43262273:188345 x1,164:13289193,43262273 k1,164:13477538,43262273:188345 x1,164:15588424,43262273 k1,164:15776768,43262273:188344 x1,164:16723104,43262273 k1,164:16911449,43262273:188345 x1,164:19059695,43262273 k1,164:19248040,43262273:188345 x1,164:21086303,43262273 k1,164:21362977,43262273:276674 x1,164:22637640,43262273 k1,164:22825984,43262273:188344 x1,164:23917799,43262273 k1,164:24106144,43262273:188345 x1,164:25089159,43262273 k1,164:25277504,43262273:188345 x1,164:25896160,43262273 x1,164:26210736,43262273 x1,164:26491888,43262273 k1,164:26680232,43262273:188344 x1,164:27481071,43262273 k1,164:27669416,43262273:188345 x1,164:29016159,43262273 k1,164:29204504,43262273:188345 x1,164:32517299,43262273 k1,164:32705643,43262273:188344 x1,164:33651979,43262273 k1,164:33840324,43262273:188345 x1,164:34168004,43262273 x1,164:35428916,43262273 k1,164:35428916,43262273:0 ) (1,164:3729360,44048705:31699556,446948,141880 x1,164:5749159,44048705 g1,164:5912999,44048705 x1,164:6240679,44048705 x1,164:6737438,44048705 g1,164:6901278,44048705 x1,164:7993093,44048705 g1,164:8156933,44048705 x1,164:8885020,44048705 g1,164:9048860,44048705 x1,164:10031900,44048705 g1,164:10195740,44048705 x1,164:10668908,44048705 x1,164:11208259,44048705 g1,164:11372099,44048705 x1,164:12172938,44048705 x1,164:12785041,44048705 g1,164:12948881,44048705 x1,164:14114095,44048705 g1,164:14277935,44048705 x1,164:14568911,44048705 g1,164:14732751,44048705 x1,164:16880997,44048705 g1,164:17044837,44048705 x1,164:18464332,44048705 g1,164:18628172,44048705 x1,164:20084362,44048705 g1,164:20248202,44048705 x1,164:21340017,44048705 g1,164:21503857,44048705 x1,164:22231944,44048705 g1,164:22395784,44048705 x1,164:23378824,44048705 g1,164:23542664,44048705 x1,164:24015832,44048705 x1,164:24555183,44048705 g1,164:24719023,44048705 x1,164:25519862,44048705 x1,164:26131965,44048705 g1,164:26295805,44048705 x1,164:27461019,44048705 g1,164:27624859,44048705 x1,164:28243515,44048705 g1,164:28407355,44048705 x1,164:30518241,44048705 g1,164:30682081,44048705 x1,164:32265416,44048705 k1,164:35428916,44048705:3163500 g1,164:35428916,44048705 ) ] (1,178:3729360,48295958:31699556,2794652,0 (1,178:3729360,48295958:15383608,2794652,0 [1,178:3729360,48295958:15383608,2794652,0 (1,176:3729360,46156666:15383608,446948,141880 x1,176:4676334,46156666 k1,175:4802926,46156666:126592 x1,175:6768333,46156666 k1,175:6894925,46156666:126592 x1,175:8351124,46156666 k1,175:8477716,46156666:126592 x1,175:8768692,46156666 x1,175:9079988,46156666 x1,175:9262171,46156666 x1,175:10900553,46156666 k1,175:11027145,46156666:126592 x1,175:12373888,46156666 x1,175:12883751,46156666 x1,175:14084997,46156666 k1,175:14211589,46156666:126592 x1,175:14739795,46156666 k1,175:14873837,46156666:134042 x1,175:16038397,46156666 x1,175:16875940,46156666 k1,175:17002532,46156666:126592 x1,175:17366898,46156666 k1,175:17493490,46156666:126592 x1,175:18003353,46156666 k1,175:18129945,46156666:126592 x1,175:18894736,46156666 x1,175:19112968,46156666 k1,176:19112968,46156666:0 ) (1,176:3729360,46943098:15383608,446948,141880 x1,176:5275992,46943098 k1,175:5550066,46943098:274074 x1,175:6132674,46943098 k1,175:6384701,46943098:252027 x1,175:7367723,46943098 k1,175:7619750,46943098:252027 x1,175:7947430,46943098 x1,175:9208342,46943098 k1,175:9460370,46943098:252028 x1,175:9788050,46943098 k1,175:10040077,46943098:252027 x1,175:11241341,46943098 k1,175:11493369,46943098:252028 x1,175:12330912,46943098 k1,175:12582939,46943098:252027 x1,175:13165547,46943098 x1,175:14094835,46943098 k1,175:14346863,46943098:252028 x1,175:15620870,46943098 k1,175:15872897,46943098:252027 x1,175:17110865,46943098 x1,175:17584024,46943098 x1,175:19112968,46943098 k1,175:19112968,46943098:0 ) (1,176:3729360,47729530:15383608,446948,7531 x1,176:5184904,47729530 x1,176:6022429,47729530 g1,175:6186269,47729530 x1,175:7023812,47729530 x1,175:8207378,47729530 k1,176:19112968,47729530:10905590 g1,176:19112968,47729530 ) ] g1,178:19112968,48295958 ) k1,178:19579138,48295958:466170 r1,178:19579138,48295958:0,2794652,0 k1,178:20045308,48295958:466170 (1,178:20045308,48295958:15383608,2794652,0 [1,178:20045308,48295958:15383608,2794652,0 (1,178:20045308,46156666:15383608,446948,9168 h1,177:20045308,46156666:655361,0,0 x1,177:23832584,46156666 k1,177:23995854,46156666:163270 x1,177:24942190,46156666 k1,177:25105460,46156666:163270 x1,177:25942994,46156666 x1,177:27071506,46156666 k1,177:27234776,46156666:163270 x1,177:27562456,46156666 x1,177:28823368,46156666 k1,177:28986638,46156666:163270 x1,177:30115157,46156666 k1,177:30278427,46156666:163270 x1,177:31370242,46156666 x1,177:32462058,46156666 x1,177:34027688,46156666 k1,177:34190958,46156666:163270 x1,177:34700821,46156666 x1,177:35210684,46156666 x1,177:35428916,46156666 k1,178:35428916,46156666:0 ) (1,178:20045308,46943098:15383608,446948,141880 x1,178:21100419,46943098 k1,177:21345511,46943098:245092 x1,177:22328533,46943098 k1,177:22573624,46943098:245091 x1,177:23374463,46943098 k1,177:23619555,46943098:245092 x1,177:24674666,46943098 x1,177:25730441,46943098 k1,177:25975533,46943098:245092 x1,177:27431732,46943098 k1,177:27676823,46943098:245091 x1,177:28550397,46943098 x1,177:29715602,46943098 k1,177:29960694,46943098:245092 x1,177:30470557,46943098 k1,177:30715649,46943098:245092 x1,177:32608959,46943098 k1,177:32854050,46943098:245091 x1,177:33800386,46943098 k1,177:34045478,46943098:245092 x1,177:35428916,46943098 k1,177:35428916,46943098:0 ) (1,178:20045308,47729530:15383608,446948,9168 x1,178:20482419,47729530 k1,177:20816645,47729530:334226 x1,177:22272844,47729530 k1,177:22607069,47729530:334225 x1,177:24426980,47729530 k1,177:24761206,47729530:334226 x1,177:25889725,47729530 k1,177:26223951,47729530:334226 x1,177:27170287,47729530 k1,177:27504512,47729530:334225 x1,177:28560287,47729530 k1,177:28894513,47729530:334226 x1,177:29404376,47729530 k1,177:29738602,47729530:334226 x1,177:31417490,47729530 x1,177:31993554,47729530 k1,177:32327779,47729530:334225 x1,177:33930122,47729530 k1,177:34264348,47729530:334226 x1,177:35210684,47729530 x1,177:35428916,47729530 k1,178:35428916,47729530:0 ) ] g1,178:35428916,48295958 ) ) ] h1,178:3729360,50262038:31699556,0,0 ] ] !13498 5 !10 6 [1,246:1,50262036:35428915,50262035,0 [1,246:3729360,50262036:31699556,49329695,0 [1,246:3729360,1718773:31699556,786432,0 h1,246:3729360,1718773:31699556,0,0 ] [1,246:3729360,48295956:31699556,44938783,0 (1,246:3729360,48295956:31699556,44938783,0 (1,246:3729360,48295956:15383608,44938783,0 [1,242:3729360,48295956:15383608,44938783,0 (1,178:3729360,4012533:15383608,446948,141880 x1,178:4676334,4012533 k1,177:4894225,4012533:217891 x1,177:6495912,4012533 k1,177:6713802,4012533:217890 x1,177:7332458,4012533 x1,177:8679840,4012533 k1,177:8897731,4012533:217891 x1,177:9661875,4012533 k1,177:9879765,4012533:217890 x1,177:11008284,4012533 x1,177:11326140,4012533 k1,177:11544031,4012533:217891 x1,177:12854734,4012533 k1,177:13072625,4012533:217891 x1,177:14201144,4012533 k1,177:14419034,4012533:217890 x1,177:15037690,4012533 k1,177:15255581,4012533:217891 x1,177:16201917,4012533 k1,177:16419807,4012533:217890 x1,177:16929670,4012533 k1,177:17147561,4012533:217891 x1,177:19112968,4012533 k1,177:19112968,4012533:0 ) (1,178:3729360,4798965:15383608,446948,141880 x1,178:5185559,4798965 k1,177:5455559,4798965:270000 x1,177:5965422,4798965 k1,177:6235423,4798965:270001 x1,177:9438798,4798965 k1,177:9708798,4798965:270000 x1,177:10655134,4798965 k1,177:10925135,4798965:270001 x1,177:11908166,4798965 x1,177:13346005,4798965 k1,177:13867646,4798965:521641 x1,177:14669142,4798965 x1,177:15093813,4798965 k1,177:15363707,4798965:269894 x1,177:16455522,4798965 x1,177:17547338,4798965 x1,177:19112968,4798965 k1,177:19112968,4798965:0 ) (1,178:3729360,5585397:15383608,446948,141880 x1,178:4675696,5585397 k1,177:4888308,5585397:212612 x1,177:5835282,5585397 x1,177:6162962,5585397 x1,177:6636121,5585397 x1,177:7910775,5585397 k1,177:8123388,5585397:212613 x1,177:8633251,5585397 x1,177:9143114,5585397 x1,177:10198225,5585397 k1,177:10410837,5585397:212612 x1,177:11357811,5585397 k1,177:11570423,5585397:212612 x1,177:13535830,5585397 k1,177:13748443,5585397:212613 x1,177:15204642,5585397 k1,177:15417254,5585397:212612 x1,177:16545773,5585397 k1,177:16758385,5585397:212612 x1,177:17377041,5585397 k1,177:17589654,5585397:212613 x1,177:18099517,5585397 k1,177:18312129,5585397:212612 x1,177:19112968,5585397 k1,177:19112968,5585397:0 ) (1,178:3729360,6371829:15383608,446948,141880 x1,178:6932735,6371829 g1,177:7096575,6371829 x1,177:7314807,6371829 (1,177:7314807,6371829:655360,446948,4912 x1,177:7970167,6371829 ) x1,177:8188399,6371829 g1,177:8352239,6371829 x1,177:9298575,6371829 g1,177:9462415,6371829 x1,177:10445446,6371829 x1,177:11719445,6371829 g1,177:11883285,6371829 x1,177:12101517,6371829 (1,177:12101517,6371829:655360,446948,9168 x1,177:12756877,6371829 ) x1,177:13138949,6371829 k1,178:19112968,6371829:5974019 g1,178:19112968,6371829 ) (1,179:3729360,7158261:15383608,446948,141880 h1,179:3729360,7158261:655361,0,0 x1,179:4785145,7158261 x1,179:5060400,7158261 k1,179:5240354,7158261:179954 x1,179:6150624,7158261 k1,179:6330577,7158261:179953 x1,179:7277551,7158261 k1,179:7457505,7158261:179954 x1,179:7785185,7158261 x1,179:8109592,7158261 x1,179:9274815,7158261 x1,179:9565791,7158261 x1,179:10421670,7158261 k1,179:10605652,7158261:183982 x1,179:11115515,7158261 k1,179:11295469,7158261:179954 x1,179:12387284,7158261 k1,179:12567237,7158261:179953 x1,179:13295324,7158261 k1,179:13475278,7158261:179954 x1,179:14622158,7158261 k1,179:14806140,7158261:183982 x1,179:15097116,7158261 k1,179:15277070,7158261:179954 x1,179:16423950,7158261 k1,179:16603904,7158261:179954 x1,179:17586935,7158261 k1,179:17766872,7158261:179937 x1,179:18894736,7158261 x1,179:19112968,7158261 k1,179:19112968,7158261:0 ) (1,179:3729360,7944693:15383608,446948,141880 x1,179:4348016,7944693 x1,179:5367088,7944693 x1,179:5658064,7944693 k1,179:5877589,7944693:219525 x1,179:6205269,7944693 x1,179:7466181,7944693 k1,179:7685707,7944693:219526 x1,179:8158875,7944693 x1,179:8698226,7944693 k1,179:8917751,7944693:219525 x1,179:10628222,7944693 k1,179:10847747,7944693:219525 x1,179:11830787,7944693 k1,179:12050313,7944693:219526 x1,179:12632921,7944693 k1,179:12852446,7944693:219525 x1,179:13471102,7944693 x1,179:14526222,7944693 k1,179:14745748,7944693:219526 x1,179:15073428,7944693 x1,179:16334340,7944693 k1,179:16553865,7944693:219525 x1,179:17045385,7944693 k1,179:17415601,7944693:370216 x1,179:18398632,7944693 x1,179:18689608,7944693 x1,179:18907840,7944693 x1,179:19112968,7944693 k1,179:19112968,7944693:0 ) (1,179:3729360,8731125:15383608,446948,92400 x1,179:5094455,8731125 k1,179:5349427,8731125:254972 x1,179:6295763,8731125 x1,179:6805626,8731125 x1,179:8080280,8731125 k1,179:8317026,8731125:236746 x1,179:9590378,8731125 k1,179:9827123,8731125:236745 x1,179:10627962,8731125 k1,179:10864708,8731125:236746 x1,179:12211451,8731125 k1,179:12448197,8731125:236746 x1,179:12994101,8731125 k1,179:13230847,8731125:236746 x1,179:13994991,8731125 k1,179:14231737,8731125:236746 x1,179:15323552,8731125 k1,179:15560298,8731125:236746 x1,179:16288385,8731125 k1,179:16525130,8731125:236745 x1,179:17672010,8731125 k1,179:18093888,8731125:421878 x1,179:19112968,8731125 k1,179:19112968,8731125:0 ) (1,179:3729360,9517557:15383608,446948,141880 x1,179:4893920,9517557 x1,179:6131878,9517557 k1,179:6311908,9517557:180030 x1,179:6857820,9517557 k1,179:7037850,9517557:180030 x1,179:7838689,9517557 k1,179:8018718,9517557:180029 x1,179:9038437,9517557 k1,179:9218467,9517557:180030 x1,179:9546147,9517557 x1,179:10807059,9517557 k1,179:10987089,9517557:180030 x1,179:11751233,9517557 k1,179:11931263,9517557:180030 x1,179:13023078,9517557 k1,179:13203108,9517557:180030 x1,179:13931195,9517557 k1,179:14111225,9517557:180030 x1,179:15094265,9517557 k1,179:15274294,9517557:180029 x1,179:15711405,9517557 k1,179:15891435,9517557:180030 x1,179:16801723,9517557 x1,179:17586187,9517557 k1,179:17766217,9517557:180030 x1,179:19112968,9517557 k1,179:19112968,9517557:0 ) (1,179:3729360,10303989:15383608,446948,141880 x1,179:5220935,10303989 k1,179:5434924,10303989:213989 x1,179:5944787,10303989 k1,179:6158777,10303989:213990 x1,179:7196856,10303989 k1,179:7410845,10303989:213989 x1,179:7738525,10303989 h1,179:7738525,10303989:0,0,0 x1,179:7902365,10303989 k1,179:8255973,10303989:353608 x1,179:9529980,10303989 k1,179:9743970,10303989:213990 x1,179:10726992,10303989 k1,179:10940981,10303989:213989 x1,179:11741820,10303989 k1,179:11955809,10303989:213989 x1,179:12465672,10303989 x1,179:13431664,10303989 k1,179:13645654,10303989:213990 x1,179:15101198,10303989 x1,179:15938723,10303989 k1,179:16152712,10303989:213989 x1,179:17135734,10303989 k1,179:17349724,10303989:213990 x1,179:17786835,10303989 k1,179:18000824,10303989:213989 x1,179:18656184,10303989 x1,179:19112968,10303989 k1,179:19112968,10303989:0 ) (1,179:3729360,11090421:15383608,446948,141880 x1,179:4348016,11090421 x1,179:5567623,11090421 k1,179:5775556,11090421:207933 x1,179:6321468,11090421 k1,179:6486898,11090421:165430 x1,179:7105554,11090421 x1,179:7615417,11090421 x1,179:8798983,11090421 k1,179:8964812,11090421:165829 x1,179:9765651,11090421 k1,179:9931082,11090421:165431 x1,179:11205081,11090421 x1,179:12224152,11090421 k1,179:12389583,11090421:165431 x1,179:13554143,11090421 k1,179:13719574,11090421:165431 x1,179:14374934,11090421 k1,179:14540364,11090421:165430 x1,179:15341203,11090421 k1,179:15506634,11090421:165431 x1,179:16161994,11090421 x1,179:16998882,11090421 k1,179:17164313,11090421:165431 x1,179:18401625,11090421 k1,179:18567056,11090421:165431 x1,179:19112968,11090421 k1,179:19112968,11090421:0 ) (1,179:3729360,11876853:15383608,446948,141880 x1,179:4530199,11876853 g1,179:4694039,11876853 x1,179:5021719,11876853 x1,179:5302871,11876853 x1,179:5885469,11876853 x1,179:6649604,11876853 g1,179:6813444,11876853 x1,179:7432100,11876853 x1,179:7905259,11876853 x1,179:8961034,11876853 g1,179:9124874,11876853 x1,179:10107896,11876853 g1,179:10271736,11876853 x1,179:10781599,11876853 g1,179:10945439,11876853 x1,179:11983518,11876853 g1,179:12147358,11876853 x1,179:12475038,11876853 h1,179:12475038,11876853:0,0,0 g1,179:12638878,11876853 x1,179:13075989,11876853 g1,179:13239829,11876853 x1,179:14004620,11876853 x1,179:14623276,11876853 x1,179:16169908,11876853 k1,179:19112968,11876853:2943060 g1,179:19112968,11876853 ) (1,179:3729360,13185650:15383608,538305,14150 (1,179:3729360,13185650:0,0,0 g1,179:3729360,13185650 ) x1,179:5564082,13185650 g1,179:5760690,13185650 x1,179:9780094,13185650 k1,179:19112968,13185650:9332874 g1,179:19112968,13185650 ) (1,182:3729360,14208011:15383608,446948,141880 h1,180:3729360,14208011:655361,0,0 x1,180:5876960,14208011 g1,180:6040800,14208011 x1,180:7787976,14208011 x1,180:9337222,14208011 g1,180:9501062,14208011 x1,180:10775069,14208011 g1,180:10938909,14208011 x1,180:13705785,14208011 g1,180:13869625,14208011 x1,180:15690183,14208011 g1,180:15854023,14208011 x1,180:16836390,14208011 k1,182:19112968,14208011:2276578 g1,182:19112968,14208011 ) (1,185:5040082,15405889:14072886,446948,141880 (1,183:5040082,15405889:0,291271,0 g1,183:5040082,15405889 g1,183:3729360,15405889 g1,183:3401680,15405889 (1,183:3401680,15405889:1310722,291271,0 k1,183:4712402,15405889:1310722 (1,183:4712402,15405889:0,291271,0 k1,183:4384721,15405889:-327681 x1,183:4712402,15405889 ) ) g1,183:5040082,15405889 ) x1,183:6059162,15405889 k1,183:6201947,15405889:142785 x1,183:6857307,15405889 x1,183:7585394,15405889 k1,183:7728179,15405889:142785 x1,183:8274091,15405889 k1,183:8416876,15405889:142785 x1,183:9217715,15405889 k1,183:9360500,15405889:142785 x1,183:10234092,15405889 k1,183:10376856,15405889:142764 x1,183:11614806,15405889 k1,183:11757591,15405889:142785 x1,183:12376247,15405889 x1,183:13905191,15405889 k1,183:14047976,15405889:142785 x1,183:16196222,15405889 k1,183:16339007,15405889:142785 x1,183:16994367,15405889 k1,183:17137152,15405889:142785 x1,183:17937991,15405889 k1,183:18080776,15405889:142785 x1,183:18408456,15405889 x1,183:18689608,15405889 x1,183:18907840,15405889 x1,183:19112968,15405889 k1,185:19112968,15405889:0 ) (1,185:5040082,16192321:14072886,446948,141880 x1,185:5404448,16192321 x1,185:6168583,16192321 k1,183:6435685,16192321:267102 x1,183:7600253,16192321 x1,183:9056434,16192321 x1,183:9748480,16192321 x1,183:9869728,16192321 k1,183:10162646,16192321:292918 x1,183:10963485,16192321 k1,183:11230587,16192321:267102 x1,183:12650082,16192321 k1,183:12917185,16192321:267103 x1,183:13535841,16192321 x1,183:14936984,16192321 k1,183:15229901,16192321:292917 x1,183:16030740,16192321 k1,183:16297843,16192321:267103 x1,183:17463066,16192321 x1,183:18299954,16192321 k1,183:18567056,16192321:267102 x1,183:19112968,16192321 k1,183:19112968,16192321:0 ) (1,185:5040082,16978753:14072886,446948,141880 x1,185:5367762,16978753 x1,183:7338418,16978753 g1,183:7502258,16978753 x1,183:8448594,16978753 g1,183:8612434,16978753 x1,183:9413273,16978753 g1,183:9577113,16978753 x1,183:9904793,16978753 x1,183:11165705,16978753 g1,183:11329545,16978753 x1,183:12494105,16978753 x1,183:13495488,16978753 k1,185:19112968,16978753:5617480 g1,185:19112968,16978753 ) (1,186:5040082,18233052:14072886,446948,92400 (1,185:5040082,18233052:0,291271,0 g1,185:5040082,18233052 g1,185:3729360,18233052 g1,185:3401680,18233052 (1,185:3401680,18233052:1310722,291271,0 k1,185:4712402,18233052:1310722 (1,185:4712402,18233052:0,291271,0 k1,185:4384721,18233052:-327681 x1,185:4712402,18233052 ) ) g1,185:5040082,18233052 ) x1,185:5404458,18233052 x1,185:5940546,18233052 k1,185:6158845,18233052:218299 x1,185:6449821,18233052 k1,185:6668120,18233052:218299 x1,185:7323480,18233052 k1,185:7541779,18233052:218299 x1,185:7869459,18233052 x1,185:9130371,18233052 k1,185:9348671,18233052:218300 x1,185:10713765,18233052 k1,185:10945678,18233052:231913 x1,185:11746517,18233052 k1,185:11964816,18233052:218299 x1,185:12838408,18233052 k1,185:13056653,18233052:218245 x1,185:14294603,18233052 k1,185:14512902,18233052:218299 x1,185:15131558,18233052 x1,185:15446134,18233052 x1,185:15727286,18233052 k1,185:15945585,18233052:218299 x1,185:16236561,18233052 k1,185:16454860,18233052:218299 x1,185:19112968,18233052 k1,185:19112968,18233052:0 ) (1,186:5040082,19019484:14072886,446948,141880 x1,186:5695442,19019484 x1,185:6661434,19019484 k1,185:6830563,19019484:169129 x1,185:8650491,19019484 k1,185:8819619,19019484:169128 x1,185:9147299,19019484 x1,185:10408211,19019484 k1,185:10577340,19019484:169129 x1,185:11414883,19019484 k1,185:11586656,19019484:171773 x1,185:12387495,19019484 k1,185:12556624,19019484:169129 x1,185:13539646,19019484 k1,185:13708775,19019484:169129 x1,185:14364135,19019484 x1,185:15330127,19019484 k1,185:15499256,19019484:169129 x1,185:15936367,19019484 k1,185:16105495,19019484:169128 x1,185:18179695,19019484 k1,185:18348824,19019484:169129 x1,185:19112968,19019484 k1,185:19112968,19019484:0 ) (1,186:5040082,19805916:14072886,446948,7531 x1,186:6204632,19805916 g1,185:6368472,19805916 x1,185:6696152,19805916 x1,185:8666808,19805916 k1,186:19112968,19805916:10446160 g1,186:19112968,19805916 ) (1,187:5040082,21060216:14072886,446948,141880 (1,186:5040082,21060216:0,291271,0 g1,186:5040082,21060216 g1,186:3729360,21060216 g1,186:3401680,21060216 (1,186:3401680,21060216:1310722,291271,0 k1,186:4712402,21060216:1310722 (1,186:4712402,21060216:0,291271,0 k1,186:4384721,21060216:-327681 x1,186:4712402,21060216 ) ) g1,186:5040082,21060216 ) x1,186:6568370,21060216 k1,186:6807359,21060216:238989 x1,186:7244470,21060216 k1,186:7483459,21060216:238989 x1,186:7774435,21060216 k1,186:8013424,21060216:238989 x1,186:9396224,21060216 k1,186:9635213,21060216:238989 x1,186:10253869,21060216 x1,186:11200205,21060216 x1,186:12728493,21060216 k1,186:12967482,21060216:238989 x1,186:13513394,21060216 k1,186:13752383,21060216:238989 x1,186:14553222,21060216 k1,186:14792211,21060216:238989 x1,186:15447571,21060216 x1,186:16175658,21060216 k1,186:16414647,21060216:238989 x1,186:16960559,21060216 k1,186:17199548,21060216:238989 x1,186:18000387,21060216 k1,186:18239376,21060216:238989 x1,186:19112968,21060216 k1,186:19112968,21060216:0 ) (1,187:5040082,21846648:14072886,446948,141880 x1,187:6278032,21846648 k1,186:6473032,21846648:195000 x1,186:7128392,21846648 k1,186:7323393,21846648:195001 x1,186:8124232,21846648 k1,186:8319232,21846648:195000 x1,186:9229520,21846648 x1,186:11449863,21846648 k1,186:11644863,21846648:195000 x1,186:12937222,21846648 k1,186:13140012,21846648:202790 x1,186:13940851,21846648 k1,186:14135852,21846648:195001 x1,186:15919714,21846648 x1,186:16247394,21846648 x1,186:17508306,21846648 k1,186:17703306,21846648:195000 x1,186:18030986,21846648 x1,186:18312138,21846648 x1,186:18894736,21846648 x1,186:19112968,21846648 k1,187:19112968,21846648:0 ) (1,187:5040082,22633080:14072886,446948,92400 x1,187:5804217,22633080 g1,186:5968057,22633080 x1,186:7133262,22633080 x1,186:8679893,22633080 g1,186:8843733,22633080 x1,186:9790069,22633080 g1,186:9953909,22633080 x1,186:10754748,22633080 g1,186:10918588,22633080 x1,186:11246268,22633080 x1,186:11527420,22633080 x1,186:12110018,22633080 x1,186:12874153,22633080 g1,186:13037993,22633080 x1,186:14165857,22633080 x1,186:15567009,22633080 k1,187:19112968,22633080:3545959 g1,187:19112968,22633080 ) (1,189:5040082,23887379:14072886,446948,141880 (1,187:5040082,23887379:0,291271,0 g1,187:5040082,23887379 g1,187:3729360,23887379 g1,187:3401680,23887379 (1,187:3401680,23887379:1310722,291271,0 k1,187:4712402,23887379:1310722 (1,187:4712402,23887379:0,291271,0 k1,187:4384721,23887379:-327681 x1,187:4712402,23887379 ) ) g1,187:5040082,23887379 ) x1,187:6677818,23887379 k1,187:6939998,23887379:262180 x1,187:8068517,23887379 x1,187:8432883,23887379 x1,187:9197018,23887379 k1,187:9459198,23887379:262180 x1,187:10405534,23887379 x1,187:10915397,23887379 x1,187:12353891,23887379 k1,187:12640655,23887379:286764 x1,187:13514247,23887379 k1,187:13776328,23887379:262081 x1,187:15014278,23887379 k1,187:15276458,23887379:262180 x1,187:16332225,23887379 k1,187:16594405,23887379:262180 x1,187:17213061,23887379 x1,187:18340925,23887379 k1,187:18603105,23887379:262180 x1,187:19112968,23887379 k1,187:19112968,23887379:0 ) (1,189:5040082,24673811:14072886,446948,92400 x1,189:6095193,24673811 x1,189:7150968,24673811 k1,187:7370095,24673811:219127 x1,187:8990117,24673811 k1,187:9223064,24673811:232947 x1,187:9550744,24673811 x1,187:10047503,24673811 k1,187:10266630,24673811:219127 x1,187:11067469,24673811 k1,187:11286595,24673811:219126 x1,187:11941955,24673811 x1,187:12670042,24673811 k1,187:12889169,24673811:219127 x1,187:13326280,24673811 k1,187:13545407,24673811:219127 x1,187:14709957,24673811 k1,187:14929083,24673811:219126 x1,187:15438946,24673811 k1,187:15658073,24673811:219127 x1,187:16458912,24673811 k1,187:16678038,24673811:219126 x1,187:18043132,24673811 k1,187:18276080,24673811:232948 x1,187:18603760,24673811 x1,187:19112968,24673811 k1,189:19112968,24673811:0 ) (1,189:5040082,25460243:14072886,446948,141880 x1,189:5367762,25460243 x1,189:5831097,25460243 x1,189:6741385,25460243 k1,187:7035347,25460243:293962 x1,187:8382098,25460243 k1,187:8676059,25460243:293961 x1,187:9440850,25460243 x1,187:10407497,25460243 x1,187:10528745,25460243 k1,187:10855236,25460243:326491 x1,187:11801572,25460243 k1,187:12095534,25460243:293962 x1,187:12532645,25460243 k1,187:12826606,25460243:293961 x1,187:14282805,25460243 k1,187:14576767,25460243:293962 x1,187:15959567,25460243 x1,187:16462222,25460243 x1,187:16599853,25460243 k1,187:16926343,25460243:326490 x1,187:18054862,25460243 k1,187:18348824,25460243:293962 x1,187:19112968,25460243 k1,187:19112968,25460243:0 ) (1,189:5040082,26246675:14072886,446948,141880 x1,189:7607096,26246675 g1,187:7810256,26246675 x1,187:8975471,26246675 g1,188:9139311,26246675 x1,188:9576422,26246675 g1,188:9740262,26246675 x1,188:10104628,26246675 x1,188:10869419,26246675 x1,188:12361649,26246675 g1,188:12525489,26246675 x1,188:13035352,26246675 g1,188:13199192,26246675 x1,188:14237271,26246675 g1,188:14401111,26246675 x1,188:14728791,26246675 h1,188:14728791,26246675:0,0,0 x1,188:14892631,26246675 k1,189:19112968,26246675:4220337 g1,189:19112968,26246675 ) (1,192:5040082,27500974:14072886,446948,9168 (1,191:5040082,27500974:0,291271,0 g1,191:5040082,27500974 g1,191:3729360,27500974 g1,191:3401680,27500974 (1,191:3401680,27500974:1310722,291271,0 k1,191:4712402,27500974:1310722 (1,191:4712402,27500974:0,291271,0 k1,191:4384721,27500974:-327681 x1,191:4712402,27500974 ) ) g1,191:5040082,27500974 ) x1,191:6059162,27500974 k1,191:6278547,27500974:219385 x1,191:7152139,27500974 k1,191:7371467,27500974:219328 x1,191:8354489,27500974 k1,191:8573874,27500974:219385 x1,191:9847873,27500974 x1,191:10866944,27500974 k1,191:11086329,27500974:219385 x1,191:11741689,27500974 x1,191:12833505,27500974 x1,191:13442337,27500974 k1,191:13661721,27500974:219384 x1,191:14425865,27500974 k1,191:14645250,27500974:219385 x1,191:15446089,27500974 k1,191:15665474,27500974:219385 x1,191:16575744,27500974 k1,191:16795128,27500974:219384 x1,191:17122808,27500974 x1,191:18383720,27500974 k1,191:18603105,27500974:219385 x1,191:19112968,27500974 k1,191:19112968,27500974:0 ) (1,192:5040082,28287406:14072886,446948,141880 x1,192:5331058,28287406 k1,191:5558252,28287406:227194 x1,191:6213612,28287406 k1,191:6440806,28287406:227194 x1,191:6768486,28287406 x1,191:8029398,28287406 k1,191:8256593,28287406:227195 x1,191:9457847,28287406 k1,191:9685041,28287406:227194 x1,191:10012721,28287406 x1,191:11233632,28287406 k1,191:11460826,28287406:227194 x1,191:13609072,28287406 k1,191:13836266,28287406:227194 x1,191:14382170,28287406 k1,191:14609364,28287406:227194 x1,191:14900340,28287406 k1,191:15127535,28287406:227195 x1,191:16292103,28287406 x1,191:17311829,28287406 k1,191:17539023,28287406:227194 x1,191:18084935,28287406 k1,191:18312129,28287406:227194 x1,191:19112968,28287406 k1,191:19112968,28287406:0 ) (1,192:5040082,29073838:14072886,446948,141880 x1,192:5658738,29073838 x1,192:6896041,29073838 k1,191:7125397,29073838:229356 x1,191:7635260,29073838 k1,191:7864616,29073838:229356 x1,191:8847638,29073838 k1,191:9076994,29073838:229356 x1,191:9404674,29073838 x1,191:10829426,29073838 k1,191:11229135,29073838:399709 x1,191:12176118,29073838 k1,191:12405474,29073838:229356 x1,191:13206313,29073838 k1,191:13435669,29073838:229356 x1,191:14163109,29073838 x1,191:15728101,29073838 x1,191:16045957,29073838 k1,191:16275313,29073838:229356 x1,191:16821225,29073838 k1,191:17050581,29073838:229356 x1,191:17851420,29073838 k1,191:18080776,29073838:229356 x1,191:18408456,29073838 x1,191:18689608,29073838 x1,191:18907840,29073838 x1,191:19112968,29073838 k1,192:19112968,29073838:0 ) (1,192:5040082,29860270:14072886,446948,141880 x1,192:5404448,29860270 x1,192:6168583,29860270 k1,191:6399978,29860270:231395 x1,191:7018634,29860270 x1,191:7491793,29860270 x1,191:8547568,29860270 k1,191:8778963,29860270:231395 x1,191:9761985,29860270 k1,191:9993380,29860270:231395 x1,191:10539292,29860270 k1,191:10770687,29860270:231395 x1,191:11717661,29860270 k1,191:11949055,29860270:231394 x1,191:12276735,29860270 x1,191:13537647,29860270 k1,191:13769042,29860270:231395 x1,191:14206153,29860270 k1,191:14437548,29860270:231395 x1,191:15602771,29860270 k1,191:15834166,29860270:231395 x1,191:16161846,29860270 x1,191:16988910,29860270 k1,191:17220305,29860270:231395 x1,191:19112968,29860270 k1,191:19112968,29860270:0 ) (1,192:5040082,30646702:14072886,446948,141880 x1,192:5658738,30646702 x1,192:6605074,30646702 x1,192:7733593,30646702 g1,191:7897433,30646702 x1,191:8552793,30646702 g1,191:8716633,30646702 x1,191:9517472,30646702 g1,191:9681312,30646702 x1,191:10299968,30646702 x1,191:11537271,30646702 g1,191:11701111,30646702 x1,191:12210974,30646702 g1,191:12374814,30646702 x1,191:13175653,30646702 g1,191:13339493,30646702 x1,191:13667173,30646702 x1,191:15091925,30646702 k1,192:19112968,30646702:4021043 g1,192:19112968,30646702 ) (1,194:3729360,32141333:15383608,541848,14544 k1,194:4924501,32141333:1195141 (1,194:4924501,32141333:0,0,0 g1,194:4924501,32141333 ) x1,194:11653157,32141333 g1,194:11849765,32141333 x1,194:12942110,32141333 g1,194:13138718,32141333 x1,194:15760651,32141333 x1,194:17917827,32141333 k1,194:19112968,32141333:1195141 ) (1,194:3729360,33058837:15383608,541848,14544 k1,194:4957140,33058837:1227780 x1,194:6136777,33058837 x1,194:6725025,33058837 x1,194:8608515,33058837 g1,194:8805123,33058837 x1,194:11688941,33058837 g1,194:11885549,33058837 x1,194:12410093,33058837 x1,194:14717469,33058837 g1,194:14914077,33058837 x1,194:17885189,33058837 g1,194:17885189,33058837 k1,194:19112968,33058837:1227779 ) (1,196:3729360,34140180:15383608,446948,141880 h1,195:3729360,34140180:655361,0,0 x1,195:5950360,34140180 k1,195:6176089,34140180:225729 x1,195:6794745,34140180 k1,195:7020473,34140180:225728 x1,195:7566385,34140180 x1,195:8403273,34140180 k1,195:8629002,34140180:225729 x1,195:9794225,34140180 x1,195:9976408,34140180 x1,195:10704503,34140180 k1,195:10930231,34140180:225728 x1,195:11148463,34140180 (1,195:11148463,34140180:327680,437780,4912 x1,195:11476143,34140180 ) x1,195:11858215,34140180 k1,195:12099416,34140180:241201 x1,195:12427096,34140180 x1,195:13906240,34140180 x1,195:14779832,34140180 x1,195:15107512,34140180 x1,195:16368424,34140180 k1,195:16594152,34140180:225728 x1,195:18013647,34140180 k1,195:18239376,34140180:225729 x1,195:18894736,34140180 x1,195:19112968,34140180 k1,196:19112968,34140180:0 ) (1,196:3729360,34926612:15383608,446948,9168 x1,196:4166471,34926612 x1,196:5186197,34926612 k1,195:5421817,34926612:235620 x1,195:6368153,34926612 k1,195:6603772,34926612:235619 x1,195:7622835,34926612 k1,195:7858455,34926612:235620 x1,195:8950270,34926612 x1,195:10042086,34926612 x1,195:11607716,34926612 k1,195:11843336,34926612:235620 x1,195:12461992,34926612 x1,195:12776568,34926612 x1,195:13057720,34926612 k1,195:13293339,34926612:235619 x1,195:14530651,34926612 k1,195:14766271,34926612:235620 x1,195:16185766,34926612 k1,195:16421386,34926612:235620 x1,195:17076746,34926612 k1,195:17312365,34926612:235619 x1,195:17603341,34926612 k1,195:17838961,34926612:235620 x1,195:19112968,34926612 k1,196:19112968,34926612:0 ) (1,196:3729360,35713044:15383608,446948,141880 x1,196:4602952,35713044 x1,196:5658727,35713044 k1,195:5811697,35713044:152970 x1,195:6430353,35713044 x1,195:7122392,35713044 k1,195:7275362,35713044:152970 x1,195:8039506,35713044 k1,195:8192476,35713044:152970 x1,195:10340722,35713044 k1,195:10493693,35713044:152971 x1,195:12331956,35713044 k1,195:12531492,35713044:199536 x1,195:13951643,35713044 k1,195:14104614,35713044:152971 x1,195:14905453,35713044 k1,195:15058423,35713044:152970 x1,195:16477918,35713044 k1,195:16630888,35713044:152970 x1,195:17649951,35713044 k1,195:17802921,35713044:152970 x1,195:18894736,35713044 x1,195:19112968,35713044 k1,196:19112968,35713044:0 ) (1,196:3729360,36499476:15383608,446948,141880 x1,196:4821176,36499476 x1,196:6550646,36499476 k1,195:6691686,36499476:141040 x1,195:7492525,36499476 k1,195:7627866,36499476:135341 x1,195:7918842,36499476 x1,195:8230138,36499476 x1,195:8544714,36499476 x1,195:9054577,36499476 x1,195:10074303,36499476 k1,195:10209643,36499476:135340 x1,195:10755555,36499476 k1,195:10890895,36499476:135340 x1,195:11691734,36499476 k1,195:11827075,36499476:135341 x1,195:13246570,36499476 k1,195:13381910,36499476:135340 x1,195:14547115,36499476 x1,195:15929906,36499476 k1,195:16065247,36499476:135341 x1,195:17521446,36499476 k1,195:17656786,36499476:135340 x1,195:19112968,36499476 k1,195:19112968,36499476:0 ) (1,196:3729360,37285908:15383608,446948,7531 x1,196:4275272,37285908 g1,195:4439112,37285908 x1,195:4766792,37285908 x1,195:6573608,37285908 g1,195:6737448,37285908 x1,195:7611032,37285908 g1,195:7774872,37285908 x1,195:9012184,37285908 g1,195:9176024,37285908 x1,195:10267839,37285908 x1,195:11851175,37285908 k1,196:19112968,37285908:7261793 g1,196:19112968,37285908 ) (1,198:3729360,38072340:15383608,446948,9168 h1,197:3729360,38072340:655361,0,0 x1,197:5913656,38072340 k1,197:6209482,38072340:295826 x1,197:7119114,38072340 x1,197:8357072,38072340 k1,197:8652897,38072340:295825 x1,197:9490440,38072340 x1,197:10510166,38072340 k1,197:10805992,38072340:295826 x1,197:11752328,38072340 k1,197:12048154,38072340:295826 x1,197:12375834,38072340 x1,197:12656986,38072340 x1,197:13239584,38072340 x1,197:14003719,38072340 k1,197:14299545,38072340:295826 x1,197:15464750,38072340 x1,197:16847541,38072340 k1,197:17143366,38072340:295825 x1,197:17943550,38072340 k1,197:18239376,38072340:295826 x1,197:18894736,38072340 x1,197:19112968,38072340 k1,198:19112968,38072340:0 ) (1,198:3729360,38858772:15383608,446948,141880 x1,198:4821176,38858772 x1,198:5430008,38858772 k1,197:5688477,38858772:258469 x1,197:6198340,38858772 k1,197:6456810,38858772:258470 x1,197:7330402,38858772 x1,197:7787186,38858772 k1,197:8045655,38858772:258469 x1,197:9501854,38858772 k1,197:9760324,38858772:258470 x1,197:10561163,38858772 k1,197:10819632,38858772:258469 x1,197:12184726,38858772 k1,197:12671774,38858772:487048 x1,197:13690854,38858772 k1,197:13949324,38858772:258470 x1,197:14822916,38858772 x1,197:15789563,38858772 k1,197:16048032,38858772:258469 x1,197:16848871,38858772 x1,197:17867942,38858772 k1,197:18126412,38858772:258470 x1,197:18563523,38858772 k1,197:18821992,38858772:258469 x1,197:19112968,38858772 k1,197:19112968,38858772:0 ) (1,198:3729360,39645204:15383608,446948,141880 x1,198:5367743,39645204 k1,197:5574224,39645204:206481 x1,197:6738792,39645204 x1,197:7758518,39645204 k1,197:7964998,39645204:206480 x1,197:8510910,39645204 k1,197:8717391,39645204:206481 x1,197:9518230,39645204 k1,197:9724711,39645204:206481 x1,197:10052391,39645204 x1,197:11313303,39645204 k1,197:11519783,39645204:206480 x1,197:12138439,39645204 x1,197:13375742,39645204 k1,197:13582223,39645204:206481 x1,197:14637335,39645204 k1,197:14843815,39645204:206480 x1,197:15881894,39645204 k1,197:16088375,39645204:206481 x1,197:16743735,39645204 h1,197:16743735,39645204:0,0,0 x1,197:17125807,39645204 k1,197:17456889,39645204:331082 x1,197:18148289,39645204 x1,197:19112968,39645204 k1,197:19112968,39645204:0 ) (1,198:3729360,40431636:15383608,446948,141880 x1,198:4530199,40431636 k1,197:4744159,40431636:213960 x1,197:5763878,40431636 k1,197:5977837,40431636:213959 x1,197:6305517,40431636 x1,197:7566429,40431636 k1,197:7780389,40431636:213960 x1,197:8290252,40431636 k1,197:8504211,40431636:213959 x1,197:9305050,40431636 k1,197:9519010,40431636:213960 x1,197:10720264,40431636 k1,197:10934223,40431636:213959 x1,197:11807807,40431636 k1,197:12021767,40431636:213960 x1,197:12640423,40431636 k1,197:12854382,40431636:213959 x1,197:14601557,40431636 x1,197:16203891,40431636 k1,197:16417851,40431636:213960 x1,197:17109242,40431636 x1,197:17716106,40431636 k1,197:17930065,40431636:213959 x1,197:19112968,40431636 k1,197:19112968,40431636:0 ) (1,198:3729360,41218068:15383608,446948,9168 x1,198:5331047,41218068 k1,197:5490283,41218068:159236 x1,197:6108939,41218068 x1,197:6836379,41218068 x1,197:8037625,41218068 k1,197:8196861,41218068:159236 x1,197:9362075,41218068 k1,197:9521311,41218068:159236 x1,197:10067223,41218068 k1,197:10226459,41218068:159236 x1,197:11027298,41218068 k1,197:11186533,41218068:159235 x1,197:11841893,41218068 x1,197:12387797,41218068 x1,197:12933709,41218068 x1,197:13208301,41218068 x1,197:14228027,41218068 k1,197:14387263,41218068:159236 x1,197:14933175,41218068 k1,197:15092411,41218068:159236 x1,197:15383387,41218068 k1,197:15542623,41218068:159236 x1,197:15870303,41218068 x1,197:16151455,41218068 x1,197:16734053,41218068 x1,197:17498188,41218068 k1,197:17657424,41218068:159236 x1,197:19112968,41218068 k1,198:19112968,41218068:0 ) (1,198:3729360,42004500:15383608,446948,141880 x1,198:4566885,42004500 g1,197:4730725,42004500 x1,197:5713747,42004500 g1,197:5877587,42004500 x1,197:6387450,42004500 g1,197:6551290,42004500 x1,197:7352129,42004500 g1,197:7515969,42004500 x1,197:8680529,42004500 x1,197:9918487,42004500 g1,197:10082327,42004500 x1,197:10628239,42004500 g1,197:10792079,42004500 x1,197:11739053,42004500 g1,197:11902893,42004500 x1,197:12230573,42004500 x1,197:13655325,42004500 g1,197:13819165,42004500 x1,197:14365069,42004500 g1,197:14528909,42004500 x1,197:15293700,42004500 x1,197:17040868,42004500 g1,197:17204708,42004500 x1,197:18151044,42004500 x1,197:18468900,42004500 x1,197:18913892,42004500 k1,198:19112968,42004500:199076 g1,198:19112968,42004500 ) (1,242:3729360,42790932:15383608,446948,141880 h1,241:3729360,42790932:655361,0,0 x1,241:4930633,42790932 k1,241:5133063,42790932:202430 x1,241:6171142,42790932 k1,241:6373572,42790932:202430 x1,241:7028932,42790932 h1,241:7028932,42790932:0,0,0 x1,241:7192772,42790932 k1,241:7404849,42790932:212077 x1,241:8205688,42790932 k1,241:8408118,42790932:202430 x1,241:8735798,42790932 x1,241:10214942,42790932 x1,241:11088534,42790932 x1,241:11416214,42790932 x1,241:12677126,42790932 k1,241:12879556,42790932:202430 x1,241:14299051,42790932 k1,241:14501481,42790932:202430 x1,241:15520544,42790932 k1,241:15722974,42790932:202430 x1,241:16669310,42790932 k1,241:16871740,42790932:202430 x1,241:17854763,42790932 k1,241:18057193,42790932:202430 x1,241:18894736,42790932 x1,241:19112968,42790932 k1,242:19112968,42790932:0 ) (1,242:3729360,43577364:15383608,446948,92400 x1,242:4749086,43577364 k1,241:5000468,43577364:251382 x1,241:5800652,43577364 k1,241:6052033,43577364:251381 x1,241:6962321,43577364 x1,241:7746785,43577364 k1,241:7998167,43577364:251382 x1,241:8762311,43577364 k1,241:9013692,43577364:251381 x1,241:9304668,43577364 k1,241:9556050,43577364:251382 x1,241:10647865,43577364 x1,241:11739681,43577364 x1,241:13050383,43577364 k1,241:13301764,43577364:251381 x1,241:14466978,43577364 k1,241:14718360,43577364:251382 x1,241:15009336,43577364 k1,241:15260718,43577364:251382 x1,241:15916078,43577364 k1,241:16167459,43577364:251381 x1,241:16750067,43577364 k1,241:17001449,43577364:251382 x1,241:18366543,43577364 k1,241:18639809,43577364:273266 x1,241:19112968,43577364 k1,241:19112968,43577364:0 ) (1,242:3729360,44363796:15383608,446948,92400 x1,242:4057040,44363796 k1,241:4271087,44363796:214047 x1,241:5508399,44363796 x1,241:5587702,44363796 k1,241:5801749,44363796:214047 x1,241:6966963,44363796 k1,241:7181009,44363796:214046 x1,241:8000209,44363796 k1,241:8214256,44363796:214047 x1,241:9707150,44363796 x1,241:11131902,44363796 k1,241:11485683,44363796:353781 x1,241:12504763,44363796 k1,241:12718810,44363796:214047 x1,241:13665146,44363796 x1,241:14175009,44363796 x1,241:15449663,44363796 k1,241:15663709,44363796:214046 x1,241:16427853,44363796 k1,241:16641900,44363796:214047 x1,241:17588874,44363796 k1,241:17802921,44363796:214047 x1,241:18894736,44363796 x1,241:19112968,44363796 k1,242:19112968,44363796:0 ) (1,242:3729360,45150228:15383608,446948,141880 x1,242:4821176,45150228 x1,242:6131878,45150228 k1,241:6351405,45150228:219527 x1,241:7151589,45150228 k1,241:7371116,45150228:219527 x1,241:7662092,45150228 x1,241:7979948,45150228 x1,241:9253946,45150228 k1,241:9473473,45150228:219527 x1,241:10274312,45150228 k1,241:10493839,45150228:219527 x1,241:11840582,45150228 k1,241:12060110,45150228:219528 x1,241:12606014,45150228 k1,241:12825541,45150228:219527 x1,241:14208988,45150228 k1,241:14428515,45150228:219527 x1,241:15192659,45150228 k1,241:15412186,45150228:219527 x1,241:16213025,45150228 k1,241:16432552,45150228:219527 x1,241:16760232,45150228 x1,241:18239376,45150228 x1,241:19112968,45150228 k1,242:19112968,45150228:0 ) (1,242:3729360,45936660:15383608,446948,141880 x1,242:4057040,45936660 x1,241:5317952,45936660 k1,241:5467891,45936660:149939 x1,241:6195331,45936660 x1,241:7760323,45936660 x1,241:8078179,45936660 k1,241:8228118,45936660:149939 x1,241:9319933,45936660 x1,241:10411749,45936660 x1,241:11722451,45936660 k1,241:11872389,45936660:149938 x1,241:13546821,45936660 k1,241:13696760,45936660:149939 x1,241:15152304,45936660 x1,241:15989829,45936660 k1,241:16139768,45936660:149939 x1,241:17122790,45936660 k1,241:17272729,45936660:149939 x1,241:18146321,45936660 x1,241:19112968,45936660 k1,241:19112968,45936660:0 ) (1,242:3729360,46723092:15383608,446948,141880 x1,242:4166471,46723092 k1,241:4338175,46723092:171704 x1,241:5248463,46723092 x1,241:6032927,46723092 k1,241:6204631,46723092:171704 x1,241:6714494,46723092 k1,241:6886198,46723092:171704 x1,241:7924277,46723092 k1,241:8095981,46723092:171704 x1,241:8423661,46723092 h1,241:8423661,46723092:0,0,0 x1,241:8587501,46723092 k1,241:8814254,46723092:226753 x1,241:9833334,46723092 k1,241:10005038,46723092:171704 $1,241:10005038,46723092x1,241:10580453,46723092k1,241:10777056,46723092:196603x1,241:11286782,46723092k1,241:11483385,46723092:196603x1,241:11811066,46723092$1,241:11811066,46723092 k1,241:11982770,46723092:171704 x1,241:12310450,46723092 x1,241:12591602,46723092 x1,241:13174200,46723092 x1,241:13938335,46723092 k1,241:14110039,46723092:171704 x1,241:15565583,46723092 x1,241:16403108,46723092 k1,241:16574812,46723092:171704 x1,241:17557834,46723092 k1,241:17729538,46723092:171704 x1,241:19112968,46723092 k1,241:19112968,46723092:0 ) (1,242:3729360,47509524:15383608,446948,141880 x1,242:4602952,47509524 x1,242:5059736,47509524 x1,242:5897279,47509524 k1,241:6136165,47509524:238886 x1,241:6609324,47509524 k1,241:6848210,47509524:238886 x1,241:7175890,47509524 x1,241:8436802,47509524 k1,241:8675688,47509524:238886 x1,241:9331048,47509524 k1,241:9569934,47509524:238886 x1,241:10516270,47509524 k1,241:10755156,47509524:238886 x1,241:12247396,47509524 k1,241:12486283,47509524:238887 x1,241:14306211,47509524 k1,241:14545097,47509524:238886 x1,241:14872777,47509524 x1,241:16133689,47509524 k1,241:16372575,47509524:238886 x1,241:17191775,47509524 k1,241:17620073,47509524:428298 x1,241:18894736,47509524 x1,241:19112968,47509524 k1,242:19112968,47509524:0 ) (1,242:3729360,48295956:15383608,446948,141880 x1,242:4348016,48295956 x1,242:5403791,48295956 k1,241:5590431,48295956:186640 x1,241:6755645,48295956 k1,241:6942284,48295956:186639 x1,241:7980363,48295956 k1,241:8167003,48295956:186640 x1,241:8822363,48295956 h1,241:8822363,48295956:0,0,0 x1,241:8986203,48295956 k1,241:9178542,48295956:192339 x1,241:9979381,48295956 k1,241:10166021,48295956:186640 x1,241:10493701,48295956 x1,241:10774853,48295956 x1,241:11357451,48295956 x1,241:12121586,48295956 k1,241:12308225,48295956:186639 x1,241:13473430,48295956 x1,241:14856221,48295956 k1,241:15042861,48295956:186640 x1,241:15916453,48295956 x1,241:16883100,48295956 k1,241:17069739,48295956:186639 x1,241:18453169,48295956 k1,241:18639809,48295956:186640 x1,241:19112968,48295956 k1,241:19112968,48295956:0 ) ] g1,246:19112968,48295956 ) k1,246:19579138,48295956:466170 r1,246:19579138,48295956:0,44938783,0 k1,246:20045308,48295956:466170 (1,246:20045308,48295956:15383608,44938783,0 [1,246:20045308,48295956:15383608,44938783,0 [1,205:20045308,41572913:15383608,38215740,0 (1,204:20045308,14781977:15383608,11424804,0 k1,204:20159061,14781977:113753 h1,201:20159061,14781977:0,0,0 (1,201:20159061,14781977:15156103,11424804,0 (1,201:20159061,14781977:15156944,11424833,0 (1,201:20159061,14781977:15156944,11424833,0 (1,201:20159061,14781977:0,9867264,0 (1,201:20159061,14781977:13090570,9867264,0 ) k1,201:20159061,14781977:-13090570 ) ) ) ) k1,201:35428916,14781977:113752 ) (1,204:20045308,26272317:15383608,11424804,0 k1,204:20159061,26272317:113753 (1,202:20159061,26272317:15156103,11424804,0 (1,202:20159061,26272317:15156944,11424833,0 (1,202:20159061,26272317:15156944,11424833,0 (1,202:20159061,26272317:0,9867264,0 (1,202:20159061,26272317:13090570,9867264,0 ) k1,202:20159061,26272317:-13090570 ) ) ) ) k1,202:35428916,26272317:113752 ) (1,204:20045308,37762657:15383608,11424804,0 k1,204:20159061,37762657:113753 (1,203:20159061,37762657:15156103,11424804,0 (1,203:20159061,37762657:15156944,11424833,0 (1,203:20159061,37762657:15156944,11424833,0 (1,203:20159061,37762657:0,9867264,0 (1,203:20159061,37762657:13090570,9867264,0 ) k1,203:20159061,37762657:-13090570 ) ) ) ) g1,204:35315164,37762657 k1,204:35428916,37762657:113752 ) (1,204:20045308,39204449:15383608,446948,141880 h1,204:20045308,39204449:0,0,0 x1,204:21756435,39204449 k1,204:22049275,39204449:292840 x1,204:22886818,39204449 k1,204:23347978,39204449:461160 k1,204:23347978,39204449:0 x1,204:25277346,39204449 x1,204:26150938,39204449 x1,204:26478618,39204449 x1,204:27739530,39204449 k1,204:28032370,39204449:292840 x1,204:29451865,39204449 k1,204:29744705,39204449:292840 x1,204:30763768,39204449 k1,204:31056608,39204449:292840 x1,204:32002944,39204449 k1,204:32295784,39204449:292840 x1,204:33278807,39204449 k1,204:33571647,39204449:292840 x1,204:34409190,39204449 x1,204:35428916,39204449 k1,204:35428916,39204449:0 ) (1,204:20045308,39990881:15383608,446948,141880 x1,204:20518467,39990881 k1,204:20815145,39990881:296678 x1,204:21470505,39990881 k1,204:21767182,39990881:296677 x1,204:22349790,39990881 k1,204:22646468,39990881:296678 x1,204:23811028,39990881 x1,204:24648571,39990881 k1,204:24945249,39990881:296678 x1,204:25891585,39990881 k1,204:26188262,39990881:296677 x1,204:27007462,39990881 k1,204:27304140,39990881:296678 x1,204:28541452,39990881 x1,204:28620755,39990881 k1,204:28917432,39990881:296677 x1,204:29317856,39990881 x1,204:30102974,39990881 k1,204:30571810,39990881:468836 x1,204:30899490,39990881 x1,204:32160402,39990881 k1,204:32457079,39990881:296677 x1,204:33075735,39990881 x1,204:34313038,39990881 k1,204:34609716,39990881:296678 x1,204:35428916,39990881 k1,204:35428916,39990881:0 ) (1,204:20045308,40777313:15383608,446948,9168 x1,204:21538202,40777313 x1,204:22962954,40777313 k1,204:23328889,40777313:365935 x1,204:24384665,40777313 x1,204:25258239,40777313 k1,204:25569916,40777313:311677 x1,204:25897596,40777313 x1,204:27158508,40777313 k1,204:27376607,40777313:218099 x1,204:27995263,40777313 x1,204:29232566,40777313 k1,204:29450664,40777313:218098 x1,204:29778344,40777313 k1,204:29996443,40777313:218099 x1,204:31489337,40777313 x1,204:32914089,40777313 k1,204:33280024,40777313:365935 x1,204:34227007,40777313 x1,204:35428916,40777313 k1,204:35428916,40777313:0 ) (1,204:20045308,41563745:15383608,446948,9168 x1,204:20372988,41563745 x1,204:21633900,41563745 g1,204:21797740,41563745 x1,204:22416396,41563745 x1,204:23653699,41563745 g1,204:23817539,41563745 x1,204:24636739,41563745 g1,204:24800579,41563745 x1,204:26293473,41563745 x1,204:27718225,41563745 k1,204:35428916,41563745:7710691 g1,204:35428916,41563745 ) ] (1,242:20045308,43639165:15383608,446948,141880 x1,242:20372988,43639165 x1,241:21633900,43639165 k1,241:21800070,43639165:166170 x1,241:22455430,43639165 k1,241:22621599,43639165:166169 x1,241:23567935,43639165 k1,241:23734105,43639165:166170 x1,241:25698849,43639165 k1,241:25865018,43639165:166169 x1,241:26155994,43639165 k1,241:26322164,43639165:166170 x1,241:28214818,43639165 k1,241:28380988,43639165:166170 x1,241:30200916,43639165 k1,241:30367085,43639165:166169 x1,241:30694765,43639165 x1,241:31955677,43639165 k1,241:32121847,43639165:166170 x1,241:32941047,43639165 k1,241:33151195,43639165:210148 x1,241:34461907,43639165 k1,241:34628077,43639165:166170 x1,241:35428916,43639165 k1,241:35428916,43639165:0 ) (1,242:20045308,44425597:15383608,446948,141880 x1,242:20700668,44425597 x1,242:21428755,44425597 k1,241:21593013,44425597:164258 x1,241:22539349,44425597 k1,241:22703607,44425597:164258 x1,241:23031287,44425597 x1,241:23322263,44425597 x1,241:23633559,44425597 x1,241:24096894,44425597 x1,241:24752254,44425597 x1,241:26062956,44425597 k1,241:26227215,44425597:164259 x1,241:27683414,44425597 k1,241:27847672,44425597:164258 x1,241:28648511,44425597 k1,241:28812769,44425597:164258 x1,241:30014023,44425597 k1,241:30178281,44425597:164258 x1,241:30724193,44425597 k1,241:30888452,44425597:164259 x1,241:31689291,44425597 k1,241:31853549,44425597:164258 x1,241:32181229,44425597 x1,241:32462381,44425597 x1,241:33044979,44425597 x1,241:33809114,44425597 k1,241:33973372,44425597:164258 x1,241:35428916,44425597 k1,242:35428916,44425597:0 ) (1,242:20045308,45212029:15383608,446948,141880 x1,242:20882833,45212029 k1,241:21052296,45212029:169463 x1,241:22035318,45212029 k1,241:22204780,45212029:169462 x1,241:22641891,45212029 k1,241:22811354,45212029:169463 x1,241:23139034,45212029 x1,241:23966098,45212029 k1,241:24135560,45212029:169462 x1,241:25082534,45212029 x1,241:25264717,45212029 x1,241:25956108,45212029 k1,241:26125571,45212029:169463 x1,241:26635434,45212029 k1,241:26804896,45212029:169462 x1,241:27605735,45212029 k1,241:27775198,45212029:169463 x1,241:28430558,45212029 x1,241:29158645,45212029 k1,241:29328107,45212029:169462 x1,241:30274443,45212029 k1,241:30443906,45212029:169463 x1,241:30771586,45212029 x1,241:31062562,45212029 x1,241:31373858,45212029 x1,241:31837193,45212029 x1,241:32492553,45212029 x1,241:33803255,45212029 k1,241:33972717,45212029:169462 x1,241:35428916,45212029 k1,241:35428916,45212029:0 ) (1,242:20045308,45998461:15383608,446948,141880 x1,242:20846147,45998461 g1,241:21009987,45998461 x1,241:22211241,45998461 g1,241:22375081,45998461 x1,241:22920993,45998461 g1,241:23084833,45998461 x1,241:23412513,45998461 x1,241:23693665,45998461 x1,241:24276263,45998461 x1,241:25040398,45998461 g1,241:25204238,45998461 x1,241:26369443,45998461 x1,241:27752234,45998461 g1,241:27916074,45998461 x1,241:28789666,45998461 x1,241:29920153,45998461 k1,242:35428916,45998461:5508763 g1,242:35428916,45998461 ) (1,244:20045308,46834979:15383608,446948,141880 h1,243:20045308,46834979:655361,0,0 x1,243:22192908,46834979 g1,243:22356748,46834979 x1,243:22684428,46834979 x1,243:22968851,46834979 x1,243:23286707,46834979 g1,243:23450547,46834979 x1,243:25197723,46834979 x1,243:26746969,46834979 g1,243:26910809,46834979 x1,243:27893176,46834979 k1,244:35428916,46834979:7535740 g1,244:35428916,46834979 ) (1,246:21356030,48295956:14072886,446948,141880 (1,245:21356030,48295956:0,291271,0 g1,245:21356030,48295956 g1,245:20045308,48295956 g1,245:19717628,48295956 (1,245:19717628,48295956:1310722,291271,0 k1,245:21028350,48295956:1310722 (1,245:21028350,48295956:0,291271,0 k1,245:20700669,48295956:-327681 x1,245:21028350,48295956 ) ) g1,245:21356030,48295956 ) x1,245:22375110,48295956 k1,245:22639279,48295956:264169 x1,245:24240958,48295956 x1,245:25078501,48295956 k1,245:25342670,48295956:264169 x1,245:26106814,48295956 k1,245:26370983,48295956:264169 x1,245:27790478,48295956 k1,245:28054647,48295956:264169 x1,245:29073710,48295956 k1,245:29337878,48295956:264168 x1,245:30211470,48295956 x1,245:31178117,48295956 k1,245:31442286,48295956:264169 x1,245:32898485,48295956 k1,245:33162654,48295956:264169 x1,245:33963493,48295956 k1,245:34227662,48295956:264169 x1,245:35428916,48295956 k1,245:35428916,48295956:0 ) ] g1,246:35428916,48295956 ) ) ] h1,246:3729360,50262036:31699556,0,0 ] ] !44828 6 !10 7 [1,311:1,50262037:35428915,50262036,0 [1,311:3729360,50262037:31699556,49329696,0 [1,311:3729360,1718773:31699556,786432,0 h1,311:3729360,1718773:31699556,0,0 ] [1,311:3729360,48295957:31699556,44938784,0 (1,311:3729360,30916806:31699556,27559633,0 (1,311:3729360,30916806:15383608,27559633,0 [1,283:3729360,30916806:15383608,27559633,0 [1,211:3729360,16482492:15383608,13125319,0 (1,210:3729360,14161868:15383608,10804695,0 k1,210:3843113,14161868:113753 h1,209:3843113,14161868:0,0,0 (1,209:3843113,14161868:15156103,10804695,0 (1,209:3843113,14161868:15155617,10804722,0 (1,209:3843113,14161868:15155617,10804722,0 (1,209:3843113,14161868:0,39205929,0 (1,209:3843113,14161868:54993551,39205929,0 ) k1,209:3843113,14161868:-54993551 ) ) ) ) g1,210:18999216,14161868 k1,210:19112968,14161868:113752 ) (1,210:3729360,15603660:15383608,446948,141880 h1,210:3729360,15603660:0,0,0 x1,210:5440487,15603660 k1,210:5648216,15603660:207729 x1,210:6485759,15603660 k1,210:6776697,15603660:290938 k1,210:6776697,15603660:0 x1,210:8706065,15603660 x1,210:9579657,15603660 x1,210:9907337,15603660 x1,210:11168249,15603660 k1,210:11375978,15603660:207729 x1,210:12795473,15603660 k1,210:13003202,15603660:207729 x1,210:14022265,15603660 k1,210:14229994,15603660:207729 x1,210:15176330,15603660 k1,210:15384059,15603660:207729 x1,210:16367082,15603660 k1,210:16574811,15603660:207729 x1,210:18432080,15603660 k1,210:18639809,15603660:207729 x1,210:19112968,15603660 k1,210:19112968,15603660:0 ) (1,210:3729360,16390092:15383608,446948,92400 x1,210:4384720,16390092 g1,210:4548560,16390092 x1,210:5295008,16390092 g1,210:5458848,16390092 x1,210:5786528,16390092 g1,210:5950368,16390092 x1,210:7187680,16390092 x1,210:7266983,16390092 g1,210:7430823,16390092 x1,210:8596037,16390092 g1,210:8759877,16390092 x1,210:9579077,16390092 g1,210:9742917,16390092 x1,210:11235811,16390092 x1,210:12660563,16390092 k1,210:19112968,16390092:6452405 g1,210:19112968,16390092 ) ] (1,246:5040082,18477517:14072886,446948,141880 x1,246:5477193,18477517 k1,245:5679097,18477517:201904 x1,245:6516640,18477517 k1,245:6718545,18477517:201905 x1,245:7046225,18477517 x1,245:7873289,18477517 k1,245:8075193,18477517:201904 x1,245:8948777,18477517 x1,245:9385888,18477517 x1,245:9750254,18477517 x1,245:10061550,18477517 x1,245:10342702,18477517 k1,245:10544606,18477517:201904 x1,245:11054469,18477517 k1,245:11256374,18477517:201905 x1,245:12057213,18477517 k1,245:12259117,18477517:201904 x1,245:13423685,18477517 x1,245:14879866,18477517 x1,245:15571912,18477517 k1,245:15773816,18477517:201904 x1,245:16319728,18477517 k1,245:16521633,18477517:201905 x1,245:17322472,18477517 k1,245:17524376,18477517:201904 x1,245:17852056,18477517 x1,245:19112968,18477517 k1,245:19112968,18477517:0 ) (1,246:5040082,19263949:14072886,446948,141880 x1,246:6204642,19263949 x1,246:7206025,19263949 k1,245:7423620,19263949:217595 x1,245:7896779,19263949 x1,245:9717362,19263949 k1,245:9924205,19263949:206843 x1,245:10725044,19263949 k1,245:10931888,19263949:206844 x1,245:12060407,19263949 x1,245:12242590,19263949 x1,245:13589996,19263949 k1,245:13796840,19263949:206844 x1,245:15216335,19263949 k1,245:15423179,19263949:206844 x1,245:16442242,19263949 k1,245:16649086,19263949:206844 x1,245:18105285,19263949 k1,245:18312129,19263949:206844 x1,245:19112968,19263949 k1,245:19112968,19263949:0 ) (1,246:5040082,20050381:14072886,446948,141880 x1,246:6241336,20050381 k1,245:6425281,20050381:183945 x1,245:6862392,20050381 k1,245:7046337,20050381:183945 x1,245:7737728,20050381 x1,245:8562824,20050381 k1,245:8746769,20050381:183945 x1,245:9510913,20050381 k1,245:9694858,20050381:183945 x1,245:9985834,20050381 k1,245:10169780,20050381:183946 x1,245:12099121,20050381 k1,245:12283066,20050381:183945 x1,245:13447634,20050381 x1,245:14903815,20050381 x1,245:15595861,20050381 x1,245:15717109,20050381 k1,245:15980584,20050381:263475 x1,245:17145799,20050381 k1,245:17329744,20050381:183945 x1,245:17948400,20050381 x1,245:18894736,20050381 x1,245:19112968,20050381 k1,246:19112968,20050381:0 ) (1,246:5040082,20836813:14072886,446948,141880 x1,246:6568370,20836813 k1,245:6754256,20836813:185886 x1,245:7409616,20836813 k1,245:7595501,20836813:185885 x1,245:8396340,20836813 k1,245:8582226,20836813:185886 x1,245:9746794,20836813 x1,245:11202975,20836813 x1,245:11895021,20836813 k1,245:12080906,20836813:185885 x1,245:12518017,20836813 k1,245:12703903,20836813:185886 x1,245:13213766,20836813 k1,245:13399651,20836813:185885 x1,245:14345987,20836813 x1,245:15474489,20836813 k1,245:15660375,20836813:185886 x1,245:16170238,20836813 k1,245:16356123,20836813:185885 x1,245:17156962,20836813 k1,245:17342848,20836813:185886 x1,245:17961504,20836813 x1,245:18580160,20836813 x1,245:18894736,20836813 x1,245:19112968,20836813 k1,246:19112968,20836813:0 ) (1,246:5040082,21623245:14072886,446948,141880 x1,246:5768177,21623245 k1,245:5968631,21623245:200454 x1,245:6514543,21623245 k1,245:6714996,21623245:200453 x1,245:7515835,21623245 k1,245:7716289,21623245:200454 x1,245:9171833,21623245 x1,245:10009358,21623245 k1,245:10209812,21623245:200454 x1,245:11611602,21623245 k1,245:11821208,21623245:209606 x1,245:13422895,21623245 k1,245:13623349,21623245:200454 x1,245:14423533,21623245 k1,245:14623987,21623245:200454 x1,245:15934689,21623245 k1,245:16135142,21623245:200453 x1,245:17008726,21623245 x1,245:17445837,21623245 x1,245:17810203,21623245 x1,245:18121499,21623245 x1,245:18402651,21623245 k1,245:18603105,21623245:200454 x1,245:19112968,21623245 k1,245:19112968,21623245:0 ) (1,246:5040082,22409677:14072886,446948,141880 x1,246:6204650,22409677 x1,246:7660831,22409677 x1,246:8352877,22409677 x1,246:8474125,22409677 k1,246:19112968,22409677:10638843 g1,246:19112968,22409677 ) (1,247:5040082,23778288:14072886,446948,141880 (1,246:5040082,23778288:0,291271,0 g1,246:5040082,23778288 g1,246:3729360,23778288 g1,246:3401680,23778288 (1,246:3401680,23778288:1310722,291271,0 k1,246:4712402,23778288:1310722 (1,246:4712402,23778288:0,291271,0 k1,246:4384721,23778288:-327681 x1,246:4712402,23778288 ) ) g1,246:5040082,23778288 ) x1,246:6569017,23778288 k1,246:6733955,23778288:164938 x1,246:7753018,23778288 k1,246:7917955,23778288:164937 x1,246:8791547,23778288 x1,246:9758194,23778288 k1,246:9923132,23778288:164938 x1,246:11379331,23778288 k1,246:11544268,23778288:164937 x1,246:12345107,23778288 k1,246:12510045,23778288:164938 x1,246:13711299,23778288 k1,246:13876237,23778288:164938 x1,246:14313348,23778288 k1,246:14478285,23778288:164937 x1,246:15534052,23778288 k1,246:15698990,23778288:164938 x1,246:16536533,23778288 k1,246:16701470,23778288:164937 x1,246:17029150,23778288 x1,246:17856214,23778288 k1,246:18021152,23778288:164938 x1,246:18894736,23778288 x1,246:19112968,23778288 k1,247:19112968,23778288:0 ) (1,247:5040082,24564720:14072886,446948,141880 x1,247:5477193,24564720 x1,247:5841559,24564720 x1,247:6152855,24564720 x1,247:6434007,24564720 g1,246:6597847,24564720 x1,246:7107710,24564720 g1,246:7271550,24564720 x1,246:8072389,24564720 g1,246:8236229,24564720 x1,246:8928275,24564720 x1,246:9765800,24564720 g1,246:9929640,24564720 x1,246:11349135,24564720 g1,246:11512975,24564720 x1,246:12532038,24564720 g1,246:12695878,24564720 x1,246:13241790,24564720 g1,246:13405630,24564720 x1,246:14206469,24564720 g1,246:14370309,24564720 x1,246:15498173,24564720 x1,246:16735485,24564720 g1,246:16899325,24564720 x1,246:17227005,24564720 x1,246:18188411,24564720 k1,247:19112968,24564720:924557 g1,247:19112968,24564720 ) (1,254:3729360,26138620:15383608,541848,14544 k1,254:5137229,26138620:1407869 (1,254:5137229,26138620:0,0,0 g1,254:5137229,26138620 ) x1,254:7539762,26138620 x1,254:10977221,26138620 g1,254:11173829,26138620 x1,254:13402555,26138620 g1,254:13599163,26138620 x1,254:14734765,26138620 x1,254:15811385,26138620 x1,254:17705099,26138620 k1,254:19112968,26138620:1407869 ) (1,254:3729360,27056124:15383608,541848,14544 k1,254:5356651,27056124:1627291 x1,254:8459092,27056124 x1,254:9870723,27056124 g1,254:10067331,27056124 x1,254:11770734,27056124 g1,254:11967342,27056124 x1,254:15113040,27056124 x1,254:17485678,27056124 g1,254:17485678,27056124 k1,254:19112968,27056124:1627290 ) (1,280:3729360,28296037:15383608,538305,162393 (1,280:3729360,28296037:0,0,0 g1,280:3729360,28296037 ) x1,280:6350503,28296037 g1,280:6547111,28296037 x1,280:9474158,28296037 k1,280:19112968,28296037:9638810 g1,280:19112968,28296037 ) (1,282:3729360,29343942:15383608,446948,141880 h1,281:3729360,29343942:655361,0,0 x1,281:5403801,29343942 k1,281:5625368,29343942:221567 x1,281:6244024,29343942 x1,281:6535000,29343942 x1,281:7700205,29343942 k1,281:7921772,29343942:221567 x1,281:8722611,29343942 x1,281:9596203,29343942 k1,281:9817771,29343942:221568 x1,281:10582562,29343942 x1,281:12329730,29343942 k1,281:12551297,29343942:221567 x1,281:13061160,29343942 k1,281:13282727,29343942:221567 x1,281:13500959,29343942 (1,281:13500959,29343942:655360,446948,14416 x1,281:14156319,29343942 ) x1,281:14374551,29343942 k1,281:14596118,29343942:221567 x1,281:15069286,29343942 x1,281:15608637,29343942 k1,281:15830205,29343942:221568 x1,281:17031469,29343942 k1,281:17253036,29343942:221567 x1,281:17762899,29343942 k1,281:17984466,29343942:221567 x1,281:18530370,29343942 x1,281:18894736,29343942 x1,281:19112968,29343942 k1,282:19112968,29343942:0 ) (1,282:3729360,30130374:15383608,446948,141880 x1,282:5003358,30130374 k1,281:5133352,30130374:129994 x1,281:5934191,30130374 k1,281:6064186,30130374:129995 x1,281:7519730,30130374 x1,281:8357255,30130374 k1,281:8487249,30130374:129994 x1,281:8997112,30130374 x1,281:9725199,30130374 x1,281:10417245,30130374 x1,281:11109291,30130374 k1,281:11239285,30130374:129994 x1,281:12840964,30130374 x1,281:13678507,30130374 k1,281:13808502,30130374:129995 x1,281:14318365,30130374 k1,281:14448359,30130374:129994 x1,281:15249198,30130374 k1,281:15379193,30130374:129995 x1,281:16725936,30130374 x1,281:17563479,30130374 k1,281:17693473,30130374:129994 x1,281:19112968,30130374 k1,281:19112968,30130374:0 ) (1,282:3729360,30916806:15383608,446948,141880 x1,282:4348016,30916806 x1,282:5549288,30916806 x1,282:5731471,30916806 x1,282:6532310,30916806 x1,282:7715876,30916806 k1,281:8131323,30916806:415447 x1,281:8677235,30916806 k1,281:8911837,30916806:234602 x1,281:9858811,30916806 k1,281:10093413,30916806:234602 x1,281:11895635,30916806 k1,281:12147927,30916806:252292 x1,281:12948766,30916806 k1,281:13183369,30916806:234603 x1,281:14238480,30916806 x1,281:15294255,30916806 k1,281:15528857,30916806:234602 x1,281:16985056,30916806 k1,281:17219658,30916806:234602 x1,281:18165994,30916806 x1,281:19112968,30916806 k1,281:19112968,30916806:0 ) ] g1,311:19112968,30916806 ) k1,311:19579138,30916806:466170 r1,311:19579138,30916806:0,27559633,0 k1,311:20045308,30916806:466170 (1,311:20045308,30916806:15383608,27559633,0 [1,311:20045308,30916806:15383608,27559633,0 (1,282:20045308,4012533:15383608,446948,141880 x1,282:20663964,4012533 x1,282:21391404,4012533 x1,282:22957052,4012533 x1,282:23794595,4012533 g1,281:23958435,4012533 x1,281:24468298,4012533 g1,281:24632138,4012533 x1,281:25432977,4012533 g1,281:25596817,4012533 x1,281:26106680,4012533 x1,281:26834767,4012533 x1,281:27526813,4012533 x1,281:28218859,4012533 g1,281:28382699,4012533 x1,281:29984378,4012533 x1,281:30821921,4012533 g1,281:30985761,4012533 x1,281:31422872,4012533 g1,281:31586712,4012533 x1,281:32096575,4012533 x1,281:32407871,4012533 x1,281:33016703,4012533 g1,281:33180543,4012533 x1,281:33835903,4012533 k1,282:35428916,4012533:1593013 g1,282:35428916,4012533 ) (1,283:20045308,4798965:15383608,0,0 h1,283:20045308,4798965:655361,0,0 k1,283:35428916,4798965:14728247 g1,283:35428916,4798965 ) (1,287:24239379,5730200:11189537,865699,709964 (1,287:24239379,5730200:6995466,865699,709964 x1,287:24578210,5730200 (1,287:24578210,5828503:941864,318577,89201 x1,287:25487306,5828503 ) g1,287:25702114,5730200 x1,287:26211840,5730200 g1,287:26393880,5730200 (1,287:26393880,5730200:4658920,865699,709964 (1,287:26393880,5730200:4658920,865699,709964 h1,287:26393880,5730200:78643,0,0 [1,287:26472523,5730200:4501634,865699,709964 (1,287:26472523,5286844:4501634,422343,127431 k1,287:27864690,5286844:1392167 x1,287:28531671,5286844 x1,287:28891815,5286844 (1,287:28891815,5385147:280122,197518,0 x1,287:29139169,5385147 ) x1,287:29581990,5286844 k1,287:30974157,5286844:1392167 ) (1,287:26472523,6252659:4501634,594554,187504 (1,287:26472523,6252659:873816,594554,87018 (1,287:26472523,5710531:546135,26213,629146 x1,287:27018658,5710531 ) [1,287:27018658,6252659:327681,594554,0 (1,287:27018658,6252659:327681,422343,0 x1,287:27346339,6252659 ) ] ) x1,287:27902940,6252659 x1,287:28763585,6252659 (1,287:28763585,6350962:281419,197518,0 x1,287:29012236,6350962 ) g1,287:29154228,6252659 (1,287:29154228,6252659:1195429,491520,187504 (1,287:29154228,6252659:254863,491520,163840 x1,287:29409091,6252659 ) x1,287:29779779,6252659 (1,287:29779779,6350962:315015,197518,89201 x1,287:30045566,6350962 ) (1,287:30094794,6252659:254863,491520,163840 x1,287:30349657,6252659 ) ) g1,287:30458881,6252659 x1,287:30927281,6252659 ) ] h1,287:30974157,5730200:78643,0,0 ) ) x1,287:31234845,5730200 ) (1,287:34664772,5730200:764144,446948,116320 (1,287:34664772,5730200:764144,446948,116320 x1,287:35428916,5730200 ) ) ) (1,290:20045308,7297468:15383608,447828,141880 x1,290:20555171,7297468 k1,288:20725819,7297468:170648 x1,288:22327506,7297468 k1,288:22498155,7297468:170649 $1,288:22498155,7297468x1,288:22821891,7297468(1,288:22821891,7395771:280122,197518,0x1,288:23069245,7395771)$1,288:23102013,7297468 k1,288:23272661,7297468:170648 x1,288:23709772,7297468 k1,288:23880421,7297468:170649 x1,288:24681260,7297468 k1,288:24851908,7297468:170648 x1,288:27982539,7297468 k1,288:28153188,7297468:170649 x1,288:29445547,7297468 k1,288:29617898,7297468:172351 $1,288:29617898,7297468x1,288:30174499,7297468$1,288:30221375,7297468 k1,288:30392023,7297468:170648 x1,288:30829134,7297468 k1,288:30999783,7297468:170649 x1,288:31800622,7297468 k1,288:31971270,7297468:170648 x1,288:35428916,7297468 k1,288:35428916,7297468:0 ) (1,290:20045308,8083900:15383608,447828,141880 x1,290:20263540,8083900 x1,288:21567034,8083900 k1,288:21733987,8083900:166953 x1,288:22279899,8083900 k1,288:22446851,8083900:166952 x1,288:23247690,8083900 k1,288:23414643,8083900:166953 x1,288:24870842,8083900 k1,288:25037794,8083900:166952 x1,288:26931104,8083900 k1,288:27098057,8083900:166953 x1,288:28044393,8083900 k1,288:28211345,8083900:166952 x1,288:28721208,8083900 k1,288:28888161,8083900:166953 x1,288:29689000,8083900 k1,288:29855953,8083900:166953 x1,288:31603128,8083900 x1,288:32057944,8083900 k1,288:32225674,8083900:167730 x1,288:33172010,8083900 k1,288:33338963,8083900:166953 $1,288:33338963,8083900x1,288:33857061,8083900$1,288:33857061,8083900 k1,288:34024013,8083900:166952 x1,288:34461124,8083900 k1,288:34628077,8083900:166953 x1,288:35428916,8083900 k1,288:35428916,8083900:0 ) (1,290:20045308,8870332:15383608,446948,141880 x1,290:21392706,8870332 k1,288:21605601,8870332:212895 x1,288:23426166,8870332 k1,288:23639061,8870332:212895 x1,288:23857293,8870332 x1,288:25160787,8870332 k1,288:25373682,8870332:212895 x1,288:28067168,8870332 k1,288:28280063,8870332:212895 x1,288:29445277,8870332 k1,288:29658171,8870332:212894 x1,288:30459010,8870332 k1,288:30671905,8870332:212895 x1,288:32382377,8870332 k1,288:32595272,8870332:212895 x1,288:33105135,8870332 k1,288:33318030,8870332:212895 x1,288:35428916,8870332 k1,288:35428916,8870332:0 ) (1,290:20045308,9656764:15383608,446948,141880 x1,290:22776143,9656764 k1,288:22932442,9656764:156299 x1,288:26116801,9656764 k1,288:26317448,9656764:200647 $1,288:26317448,9656764x1,288:26727501,9656764$1,288:26727501,9656764 k1,288:26883800,9656764:156299 x1,288:27320911,9656764 k1,288:27477210,9656764:156299 x1,288:28278049,9656764 k1,288:28434348,9656764:156299 x1,288:29926588,9656764 k1,288:30082887,9656764:156299 x1,288:31502382,9656764 k1,288:31658681,9656764:156299 x1,288:32204593,9656764 k1,289:32360892,9656764:156299 x1,289:33161731,9656764 k1,289:33318030,9656764:156299 x1,289:35428916,9656764 k1,289:35428916,9656764:0 ) (1,290:20045308,10443196:15383608,446948,7531 x1,290:22776143,10443196 g1,289:22939983,10443196 x1,289:23449846,10443196 g1,289:23613686,10443196 x1,289:24414525,10443196 g1,289:24578365,10443196 x1,289:25743580,10443196 g1,289:25907420,10443196 x1,289:26453332,10443196 g1,289:26617172,10443196 x1,289:27418011,10443196 g1,289:27581851,10443196 x1,289:27909531,10443196 x1,289:29352626,10443196 k1,290:35428916,10443196:6076290 g1,290:35428916,10443196 ) (1,293:26023670,11802861:9405246,725524,449545 (1,293:26023670,11802861:3426885,725524,449545 x1,293:26433723,11802861 g1,293:26615763,11802861 x1,293:27125489,11802861 g1,293:27307529,11802861 (1,293:27307529,11802861:1960981,725524,449545 (1,293:27307529,11802861:1960981,725524,449545 h1,293:27307529,11802861:78643,0,0 [1,293:27386172,11802861:1803695,725524,449545 (1,293:27386172,11359505:1803695,282168,187504 x1,293:27794103,11359505 (1,293:27794103,11457808:719314,197518,89201 x1,293:28464189,11457808 ) x1,293:28887898,11359505 (1,293:28887898,11457808:301969,197518,0 x1,293:29138288,11457808 ) ) (1,293:27386172,12252405:1803695,282168,0 k1,293:28146214,12252405:760042 x1,293:28429825,12252405 k1,293:29189867,12252405:760042 ) ] h1,293:29189867,11802861:78643,0,0 ) ) x1,293:29450555,11802861 ) (1,293:34664772,11802861:764144,446948,116320 (1,293:34664772,11802861:764144,446948,116320 x1,293:35428916,11802861 ) ) ) (1,295:20045308,13338124:15383608,446948,187504 x1,295:21646340,13338124 k1,294:21828615,13338124:182275 $1,294:21828615,13338124x1,294:22203096,13338124(1,294:22203096,13436427:301969,197518,0x1,294:22453486,13436427)$1,294:22505065,13338124 k1,294:22687340,13338124:182275 x1,294:23124451,13338124 k1,294:23306725,13338124:182274 x1,294:24107564,13338124 k1,294:24289839,13338124:182275 x1,294:24617519,13338124 x1,294:25878431,13338124 k1,294:26060706,13338124:182275 x1,294:27862928,13338124 k1,294:28049811,13338124:186883 x1,294:28996147,13338124 k1,294:29178422,13338124:182275 $1,294:29178422,13338124x1,294:29586353,13338124(1,294:29586353,13436427:719314,197518,89201x1,294:30256439,13436427)$1,294:30305667,13338124 k1,294:30487942,13338124:182275 x1,294:30925053,13338124 k1,294:31107327,13338124:182274 x1,294:31908166,13338124 k1,294:32090441,13338124:182275 x1,294:32817881,13338124 x1,294:34382873,13338124 x1,294:34700729,13338124 k1,294:34883004,13338124:182275 x1,294:35428916,13338124 k1,294:35428916,13338124:0 ) (1,295:20045308,14124556:15383608,446948,92400 x1,295:21100419,14124556 x1,295:22156194,14124556 k1,294:22402413,14124556:246219 x1,294:23239956,14124556 x1,294:24259682,14124556 k1,294:24505901,14124556:246219 x1,294:26180978,14124556 k1,294:26427197,14124556:246219 x1,294:27228036,14124556 k1,294:27474255,14124556:246219 x1,294:28639470,14124556 k1,294:28885688,14124556:246218 x1,294:29431600,14124556 k1,294:29677819,14124556:246219 x1,294:30478658,14124556 k1,294:30724877,14124556:246219 x1,294:31052557,14124556 x1,294:32477309,14124556 k1,294:32744122,14124556:266813 x1,294:34345809,14124556 k1,294:34592028,14124556:246219 x1,294:35210684,14124556 x1,294:35428916,14124556 k1,295:35428916,14124556:0 ) (1,295:20045308,14910988:15383608,446948,141880 x1,295:21574252,14910988 g1,294:21738092,14910988 x1,294:22393452,14910988 g1,294:22557292,14910988 x1,294:23358131,14910988 g1,294:23521971,14910988 x1,294:23849651,14910988 x1,294:25110563,14910988 g1,294:25274403,14910988 x1,294:25784266,14910988 x1,294:26585105,14910988 x1,294:27532079,14910988 g1,294:27695919,14910988 x1,294:28642255,14910988 g1,294:28806095,14910988 x1,294:30080094,14910988 x1,294:30407774,14910988 x1,294:31453062,14910988 g1,294:31616902,14910988 x1,294:32799805,14910988 k1,295:35428916,14910988:2629111 g1,295:35428916,14910988 ) (1,298:20045308,15697420:15383608,447828,141880 h1,296:20045308,15697420:655361,0,0 x1,296:22630036,15697420 k1,296:22907324,15697420:277288 x1,296:23562684,15697420 k1,296:23839971,15697420:277287 x1,296:24495331,15697420 x1,296:25296170,15697420 x1,296:25696585,15697420 x1,296:26460720,15697420 k1,296:26738008,15697420:277288 x1,296:27684982,15697420 x1,296:28012662,15697420 x1,296:28485821,15697420 x1,296:29924315,15697420 k1,296:30229964,15697420:305649 $1,297:30229964,15697420x1,297:30786565,15697420$1,297:30833441,15697420 k1,297:31110728,15697420:277287 x1,297:31547839,15697420 k1,297:31825127,15697420:277288 x1,297:32443783,15697420 x1,297:33645055,15697420 x1,297:33827238,15697420 x1,297:35428916,15697420 k1,297:35428916,15697420:0 ) (1,298:20045308,16483852:15383608,447828,127431 x1,298:20591212,16483852 k1,297:20836268,16483852:245056 $1,297:20836268,16483852x1,297:21246321,16483852$1,297:21246321,16483852 x1,297:21410161,16483852 k1,297:21675521,16483852:265360 x1,297:22621857,16483852 k1,297:22866914,16483852:245057 $1,297:22866914,16483852x1,297:23385012,16483852$1,297:23385012,16483852 k1,297:23630068,16483852:245056 x1,297:24067179,16483852 k1,297:24312236,16483852:245057 x1,297:25113075,16483852 x1,297:25725178,16483852 k1,297:25970234,16483852:245056 x1,297:26480097,16483852 k1,297:26725154,16483852:245057 x1,297:27343810,16483852 k1,297:27588866,16483852:245056 x1,297:28389705,16483852 k1,297:28634761,16483852:245056 x1,297:30564102,16483852 k1,297:30809159,16483852:245057 x1,297:31355071,16483852 k1,297:31600127,16483852:245056 $1,297:31600127,16483852x1,297:32118225,16483852(1,297:32118225,16582155:525126,318577,0x1,297:32607927,16582155)$1,297:32643351,16483852 k1,297:32888408,16483852:245057 x1,297:33834744,16483852 k1,297:34079800,16483852:245056 $1,297:34079800,16483852x1,297:34489853,16483852$1,297:34489853,16483852 x1,297:34653693,16483852 k1,297:34919053,16483852:265360 x1,297:35428916,16483852 k1,297:35428916,16483852:0 ) (1,298:20045308,17270284:15383608,446948,141880 x1,298:21646995,17270284 k1,297:21805249,17270284:158254 x1,297:22606088,17270284 k1,297:22764342,17270284:158254 x1,297:23747382,17270284 x1,297:24439428,17270284 x1,297:25239612,17270284 k1,297:25397866,17270284:158254 x1,297:26162002,17270284 x1,297:27836433,17270284 k1,297:27994687,17270284:158254 x1,297:29123206,17270284 x1,297:29815252,17270284 k1,297:29973506,17270284:158254 x1,297:30191738,17270284 x1,297:30767137,17270284 x1,297:31495232,17270284 k1,297:31653486,17270284:158254 x1,297:32090597,17270284 k1,297:32248851,17270284:158254 x1,297:32794755,17270284 x1,297:33159121,17270284 x1,297:34760799,17270284 k1,297:34919053,17270284:158254 x1,297:35428916,17270284 k1,297:35428916,17270284:0 ) (1,298:20045308,18056716:15383608,447828,127431 x1,298:20663964,18056716 g1,297:20827804,18056716 $1,297:20827804,18056716x1,297:21345902,18056716(1,297:21345902,18155019:525126,318577,0x1,297:21835604,18155019)g1,297:22053068,18056716x1,297:22562794,18056716g1,297:22744834,18056716x1,297:23072515,18056716$1,297:23072515,18056716 x1,297:23236355,18056716 k1,298:35428916,18056716:12192561 g1,298:35428916,18056716 ) (1,300:20045308,18843148:15383608,446948,141880 h1,299:20045308,18843148:655361,0,0 x1,299:21319325,18843148 x1,299:21985162,18843148 k1,299:22265952,18843148:280790 x1,299:23612695,18843148 k1,299:23893484,18843148:280789 x1,299:24439388,18843148 x1,299:25859539,18843148 x1,299:27298033,18843148 k1,299:27608059,18843148:310026 x1,299:28408898,18843148 k1,299:28689688,18843148:280790 x1,299:30291367,18843148 x1,299:31128910,18843148 k1,299:31409700,18843148:280790 x1,299:31955612,18843148 k1,299:32236401,18843148:280789 x1,299:33037240,18843148 k1,299:33318030,18843148:280790 x1,299:34373141,18843148 x1,299:35428916,18843148 k1,299:35428916,18843148:0 ) (1,300:20045308,19629580:15383608,446948,141880 x1,300:21501507,19629580 g1,299:21665347,19629580 x1,299:22611683,19629580 x1,299:23558657,19629580 g1,299:23722497,19629580 x1,299:24486641,19629580 g1,299:24650481,19629580 x1,299:25451320,19629580 g1,299:25615160,19629580 x1,299:26125023,19629580 x1,299:26853110,19629580 x1,299:27545156,19629580 x1,299:28237202,19629580 g1,299:28401042,19629580 x1,299:29019698,19629580 x1,299:30694121,19629580 k1,300:35428916,19629580:4734795 g1,300:35428916,19629580 ) (1,303:23126795,21275928:12302121,865699,709964 (1,303:23126795,21275928:9220634,865699,709964 x1,303:23465626,21275928 (1,303:23465626,21374231:941864,318577,89201 x1,303:24374722,21374231 ) g1,303:24589530,21275928 x1,303:25099256,21275928 g1,303:25281296,21275928 (1,303:25281296,21275928:6884088,865699,709964 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x1,304:35428916,24644473 k1,304:35428916,24644473:0 ) (1,305:20045308,25430905:15383608,533465,92400 x1,305:21173827,25430905 k1,304:21371934,25430905:198107 x1,304:22172773,25430905 k1,304:22370880,25430905:198107 x1,304:24882191,25430905 k1,304:25080298,25430905:198107 x1,304:27519520,25430905 k1,304:27717627,25430905:198107 x1,304:28263539,25430905 k1,304:28461646,25430905:198107 $1,304:28461646,25430905x1,304:28789327,25430905k1,304:28960341,25430905:171014x1,304:29470067,25430905k1,304:29641082,25430905:171015x1,304:30296444,25430905(1,304:30296444,25193080:555238,295640,0x1,304:30818914,25193080)$1,304:30851682,25430905 k1,304:31049788,25430905:198106 x1,304:31559651,25430905 $1,304:31559651,25430905(1,304:31559651,25193080:703604,295640,54614x1,304:32230487,25193080)$1,304:32263255,25430905 x1,304:32427095,25430905 k1,304:32733056,25430905:305961 x1,304:33388407,25430905 k1,304:33586514,25430905:198107 x1,304:33914194,25430905 k1,304:34112301,25430905:198107 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x1,276:8467580,33602437 x1,276:10095465,33602437 g1,276:10259305,33602437 x1,276:12807301,33602437 g1,276:12971141,33602437 x1,276:13444300,33602437 g1,276:13608140,33602437 x1,276:13935820,33602437 g1,276:14099660,33602437 x1,276:15336972,33602437 x1,276:15416275,33602437 g1,276:15580115,33602437 x1,276:16599195,33602437 g1,276:16763035,33602437 x1,276:18219234,33602437 g1,276:18383074,33602437 x1,276:20276384,33602437 g1,276:20440224,33602437 $1,276:20440224,33602437x1,276:20779055,33602437(1,276:20779055,33700740:281419,197518,0x1,276:21027706,33700740)$1,276:21060474,33602437 g1,276:21224314,33602437 x1,276:21661425,33602437 g1,276:21825265,33602437 x1,276:22335128,33602437 g1,276:22498968,33602437 x1,276:23773622,33602437 g1,276:23937462,33602437 x1,276:24483374,33602437 g1,276:24647214,33602437 x1,276:25157077,33602437 $1,276:25157077,33602437(1,276:25157077,33364612:703604,295640,54614x1,276:25827913,33364612)$1,276:25860681,33602437 x1,276:26024521,33602437 g1,276:26227681,33602437 x1,276:27502344,33602437 g1,276:27666184,33602437 x1,276:28940838,33602437 g1,276:29104678,33602437 x1,276:29904862,33602437 g1,276:30068702,33602437 x1,276:32034766,33602437 g1,276:32198606,33602437 x1,276:35384913,33602437 k1,276:35428916,33602437:44003 g1,276:35428916,33602437 ) ] (1,323:3729360,48295958:31699556,13240920,0 (1,323:3729360,48295958:15383608,13240920,0 [1,313:3729360,48295958:15383608,13240920,0 (1,311:3729360,35710398:15383608,446948,141880 x1,311:4275272,35710398 k1,309:4487227,35710398:211955 x1,309:5288066,35710398 k1,309:5500021,35710398:211955 x1,309:7101700,35710398 x1,309:7939243,35710398 k1,309:8151198,35710398:211955 x1,309:9097534,35710398 x1,309:10044508,35710398 k1,309:10256463,35710398:211955 x1,309:10729622,35710398 k1,309:10941578,35710398:211956 x1,309:11269258,35710398 k1,309:11481213,35710398:211955 x1,309:12718525,35710398 x1,309:12797828,35710398 k1,309:13009783,35710398:211955 x1,309:14574767,35710398 k1,309:14786722,35710398:211955 x1,309:15733696,35710398 x1,309:16061376,35710398 x1,309:16534535,35710398 x1,309:17809189,35710398 k1,309:18021144,35710398:211955 x1,309:18239376,35710398 (1,309:18239376,35710398:655360,446948,14416 x1,309:18894736,35710398 ) x1,309:19112968,35710398 k1,309:19112968,35710398:0 ) (1,311:3729360,36496830:15383608,446948,141880 x1,311:4639648,36496830 x1,309:5096432,36496830 k1,309:5314507,36496830:218075 x1,309:6479730,36496830 k1,309:6697806,36496830:218076 x1,309:7025486,36496830 x1,309:7306638,36496830 x1,309:7889236,36496830 x1,309:8653371,36496830 k1,309:8871446,36496830:218075 x1,309:10036651,36496830 x1,309:11419442,36496830 k1,309:11637518,36496830:218076 x1,309:12511110,36496830 x1,309:13641597,36496830 k1,309:13873231,36496830:231634 x1,309:14819567,36496830 k1,309:15037642,36496830:218075 x1,309:15838481,36496830 k1,309:16056557,36496830:218076 x1,309:17148373,36496830 x1,309:18348981,36496830 k1,309:18567056,36496830:218075 x1,309:19112968,36496830 k1,309:19112968,36496830:0 ) (1,311:3729360,37283262:15383608,446948,141880 x1,311:5184904,37283262 x1,311:6022429,37283262 k1,309:6156423,37283262:133994 x1,309:7394373,37283262 k1,309:7528366,37283262:133993 x1,309:8038229,37283262 k1,309:8172223,37283262:133994 x1,309:8973062,37283262 k1,309:9107056,37283262:133994 x1,309:10744783,37283262 x1,309:10963671,37283262 k1,309:11097665,37283262:133994 x1,309:12736047,37283262 k1,309:12870040,37283262:133993 x1,309:13707583,37283262 x1,309:14727309,37283262 k1,309:14861303,37283262:133994 x1,309:16025863,37283262 x1,309:17427661,37283262 k1,309:17567624,37283262:139963 x1,309:18513960,37283262 x1,309:18831816,37283262 x1,309:19112968,37283262 k1,309:19112968,37283262:0 ) (1,311:3729360,38069694:15383608,446948,141880 x1,311:4530199,38069694 k1,309:4761416,38069694:231217 x1,309:6363095,38069694 x1,309:7200638,38069694 k1,309:7431855,38069694:231217 x1,309:8378191,38069694 x1,309:9325165,38069694 k1,309:9556382,38069694:231217 x1,309:10611494,38069694 k1,309:10842711,38069694:231217 x1,309:11880790,38069694 k1,309:12112007,38069694:231217 x1,309:12767367,38069694 h1,309:12767367,38069694:0,0,0 x1,309:12931207,38069694 k1,309:13179267,38069694:248060 x1,309:14434913,38069694 k1,309:14840204,38069694:405291 x1,309:15859284,38069694 k1,309:16090501,38069694:231217 x1,309:17037475,38069694 x1,309:17365155,38069694 x1,309:17838314,38069694 x1,309:19112968,38069694 k1,309:19112968,38069694:0 ) (1,311:3729360,38856126:15383608,446948,141880 x1,311:4639648,38856126 x1,309:5096432,38856126 k1,309:5267664,38856126:171232 x1,309:6250686,38856126 k1,309:6421917,38856126:171231 x1,309:8060299,38856126 k1,309:8231531,38856126:171232 x1,309:9359395,38856126 x1,309:10596707,38856126 k1,309:10767938,38856126:171231 x1,309:11205049,38856126 k1,309:11376281,38856126:171232 x1,309:12213824,38856126 k1,309:12385056,38856126:171232 x1,309:13476871,38856126 k1,309:13648102,38856126:171231 x1,309:14157965,38856126 k1,309:14329197,38856126:171232 x1,309:15239485,38856126 x1,309:16586229,38856126 k1,309:16757460,38856126:171231 x1,309:17558299,38856126 k1,309:17729531,38856126:171232 x1,309:18894736,38856126 x1,309:19112968,38856126 k1,311:19112968,38856126:0 ) (1,311:3729360,39642558:15383608,446948,141880 x1,311:5112151,39642558 k1,309:5288796,39642558:176645 x1,309:6162388,39642558 x1,309:7292875,39642558 k1,309:7534448,39642558:241573 x1,309:8553528,39642558 k1,309:8730173,39642558:176645 x1,309:9021149,39642558 x1,309:10460292,39642558 k1,309:10636936,39642558:176644 x1,309:11182848,39642558 k1,309:11359493,39642558:176645 x1,309:12124284,39642558 x1,309:12742940,39642558 x1,309:14217483,39642558 k1,309:14397328,39642558:179845 x1,309:15343664,39642558 k1,309:15520309,39642558:176645 x1,309:15811285,39642558 k1,309:15987929,39642558:176644 x1,309:16497137,39642558 x1,309:16970296,39642558 x1,309:17407407,39642558 x1,309:18062749,39642558 k1,309:18239394,39642558:176645 x1,309:18894736,39642558 x1,309:19112968,39642558 k1,311:19112968,39642558:0 ) (1,311:3729360,40428990:15383608,446948,141880 x1,311:4675678,40428990 k1,309:4839899,40428990:164221 x1,309:6005113,40428990 k1,309:6169334,40428990:164221 x1,309:6824694,40428990 k1,309:6988915,40428990:164221 x1,309:8626642,40428990 x1,309:10082841,40428990 k1,309:10247061,40428990:164220 x1,309:10756924,40428990 x1,309:11230083,40428990 x1,309:12030267,40428990 x1,309:13049993,40428990 k1,309:13214214,40428990:164221 x1,309:14980388,40428990 k1,309:15144704,40428990:164316 x1,309:16418056,40428990 k1,309:16582277,40428990:164221 x1,309:17638044,40428990 k1,309:17802265,40428990:164221 x1,309:18894736,40428990 x1,309:19112968,40428990 k1,311:19112968,40428990:0 ) (1,311:3729360,41215422:15383608,446948,7531 x1,311:4694039,41215422 k1,309:5061136,41215422:367097 x1,309:6080216,41215422 k1,309:6298701,41215422:218485 x1,309:7900380,41215422 x1,309:8737923,41215422 k1,309:8956409,41215422:218486 x1,309:9902745,41215422 x1,309:10339856,41215422 x1,309:11250126,41215422 k1,309:11468611,41215422:218485 x1,309:12997563,41215422 k1,309:13216049,41215422:218486 x1,309:13980193,41215422 k1,309:14198678,41215422:218485 x1,309:14999517,41215422 k1,309:15218002,41215422:218485 x1,309:15727210,41215422 x1,309:16200369,41215422 x1,309:16637480,41215422 x1,309:17292822,41215422 k1,309:17511308,41215422:218486 x1,309:18166650,41215422 x1,309:19112968,41215422 k1,309:19112968,41215422:0 ) (1,311:3729360,42001854:15383608,533465,141880 x1,311:5403791,42001854 k1,309:5563079,42001854:159288 $1,309:5563079,42001854x1,309:6400486,42001854k1,309:6529261,42001854:128775x1,309:7038987,42001854k1,309:7167761,42001854:128774x1,309:7823123,42001854(1,309:7823123,41764029:555238,295640,0x1,309:8345593,41764029)$1,309:8378361,42001854 k1,309:8537649,42001854:159288 x1,309:9047512,42001854 $1,309:9047512,42001854(1,309:9047512,41764029:703604,295640,54614x1,309:9718348,41764029)$1,309:9751116,42001854 x1,309:9914956,42001854 k1,309:10075155,42001854:160199 x1,309:10912035,42001854 k1,309:11071323,42001854:159288 x1,309:12109402,42001854 k1,309:12268691,42001854:159289 x1,309:12924051,42001854 h1,309:12924051,42001854:0,0,0 x1,309:13087891,42001854 k1,309:13248090,42001854:160199 x1,309:14704280,42001854 k1,309:14863568,42001854:159288 x1,309:16136920,42001854 k1,310:16296209,42001854:159289 x1,310:17752408,42001854 k1,310:17911696,42001854:159288 x1,310:19112968,42001854 k1,310:19112968,42001854:0 ) (1,311:3729360,42788286:15383608,550688,141880 x1,311:5403791,42788286 g1,310:5567631,42788286 x1,310:6696150,42788286 g1,310:6859990,42788286 x1,310:7660829,42788286 g1,310:7824669,42788286 x1,310:8443325,42788286 x1,310:8734301,42788286 x1,310:9426347,42788286 x1,310:9608530,42788286 x1,310:10372665,42788286 g1,310:10536505,42788286 x1,310:11082409,42788286 x1,310:11446775,42788286 x1,310:12975701,42788286 (1,310:12975701,42788286:262144,550688,0 $1,310:12975701,42788286(1,310:12975701,42550461:262144,312863,0x1,310:13205077,42550461)$1,310:13237845,42788286 ) x1,310:13401685,42788286 k1,311:19112968,42788286:5711283 g1,311:19112968,42788286 ) (1,313:3729360,43574718:15383608,446948,141880 h1,312:3729360,43574718:655361,0,0 x1,312:7150958,43574718 k1,312:7338556,43574718:187598 x1,312:8285530,43574718 x1,312:8613210,43574718 x1,312:9086369,43574718 x1,312:10361023,43574718 k1,312:10548621,43574718:187598 x1,312:11131229,43574718 x1,312:11968772,43574718 k1,312:12156370,43574718:187598 x1,312:14122434,43574718 k1,312:14310031,43574718:187597 x1,312:14528263,43574718 (1,312:14528263,43574718:655360,446948,0 x1,312:15183623,43574718 ) x1,312:15347463,43574718 k1,312:15535061,43574718:187598 (1,312:15535061,43574718:655360,446948,0 x1,312:16190421,43574718 ) x1,312:16408653,43574718 k1,312:16596251,43574718:187598 x1,312:17869603,43574718 k1,312:18057201,43574718:187598 x1,312:19112968,43574718 k1,312:19112968,43574718:0 ) (1,313:3729360,44361150:15383608,446948,141880 x1,313:5003376,44361150 k1,312:5193144,44361150:189768 x1,312:5957288,44361150 k1,312:6147056,44361150:189768 x1,312:6474736,44361150 k1,312:6664504,44361150:189768 x1,312:7901816,44361150 k1,312:8091558,44361150:189742 x1,312:9511053,44361150 k1,312:9700821,44361150:189768 x1,312:10319477,44361150 x1,312:10828685,44361150 x1,312:11472253,44361150 x1,312:11593501,44361150 k1,312:11874445,44361150:280944 x1,312:13439429,44361150 k1,312:13629197,44361150:189768 x1,312:14576171,44361150 x1,312:14903851,44361150 x1,312:15377010,44361150 x1,312:16651664,44361150 k1,312:16841432,44361150:189768 x1,312:17715016,44361150 k1,312:17904784,44361150:189768 x1,312:18195760,44361150 k1,312:18385528,44361150:189768 x1,312:18894736,44361150 x1,312:19112968,44361150 k1,313:19112968,44361150:0 ) (1,313:3729360,45147582:15383608,446948,141880 x1,313:4202519,45147582 x1,313:4639630,45147582 x1,313:5294972,45147582 k1,312:5505531,45147582:210559 x1,312:6160873,45147582 x1,312:7271031,45147582 k1,312:7493270,45147582:222239 x1,312:8658484,45147582 k1,312:8869044,45147582:210560 x1,312:10506771,45147582 x1,312:11962970,45147582 k1,312:12173529,45147582:210559 x1,312:12683392,45147582 x1,312:13156551,45147582 x1,312:13956735,45147582 x1,312:14976461,45147582 k1,312:15187020,45147582:210559 x1,312:16789354,45147582 k1,312:16999914,45147582:210560 x1,312:17473073,45147582 k1,312:17683632,45147582:210559 x1,312:17974608,45147582 x1,312:18285904,45147582 x1,312:19112968,45147582 k1,312:19112968,45147582:0 ) (1,313:3729360,45934014:15383608,446948,141880 x1,313:4384702,45934014 x1,313:5331020,45934014 k1,312:5547536,45934014:216516 x1,312:6020695,45934014 x1,312:6311671,45934014 x1,312:7786213,45934014 k1,312:8147400,45934014:361187 x1,312:8875496,45934014 k1,312:9092012,45934014:216516 x1,312:10257226,45934014 k1,312:10473742,45934014:216516 x1,312:11238542,45934014 x1,312:12640350,45934014 k1,312:12856866,45934014:216516 x1,312:13657705,45934014 x1,312:13975561,45934014 k1,312:14192076,45934014:216515 x1,312:15102364,45934014 x1,312:15559148,45934014 k1,312:15775664,45934014:216516 x1,312:16940887,45934014 k1,312:17157403,45934014:216516 x1,312:17485083,45934014 x1,312:17766235,45934014 x1,312:18348833,45934014 x1,312:19112968,45934014 k1,312:19112968,45934014:0 ) (1,310:3729360,47050774:15383608,453198,113504 (1,310:3729360,47050774:943720,453198,0 k1,310:4443704,47050774:714344 (1,310:4443704,47050774:229376,453198,0 $1,310:4443704,47050774(1,310:4443704,46865744:229376,268168,0x1,310:4640312,46865744)$1,310:4673080,47050774 ) ) (1,310:4673080,47050774:0,435814,0 r1,313:4673080,47050774:0,435814,0 ) x1,310:5488344,47050774 k1,310:5617517,47050774:129173 x1,310:6753114,47050774 k1,310:6882287,47050774:129173 x1,310:9182308,47050774 k1,310:9311480,47050774:129172 x1,310:10272492,47050774 k1,310:10401665,47050774:129173 x1,310:11012981,47050774 k1,310:11142154,47050774:129173 x1,310:11782826,47050774 k1,310:11911999,47050774:129173 x1,310:14329977,47050774 k1,310:14459150,47050774:129173 x1,310:15187381,47050774 x1,310:15814952,47050774 k1,310:15944125,47050774:129173 x1,310:16352016,47050774 k1,310:16481188,47050774:129172 x1,310:17311652,47050774 k1,310:17440825,47050774:129173 x1,310:17965113,47050774 h1,310:17965113,47050774:0,0,0 k1,310:18094286,47050774:129173 x1,310:19112968,47050774 k1,310:19112968,47050774:0 ) (1,310:3729360,47673366:15383608,357558,113504 x1,310:4137251,47673366 x1,310:5521354,47673366 k1,310:5662543,47673366:141189 x1,310:6565359,47673366 k1,310:6706549,47673366:141190 x1,310:7813308,47673366 k1,310:7954497,47673366:141189 x1,310:9963548,47673366 k1,310:10104738,47673366:141190 x1,310:10512629,47673366 k1,310:10653818,47673366:141189 x1,310:10974157,47673366 x1,310:11805670,47673366 k1,310:11946860,47673366:141190 x1,310:12209004,47673366 h1,310:12209004,47673366:0,0,0 x1,310:12340076,47673366 k1,310:12532956,47673366:192880 x1,310:12824457,47673366 x1,310:13253328,47673366 k1,310:13394517,47673366:141189 x1,310:14035189,47673366 k1,310:14176379,47673366:141190 x1,310:16476400,47673366 k1,310:16617589,47673366:141189 x1,310:17578601,47673366 k1,310:17719791,47673366:141190 x1,310:18331107,47673366 k1,310:18472296,47673366:141189 x1,310:19112968,47673366 k1,310:19112968,47673366:0 ) (1,310:3729360,48295958:15383608,438435,186779 x1,310:6278410,48295958 g1,310:6409482,48295958 x1,310:7050154,48295958 g1,310:7181226,48295958 x1,310:9219626,48295958 g1,310:9350698,48295958 x1,310:11302059,48295958 g1,310:11433131,48295958 x1,310:12451813,48295958 g1,310:12582885,48295958 x1,310:13747845,48295958 g1,310:13878917,48295958 $1,310:13878917,48295958x1,310:14590721,48295958g1,310:14714513,48295958x1,310:15147785,48295958g1,310:15271577,48295958x1,310:15828641,48295958(1,310:15828641,48110928:513358,253405,0x1,310:16309231,48110928)$1,310:16341999,48295958 g1,310:16473071,48295958 x1,310:16880962,48295958 $1,310:16880962,48295958(1,310:16880962,48110928:651712,253405,54612x1,310:17499906,48110928)$1,310:17532674,48295958 x1,310:17663746,48295958 r1,313:17663746,48295958:0,186779,186779 k1,310:19112968,48295958:1449222 g1,310:19112968,48295958 ) ] g1,323:19112968,48295958 ) k1,323:19579138,48295958:466170 r1,323:19579138,48295958:0,13240920,0 k1,323:20045308,48295958:466170 (1,323:20045308,48295958:15383608,13240920,0 [1,323:20045308,48295958:15383608,13240920,0 (1,313:20045308,35710398:15383608,446948,141880 x1,313:21210513,35710398 x1,313:22593304,35710398 k1,312:22749160,35710398:155856 x1,312:23622752,35710398 x1,312:24753239,35710398 k1,312:24910692,35710398:157453 x1,312:25857028,35710398 k1,312:26012884,35710398:155856 x1,312:26813723,35710398 k1,312:26969579,35710398:155856 x1,312:28061395,35710398 x1,312:29262003,35710398 k1,312:29417859,35710398:155856 x1,312:29963771,35710398 k1,312:30119627,35710398:155856 x1,312:31575171,35710398 x1,312:32412696,35710398 k1,312:32568552,35710398:155856 x1,312:33806502,35710398 k1,312:33962358,35710398:155856 x1,312:34472221,35710398 k1,312:34628077,35710398:155856 x1,312:35428916,35710398 k1,312:35428916,35710398:0 ) (1,313:20045308,36496830:15383608,446948,141880 x1,313:21683035,36496830 x1,312:21901923,36496830 k1,312:22087594,36496830:185671 x1,312:23725976,36496830 k1,312:23911648,36496830:185672 x1,312:24749191,36496830 x1,312:25768917,36496830 k1,312:25954588,36496830:185671 x1,312:27119148,36496830 x1,312:28357106,36496830 k1,312:28542778,36496830:185672 x1,312:29597890,36496830 k1,312:29783561,36496830:185671 x1,312:30821640,36496830 k1,312:31007312,36496830:185672 x1,312:31662672,36496830 h1,312:31662672,36496830:0,0,0 x1,312:31826512,36496830 k1,312:32017641,36496830:191129 x1,312:33273287,36496830 k1,312:33541942,36496830:268655 x1,312:33906318,36496830 x1,312:34442406,36496830 k1,312:34628077,36496830:185671 x1,312:35428916,36496830 k1,312:35428916,36496830:0 ) (1,313:20045308,37283262:15383608,446948,141880 x1,313:21392051,37283262 k1,312:21601698,37283262:209647 x1,312:23057897,37283262 k1,312:23267543,37283262:209646 x1,312:24213879,37283262 x1,312:25160853,37283262 k1,312:25370500,37283262:209647 x1,312:26316836,37283262 x1,312:26634692,37283262 x1,312:26915844,37283262 k1,312:27125491,37283262:209647 x1,312:27926330,37283262 k1,312:28135976,37283262:209646 x1,312:29591520,37283262 x1,312:30429045,37283262 k1,312:30638692,37283262:209647 x1,312:32240371,37283262 x1,312:33241754,37283262 k1,312:33462852,37283262:221098 x1,312:35428916,37283262 k1,312:35428916,37283262:0 ) (1,313:20045308,38069694:15383608,446948,141880 x1,313:20991644,38069694 k1,312:21262876,38069694:271232 x1,312:22027676,38069694 x1,312:23265644,38069694 k1,312:23536768,38069694:271124 x1,312:24373656,38069694 x1,312:25356678,38069694 k1,312:25627910,38069694:271232 x1,312:26428749,38069694 k1,312:26699981,38069694:271232 x1,312:28046724,38069694 k1,312:28317957,38069694:271233 x1,312:28791116,38069694 x1,312:29102412,38069694 x1,312:29565747,38069694 k1,312:29836979,38069694:271232 x1,312:30382891,38069694 k1,312:30654123,38069694:271232 x1,312:30981803,38069694 x1,312:31262955,38069694 x1,312:31845553,38069694 x1,312:32609688,38069694 k1,312:32880920,38069694:271232 x1,312:34046125,38069694 x1,312:35428916,38069694 k1,312:35428916,38069694:0 ) (1,313:20045308,38856126:15383608,446948,141880 x1,313:20918900,38856126 x1,312:21885547,38856126 k1,312:22121315,38856126:235768 x1,312:22630523,38856126 x1,312:23321914,38856126 k1,312:23557681,38856126:235767 x1,312:24540721,38856126 k1,312:24776489,38856126:235768 x1,312:26251032,38856126 k1,312:26504781,38856126:253749 x1,312:28179858,38856126 k1,312:28415625,38856126:235767 x1,312:28706601,38856126 k1,312:28942369,38856126:235768 x1,312:29160601,38856126 x1,312:29736000,38856126 x1,312:30464095,38856126 k1,312:30699863,38856126:235768 x1,312:31245775,38856126 k1,312:31481542,38856126:235767 x1,312:31809222,38856126 k1,312:32044990,38856126:235768 x1,312:33100102,38856126 k1,312:33335869,38856126:235767 x1,312:34373948,38856126 k1,312:34609716,38856126:235768 x1,312:35265076,38856126 h1,312:35265076,38856126:0,0,0 x1,312:35428916,38856126 k1,312:35428916,38856126:0 ) (1,313:20045308,39642558:15383608,446948,141880 x1,313:21665338,39642558 k1,313:35428916,39642558:13763578 g1,313:35428916,39642558 ) (1,323:20045308,41218070:15383608,451540,133360 k1,316:20700669,41218070:655361 x1,316:21319973,41218070 x1,316:21938629,41218070 x1,316:22593981,41218070 x1,316:22884957,41218070 x1,316:23746749,41218070 k1,316:23901124,41218070:154375 x1,316:25138427,41218070 x1,316:26630667,41218070 k1,316:26785042,41218070:154375 x1,316:27403698,41218070 x1,316:27719586,41218070 x1,316:28768809,41218070 k1,316:29424169,41218070:655360 x1,316:29788545,41218070 x1,316:30324633,41218070 k1,316:30479007,41218070:154374 x1,316:31279846,41218070 k1,316:31434221,41218070:154375 x1,316:31943429,41218070 x1,316:32416588,41218070 x1,316:32853699,41218070 x1,316:33509041,41218070 k1,316:33663416,41218070:154375 x1,316:34318758,41218070 x1,316:35428916,41218070 k1,316:35428916,41218070:0 ) (1,323:20045308,42004502:15383608,446948,9168 x1,323:20810108,42004502 x1,323:22048076,42004502 k1,316:22348375,42004502:300299 x1,316:22821543,42004502 x1,316:23360894,42004502 k1,316:23661330,42004502:300436 x1,316:24717097,42004502 k1,316:25017534,42004502:300437 x1,316:26218798,42004502 k1,316:26519234,42004502:300436 x1,316:27029097,42004502 k1,316:27329533,42004502:300436 x1,316:27875437,42004502 x1,316:28239803,42004502 x1,316:29513801,42004502 k1,316:29814238,42004502:300437 x1,316:30324101,42004502 x1,316:30942757,42004502 x1,316:31561413,42004502 x1,316:32580484,42004502 k1,316:32880920,42004502:300436 x1,316:34046125,42004502 x1,316:35428916,42004502 k1,316:35428916,42004502:0 ) (1,323:20045308,42790934:15383608,446948,141880 x1,323:20918900,42790934 x1,316:21885547,42790934 k1,316:22023080,42790934:137533 x1,316:22641736,42790934 x1,316:23151599,42790934 x1,316:23608383,42790934 k1,316:23745916,42790934:137533 x1,316:24546755,42790934 k1,316:24684288,42790934:137533 x1,316:26139832,42790934 x1,316:26977357,42790934 k1,316:27114890,42790934:137533 x1,316:28716569,42790934 x1,316:29717952,42790934 k1,316:29912343,42790934:194391 x1,316:30567694,42790934 k1,316:30705227,42790934:137533 x1,316:31506066,42790934 k1,316:31643599,42790934:137533 x1,316:33972717,42790934 x1,316:35428916,42790934 k1,317:35428916,42790934:0 ) (1,323:20045308,43577366:15383608,533465,141880 x1,323:20991644,43577366 x1,323:21938618,43577366 k1,317:22082153,43577366:143535 $1,317:22082153,43577366x1,317:22919560,43577366k1,317:22989987,43577366:70427x1,317:23499713,43577366k1,317:23570140,43577366:70427x1,317:24225502,43577366(1,317:24225502,43339541:555238,295640,0x1,317:24747972,43339541)$1,317:24780740,43577366 k1,317:24924275,43577366:143535 x1,317:25434138,43577366 $1,317:25434138,43577366(1,317:25434138,43339541:703604,295640,54614x1,317:26104974,43339541)$1,317:26137742,43577366 k1,317:26281276,43577366:143534 x1,317:27045420,43577366 k1,317:27188955,43577366:143535 x1,317:27516635,43577366 k1,317:27660170,43577366:143535 x1,317:28897482,43577366 k1,317:29040997,43577366:143515 x1,317:30460492,43577366 k1,317:30604027,43577366:143535 x1,317:31222683,43577366 x1,317:31731891,43577366 x1,317:32375459,43577366 x1,317:32496707,43577366 k1,317:32644303,43577366:147596 x1,317:33445142,43577366 k1,318:33588677,43577366:143535 x1,318:34462269,43577366 x1,318:35428916,43577366 k1,318:35428916,43577366:0 ) (1,323:20045308,44363798:15383608,533465,127431 x1,323:21027675,44363798 k1,318:21250772,44363798:223097 x1,318:21796684,44363798 k1,318:22019781,44363798:223097 x1,318:22820620,44363798 k1,318:23043718,44363798:223098 x1,318:24463213,44363798 k1,318:24686310,44363798:223097 x1,318:25705373,44363798 k1,318:25928470,44363798:223097 x1,318:26365581,44363798 k1,318:26588678,44363798:223097 x1,318:28044877,44363798 k1,318:28267975,44363798:223098 $1,318:28267975,44363798x1,318:29105382,44363798k1,318:29294907,44363798:189525x1,318:29804633,44363798k1,318:29994159,44363798:189526x1,318:30649521,44363798(1,318:30649521,44125973:703604,295640,54614x1,318:31320357,44125973)$1,318:31353125,44363798 k1,318:31576222,44363798:223097 $1,318:31576222,44363798x1,318:31971108,44363798$1,318:31971108,44363798 x1,318:33882769,44363798 k1,318:34263701,44363798:380932 x1,318:35428916,44363798 k1,318:35428916,44363798:0 ) (1,323:20045308,45150230:15383608,446948,141880 x1,323:20482419,45150230 k1,319:20749962,45150230:267543 x1,319:22679303,45150230 k1,319:22946846,45150230:267543 x1,319:24075365,45150230 k1,319:24342908,45150230:267543 x1,319:25143747,45150230 k1,319:25411290,45150230:267543 x1,319:25920498,45150230 x1,319:26430361,45150230 x1,319:26721337,45150230 x1,319:27741063,45150230 k1,319:28008606,45150230:267543 x1,319:29464805,45150230 x1,319:30302348,45150230 k1,319:30569891,45150230:267543 x1,319:31552258,45150230 k1,319:31819801,45150230:267543 x1,319:32329664,45150230 k1,319:32597207,45150230:267543 x1,319:34276095,45150230 x1,319:34688319,45150230 k1,319:34955757,45150230:267438 x1,319:35428916,45150230 k1,319:35428916,45150230:0 ) (1,323:20045308,45936662:15383608,533465,187504 x1,323:20372988,45936662 k1,319:20600348,45936662:227360 x1,319:21837660,45936662 x1,319:21916963,45936662 k1,319:22144323,45936662:227360 $1,320:22144323,45936662x1,320:22981730,45936662k1,320:23174413,45936662:192683x1,320:23684139,45936662k1,320:23876822,45936662:192683x1,320:24532184,45936662(1,320:24532184,45698837:703604,295640,54614x1,320:25203020,45698837)x1,320:25610269,45936662(1,320:25610269,46034965:315015,197518,89201x1,320:25876056,46034965)$1,320:25925284,45936662 x1,320:27327082,45936662 k1,320:27720802,45936662:393720 x1,320:28448898,45936662 x1,320:29541369,45936662 x1,320:30378912,45936662 k1,320:30606272,45936662:227360 x1,320:31407111,45936662 k1,320:31634471,45936662:227360 x1,320:32689582,45936662 x1,320:33745357,45936662 k1,320:33972717,45936662:227360 x1,320:35428916,45936662 k1,320:35428916,45936662:0 ) (1,323:20045308,46723094:15383608,446948,141880 x1,323:20918900,46723094 x1,320:21885547,46723094 k1,320:22122372,46723094:236825 x1,320:23104739,46723094 k1,320:23341564,46723094:236825 x1,320:23778675,46723094 k1,320:24015500,46723094:236825 x1,320:24525363,46723094 x1,320:25144019,46723094 x1,320:26090355,46723094 x1,320:27218874,46723094 k1,320:27455699,46723094:236825 x1,320:28001611,46723094 k1,320:28238436,46723094:236825 x1,320:29657931,46723094 k1,320:29894755,46723094:236824 x1,320:31077658,46723094 k1,320:31332729,46723094:255071 x1,320:32133568,46723094 k1,320:32370393,46723094:236825 x1,320:33498912,46723094 x1,320:34190958,46723094 x1,320:35428916,46723094 k1,320:35428916,46723094:0 ) (1,323:20045308,47509526:15383608,446948,141880 x1,323:20372988,47509526 x1,323:20654140,47509526 x1,323:21236738,47509526 x1,323:22000873,47509526 k1,320:22136856,47509526:135983 x1,320:23556351,47509526 k1,320:23692334,47509526:135983 x1,320:24711397,47509526 k1,320:24847380,47509526:135983 x1,320:25867097,47509526 k1,320:26003080,47509526:135983 x1,320:26512943,47509526 x1,320:28150007,47509526 k1,320:28285990,47509526:135983 x1,320:28941350,47509526 k1,320:29077334,47509526:135984 x1,320:29368310,47509526 k1,320:29504293,47509526:135983 x1,320:29722525,47509526 x1,320:30297924,47509526 x1,320:31026019,47509526 k1,320:31162002,47509526:135983 x1,320:31707914,47509526 k1,320:31843897,47509526:135983 $1,321:31843897,47509526x1,321:32353623,47509526k1,321:32535663,47509526:182040x1,321:33700751,47509526$1,321:33700751,47509526 k1,321:33836734,47509526:135983 x1,321:34783070,47509526 k1,321:34919053,47509526:135983 x1,321:35428916,47509526 k1,321:35428916,47509526:0 ) (1,323:20045308,48295958:15383608,446948,141880 x1,323:21173828,48295958 x1,323:23613050,48295958 k1,321:23801777,48295958:188727 x1,321:24311640,48295958 x1,321:24930296,48295958 x1,321:25548952,48295958 x1,321:26568023,48295958 k1,321:26756749,48295958:188726 x1,321:27921954,48295958 x1,321:29304745,48295958 k1,321:29493472,48295958:188727 x1,321:30367064,48295958 x1,321:31497551,48295958 k1,321:31775370,48295958:277819 x1,321:33304305,48295958 k1,322:33493032,48295958:188727 x1,322:34512095,48295958 k1,322:34700821,48295958:188726 x1,322:35210684,48295958 x1,322:35428916,48295958 k1,323:35428916,48295958:0 ) ] g1,323:35428916,48295958 ) ) ] h1,323:3729360,50262038:31699556,0,0 ] ] !35383 8 !10 9 [1,386:1,50262036:35428915,50262035,0 [1,386:3729360,50262036:31699556,49329695,0 [1,386:3729360,1718773:31699556,786432,0 h1,386:3729360,1718773:31699556,0,0 ] [1,386:3729360,48295956:31699556,44938783,0 (1,386:3729360,48295956:31699556,44938783,0 (1,386:3729360,48295956:15383608,44938783,0 [1,346:3729360,48295956:15383608,44938783,0 (1,323:3729360,4012533:15383608,446948,141880 x1,323:5621352,4012533 k1,322:5785680,4012533:164328 x1,322:6331592,4012533 k1,322:6495920,4012533:164328 x1,322:7442894,4012533 k1,322:7607222,4012533:164328 x1,322:8735741,4012533 x1,322:9245604,4012533 x1,322:10374123,4012533 k1,322:10538451,4012533:164328 x1,322:12286282,4012533 k1,322:12450610,4012533:164328 x1,322:13069266,4012533 k1,322:13233594,4012533:164328 x1,322:13888954,4012533 x1,322:14653090,4012533 x1,322:14980770,4012533 x1,322:16056201,4012533 k1,322:16220529,4012533:164328 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x1,322:17104736,7158261 x1,322:18071383,7158261 k1,322:18312784,7158261:241401 x1,322:19112968,7158261 k1,322:19112968,7158261:0 ) (1,323:3729360,7944693:15383608,446948,141880 x1,323:5804215,7944693 g1,322:5968055,7944693 x1,322:6732199,7944693 g1,322:6896039,7944693 x1,322:7696878,7944693 g1,322:7860718,7944693 x1,322:8406630,7944693 x1,322:9589541,7944693 k1,323:19112968,7944693:9523427 g1,323:19112968,7944693 ) (1,330:3729360,9546482:15383608,541848,14544 k1,330:6734471,9546482:3005111 (1,330:6734471,9546482:0,0,0 g1,330:6734471,9546482 ) x1,330:10798704,9546482 g1,330:10995312,9546482 x1,330:12655458,9546482 g1,330:12852066,9546482 x1,330:14512990,9546482 g1,330:14709598,9546482 x1,330:16107857,9546482 g1,330:16107857,9546482 k1,330:19112968,9546482:3005111 ) (1,333:3729360,10666842:15383608,446948,141880 h1,332:3729360,10666842:655361,0,0 x1,332:4930633,10666842 k1,332:5194777,10666842:264144 x1,332:5595201,10666842 x1,332:5833752,10666842 x1,332:6634591,10666842 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x1,333:5659356,13026138 x1,333:6351402,13026138 x1,333:6472650,13026138 k1,332:6686074,13026138:213424 x1,332:7268682,13026138 x1,332:8106225,13026138 k1,332:8309732,13026138:203507 x1,332:9110571,13026138 k1,332:9314078,13026138:203507 x1,332:10261052,13026138 x1,332:11061891,13026138 k1,332:11265398,13026138:203507 x1,332:11884054,13026138 x1,332:12175030,13026138 x1,332:12867076,13026138 x1,332:13049259,13026138 x1,332:13813394,13026138 k1,332:14016902,13026138:203508 x1,332:14562814,13026138 x1,332:16091748,13026138 k1,332:16295255,13026138:203507 x1,332:17896942,13026138 k1,332:18100449,13026138:203507 x1,332:18573617,13026138 x1,332:19112968,13026138 k1,332:19112968,13026138:0 ) (1,333:3729360,13812570:15383608,446948,141880 x1,333:4639648,13812570 x1,332:5424112,13812570 k1,332:5616871,13812570:192759 x1,332:6126734,13812570 k1,332:6319494,13812570:192760 x1,332:6829357,13812570 x1,332:7140653,13812570 x1,332:7421805,13812570 k1,332:7614564,13812570:192759 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$1,332:13826433,14599002x1,332:14663840,14599002k1,332:14868120,14599002:204280x1,332:15377846,14599002k1,332:15582125,14599002:204279x1,332:16237487,14599002(1,332:16237487,14361177:555238,295640,0x1,332:16759957,14361177)$1,332:16792725,14599002 k1,332:17035741,14599002:243016 x1,332:17545604,14599002 $1,332:17545604,14599002(1,332:17545604,14361177:703604,295640,54614x1,332:18216440,14361177)$1,332:18249208,14599002 x1,332:18413048,14599002 k1,332:18675857,14599002:262809 x1,332:19112968,14599002 k1,332:19112968,14599002:0 ) (1,333:3729360,15385434:15383608,446948,141880 x1,333:5040062,15385434 k1,332:5266320,15385434:226258 x1,332:5776183,15385434 x1,332:6232967,15385434 x1,332:6354215,15385434 k1,332:6596078,15385434:241863 x1,332:7324173,15385434 x1,332:8125012,15385434 x1,332:9035290,15385434 k1,332:9261548,15385434:226258 x1,332:9880204,15385434 x1,332:11335740,15385434 k1,332:11561998,15385434:226258 x1,332:12107910,15385434 k1,332:12334168,15385434:226258 x1,332:13135007,15385434 k1,332:13361266,15385434:226259 x1,332:13871129,15385434 x1,332:14327913,15385434 k1,332:14554171,15385434:226258 x1,332:15391714,15385434 x1,332:16520233,15385434 x1,332:17539959,15385434 k1,332:17766217,15385434:226258 x1,332:18894736,15385434 x1,332:19112968,15385434 k1,333:19112968,15385434:0 ) (1,333:3729360,16171866:15383608,446948,141880 x1,333:5822558,16171866 k1,332:6167432,16171866:344874 x1,332:7641984,16171866 k1,332:7864872,16171866:222888 x1,332:8592967,16171866 x1,332:8920647,16171866 x1,332:10049167,16171866 k1,332:10260245,16171866:211078 x1,332:11170533,16171866 x1,332:12262349,16171866 x1,332:13354820,16171866 k1,332:13565898,16171866:211078 x1,332:14111810,16171866 k1,332:14322888,16171866:211078 x1,332:15123727,16171866 k1,332:15334805,16171866:211078 x1,332:16389916,16171866 x1,332:17445691,16171866 k1,332:17656769,16171866:211078 x1,332:19112968,16171866 k1,332:19112968,16171866:0 ) (1,333:3729360,16958298:15383608,446948,141880 x1,333:4749077,16958298 k1,332:4883085,16958298:134008 x1,332:5501741,16958298 k1,332:5635749,16958298:134008 x1,332:6144957,16958298 x1,332:7819388,16958298 k1,332:7953396,16958298:134008 x1,332:8463259,16958298 k1,332:8597267,16958298:134008 x1,332:9980058,16958298 x1,332:10291354,16958298 x1,332:10572506,16958298 k1,332:10706514,16958298:134008 x1,332:11507353,16958298 k1,332:11641361,16958298:134008 x1,332:12260017,16958298 x1,332:13352488,16958298 k1,332:13486496,16958298:134008 x1,332:14651701,16958298 x1,332:16034492,16958298 k1,332:16168500,16958298:134008 x1,332:16678363,16958298 k1,332:16812371,16958298:134008 x1,332:17759345,16958298 k1,332:17893353,16958298:134008 x1,332:19112968,16958298 k1,332:19112968,16958298:0 ) (1,333:3729360,17744730:15383608,446948,141880 x1,333:4894575,17744730 g1,332:5058415,17744730 x1,332:5968703,17744730 x1,332:7060519,17744730 x1,332:8152990,17744730 g1,332:8316830,17744730 x1,332:8753941,17744730 g1,332:8917781,17744730 x1,332:9499733,17744730 x1,332:10628252,17744730 x1,332:12521562,17744730 g1,332:12685402,17744730 x1,332:13340762,17744730 k1,333:19112968,17744730:5772206 g1,333:19112968,17744730 ) (1,335:5040082,19117056:14072886,446948,141880 (1,334:5040082,19117056:0,291271,0 g1,334:5040082,19117056 g1,334:3729360,19117056 g1,334:3401680,19117056 (1,334:3401680,19117056:1310722,291271,0 k1,334:4712402,19117056:1310722 (1,334:4712402,19117056:0,291271,0 k1,334:4384721,19117056:-327681 x1,334:4712402,19117056 ) ) g1,334:5040082,19117056 ) x1,334:8244105,19117056 k1,334:8517748,19117056:273643 x1,334:9318587,19117056 k1,334:9592230,19117056:273643 x1,334:10320325,19117056 x1,334:11667076,19117056 k1,334:11940718,19117056:273642 x1,334:12923758,19117056 x1,334:13433621,19117056 x1,334:14488732,19117056 x1,334:15799435,19117056 k1,334:16073078,19117056:273643 x1,334:17674765,19117056 k1,334:17948408,19117056:273643 x1,334:19112968,19117056 k1,334:19112968,19117056:0 ) (1,335:5040082,19903488:14072886,446948,141880 x1,335:5840921,19903488 k1,334:6044931,19903488:204010 x1,334:7664970,19903488 k1,334:7879023,19903488:214053 x1,334:9917838,19903488 k1,334:10121848,19903488:204010 x1,334:10922687,19903488 k1,334:11126697,19903488:204010 x1,334:12000281,19903488 k1,334:12204291,19903488:204010 x1,334:12750203,19903488 k1,334:12954214,19903488:204011 x1,334:13572870,19903488 x1,334:15465524,19903488 x1,334:16557340,19903488 k1,334:16761350,19903488:204010 x1,334:17707686,19903488 k1,334:17911696,19903488:204010 x1,334:18894736,19903488 x1,334:19112968,19903488 k1,335:19112968,19903488:0 ) (1,335:5040082,20689920:14072886,446948,141880 x1,335:5877625,20689920 g1,334:6041465,20689920 x1,334:7571047,20689920 g1,334:7734887,20689920 x1,334:8681223,20689920 k1,335:19112968,20689920:10431745 g1,335:19112968,20689920 ) (1,336:5040082,22082781:14072886,446948,141880 (1,335:5040082,22082781:0,291271,0 g1,335:5040082,22082781 g1,335:3729360,22082781 g1,335:3401680,22082781 (1,335:3401680,22082781:1310722,291271,0 k1,335:4712402,22082781:1310722 (1,335:4712402,22082781:0,291271,0 k1,335:4384721,22082781:-327681 x1,335:4712402,22082781 ) ) g1,335:5040082,22082781 ) x1,335:8244105,22082781 k1,335:8486447,22082781:242342 x1,335:9287286,22082781 k1,335:9529628,22082781:242342 x1,335:10366508,22082781 x1,335:12186436,22082781 k1,335:12428779,22082781:242343 x1,335:13483890,22082781 x1,335:14794593,22082781 k1,335:15036935,22082781:242342 x1,335:16638622,22082781 k1,335:16880964,22082781:242342 x1,335:17208644,22082781 x1,335:18215266,22082781 k1,335:18457608,22082781:242342 x1,335:19112968,22082781 k1,335:19112968,22082781:0 ) (1,336:5040082,22869213:14072886,446948,141880 x1,336:5840921,22869213 k1,335:6140983,22869213:300062 x1,335:7597182,22869213 k1,335:7897244,22869213:300062 x1,335:9025108,22869213 k1,335:9325170,22869213:300062 x1,335:10126009,22869213 k1,335:10426071,22869213:300062 x1,335:12009406,22869213 k1,335:12343523,22869213:334117 x1,335:14382338,22869213 k1,335:14682400,22869213:300062 x1,335:15483239,22869213 k1,335:15783301,22869213:300062 x1,335:16656885,22869213 k1,335:16956947,22869213:300062 x1,335:17502859,22869213 k1,335:17802921,22869213:300062 x1,335:18894736,22869213 x1,335:19112968,22869213 k1,336:19112968,22869213:0 ) (1,336:5040082,23655645:14072886,446948,141880 x1,336:6132553,23655645 k1,335:6339037,23655645:206484 x1,335:7504251,23655645 k1,335:7710735,23655645:206484 x1,335:8001711,23655645 k1,335:8208195,23655645:206484 x1,335:8718058,23655645 x1,335:9174842,23655645 k1,335:9381326,23655645:206484 x1,335:10218206,23655645 x1,335:12038134,23655645 k1,335:12244618,23655645:206484 x1,335:13482568,23655645 x1,335:14575039,23655645 k1,335:14781523,23655645:206484 x1,335:16256065,23655645 k1,335:16473210,23655645:217145 x1,335:19112968,23655645 k1,335:19112968,23655645:0 ) (1,336:5040082,24442077:14072886,446948,141880 x1,336:6241354,24442077 x1,335:6559210,24442077 x1,335:7259130,24442077 g1,335:7422970,24442077 x1,335:8369306,24442077 g1,335:8533146,24442077 x1,335:9588257,24442077 x1,335:10644032,24442077 g1,335:10807872,24442077 x1,335:11862983,24442077 x1,335:13628502,24442077 k1,336:19112968,24442077:5484466 g1,336:19112968,24442077 ) (1,338:3729360,25814404:15383608,446948,141880 x1,338:4748440,25814404 k1,337:4940276,25814404:191836 x1,337:5267956,25814404 x1,337:5585812,25814404 x1,337:6740538,25814404 k1,337:6932374,25814404:191836 x1,337:10026949,25814404 k1,337:10218785,25814404:191836 x1,337:10655896,25814404 k1,337:10847733,25814404:191837 x1,337:13177516,25814404 k1,337:13369352,25814404:191836 x1,337:13879215,25814404 k1,337:14071051,25814404:191836 x1,337:15817571,25814404 k1,337:16009407,25814404:191836 x1,337:16810246,25814404 k1,337:17002082,25814404:191836 x1,337:19112968,25814404 k1,337:19112968,25814404:0 ) (1,338:3729360,26600836:15383608,446948,141880 x1,338:5185559,26600836 k1,337:5356892,26600836:171333 x1,337:7250202,26600836 k1,337:7421535,26600836:171333 x1,337:8549399,26600836 k1,337:8720731,26600836:171332 x1,337:9521570,26600836 k1,337:9692903,26600836:171333 x1,337:11112398,26600836 k1,337:11283731,26600836:171333 x1,337:11793594,26600836 k1,337:11964927,26600836:171333 x1,337:12255903,26600836 k1,337:12427236,26600836:171333 x1,337:12900395,26600836 x1,337:13211691,26600836 x1,337:13675026,26600836 k1,337:13846359,26600836:171333 x1,337:15448046,26600836 k1,337:15619378,26600836:171332 x1,337:16056489,26600836 k1,337:16227822,26600836:171333 x1,337:17356332,26600836 k1,337:17527665,26600836:171333 x1,337:18656184,26600836 x1,337:19112968,26600836 k1,337:19112968,26600836:0 ) (1,338:3729360,27387268:15383608,446948,141880 x1,338:4530199,27387268 g1,337:4694039,27387268 x1,337:7316081,27387268 g1,337:7479921,27387268 x1,337:10082983,27387268 k1,338:19112968,27387268:9029985 g1,338:19112968,27387268 ) (1,341:3729360,28812107:15383608,538305,162393 (1,341:3729360,28812107:0,0,0 g1,341:3729360,28812107 ) x1,341:7398804,28812107 x1,341:8149850,28812107 g1,341:8346458,28812107 x1,341:10137934,28812107 x1,341:12074887,28812107 k1,341:19112968,28812107:7038081 g1,341:19112968,28812107 ) (1,343:3729360,29870713:15383608,446948,145636 h1,342:3729360,29870713:655361,0,0 x1,342:5403801,29870713 k1,342:5692035,29870713:288234 x1,342:6201243,29870713 x1,342:7802921,29870713 k1,342:8091154,29870713:288233 x1,342:8418834,29870713 x1,342:8709810,29870713 x1,342:9021106,29870713 x1,342:9484441,29870713 x1,342:10758457,29870713 k1,342:11046691,29870713:288234 x1,342:12284003,29870713 k1,342:12572236,29870713:288233 x1,342:15324724,29870713 k1,342:15612958,29870713:288234 x1,342:15831190,29870713 (1,342:15831190,29870713:655360,446948,0 x1,342:16486550,29870713 ) x1,342:16650390,29870713 k1,342:16938623,29870713:288233 (1,342:16938623,29870713:655360,446948,9168 x1,342:17593983,29870713 ) x1,342:17812215,29870713 k1,342:18100449,29870713:288234 x1,342:18573617,29870713 x1,342:19112968,29870713 k1,342:19112968,29870713:0 ) (1,343:3729360,30657145:15383608,446948,141880 x1,343:4930624,30657145 k1,342:5169436,30657145:238812 x1,342:5679299,30657145 k1,342:5918112,30657145:238813 x1,342:6464016,30657145 x1,342:6828382,30657145 x1,342:8102380,30657145 k1,342:8341192,30657145:238812 x1,342:9142031,30657145 k1,342:9380844,30657145:238813 x1,342:10145635,30657145 x1,342:10728233,30657145 x1,342:11055913,30657145 x1,342:11370489,30657145 x1,342:12390215,30657145 k1,342:12629027,30657145:238812 x1,342:13174939,30657145 k1,342:13413752,30657145:238813 x1,342:14324040,30657145 x1,342:16544383,30657145 k1,342:16783195,30657145:238812 x1,342:17292403,30657145 x1,342:17802266,30657145 x1,342:18093242,30657145 x1,342:19112968,30657145 k1,342:19112968,30657145:0 ) (1,343:3729360,31443577:15383608,446948,141880 x1,343:4712400,31443577 x1,343:5549943,31443577 k1,342:5736721,31443577:186778 x1,342:6355377,31443577 x1,342:7483897,31443577 x1,342:8503623,31443577 k1,342:8690401,31443577:186778 x1,342:9855599,31443577 k1,342:10042376,31443577:186777 x1,342:11862304,31443577 k1,342:12049082,31443577:186778 x1,342:12849921,31443577 k1,342:13036699,31443577:186778 x1,342:14256314,31443577 k1,342:14448826,31443577:192512 x1,342:16050513,31443577 k1,342:16237291,31443577:186778 x1,342:16855947,31443577 x1,342:17329106,31443577 x1,342:19112968,31443577 k1,342:19112968,31443577:0 ) (1,343:3729360,32230009:15383608,446948,141880 x1,343:4530199,32230009 k1,342:4743758,32230009:213559 x1,342:5435804,32230009 x1,342:6273329,32230009 k1,342:6486888,32230009:213559 x1,342:7251679,32230009 x1,342:7834277,32230009 x1,342:8161957,32230009 x1,342:8476533,32230009 x1,342:9496259,32230009 k1,342:9709818,32230009:213559 x1,342:10255730,32230009 k1,342:10469289,32230009:213559 x1,342:11452329,32230009 x1,342:11962192,32230009 x1,342:13017303,32230009 x1,342:14491846,32230009 k1,342:14844163,32230009:352317 x1,342:15863243,32230009 k1,342:16076802,32230009:213559 x1,342:16695458,32230009 x1,342:17787929,32230009 k1,342:18001488,32230009:213559 x1,342:18329168,32230009 x1,342:18894736,32230009 x1,342:19112968,32230009 k1,343:19112968,32230009:0 ) (1,343:3729360,33016441:15383608,446948,141880 x1,343:4894583,33016441 k1,342:5107486,33016441:212903 x1,342:6526981,33016441 x1,342:7363869,33016441 k1,342:7576773,33016441:212904 x1,342:8195429,33016441 x1,342:9032972,33016441 x1,342:9323948,33016441 x1,342:10052043,33016441 x1,342:10173291,33016441 k1,342:10398460,33016441:225169 x1,342:10908323,33016441 x1,342:12036842,33016441 x1,342:12874385,33016441 k1,342:13087288,33016441:212903 x1,342:13742630,33016441 k1,342:13955533,33016441:212903 x1,342:14755717,33016441 x1,342:15774788,33016441 k1,342:15987692,33016441:212904 x1,342:16606348,33016441 x1,342:17698801,33016441 k1,342:17911704,33016441:212903 x1,342:19112968,33016441 k1,342:19112968,33016441:0 ) (1,343:3729360,33802873:15383608,446948,141880 x1,343:4275264,33802873 k1,342:4488683,33802873:213419 x1,342:5107339,33802873 x1,342:6999993,33802873 x1,342:8091809,33802873 k1,342:8305227,33802873:213418 x1,342:9251563,33802873 k1,342:9464982,33802873:213419 x1,342:10448022,33802873 x1,342:11285565,33802873 k1,342:11498983,33802873:213418 x1,342:13010222,33802873 k1,342:13236035,33802873:225813 x1,342:13673146,33802873 k1,342:13886564,33802873:213418 x1,342:15087828,33802873 k1,342:15301247,33802873:213419 x1,342:15811110,33802873 k1,342:16024528,33802873:213418 x1,342:16825367,33802873 k1,342:17038786,33802873:213419 x1,342:17802921,33802873 x1,342:18421577,33802873 x1,342:18894736,33802873 x1,342:19112968,33802873 k1,343:19112968,33802873:0 ) (1,343:3729360,34589305:15383608,446948,141880 x1,343:4912926,34589305 k1,342:5274115,34589305:361189 x1,342:6548778,34589305 k1,342:6765294,34589305:216516 x1,342:7929854,34589305 x1,342:8257534,34589305 x1,342:8948925,34589305 k1,342:9165442,34589305:216517 x1,342:10111778,34589305 k1,342:10328294,34589305:216516 x1,342:10838157,34589305 x1,342:12131821,34589305 k1,342:12348337,34589305:216516 x1,342:13367400,34589305 x1,342:13840559,34589305 x1,342:14896334,34589305 k1,342:15112850,34589305:216516 x1,342:15913034,34589305 k1,342:16129551,34589305:216517 x1,342:16639414,34589305 x1,342:18386589,34589305 k1,342:18603105,34589305:216516 x1,342:19112968,34589305 k1,342:19112968,34589305:0 ) (1,343:3729360,35375737:15383608,446948,141880 x1,343:4530199,35375737 k1,342:4747022,35375737:216823 x1,342:5693996,35375737 x1,342:6021676,35375737 x1,342:6494835,35375737 x1,342:7678401,35375737 k1,342:8040508,35375737:362107 x1,342:8404884,35375737 x1,342:8940972,35375737 k1,342:9157795,35375737:216823 x1,342:9958634,35375737 k1,342:10175456,35375737:216822 x1,342:11194519,35375737 x1,342:11667678,35375737 x1,342:12723453,35375737 k1,342:12940276,35375737:216823 x1,342:13704411,35375737 x1,342:14323067,35375737 x1,342:14796226,35375737 x1,342:15979792,35375737 k1,342:16209859,35375737:230067 x1,342:17010698,35375737 k1,342:17227521,35375737:216823 x1,342:17810129,35375737 x1,342:18246593,35375737 x1,342:19112968,35375737 k1,342:19112968,35375737:0 ) (1,343:3729360,36162169:15383608,446948,141880 x1,343:4530199,36162169 x1,343:5003358,36162169 x1,343:5876932,36162169 k1,342:6130786,36162169:253854 x1,342:6567897,36162169 k1,342:6821751,36162169:253854 x1,342:7440407,36162169 x1,342:8641679,36162169 x1,342:8823862,36162169 x1,342:10425540,36162169 k1,342:10679394,36162169:253854 x1,342:11225298,36162169 k1,342:11479152,36162169:253854 x1,342:12097808,36162169 k1,342:12351661,36162169:253853 x1,342:13152500,36162169 x1,342:15009769,36162169 k1,342:15263623,36162169:253854 x1,342:15846231,36162169 x1,342:16282695,36162169 x1,342:17149070,36162169 k1,342:17402924,36162169:253854 x1,342:18568138,36162169 k1,342:18821992,36162169:253854 x1,342:19112968,36162169 k1,342:19112968,36162169:0 ) (1,343:3729360,36948601:15383608,550688,141880 x1,343:4749086,36948601 k1,342:5001397,36948601:252311 x1,342:5583349,36948601 x1,342:6784621,36948601 k1,342:7036932,36948601:252311 x1,342:8128747,36948601 x1,342:8966290,36948601 (1,342:8966290,36948601:262144,550688,0 $1,342:8966290,36948601(1,342:8966290,36710776:262144,312863,0x1,342:9195666,36710776)$1,342:9228434,36948601 ) x1,342:9392274,36948601 k1,342:9666702,36948601:274428 x1,342:10613038,36948601 k1,342:10865350,36948601:252312 x1,342:11666189,36948601 k1,342:11918500,36948601:252311 x1,342:12501108,36948601 x1,342:12937572,36948601 x1,342:13803947,36948601 k1,342:14056258,36948601:252311 x1,342:15585210,36948601 x1,342:16713722,36948601 k1,342:16966033,36948601:252311 x1,342:17584689,36948601 x1,342:18603760,36948601 x1,342:18894736,36948601 x1,342:19112968,36948601 k1,343:19112968,36948601:0 ) (1,343:3729360,37735033:15383608,446948,141880 x1,343:4675679,37735033 k1,342:4854435,37735033:178756 x1,342:5837457,37735033 k1,342:6016213,37735033:178756 x1,342:7289565,37735033 k1,342:7468320,37735033:178755 x1,342:8014224,37735033 x1,342:9725351,37735033 k1,342:9904107,37735033:178756 x1,342:11105362,37735033 k1,342:11284118,37735033:178756 x1,342:12813070,37735033 x1,342:13941582,37735033 k1,342:14120338,37735033:178756 x1,342:15103378,37735033 k1,342:15282133,37735033:178755 x1,342:16283516,37735033 k1,342:16466001,37735033:182485 x1,342:17740000,37735033 x1,342:18067680,37735033 x1,342:19112968,37735033 k1,342:19112968,37735033:0 ) (1,343:3729360,38521465:15383608,446948,141880 x1,343:4348016,38521465 x1,343:5075456,38521465 x1,343:5548615,38521465 x1,343:6568341,38521465 k1,342:6751987,38521465:183646 x1,342:8390369,38521465 k1,342:8574014,38521465:183645 x1,342:9884734,38521465 k1,342:10068380,38521465:183646 x1,342:11124155,38521465 k1,342:11307800,38521465:183645 x1,342:12107984,38521465 k1,342:12291630,38521465:183646 x1,342:13128518,38521465 x1,342:13637726,38521465 x1,342:14511310,38521465 x1,342:14984469,38521465 x1,342:15348835,38521465 x1,342:15660131,38521465 x1,342:15941283,38521465 k1,342:16124928,38521465:183645 x1,342:16670840,38521465 k1,342:16854486,38521465:183646 x1,342:17145462,38521465 k1,342:17329107,38521465:183645 x1,342:18166650,38521465 x1,342:18348833,38521465 x1,342:19112968,38521465 k1,342:19112968,38521465:0 ) (1,343:3729360,39307897:15383608,446948,141880 x1,343:4821175,39307897 x1,343:5331038,39307897 x1,343:6095173,39307897 k1,342:6349234,39307897:254061 x1,342:6676914,39307897 x1,342:7242482,39307897 x1,342:8407705,39307897 k1,342:8661766,39307897:254061 x1,342:10081261,39307897 x1,342:10918149,39307897 x1,342:11045949,39307897 k1,342:11519772,39307897:473823 x1,342:11884148,39307897 x1,342:12420236,39307897 k1,342:12674297,39307897:254061 x1,342:14021040,39307897 k1,342:14275101,39307897:254061 x1,342:14893757,39307897 x1,342:15912828,39307897 x1,342:16203804,39307897 x1,342:16895195,39307897 x1,342:17032826,39307897 k1,342:17309441,39307897:276615 x1,342:17819304,39307897 x1,342:19112968,39307897 k1,342:19112968,39307897:0 ) (1,343:3729360,40094329:15383608,446948,141880 x1,343:4748423,40094329 x1,343:5221582,40094329 x1,343:6277357,40094329 g1,342:6441197,40094329 x1,342:7606420,40094329 x1,342:7788603,40094329 x1,342:9135346,40094329 g1,342:9299186,40094329 x1,342:9626866,40094329 x1,342:9944722,40094329 x1,342:10444106,40094329 g1,342:10607946,40094329 x1,342:11772506,40094329 x1,342:12100186,40094329 x1,342:12791577,40094329 g1,342:12955417,40094329 x1,342:13464625,40094329 x1,342:14410953,40094329 x1,342:15594519,40094329 k1,343:19112968,40094329:3518449 g1,343:19112968,40094329 ) (1,345:3729360,40901296:15383608,446948,141880 h1,344:3729360,40901296:655361,0,0 x1,344:5403801,40901296 k1,344:5636285,40901296:232484 x1,344:5963965,40901296 x1,344:6529533,40901296 x1,344:7694756,40901296 k1,344:7927240,40901296:232484 x1,344:9346735,40901296 x1,344:10183623,40901296 k1,344:10416107,40901296:232484 x1,344:11034763,40901296 x1,344:12127234,40901296 k1,344:12359718,40901296:232484 x1,344:12577950,40901296 (1,344:12577950,40901296:655360,446948,9168 x1,344:13233310,40901296 ) x1,344:13451542,40901296 k1,344:13684027,40901296:232485 x1,344:14121138,40901296 k1,344:14353622,40901296:232484 x1,344:15263910,40901296 x1,344:16048374,40901296 k1,344:16280858,40901296:232484 x1,344:16790721,40901296 k1,344:17023205,40901296:232484 x1,344:18061284,40901296 k1,344:18293768,40901296:232484 x1,344:18949128,40901296 h1,344:18949128,40901296:0,0,0 x1,344:19112968,40901296 k1,344:19112968,40901296:0 ) (1,345:3729360,41687728:15383608,446948,141880 x1,345:4748440,41687728 k1,344:4906436,41687728:157996 x1,344:5525092,41687728 x1,344:7417746,41687728 x1,344:8509562,41687728 k1,344:8667559,41687728:157997 x1,344:9577847,41687728 x1,344:10362311,41687728 k1,344:10520307,41687728:157996 x1,344:11030170,41687728 k1,344:11188167,41687728:157997 x1,344:12135141,41687728 k1,344:12293137,41687728:157996 x1,344:12985193,41687728 x1,344:13822081,41687728 k1,344:13980078,41687728:157997 x1,344:14780262,41687728 k1,344:14938258,41687728:157996 x1,344:15556914,41687728 x1,344:17268041,41687728 k1,344:17426038,41687728:157997 x1,344:17935901,41687728 k1,344:18093897,41687728:157996 x1,344:19112968,41687728 k1,344:19112968,41687728:0 ) (1,345:3729360,42474160:15383608,446948,141880 x1,345:5004014,42474160 k1,344:5281184,42474160:277170 x1,344:5827096,42474160 k1,344:6104266,42474160:277170 x1,344:6905105,42474160 k1,344:7182275,42474160:277170 x1,344:8092563,42474160 x1,344:10312906,42474160 k1,344:10590076,42474160:277170 x1,344:11099284,42474160 x1,344:11609147,42474160 x1,344:11900123,42474160 x1,344:12919849,42474160 k1,344:13197019,42474160:277170 x1,344:14180059,42474160 x1,344:15436370,42474160 k1,344:15741871,42474160:305501 x1,344:17707278,42474160 k1,344:17984448,42474160:277170 x1,344:18894736,42474160 x1,344:19112968,42474160 k1,345:19112968,42474160:0 ) (1,345:3729360,43260592:15383608,446948,141880 x1,345:5076104,43260592 x1,345:5913647,43260592 k1,344:6166468,43260592:252821 x1,344:6967307,43260592 k1,344:7220128,43260592:252821 x1,344:8203168,43260592 x1,344:9295639,43260592 k1,344:9548460,43260592:252821 x1,344:10167116,43260592 x1,344:11914292,43260592 k1,344:12167113,43260592:252821 x1,344:12676976,43260592 k1,344:12929797,43260592:252821 x1,344:13730636,43260592 k1,344:13983457,43260592:252821 x1,344:15402952,43260592 x1,344:16239840,43260592 k1,344:16492661,43260592:252821 x1,344:16857027,43260592 x1,344:17967186,43260592 k1,344:18437289,43260592:470103 x1,344:18837713,43260592 x1,344:19112968,43260592 k1,344:19112968,43260592:0 ) (1,345:3729360,44047024:15383608,446948,141880 x1,345:5112151,44047024 x1,344:5423447,44047024 x1,344:5704599,44047024 k1,344:5870751,44047024:166152 x1,344:6671590,44047024 k1,344:6837742,44047024:166152 x1,344:7346950,44047024 x1,344:9021381,44047024 k1,344:9187534,44047024:166153 x1,344:9660693,44047024 x1,344:9971989,44047024 x1,344:10435324,44047024 k1,344:10601476,44047024:166152 x1,344:11147388,44047024 k1,344:11313540,44047024:166152 x1,344:12223828,44047024 x1,344:13315644,44047024 x1,344:14408115,44047024 k1,344:14574267,44047024:166152 x1,344:15084130,44047024 k1,344:15250283,44047024:166153 x1,344:16051122,44047024 k1,344:16217274,44047024:166152 x1,344:17200305,44047024 x1,344:18219376,44047024 k1,344:18385528,44047024:166152 x1,344:18603760,44047024 x1,344:19112968,44047024 k1,345:19112968,44047024:0 ) (1,345:3729360,44833456:15383608,446948,141880 x1,345:5313351,44833456 k1,344:5571718,44833456:258367 x1,344:5936084,44833456 k1,344:6175547,44833456:239463 x1,344:6648715,44833456 x1,344:7188066,44833456 k1,344:7427528,44833456:239462 x1,344:8337160,44833456 x1,344:8883064,44833456 x1,344:9974880,44833456 k1,344:10214343,44833456:239463 x1,344:10724206,44833456 k1,344:10963668,44833456:239462 x1,344:11837252,44833456 k1,344:12076714,44833456:239462 x1,344:12695370,44833456 k1,344:12934833,44833456:239463 x1,344:13553489,44833456 x1,344:15446143,44833456 x1,344:16283031,44833456 k1,344:16522493,44833456:239462 x1,344:17687707,44833456 k1,344:17927170,44833456:239463 x1,344:18218146,44833456 k1,344:18457608,44833456:239462 x1,344:19112968,44833456 k1,344:19112968,44833456:0 ) (1,345:3729360,45619888:15383608,446948,141880 x1,345:4749086,45619888 k1,344:4995906,45619888:246820 x1,344:5323586,45619888 x1,344:6102809,45619888 k1,344:6554911,45619888:452102 x1,344:7100823,45619888 k1,344:7347643,45619888:246820 x1,344:8148482,45619888 k1,344:8395303,45619888:246821 x1,344:9195487,45619888 k1,344:9442307,45619888:246820 x1,344:10388643,45619888 k1,344:10635463,45619888:246820 x1,344:12273845,45619888 k1,344:12520666,45619888:246821 x1,344:13940161,45619888 x1,344:15195817,45619888 k1,344:15463382,45619888:267565 x1,344:15754358,45619888 k1,344:16001178,45619888:246820 x1,344:16656538,45619888 k1,344:16903359,45619888:246821 x1,344:17923085,45619888 k1,344:18169905,45619888:246820 x1,344:18497585,45619888 x1,344:19112968,45619888 k1,344:19112968,45619888:0 ) (1,345:3729360,46406320:15383608,451272,141880 x1,345:4202528,46406320 x1,344:4741879,46406320 k1,344:4983083,46406320:241204 x1,344:5565691,46406320 x1,344:6131915,46406320 x1,344:7569753,46406320 k1,344:7830298,46406320:260545 x1,344:8157978,46406320 x1,344:8654737,46406320 k1,344:8895941,46406320:241204 x1,344:9186917,46406320 k1,344:9428121,46406320:241204 x1,344:10083481,46406320 $1,344:10083481,46406320(1,344:10083481,46168495:301284,213447,0x1,344:10351997,46168495)$1,344:10384765,46406320 k1,344:10625969,46406320:241204 x1,344:12009416,46406320 k1,344:12250620,46406320:241204 x1,344:13196956,46406320 k1,344:13438161,46406320:241205 x1,344:13948024,46406320 k1,344:14189228,46406320:241204 x1,344:14807884,46406320 k1,344:15049088,46406320:241204 x1,344:15558951,46406320 x1,344:16287046,46406320 x1,344:17852038,46406320 k1,344:18093242,46406320:241204 x1,344:19112968,46406320 k1,344:19112968,46406320:0 ) (1,342:3729360,47673364:15383608,453198,113504 (1,342:3729360,47673364:943720,453198,0 k1,342:4443704,47673364:714344 (1,342:4443704,47673364:229376,453198,0 $1,342:4443704,47673364(1,342:4443704,47488334:229376,268168,0x1,342:4640312,47488334)$1,342:4673080,47673364 ) ) (1,342:4673080,47673364:0,435814,0 r1,346:4673080,47673364:0,435814,0 ) x1,342:5488344,47673364 k1,342:5686009,47673364:197665 x1,342:7519950,47673364 k1,342:7717615,47673364:197665 x1,342:8154345,47673364 k1,342:8352010,47673364:197665 x1,342:9109591,47673364 k1,342:9307256,47673364:197665 x1,342:10850744,47673364 k1,342:11048409,47673364:197665 x1,342:11398099,47673364 k1,342:11595764,47673364:197665 x1,342:13663535,47673364 k1,342:13861200,47673364:197665 x1,342:14385488,47673364 k1,342:14583153,47673364:197665 x1,342:15136273,47673364 x1,342:15805784,47673364 k1,342:16003449,47673364:197665 x1,342:18041850,47673364 k1,342:18239515,47673364:197665 x1,342:19112968,47673364 k1,342:19112968,47673364:0 ) (1,342:3729360,48295956:15383608,357558,186779 x1,342:4748567,48295956 g1,342:4879639,48295956 x1,342:5112420,48295956 g1,342:5243492,48295956 x1,342:7719159,48295956 g1,342:7850231,48295956 x1,342:9742884,48295956 g1,342:9873956,48295956 x1,342:10136100,48295956 x1,342:11520734,48295956 g1,342:11651806,48295956 x1,342:13456914,48295956 g1,342:13587986,48295956 x1,342:14403245,48295956 x1,342:15227418,48295956 r1,346:15227418,48295956:0,186779,186779 k1,342:19112968,48295956:3885550 g1,342:19112968,48295956 ) ] g1,386:19112968,48295956 ) k1,386:19579138,48295956:466170 r1,386:19579138,48295956:0,44938783,0 k1,386:20045308,48295956:466170 (1,386:20045308,48295956:15383608,44938783,0 [1,386:20045308,48295956:15383608,44938783,0 (1,345:20045308,4012533:15383608,446948,141880 x1,345:20846147,4012533 k1,344:21008810,4012533:162663 x1,344:21627466,4012533 x1,344:23520120,4012533 x1,344:24357008,4012533 k1,344:24519671,4012533:162663 x1,344:25756983,4012533 k1,344:25919646,4012533:162663 x1,344:26392814,4012533 x1,344:27460371,4012533 k1,344:27623269,4012533:162898 x1,344:28169173,4012533 k1,344:28331836,4012533:162663 x1,344:29242124,4012533 x1,344:30026588,4012533 k1,344:30189252,4012533:162664 x1,344:30699115,4012533 k1,344:30861778,4012533:162663 x1,344:31899857,4012533 k1,344:32062520,4012533:162663 x1,344:32717880,4012533 h1,344:32717880,4012533:0,0,0 x1,344:32881720,4012533 k1,344:33044618,4012533:162898 x1,344:33554481,4012533 k1,344:33717144,4012533:162663 x1,344:34227007,4012533 x1,344:35428916,4012533 k1,344:35428916,4012533:0 ) (1,345:20045308,4798965:15383608,446948,141880 x1,345:21064371,4798965 x1,345:21537530,4798965 x1,345:22593305,4798965 g1,344:22757145,4798965 x1,344:23594688,4798965 g1,344:23758528,4798965 x1,344:24304440,4798965 g1,344:24468280,4798965 x1,344:25269119,4798965 g1,344:25432959,4798965 x1,344:26051615,4798965 x1,344:27944269,4798965 x1,344:28781157,4798965 x1,344:28908957,4798965 k1,345:35428916,4798965:6519959 g1,345:35428916,4798965 ) (1,349:20045308,5751681:15383608,446948,141880 h1,346:20045308,5751681:655361,0,0 x1,346:21719749,5751681 k1,346:22374286,5751681:654537 x1,346:23357326,5751681 x1,346:24194869,5751681 k1,346:24849405,5751681:654536 x1,346:25359268,5751681 x1,346:26160107,5751681 x1,346:27107081,5751681 k1,346:27761618,5751681:654537 x1,346:28526409,5751681 x1,346:29109007,5751681 x1,346:29436687,5751681 x1,346:29751263,5751681 x1,346:31025917,5751681 k1,346:31680453,5751681:654536 x1,346:32517341,5751681 x1,346:34119019,5751681 k1,346:34773556,5751681:654537 x1,346:35428916,5751681 k1,349:35428916,5751681:0 ) (1,349:20045308,6538113:15383608,446948,145636 x1,349:22797796,6538113 k1,346:23101084,6538113:303288 x1,346:23865228,6538113 k1,346:24168657,6538113:303429 x1,346:25297176,6538113 x1,346:26571174,6538113 k1,347:26874603,6538113:303429 x1,347:27347762,6538113 x1,347:27638738,6538113 x1,347:29204368,6538113 k1,347:29507797,6538113:303429 x1,347:30017660,6538113 k1,347:30321089,6538113:303429 x1,347:31267425,6538113 k1,347:31570854,6538113:303429 x1,347:32116766,6538113 k1,347:32420195,6538113:303429 x1,347:33221034,6538113 k1,347:33524463,6538113:303429 x1,347:34579575,6538113 k1,347:34883004,6538113:303429 x1,347:35428916,6538113 k1,347:35428916,6538113:0 ) (1,349:20045308,7324545:15383608,446948,141880 x1,349:20846147,7324545 k1,347:21097843,7324545:251696 x1,347:22554042,7324545 x1,347:23391585,7324545 k1,347:23643281,7324545:251696 x1,347:24699056,7324545 k1,347:24950752,7324545:251696 x1,347:25750936,7324545 k1,347:26002631,7324545:251695 x1,347:26912919,7324545 x1,347:27697383,7324545 k1,347:27949079,7324545:251696 x1,347:28458942,7324545 k1,347:28710638,7324545:251696 x1,347:29748717,7324545 k1,347:30000413,7324545:251696 x1,347:30655773,7324545 h1,347:30655773,7324545:0,0,0 x1,347:30819613,7324545 k1,347:31286340,7324545:466727 x1,347:32305420,7324545 k1,347:32557116,7324545:251696 x1,347:33175772,7324545 x1,347:34631308,7324545 k1,347:34883004,7324545:251696 x1,347:35428916,7324545 k1,347:35428916,7324545:0 ) (1,349:20045308,8110977:15383608,446948,141880 x1,349:21028348,8110977 x1,349:22120819,8110977 k1,347:22285500,8110977:164681 x1,347:22758659,8110977 k1,347:22923340,8110977:164681 x1,347:24051204,8110977 k1,347:24215885,8110977:164681 x1,347:25162221,8110977 k1,347:25326902,8110977:164681 $1,347:25326902,8110977x1,347:25700475,8110977$1,347:25723988,8110977 k1,347:25888669,8110977:164681 x1,347:26397877,8110977 x1,347:26907740,8110977 x1,347:27781324,8110977 k1,347:27946004,8110977:164680 x1,347:28746188,8110977 k1,347:28910869,8110977:164681 x1,347:29857205,8110977 k1,347:30021886,8110977:164681 x1,347:30531749,8110977 k1,347:30696430,8110977:164681 x1,347:31497269,8110977 k1,347:31661950,8110977:164681 x1,347:32280606,8110977 x1,347:34173260,8110977 x1,347:35428916,8110977 k1,348:35428916,8110977:0 ) (1,349:20045308,8897409:15383608,446948,141880 x1,349:21064388,8897409 k1,348:21221818,8897409:157430 x1,348:22277593,8897409 x1,348:24134862,8897409 k1,348:24292293,8897409:157431 x1,348:25675740,8897409 x1,348:26476579,8897409 x1,348:27204674,8897409 k1,348:27362104,8897409:157430 x1,348:27799215,8897409 k1,348:27956646,8897409:157431 x1,348:28902982,8897409 k1,348:29060412,8897409:157430 x1,348:29570275,8897409 k1,348:29727706,8897409:157431 x1,348:30528545,8897409 k1,348:30685975,8897409:157430 x1,348:31304631,8897409 x1,348:33197285,8897409 x1,348:34034173,8897409 k1,348:34191604,8897409:157431 x1,348:35428916,8897409 k1,348:35428916,8897409:0 ) (1,349:20045308,9683841:15383608,446948,141880 x1,349:20518476,9683841 x1,348:21167265,9683841 g1,348:21331105,9683841 x1,348:22878392,9683841 k1,349:35428916,9683841:12550524 g1,349:35428916,9683841 ) (1,349:20045308,11779124:15383608,538305,162393 (1,349:20045308,11779124:0,0,0 g1,349:20045308,11779124 ) x1,349:23364815,11779124 g1,349:23561423,11779124 x1,349:25352090,11779124 x1,349:26103136,11779124 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(1,350:22204680,13881406:655360,446948,0 x1,350:22860040,13881406 ) x1,350:23078272,13881406 k1,350:23308284,13881406:230012 x1,350:23818147,13881406 k1,350:24048160,13881406:230013 x1,350:25248768,13881406 k1,350:25478780,13881406:230012 x1,350:26607299,13881406 x1,350:27881297,13881406 k1,350:28111310,13881406:230013 x1,350:29276525,13881406 k1,350:29506537,13881406:230012 x1,350:29724769,13881406 x1,350:30015745,13881406 x1,350:31171144,13881406 k1,351:31401157,13881406:230013 x1,351:31911020,13881406 k1,351:32141032,13881406:230012 x1,351:33051320,13881406 x1,351:34398064,13881406 k1,351:34628077,13881406:230013 x1,351:35428916,13881406 k1,351:35428916,13881406:0 ) (1,363:20045308,14667838:15383608,446948,141880 x1,363:20372988,14667838 x1,351:22034970,14667838 k1,351:22263160,14667838:228190 x1,351:22809072,14667838 k1,351:23037263,14667838:228191 x1,351:23838102,14667838 k1,351:24066292,14667838:228190 x1,351:25121403,14667838 x1,351:26177178,14667838 k1,351:26405369,14667838:228191 x1,351:27861568,14667838 k1,351:28089758,14667838:228190 x1,351:28599621,14667838 k1,351:28827812,14667838:228191 x1,351:29628651,14667838 k1,352:29856841,14667838:228190 x1,352:30184521,14667838 x1,352:30750089,14667838 x1,352:31915312,14667838 k1,352:32143503,14667838:228191 x1,352:33562998,14667838 x1,352:34654814,14667838 k1,352:34883004,14667838:228190 x1,352:35428916,14667838 k1,352:35428916,14667838:0 ) (1,363:20045308,15454270:15383608,446948,141880 x1,363:20846147,15454270 k1,352:21064956,15454270:218809 x1,352:22284571,15454270 k1,352:22652639,15454270:368068 x1,352:23671719,15454270 k1,352:23890529,15454270:218810 x1,352:24764758,15454270 x1,352:24946941,15454270 x1,352:25274621,15454270 x1,352:25562324,15454270 x1,352:26582050,15454270 k1,352:26800859,15454270:218809 x1,352:28111561,15454270 x1,352:29021849,15454270 k1,352:29240658,15454270:218809 x1,352:30514010,15454270 k1,353:30732819,15454270:218809 x1,353:31278723,15454270 x1,353:32843697,15454270 k1,353:33062507,15454270:218810 x1,353:34554747,15454270 k1,353:34773556,15454270:218809 x1,353:35428916,15454270 k1,353:35428916,15454270:0 ) (1,363:20045308,16240702:15383608,446948,141880 x1,363:20846147,16240702 k1,353:21139076,16240702:292929 x1,353:21648284,16240702 x1,353:22850186,16240702 k1,353:23143116,16240702:292930 x1,353:23689028,16240702 k1,353:23981957,16240702:292929 x1,353:24600613,16240702 k1,353:24893543,16240702:292930 x1,353:25403406,16240702 x1,353:26750149,16240702 k1,353:27043078,16240702:292929 x1,353:27334054,16240702 x1,353:27651910,16240702 x1,353:28270566,16240702 x1,353:28670981,16240702 x1,353:29799500,16240702 x1,353:30454842,16240702 k1,353:30747771,16240702:292929 x1,353:31256979,16240702 x1,353:32930747,16240702 k1,353:33223677,16240702:292930 x1,353:33514653,16240702 x1,353:34662828,16240702 k1,354:34955757,16240702:292929 x1,354:35428916,16240702 k1,354:35428916,16240702:0 ) (1,363:20045308,17027134:15383608,446948,141880 x1,363:21724196,17027134 x1,354:22300260,17027134 k1,354:22482051,17027134:181791 x1,354:24265931,17027134 k1,354:24447704,17027134:181773 x1,354:25394040,17027134 k1,354:25575831,17027134:181791 x1,354:26923239,17027134 x1,354:27523543,17027134 k1,354:27705334,17027134:181791 x1,354:28251246,17027134 k1,354:28433037,17027134:181791 x1,354:29233876,17027134 k1,354:29415667,17027134:181791 x1,354:30215851,17027134 k1,354:30397642,17027134:181791 x1,354:30615874,17027134 x1,354:30906850,17027134 x1,354:32317177,17027134 k1,354:32498968,17027134:181791 x1,354:33044880,17027134 k1,354:33226671,17027134:181791 x1,354:34027510,17027134 k1,354:34209301,17027134:181791 x1,354:35428916,17027134 k1,354:35428916,17027134:0 ) (1,363:20045308,17813566:15383608,446948,141880 x1,363:20846147,17813566 k1,354:21038471,17813566:192324 x1,354:21912063,17813566 x1,354:22931789,17813566 k1,355:23124112,17813566:192323 x1,355:24033744,17813566 x1,355:25271047,17813566 k1,355:25463371,17813566:192324 x1,355:25791051,17813566 x1,355:26356619,17813566 x1,355:27521842,17813566 k1,355:27714166,17813566:192324 x1,355:29133661,17813566 x1,355:30225477,17813566 k1,355:30417800,17813566:192323 x1,355:31109846,17813566 x1,355:32056164,17813566 k1,355:32248488,17813566:192324 x1,355:32648912,17813566 x1,355:33499558,17813566 x1,355:35428916,17813566 k1,355:35428916,17813566:0 ) (1,363:20045308,18599998:15383608,446948,141880 x1,363:20555171,18599998 k1,355:20769618,18599998:214447 x1,355:21679906,18599998 x1,355:23026650,18599998 k1,356:23241097,18599998:214447 x1,356:24041936,18599998 k1,356:24256383,18599998:214447 x1,356:25093263,18599998 x1,356:26913191,18599998 k1,356:27127638,18599998:214447 x1,356:28182749,18599998 x1,356:29238524,18599998 k1,356:29452971,18599998:214447 x1,356:30763673,18599998 k1,356:30978120,18599998:214447 x1,356:31524032,18599998 k1,356:31738479,18599998:214447 x1,356:32539318,18599998 k1,356:32753765,18599998:214447 x1,356:34173260,18599998 x1,356:35428916,18599998 k1,356:35428916,18599998:0 ) (1,363:20045308,19386430:15383608,446948,141880 x1,363:20591220,19386430 k1,356:20796773,19386430:205553 x1,356:21597612,19386430 k1,356:21803166,19386430:205554 x1,356:23860316,19386430 k1,356:24076297,19386430:215981 x1,356:24877136,19386430 k1,356:25082690,19386430:205554 x1,356:26137801,19386430 x1,356:27193576,19386430 k1,357:27399129,19386430:205553 x1,357:28855328,19386430 k1,357:29060882,19386430:205554 x1,357:29497993,19386430 k1,357:29703546,19386430:205553 x1,357:30249458,19386430 x1,357:31486761,19386430 k1,357:31692315,19386430:205554 x1,357:32602603,19386430 x1,357:34568003,19386430 k1,357:34773556,19386430:205553 x1,357:35428916,19386430 k1,357:35428916,19386430:0 ) (1,363:20045308,20172862:15383608,446948,141880 x1,363:20846147,20172862 k1,357:21029825,20172862:183678 x1,357:21903409,20172862 k1,357:22087087,20172862:183678 x1,357:22632999,20172862 k1,357:22816677,20172862:183678 x1,357:23654220,20172862 x1,357:24164083,20172862 x1,357:24673946,20172862 x1,357:25183809,20172862 x1,357:25984648,20172862 k1,357:26168326,20172862:183678 x1,357:27369598,20172862 x1,357:27687454,20172862 x1,357:28223534,20172862 k1,357:28407212,20172862:183678 x1,357:29062572,20172862 k1,358:29246249,20172862:183677 x1,358:30047088,20172862 k1,358:30230766,20172862:183678 x1,358:31068309,20172862 k1,358:31251987,20172862:183678 x1,358:32198323,20172862 k1,358:32382001,20172862:183678 x1,358:33219544,20172862 x1,358:34239270,20172862 k1,358:34422948,20172862:183678 x1,358:35005556,20172862 x1,358:35223788,20172862 x1,358:35428916,20172862 k1,363:35428916,20172862:0 ) (1,363:20045308,20959294:15383608,446948,141880 x1,363:20263540,20959294 x1,363:21548683,20959294 k1,358:21739186,20959294:190503 x1,358:22285098,20959294 k1,358:22410967,20959294:125869 x1,358:23211806,20959294 k1,358:23337674,20959294:125868 x1,358:24320705,20959294 x1,358:25339776,20959294 k1,358:25465644,20959294:125868 x1,358:25683876,20959294 x1,358:25974852,20959294 x1,358:27294091,20959294 k1,358:27427553,20959294:133462 x1,358:27718529,20959294 k1,359:27844398,20959294:125869 x1,359:28717990,20959294 x1,359:29737716,20959294 k1,359:29863584,20959294:125868 x1,359:30773216,20959294 x1,359:32010519,20959294 k1,359:32136387,20959294:125868 x1,359:33555882,20959294 x1,359:34392770,20959294 k1,359:34518638,20959294:125868 x1,359:35210684,20959294 x1,359:35428916,20959294 k1,363:35428916,20959294:0 ) (1,363:20045308,21745726:15383608,446948,141880 x1,363:21246554,21745726 k1,359:21384110,21745726:137556 x1,359:22439221,21745726 x1,359:23494996,21745726 k1,359:23632553,21745726:137557 x1,359:24687664,21745726 x1,359:26289343,21745726 k1,359:26426899,21745726:137556 x1,359:26936762,21745726 k1,359:27074318,21745726:137556 x1,359:27984606,21745726 x1,359:29331350,21745726 k1,359:29468907,21745726:137557 x1,359:30342499,21745726 x1,359:31309146,21745726 k1,359:31446702,21745726:137556 x1,359:31992614,21745726 k1,359:32130170,21745726:137556 x1,359:32931009,21745726 k1,359:33068566,21745726:137557 x1,359:34688605,21745726 k1,359:34883004,21745726:194399 x1,359:35428916,21745726 k1,359:35428916,21745726:0 ) (1,363:20045308,22532158:15383608,446948,141880 x1,363:20846147,22532158 k1,359:20997755,22532158:151608 x1,359:23218728,22532158 k1,359:23370335,22532158:151607 x1,359:23916247,22532158 k1,359:24067855,22532158:151608 x1,359:24868694,22532158 k1,359:25020301,22532158:151607 x1,359:26239916,22532158 k1,359:26393971,22532158:154055 x1,359:27194810,22532158 k1,359:27346417,22532158:151607 x1,359:28220009,22532158 x1,359:29239735,22532158 k1,359:29391343,22532158:151608 x1,359:30300975,22532158 x1,359:31538278,22532158 k1,359:31689885,22532158:151607 x1,359:33218837,22532158 k1,359:33370445,22532158:151608 x1,359:33698125,22532158 x1,359:34263693,22532158 x1,359:35428916,22532158 k1,359:35428916,22532158:0 ) (1,363:20045308,23318590:15383608,446948,141880 x1,363:21464803,23318590 x1,363:22301691,23318590 k1,359:22443673,23318590:141982 x1,359:22880784,23318590 k1,359:23022767,23318590:141983 x1,359:23423191,23318590 x1,359:24273837,23318590 x1,359:25984308,23318590 k1,359:26126290,23318590:141982 x1,359:26890434,23318590 k1,360:27032416,23318590:141982 x1,360:27869296,23318590 x1,360:29689224,23318590 k1,360:29831206,23318590:141982 x1,360:30741494,23318590 x1,360:31833310,23318590 x1,360:33089621,23318590 k1,360:33285496,23318590:195875 x1,360:34559494,23318590 k1,360:34701476,23318590:141982 x1,360:34919708,23318590 x1,360:35428916,23318590 k1,363:35428916,23318590:0 ) (1,363:20045308,24105022:15383608,446948,141880 x1,363:21465459,24105022 k1,360:21605112,24105022:139653 x1,360:23643927,24105022 x1,360:24481470,24105022 k1,360:24621123,24105022:139653 x1,360:25421962,24105022 k1,361:25561615,24105022:139653 x1,361:26617390,24105022 k1,361:26757043,24105022:139653 x1,361:27776760,24105022 k1,361:27916413,24105022:139653 x1,361:28717252,24105022 x1,361:29554795,24105022 x1,361:29875922,24105022 k1,361:30015575,24105022:139653 x1,361:32236565,24105022 k1,361:32376218,24105022:139653 x1,361:33686929,24105022 x1,361:34779400,24105022 k1,361:34919053,24105022:139653 x1,361:35428916,24105022 k1,361:35428916,24105022:0 ) (1,363:20045308,24891454:15383608,446948,141880 x1,363:20591220,24891454 x1,363:21828523,24891454 g1,361:21992363,24891454 x1,361:22902651,24891454 x1,361:24249395,24891454 g1,361:24413235,24891454 x1,361:25869434,24891454 g1,362:26033274,24891454 x1,362:26360954,24891454 x1,362:28022936,24891454 g1,362:28186776,24891454 x1,362:29314640,24891454 g1,362:29478480,24891454 x1,362:30279319,24891454 g1,362:30443159,24891454 x1,362:32026494,24891454 k1,363:35428916,24891454:3402422 g1,363:35428916,24891454 ) (1,364:20045308,26986737:15383608,538305,162393 (1,364:20045308,26986737:0,0,0 g1,364:20045308,26986737 ) x1,364:23103693,26986737 g1,364:23300301,26986737 x1,364:23912137,26986737 g1,364:24108745,26986737 x1,364:25069752,26986737 g1,364:25266360,26986737 x1,364:28062877,26986737 g1,364:28259485,26986737 x1,364:30050152,26986737 x1,364:30801198,26986737 g1,364:30997806,26986737 x1,364:32745245,26986737 g1,364:32941853,26986737 x1,364:35169781,26986737 k1,364:35428916,26986737:259135 g1,364:35428916,26986737 ) (1,366:20045308,28302587:15383608,446948,141880 h1,365:20045308,28302587:655361,0,0 x1,365:22266308,28302587 k1,365:22563898,28302587:297590 x1,365:23364737,28302587 k1,365:23662326,28302587:297589 x1,365:24645366,28302587 x1,365:25482909,28302587 k1,365:25780499,28302587:297590 x1,365:26545290,28302587 x1,365:27127888,28302587 x1,365:27455568,28302587 x1,365:27770144,28302587 x1,365:29044798,28302587 k1,365:29342388,28302587:297590 x1,365:30252676,28302587 x1,365:31037140,28302587 k1,365:31334729,28302587:297589 x1,365:31844592,28302587 k1,365:32142182,28302587:297590 x1,365:33180261,28302587 k1,365:33477851,28302587:297590 x1,365:34133211,28302587 h1,365:34133211,28302587:0,0,0 x1,365:34297051,28302587 k1,365:34628077,28302587:331026 x1,365:35428916,28302587 k1,365:35428916,28302587:0 ) (1,366:20045308,29089019:15383608,446948,141880 x1,366:21100419,29089019 x1,366:22156194,29089019 k1,365:22614852,29089019:458658 x1,365:24071051,29089019 k1,365:24529709,29089019:458658 x1,365:24857389,29089019 x1,365:26519371,29089019 k1,365:26978029,29089019:458658 x1,365:28470269,29089019 k1,365:28928928,29089019:458659 x1,365:31501195,29089019 k1,365:31959853,29089019:458658 x1,365:34024219,29089019 k1,365:34482580,29089019:458361 x1,365:35428916,29089019 k1,366:35428916,29089019:0 ) (1,366:20045308,29875451:15383608,446948,141880 x1,366:23141857,29875451 k1,365:23438971,29875451:297114 x1,365:24057627,29875451 x1,365:24372203,29875451 x1,365:24653355,29875451 k1,365:24950604,29875451:297249 x1,365:26187916,29875451 k1,365:26485165,29875451:297249 x1,365:27686429,29875451 k1,365:27983677,29875451:297248 x1,365:28493540,29875451 k1,365:28790789,29875451:297249 x1,365:29336693,29875451 x1,365:29701059,29875451 x1,365:30975057,29875451 k1,365:31272305,29875451:297248 x1,365:32073144,29875451 k1,365:32370393,29875451:297249 x1,365:33498912,29875451 x1,365:34190958,29875451 x1,365:35428916,29875451 k1,365:35428916,29875451:0 ) (1,366:20045308,30661883:15383608,446948,141880 x1,366:21100419,30661883 x1,366:22156194,30661883 k1,365:22364857,30661883:208663 x1,365:23821056,30661883 k1,365:24029718,30661883:208662 x1,365:24976054,30661883 x1,365:25923028,30661883 k1,365:26131691,30661883:208663 x1,365:26459371,30661883 x1,365:28121353,30661883 k1,365:28330016,30661883:208663 x1,365:28839879,30661883 k1,365:29048542,30661883:208663 x1,365:29558405,30661883 x1,365:30269461,30661883 x1,365:31288532,30661883 k1,365:31497194,30661883:208662 x1,365:32625713,30661883 x1,365:33899711,30661883 k1,365:34108374,30661883:208663 x1,365:34399350,30661883 x1,365:34727030,30661883 x1,365:35210684,30661883 x1,365:35428916,30661883 k1,366:35428916,30661883:0 ) (1,366:20045308,31448315:15383608,446948,141880 x1,366:20918900,31448315 x1,366:22484530,31448315 k1,365:22700130,31448315:215600 x1,365:23209993,31448315 k1,365:23425593,31448315:215600 x1,365:24226432,31448315 k1,365:24442032,31448315:215600 x1,365:25497808,31448315 k1,365:25713356,31448315:215548 x1,365:27169555,31448315 x1,365:28007098,31448315 k1,365:28222697,31448315:215599 x1,365:29442312,31448315 k1,365:29800752,31448315:358440 x1,365:30346664,31448315 k1,365:30562264,31448315:215600 x1,365:31762872,31448315 k1,365:31978472,31448315:215600 x1,365:32269448,31448315 x1,365:32597128,31448315 x1,365:33080782,31448315 x1,365:33954374,31448315 x1,365:35428916,31448315 k1,365:35428916,31448315:0 ) (1,366:20045308,32234747:15383608,446948,141880 x1,366:20846147,32234747 k1,365:20992449,32234747:146302 x1,365:21611105,32234747 x1,365:22484697,32234747 x1,365:23212792,32234747 x1,365:23976927,32234747 k1,365:24123228,32234747:146301 x1,365:24960108,32234747 x1,365:26780036,32234747 k1,365:26926338,32234747:146302 x1,365:27981449,32234747 x1,365:29037224,32234747 k1,365:29183526,32234747:146302 x1,365:30057755,32234747 x1,365:30239938,32234747 x1,365:30567618,32234747 x1,365:30855321,32234747 x1,365:31875047,32234747 k1,365:32021349,32234747:146302 x1,365:33113164,32234747 x1,365:34733203,32234747 k1,365:34883012,32234747:149809 x1,365:35428916,32234747 k1,365:35428916,32234747:0 ) (1,366:20045308,33021179:15383608,446948,141880 x1,366:21209868,33021179 x1,366:21392051,33021179 x1,366:22375083,33021179 k1,365:22626180,33021179:251097 x1,365:23136043,33021179 k1,365:23387141,33021179:251098 x1,365:24333469,33021179 x1,365:24624445,33021179 x1,365:26043285,33021179 k1,365:26294382,33021179:251097 x1,365:26512614,33021179 (1,365:26512614,33021179:655360,446948,0 x1,365:27167974,33021179 ) x1,365:27550046,33021179 k1,365:27822957,33021179:272911 x1,365:28696541,33021179 k1,365:28947638,33021179:251097 x1,365:30184950,33021179 k1,365:30436048,33021179:251098 x1,365:30981952,33021179 x1,365:32856919,33021179 k1,365:33321851,33021179:464932 x1,365:34340931,33021179 k1,365:34592028,33021179:251097 x1,365:35210684,33021179 x1,365:35428916,33021179 k1,366:35428916,33021179:0 ) (1,366:20045308,33807611:15383608,446948,141880 x1,366:21501489,33807611 g1,365:21665329,33807611 x1,365:22174537,33807611 x1,365:23376439,33807611 g1,365:23540279,33807611 x1,365:24340463,33807611 g1,365:24504303,33807611 x1,365:25341191,33807611 x1,365:27015614,33807611 g1,365:27179454,33807611 x1,365:27689317,33807611 g1,365:27853157,33807611 x1,365:28799485,33807611 x1,365:29090461,33807611 x1,365:30509301,33807611 g1,365:30673141,33807611 x1,365:30891373,33807611 (1,365:30891373,33807611:655360,446948,9168 x1,365:31546733,33807611 ) x1,365:31928805,33807611 k1,366:35428916,33807611:3500111 g1,366:35428916,33807611 ) (1,369:20045308,34760328:15383608,446948,141880 h1,367:20045308,34760328:655361,0,0 x1,367:21246581,34760328 k1,367:21485575,34760328:238994 x1,367:22286414,34760328 k1,367:22525408,34760328:238994 x1,367:23180768,34760328 k1,367:23419763,34760328:238995 x1,367:23929626,34760328 k1,367:24168620,34760328:238994 x1,367:24714532,34760328 k1,367:24953526,34760328:238994 x1,367:25936557,34760328 x1,367:27374396,34760328 k1,367:27632178,34760328:257782 x1,367:28433017,34760328 k1,367:28672012,34760328:238995 x1,367:29727123,34760328 x1,367:30782898,34760328 k1,367:31021892,34760328:238994 x1,367:32077003,34760328 x1,367:33678682,34760328 k1,367:33917676,34760328:238994 x1,367:34208652,34760328 x1,367:35210684,34760328 x1,367:35428916,34760328 k1,369:35428916,34760328:0 ) (1,369:20045308,35546760:15383608,446948,141880 x1,369:20409674,35546760 x1,369:20720970,35546760 x1,369:21511985,35546760 k1,367:21696307,35546760:184322 x1,367:22205515,35546760 x1,367:23442827,35546760 k1,367:23627148,35546760:184321 x1,367:24427987,35546760 k1,367:24612309,35546760:184322 x1,367:26068508,35546760 k1,367:26252829,35546760:184321 x1,367:27199165,35546760 x1,367:28146139,35546760 k1,367:28330461,35546760:184322 x1,367:28840324,35546760 k1,367:29024645,35546760:184321 x1,367:29315621,35546760 k1,367:29499943,35546760:184322 x1,367:30118599,35546760 x1,367:30436455,35546760 x1,367:30800821,35546760 x1,367:31310684,35546760 x1,367:32111523,35546760 k1,367:32295844,35546760:184321 x1,367:32769003,35546760 x1,367:33080299,35546760 x1,367:33707474,35546760 k1,367:33972079,35546760:264605 x1,367:35028501,35546760 x1,367:35210684,35546760 x1,367:35428916,35546760 k1,369:35428916,35546760:0 ) (1,369:20045308,36333192:15383608,446948,141880 x1,369:21246562,36333192 x1,369:21367810,36333192 k1,367:21541142,36333192:173332 x1,367:22341981,36333192 k1,367:22513415,36333192:171434 x1,367:22913839,36333192 x1,367:23546253,36333192 k1,367:23717680,36333192:171427 x1,367:24809495,36333192 x1,367:25647038,36333192 k1,367:25818472,36333192:171434 x1,367:26764808,36333192 k1,367:26936242,36333192:171434 x1,367:27737081,36333192 k1,367:27908515,36333192:171434 x1,367:30384433,36333192 k1,367:30555867,36333192:171434 x1,367:31065730,36333192 k1,367:31237164,36333192:171434 x1,367:32038003,36333192 k1,367:32209437,36333192:171434 x1,367:33337947,36333192 k1,367:33509381,36333192:171434 x1,367:33727613,36333192 x1,367:34018589,36333192 x1,367:35428916,36333192 k1,367:35428916,36333192:0 ) (1,369:20045308,37119624:15383608,446948,141880 x1,369:20773403,37119624 x1,369:21428745,37119624 k1,367:21564549,37119624:135804 x1,367:23202931,37119624 k1,367:23338735,37119624:135804 $1,367:23338735,37119624x1,367:23848461,37119624$1,367:23848461,37119624 x1,367:25159181,37119624 k1,367:25294985,37119624:135804 x1,367:26023080,37119624 k1,367:26158884,37119624:135804 x1,367:27214651,37119624 k1,367:27350456,37119624:135805 x1,367:28260744,37119624 x1,367:29607488,37119624 k1,367:29743292,37119624:135804 x1,367:30544131,37119624 k1,367:30679935,37119624:135804 x1,367:32136134,37119624 k1,367:32271938,37119624:135804 x1,367:33218274,37119624 x1,367:34165248,37119624 k1,367:34301052,37119624:135804 x1,367:35428916,37119624 k1,367:35428916,37119624:0 ) (1,369:20045308,37906056:15383608,446948,141880 x1,369:20846147,37906056 k1,367:21091168,37906056:245021 x1,367:22510663,37906056 k1,367:22755684,37906056:245021 x1,367:23265547,37906056 k1,367:23510568,37906056:245021 x1,367:24857311,37906056 k1,367:25102332,37906056:245021 x1,367:25320564,37906056 x1,367:25611540,37906056 x1,367:27021867,37906056 k1,367:27266888,37906056:245021 x1,367:28577590,37906056 k1,367:28822611,37906056:245021 x1,367:29113587,37906056 x1,367:30115619,37906056 x1,367:30479985,37906056 x1,367:30791281,37906056 x1,367:31582296,37906056 x1,367:31703544,37906056 k1,367:32150247,37906056:446703 x1,367:33169327,37906056 k1,367:33414348,37906056:245021 x1,367:33705324,37906056 x1,367:34019900,37906056 x1,367:34301052,37906056 x1,367:35428916,37906056 k1,367:35428916,37906056:0 ) (1,369:20045308,38692488:15383608,446948,141880 x1,369:21501507,38692488 k1,367:21745159,38692488:243652 x1,367:22691495,38692488 x1,367:23638469,38692488 k1,367:23882122,38692488:243653 x1,367:25557199,38692488 k1,367:25800851,38692488:243652 $1,367:25800851,38692488x1,367:26830694,38692488$1,367:26854207,38692488 k1,367:27097860,38692488:243653 x1,367:27643772,38692488 k1,367:27887424,38692488:243652 x1,367:28688263,38692488 k1,367:28931916,38692488:243653 x1,367:30351411,38692488 k1,367:30595063,38692488:243652 x1,367:31104926,38692488 k1,367:31348579,38692488:243653 x1,367:32149418,38692488 k1,367:32393070,38692488:243652 $1,367:32393070,38692488x1,367:32902796,38692488$1,367:32902796,38692488 x1,367:33885836,38692488 k1,367:34129489,38692488:243653 x1,367:34639352,38692488 k1,367:34883004,38692488:243652 x1,367:35428916,38692488 k1,368:35428916,38692488:0 ) (1,369:20045308,39478920:15383608,533465,141880 x1,369:21938618,39478920 k1,368:22146629,39478920:208011 x1,368:22583740,39478920 k1,368:22791750,39478920:208010 $1,368:22791750,39478920x1,368:23629157,39478920k1,368:23807507,39478920:178350x1,368:24317233,39478920k1,368:24495584,39478920:178351x1,368:25150946,39478920(1,368:25150946,39241095:555238,295640,0x1,368:25673416,39241095)$1,368:25706184,39478920 k1,368:25914194,39478920:208010 x1,368:26424057,39478920 $1,368:26424057,39478920(1,368:26424057,39241095:703604,295640,54614x1,368:27094893,39241095)$1,368:27127661,39478920 k1,368:27335672,39478920:208011 x1,368:28099808,39478920 x1,368:29192279,39478920 x1,368:30029822,39478920 k1,368:30237832,39478920:208010 x1,368:30893192,39478920 k1,368:31101203,39478920:208011 x1,368:32302475,39478920 x1,368:32620331,39478920 x1,368:33538483,39478920 k1,368:33874154,39478920:335671 x1,368:34420066,39478920 k1,368:34628077,39478920:208011 x1,368:35428916,39478920 k1,368:35428916,39478920:0 ) (1,369:20045308,40265352:15383608,446948,141880 x1,369:20555034,40265352 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x1,368:32085268,41051784 $1,368:32108781,41051784k1,368:32293411,41051784:184630x1,368:32839323,41051784k1,368:33023952,41051784:184629x1,368:33824791,41051784k1,368:34009421,41051784:184630x1,368:35428916,41051784k1,368:35428916,41051784:0)(1,369:20045308,41838216:15383608,533465,141880x1,369:20482419,41838216k1,368:20688718,41838216:206299x1,368:22144917,41838216k1,368:22351215,41838216:206298$1,368:22351215,41838216 x1,368:23188622,41838216 k1,368:23365705,41838216:177083 x1,368:23875431,41838216 k1,368:24052513,41838216:177082 x1,368:24707875,41838216 (1,368:24707875,41600391:555238,295640,0 x1,368:25230345,41600391 ) $1,368:25263113,41838216k1,368:25469412,41838216:206299x1,368:25979275,41838216$1,368:25979275,41838216 (1,368:25979275,41600391:703604,295640,54614 x1,368:26650111,41600391 ) 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k1,372:9202716,27519659:230644 x1,372:9712579,27519659 k1,372:9943222,27519659:230643 x1,372:11308334,27519659 k1,372:11711905,27519659:403571 x1,372:13186457,27519659 k1,372:13433801,27519659:247344 x1,372:14234640,27519659 k1,372:14465284,27519659:230644 x1,372:16066963,27519659 x1,372:16904506,27519659 k1,372:17135149,27519659:230643 x1,372:18263668,27519659 k1,372:18494312,27519659:230644 x1,372:19112968,27519659 k1,372:19112968,27519659:0 ) (1,386:3729360,28306091:15383608,533465,54613 x1,386:4275264,28306091 k1,372:4473226,28306091:197962 x1,372:4983089,28306091 x1,372:5439873,28306091 k1,372:5637834,28306091:197961 x1,372:6183738,28306091 k1,372:6381700,28306091:197962 $1,372:6381700,28306091x1,372:6891426,28306091k1,372:7136653,28306091:245227x1,372:7974060,28306091k1,372:8144967,28306091:170907x1,372:8654693,28306091k1,372:8825600,28306091:170907x1,372:9480962,28306091(1,372:9480962,28068266:555238,295640,0x1,372:10003432,28068266)$1,372:10036200,28306091 k1,372:10234162,28306091:197962 x1,372:10744025,28306091 $1,372:10744025,28306091(1,372:10744025,28068266:703604,295640,54614x1,372:11414861,28068266)$1,372:11447629,28306091 x1,372:11611469,28306091 k1,372:11916994,28306091:305525 x1,372:13082209,28306091 k1,372:13280171,28306091:197962 x1,372:13717282,28306091 k1,372:13915243,28306091:197961 x1,372:14898903,28306091 k1,372:15096865,28306091:197962 x1,372:15570024,28306091 k1,372:15767986,28306091:197962 x1,372:16969232,28306091 k1,372:17167193,28306091:197961 x1,372:17494873,28306091 k1,372:17692835,28306091:197962 x1,372:19112968,28306091 k1,372:19112968,28306091:0 ) (1,386:3729360,29092523:15383608,446948,141880 x1,386:4420751,29092523 x1,372:5245847,29092523 k1,372:5478598,29092523:232751 x1,372:6607117,29092523 k1,372:6839868,29092523:232751 x1,372:7640707,29092523 k1,372:7873457,29092523:232750 x1,372:8419361,29092523 x1,372:8783727,29092523 x1,372:10385405,29092523 k1,372:10618156,29092523:232751 x1,372:12074355,29092523 k1,372:12307106,29092523:232751 x1,372:13253442,29092523 x1,372:14200416,29092523 k1,372:14433167,29092523:232751 x1,372:14943030,29092523 k1,372:15175780,29092523:232750 x1,372:15976619,29092523 k1,372:16209370,29092523:232751 x1,372:17428985,29092523 k1,372:17678963,29092523:249978 x1,372:18079378,29092523 k1,372:18312129,29092523:232751 x1,372:19112968,29092523 k1,372:19112968,29092523:0 ) (1,386:3729360,29878955:15383608,446948,141880 x1,386:4712400,29878955 x1,386:5549943,29878955 k1,372:5677602,29878955:127659 x1,372:6587890,29878955 x1,372:7679706,29878955 x1,372:8772177,29878955 k1,372:8899836,29878955:127659 x1,372:9846172,29878955 k1,372:9973831,29878955:127659 x1,372:10810711,29878955 x1,372:12630639,29878955 k1,372:12758299,29878955:127660 x1,372:13813410,29878955 x1,372:14869185,29878955 k1,372:14996844,29878955:127659 x1,372:15871073,29878955 x1,372:16053256,29878955 x1,372:16380936,29878955 x1,372:16668639,29878955 x1,372:17943293,29878955 k1,372:18070952,29878955:127659 x1,372:18689608,29878955 x1,372:18907840,29878955 x1,372:19112968,29878955 k1,386:19112968,29878955:0 ) (1,386:3729360,30665387:15383608,446948,141880 x1,386:5003367,30665387 k1,372:5199528,30665387:196161 x1,372:5745432,30665387 k1,372:5941593,30665387:196161 x1,372:6888567,30665387 x1,372:7216247,30665387 x1,372:8654085,30665387 k1,372:8954209,30665387:300124 x1,372:10428761,30665387 k1,372:10633002,30665387:204241 x1,372:10997368,30665387 k1,372:11193529,30665387:196161 x1,372:11812185,30665387 x1,372:13194977,30665387 k1,372:13391138,30665387:196161 x1,372:14374160,30665387 k1,372:14570321,30665387:196161 x1,372:15371160,30665387 k1,372:15567321,30665387:196161 x1,372:16623097,30665387 k1,372:16819226,30665387:196129 x1,372:18275425,30665387 x1,372:19112968,30665387 k1,372:19112968,30665387:0 ) (1,386:3729360,31451819:15383608,446948,141880 x1,386:4785135,31451819 k1,372:4990240,31451819:205105 x1,372:6738071,31451819 k1,372:6943176,31451819:205105 x1,372:7780719,31451819 k1,372:7985824,31451819:205105 x1,372:8568432,31451819 x1,372:9497720,31451819 k1,372:9702825,31451819:205105 x1,372:10976832,31451819 k1,372:11181937,31451819:205105 x1,372:11800593,31451819 k1,372:12005698,31451819:205105 x1,372:14334816,31451819 x1,372:15791015,31451819 k1,372:15996120,31451819:205105 x1,372:16505983,31451819 x1,372:18070975,31451819 k1,372:18276080,31451819:205105 x1,372:18894736,31451819 x1,372:19112968,31451819 k1,386:19112968,31451819:0 ) (1,386:3729360,32238251:15383608,446948,141880 x1,386:4348016,32238251 x1,386:5367087,32238251 k1,372:5536813,32238251:169726 x1,372:6046676,32238251 x1,372:6774763,32238251 x1,372:7466809,32238251 x1,372:8158855,32238251 x1,372:8280103,32238251 k1,372:8500920,32238251:220817 x1,372:10029208,32238251 k1,372:10198934,32238251:169726 x1,372:11327453,32238251 k1,372:11497178,32238251:169725 x1,372:12480838,32238251 k1,372:12650564,32238251:169726 x1,372:13269220,32238251 k1,372:13438946,32238251:169726 x1,372:14604151,32238251 x1,372:15986942,32238251 k1,372:16156667,32238251:169725 x1,372:17030259,32238251 x1,372:17996906,32238251 k1,372:18166632,32238251:169726 x1,372:19112968,32238251 k1,372:19112968,32238251:0 ) (1,386:3729360,33024683:15383608,446948,141880 x1,386:4239223,33024683 k1,372:4421342,33024683:182119 x1,372:4931205,33024683 x1,372:5549861,33024683 x1,372:6168517,33024683 x1,372:7187588,33024683 k1,372:7369707,33024683:182119 x1,372:7660683,33024683 x1,372:9263666,33024683 k1,372:9450354,33024683:186688 x1,372:10105714,33024683 x1,372:10562498,33024683 x1,372:10853474,33024683 x1,372:11164770,33024683 x1,372:11664154,33024683 x1,372:11791954,33024683 k1,372:12049951,33024683:257997 x1,372:12450366,33024683 k1,372:12632485,33024683:182119 x1,372:13761004,33024683 k1,372:13943122,33024683:182118 x1,372:14561778,33024683 k1,372:14743897,33024683:182119 x1,372:15618126,33024683 x1,372:16601148,33024683 k1,372:16783267,33024683:182119 x1,372:18130010,33024683 k1,372:18312129,33024683:182119 x1,372:19112968,33024683 k1,372:19112968,33024683:0 ) (1,386:3729360,33811115:15383608,446948,141880 x1,386:5185559,33811115 k1,372:5405804,33811115:220245 x1,372:6352140,33811115 x1,372:7299114,33811115 k1,372:7519358,33811115:220244 x1,372:7956469,33811115 k1,372:8176714,33811115:220245 x1,372:9487416,33811115 k1,372:9707660,33811115:220244 x1,372:10217523,33811115 x1,372:10674307,33811115 x1,372:10795555,33811115 k1,372:11167929,33811115:372374 x1,372:12333144,33811115 k1,372:12553388,33811115:220244 x1,372:12990499,33811115 k1,372:13210744,33811115:220245 x1,372:13501720,33811115 k1,372:13721964,33811115:220244 x1,372:14632252,33811115 x1,372:15578570,33811115 k1,372:15798815,33811115:220245 x1,372:16781837,33811115 k1,372:17002081,33811115:220244 x1,372:17511289,33811115 x1,372:19112968,33811115 k1,372:19112968,33811115:0 ) (1,386:3729360,34597547:15383608,446948,141880 x1,386:4348016,34597547 x1,386:4857879,34597547 x1,386:6350764,34597547 g1,372:6514604,34597547 x1,372:7934755,34597547 x1,372:8056003,34597547 g1,372:8259163,34597547 k1,386:19112968,34597547:10853805 g1,386:19112968,34597547 ) (1,389:3729360,36302826:15383608,541848,14544 k1,389:5406184,36302826:1676824 (1,389:5406184,36302826:0,0,0 g1,389:5406184,36302826 ) x1,389:9031612,36302826 x1,389:9571889,36302826 g1,389:9768497,36302826 x1,389:11471900,36302826 g1,389:11668508,36302826 x1,389:17436145,36302826 g1,389:17436145,36302826 k1,389:19112968,36302826:1676823 ) (1,392:5040082,37465931:14072886,446948,141880 (1,391:5040082,37465931:0,291271,0 g1,391:5040082,37465931 g1,391:3729360,37465931 g1,391:3401680,37465931 (1,391:3401680,37465931:1310722,291271,0 k1,391:4712402,37465931:1310722 (1,391:4712402,37465931:0,291271,0 k1,391:4384721,37465931:-327681 x1,391:4712402,37465931 ) ) g1,391:5040082,37465931 ) x1,391:6059162,37465931 k1,391:6185361,37465931:126199 x1,391:7864249,37465931 x1,391:8276473,37465931 k1,391:8402635,37465931:126162 x1,391:8911843,37465931 x1,391:10585611,37465931 k1,391:10711811,37465931:126200 x1,391:11585403,37465931 x1,391:12932154,37465931 k1,391:13058353,37465931:126199 x1,391:13931937,37465931 k1,391:14058136,37465931:126199 x1,391:14240319,37465931 x1,391:14567999,37465931 x1,391:14869470,37465931 x1,391:15405550,37465931 x1,391:15769916,37465931 x1,391:16097596,37465931 x1,391:17186138,37465931 k1,391:17312338,37465931:126200 x1,391:18113177,37465931 k1,391:18239376,37465931:126199 x1,391:18894736,37465931 x1,391:19112968,37465931 k1,392:19112968,37465931:0 ) (1,392:5040082,38252363:14072886,446948,141880 x1,392:6168601,38252363 x1,392:6896688,38252363 k1,391:7119489,38252363:222801 x1,391:7665401,38252363 k1,391:7888202,38252363:222801 x1,391:9344384,38252363 k1,391:9567184,38252363:222800 x1,391:10113096,38252363 k1,391:10335897,38252363:222801 x1,391:12484143,38252363 k1,391:12706944,38252363:222801 x1,391:13034624,38252363 x1,391:14841440,38252363 k1,391:15064241,38252363:222801 x1,391:15574104,38252363 k1,391:15796904,38252363:222800 x1,391:16597743,38252363 k1,391:16820544,38252363:222801 x1,391:17912360,38252363 x1,391:19112968,38252363 k1,391:19112968,38252363:0 ) (1,392:5040082,39038795:14072886,446948,141880 x1,392:5585994,39038795 k1,391:5731457,39038795:145463 x1,391:6532296,39038795 k1,391:6677758,39038795:145462 x1,391:7732869,39038795 x1,391:8788644,39038795 k1,391:8934107,39038795:145463 x1,391:10390306,39038795 k1,391:10535768,39038795:145462 x1,391:12574583,39038795 k1,391:12720046,39038795:145463 x1,391:13811861,39038795 x1,391:14903677,39038795 x1,391:16469307,39038795 k1,391:16614769,39038795:145462 x1,391:17160681,39038795 k1,391:17306144,39038795:145463 x1,391:17633824,39038795 x1,391:19112968,39038795 k1,392:19112968,39038795:0 ) (1,392:5040082,39825227:14072886,446948,141880 x1,392:5913674,39825227 x1,392:6241354,39825227 x1,391:7502266,39825227 k1,391:7678778,39825227:176512 x1,391:8297434,39825227 x1,391:8916090,39825227 x1,391:9935161,39825227 k1,391:10111674,39825227:176513 x1,391:11240193,39825227 k1,391:11416705,39825227:176512 x1,391:13000679,39825227 k1,391:13180360,39825227:179681 x1,391:13907800,39825227 x1,391:15472792,39825227 x1,391:15790648,39825227 k1,391:15967160,39825227:176512 x1,391:17131720,39825227 x1,391:17986951,39825227 k1,391:18166632,39825227:179681 x1,391:19112968,39825227 k1,391:19112968,39825227:0 ) (1,392:5040082,40611659:14072886,446948,7531 x1,392:6459577,40611659 g1,391:6623417,40611659 x1,391:7806320,40611659 k1,392:19112968,40611659:11306648 g1,392:19112968,40611659 ) (1,393:5040082,42094512:14072886,446948,141880 (1,392:5040082,42094512:0,291271,0 g1,392:5040082,42094512 g1,392:3729360,42094512 g1,392:3401680,42094512 (1,392:3401680,42094512:1310722,291271,0 k1,392:4712402,42094512:1310722 (1,392:4712402,42094512:0,291271,0 k1,392:4384721,42094512:-327681 x1,392:4712402,42094512 ) ) g1,392:5040082,42094512 ) x1,392:7442609,42094512 k1,392:7575706,42094512:133097 x1,392:8704225,42094512 k1,392:8837321,42094512:133096 x1,392:10257455,42094512 k1,392:10390552,42094512:133097 x1,392:11009208,42094512 x1,392:11323784,42094512 x1,392:11604936,42094512 k1,392:11738033,42094512:133097 x1,392:12975345,42094512 k1,392:13108442,42094512:133097 x1,392:14236961,42094512 x1,392:15692505,42094512 k1,392:15825601,42094512:133096 x1,392:16990815,42094512 k1,392:17123912,42094512:133097 x1,392:17924751,42094512 k1,392:18057848,42094512:133097 x1,392:18894736,42094512 x1,392:19112968,42094512 k1,393:19112968,42094512:0 ) (1,393:5040082,42880944:14072886,446948,141880 x1,393:5840921,42880944 x1,393:7115575,42880944 k1,392:7338746,42880944:223171 x1,392:7884658,42880944 k1,392:8107828,42880944:223170 x1,392:9564027,42880944 k1,392:9787198,42880944:223171 x1,392:10114878,42880944 x1,392:11776860,42880944 k1,392:12000030,42880944:223170 x1,392:13165253,42880944 x1,392:13893340,42880944 k1,392:14116511,42880944:223171 x1,392:15281079,42880944 x1,392:17210437,42880944 k1,392:17433607,42880944:223170 x1,392:18598821,42880944 k1,392:18821992,42880944:223171 x1,392:19112968,42880944 k1,392:19112968,42880944:0 ) (1,393:5040082,43667376:14072886,446948,145636 x1,393:5658738,43667376 x1,392:6115522,43667376 k1,392:6352104,43667376:236582 x1,392:7753256,43667376 k1,392:8008023,43667376:254767 x1,392:10924351,43667376 k1,392:11160932,43667376:236581 x1,392:11670795,43667376 k1,392:11907377,43667376:236582 x1,392:12817009,43667376 x1,392:13617193,43667376 x1,392:14090352,43667376 x1,392:15072719,43667376 k1,392:15309301,43667376:236582 x1,392:16110140,43667376 k1,392:16346722,43667376:236582 x1,392:17329762,43667376 x1,392:17839625,43667376 x1,392:18894736,43667376 x1,392:19112968,43667376 k1,393:19112968,43667376:0 ) (1,393:5040082,44453808:14072886,446948,141880 x1,393:6569017,44453808 k1,392:6748566,44453808:179549 x1,392:7258429,44453808 k1,392:7437977,44453808:179548 x1,392:7765657,44453808 x1,392:8820777,44453808 k1,392:9000326,44453808:179549 x1,392:9510189,44453808 k1,392:9689737,44453808:179548 x1,392:10017417,44453808 x1,392:10474192,44453808 x1,392:10656375,44453808 x1,392:11748190,44453808 k1,392:11927739,44453808:179549 x1,392:12728578,44453808 k1,392:12908126,44453808:179548 x1,392:13235806,44453808 x1,392:14897788,44453808 k1,392:15077337,44453808:179549 x1,392:16242560,44453808 x1,392:16970647,44453808 k1,392:17150195,44453808:179548 x1,392:18096531,44453808 k1,392:18276080,44453808:179549 x1,392:18894736,44453808 x1,392:19112968,44453808 k1,393:19112968,44453808:0 ) (1,393:5040082,45240240:14072886,446948,141880 x1,393:5513241,45240240 x1,393:7042175,45240240 g1,392:7206015,45240240 x1,392:8407269,45240240 g1,392:8571109,45240240 x1,392:9189765,45240240 x1,392:10208836,45240240 x1,392:10499812,45240240 x1,392:11609971,45240240 k1,393:19112968,45240240:7502997 g1,393:19112968,45240240 ) (1,394:5040082,46723093:14072886,446948,141880 (1,393:5040082,46723093:0,291271,0 g1,393:5040082,46723093 g1,393:3729360,46723093 g1,393:3401680,46723093 (1,393:3401680,46723093:1310722,291271,0 k1,393:4712402,46723093:1310722 (1,393:4712402,46723093:0,291271,0 k1,393:4384721,46723093:-327681 x1,393:4712402,46723093 ) ) g1,393:5040082,46723093 ) x1,393:7478658,46723093 x1,393:7796514,46723093 k1,393:8064527,46723093:268013 x1,393:9229087,46723093 x1,393:9920478,46723093 k1,393:10188491,46723093:268013 x1,393:10807147,46723093 x1,393:11121723,46723093 x1,393:11402875,46723093 k1,393:11670888,46723093:268013 x1,393:12908200,46723093 k1,393:13176214,46723093:268014 x1,393:14377478,46723093 k1,393:14645491,46723093:268013 x1,393:15155354,46723093 k1,393:15423367,46723093:268013 x1,393:16042023,46723093 x1,393:16515182,46723093 x1,393:18044116,46723093 k1,393:18312129,46723093:268013 x1,393:19112968,46723093 k1,393:19112968,46723093:0 ) (1,394:5040082,47509525:14072886,446948,141880 x1,394:6641761,47509525 x1,394:7479304,47509525 k1,393:7759221,47509525:279917 x1,393:8705557,47509525 x1,393:9652531,47509525 k1,393:9932448,47509525:279917 x1,393:10878784,47509525 x1,393:11196640,47509525 x1,393:11477792,47509525 k1,393:11757710,47509525:279918 x1,393:13359397,47509525 k1,393:13639314,47509525:279917 x1,393:14404105,47509525 x1,393:15459872,47509525 k1,393:15739789,47509525:279917 x1,393:16503933,47509525 k1,393:16783850,47509525:279917 x1,393:19112968,47509525 k1,394:19112968,47509525:0 ) (1,394:5040082,48295957:14072886,446948,141880 x1,394:6714513,48295957 x1,394:8789368,48295957 g1,393:8953208,48295957 x1,393:10408752,48295957 x1,393:11246277,48295957 g1,393:11410117,48295957 x1,393:11919980,48295957 x1,393:12648067,48295957 x1,393:13340113,48295957 x1,393:13522296,48295957 x1,393:14432566,48295957 g1,393:14596406,48295957 x1,393:14924086,48295957 x1,393:15215062,48295957 x1,393:15526358,48295957 x1,393:16808893,48295957 k1,394:19112968,48295957:2304075 g1,394:19112968,48295957 ) ] g1,396:19112968,48295957 ) k1,396:19579138,48295957:466170 r1,396:19579138,48295957:0,25020340,0 k1,396:20045308,48295957:466170 (1,396:20045308,48295957:15383608,25020340,0 [1,396:20045308,48295957:15383608,25020340,0 [1,385:20045308,40742472:15383608,17466855,0 (1,384:20045308,33000056:15383608,9724439,0 k1,384:20159061,33000056:113753 h1,375:20159061,33000056:0,0,0 (1,375:20159061,33000056:15156103,9724439,0 (1,375:20159061,33000056:15156319,9724463,0 (1,375:20159061,33000056:15156319,9724463,0 (1,375:20159061,33000056:0,18233125,0 (1,375:20159061,33000056:28417720,18233125,0 ) k1,375:20159061,33000056:-28417720 ) ) ) ) g1,384:35315164,33000056 k1,384:35428916,33000056:113752 ) (1,384:20045308,34441848:15383608,446948,141880 h1,384:20045308,34441848:0,0,0 x1,384:21756435,34441848 k1,384:21967196,34441848:210761 x1,384:22804739,34441848 k1,384:23101740,34441848:297001 k1,384:23101740,34441848:0 x1,384:25395465,34441848 k1,384:25606225,34441848:210760 x1,384:27281294,34441848 x1,384:28870524,34441848 k1,384:29081285,34441848:210761 x1,384:29845429,34441848 k1,384:30056189,34441848:210760 x1,384:32422021,34441848 k1,384:32632782,34441848:210761 x1,384:34708293,34441848 k1,384:34919053,34441848:210760 x1,384:35428916,34441848 k1,384:35428916,34441848:0 ) (1,384:20045308,35228280:15383608,446948,141880 x1,384:20991644,35228280 k1,384:21195680,35228280:204036 x1,384:21995864,35228280 k1,384:22199899,35228280:204035 x1,384:22745811,35228280 k1,384:22949847,35228280:204036 x1,384:23750686,35228280 k1,384:23954722,35228280:204036 x1,384:26248464,35228280 k1,384:26452500,35228280:204036 x1,384:27672115,35228280 k1,384:27886199,35228280:214084 x1,384:28177175,35228280 x1,384:28491751,35228280 x1,384:30228447,35228280 k1,384:30432483,35228280:204036 x1,384:30760163,35228280 x1,384:31078019,35228280 x1,384:31577403,35228280 k1,384:31781439,35228280:204036 x1,384:33091486,35228280 k1,384:33295521,35228280:204035 x1,384:34678968,35228280 k1,384:34883004,35228280:204036 x1,384:35428916,35228280 k1,384:35428916,35228280:0 ) (1,384:20045308,36014712:15383608,446948,141880 x1,384:22447825,36014712 k1,384:22652353,36014712:204528 x1,384:23271009,36014712 x1,384:26357725,36014712 k1,384:26682950,36014712:325225 x1,384:27702030,36014712 k1,384:27906558,36014712:204528 x1,384:28197534,36014712 x1,384:28512110,36014712 x1,384:29921126,36014712 k1,384:30125654,36014712:204528 x1,384:30453334,36014712 x1,384:30771190,36014712 x1,384:31270574,36014712 k1,384:31475102,36014712:204528 x1,384:32275941,36014712 k1,384:32480469,36014712:204528 x1,384:35428916,36014712 k1,384:35428916,36014712:0 ) (1,384:20045308,36801144:15383608,446948,141880 x1,384:20554516,36801144 x1,384:21964843,36801144 k1,384:22212907,36801144:248064 x1,384:22650018,36801144 k1,384:22898083,36801144:248065 x1,384:23808371,36801144 x1,384:24592835,36801144 k1,384:24840899,36801144:248064 x1,384:25350762,36801144 k1,384:25598827,36801144:248065 x1,384:26599555,36801144 k1,384:27055388,36801144:455833 x1,384:28074468,36801144 k1,384:28322533,36801144:248065 x1,384:28613509,36801144 x1,384:28928085,36801144 x1,384:30337101,36801144 k1,384:30585165,36801144:248064 x1,384:30912845,36801144 x1,384:31230701,36801144 x1,384:31730085,36801144 k1,384:31978150,36801144:248065 x1,384:32778989,36801144 k1,384:33027053,36801144:248064 x1,384:34410500,36801144 x1,384:35428916,36801144 k1,384:35428916,36801144:0 ) (1,384:20045308,37587576:15383608,446948,141880 x1,384:20554516,37587576 x1,384:21964843,37587576 k1,384:22199637,37587576:234794 x1,384:22636748,37587576 k1,384:22871543,37587576:234795 x1,384:23781831,37587576 x1,384:24566295,37587576 k1,384:24801089,37587576:234794 x1,384:25310952,37587576 k1,384:25545747,37587576:234795 x1,384:26674266,37587576 k1,384:26909060,37587576:234794 x1,384:27855396,37587576 k1,384:28090190,37587576:234794 x1,384:28891029,37587576 k1,384:29125824,37587576:234795 x1,384:29416800,37587576 x1,384:29731376,37587576 x1,384:31140392,37587576 k1,384:31375186,37587576:234794 x1,384:31702866,37587576 x1,384:32020722,37587576 x1,384:32520106,37587576 k1,384:32754901,37587576:234795 x1,384:33555740,37587576 k1,384:33790534,37587576:234794 x1,384:35428916,37587576 k1,384:35428916,37587576:0 ) (1,384:20045308,38374008:15383608,446948,141880 x1,384:20554516,38374008 x1,384:21964843,38374008 k1,384:22154530,38374008:189687 x1,384:22591641,38374008 k1,384:22781329,38374008:189688 x1,384:23691617,38374008 x1,384:24476081,38374008 k1,384:24665768,38374008:189687 x1,384:25175631,38374008 k1,384:25365318,38374008:189687 x1,384:26948653,38374008 k1,384:27229355,38374008:280702 x1,384:28248435,38374008 k1,384:28438122,38374008:189687 x1,384:29857617,38374008 k1,384:30047305,38374008:189688 x1,384:30484416,38374008 k1,384:30674103,38374008:189687 x1,384:31292759,38374008 x1,384:32603471,38374008 k1,384:32793158,38374008:189687 x1,384:33303021,38374008 k1,384:33492709,38374008:189688 x1,384:34111365,38374008 k1,384:34301052,38374008:189687 x1,384:35428916,38374008 k1,384:35428916,38374008:0 ) (1,384:20045308,39160440:15383608,446948,141880 x1,384:21646340,39160440 k1,384:21784239,39160440:137899 x1,384:22585078,39160440 k1,384:22722977,39160440:137899 x1,384:23050657,39160440 x1,384:24781448,39160440 k1,384:24919346,39160440:137898 x1,384:27175729,39160440 k1,384:27313628,39160440:137899 x1,384:29788873,39160440 k1,384:29926772,39160440:137899 x1,384:30727611,39160440 k1,384:30865510,39160440:137899 x1,384:32139526,39160440 k1,384:32277424,39160440:137898 x1,384:33696919,39160440 k1,384:33834818,39160440:137899 x1,384:34490178,39160440 k1,384:34628077,39160440:137899 x1,384:35428916,39160440 k1,384:35428916,39160440:0 ) (1,384:20045308,39946872:15383608,491520,163840 x1,384:21938618,39946872 k1,384:22139147,39946872:200529 x1,384:22685059,39946872 k1,384:22885589,39946872:200530 x1,384:23686428,39946872 k1,384:23886957,39946872:200529 x1,384:25106572,39946872 k1,384:25419800,39946872:313228 x1,384:26438880,39946872 k1,384:26639409,39946872:200529 x1,384:28058904,39946872 k1,384:28259434,39946872:200530 $1,384:28259434,39946872x1,384:28633007,39946872x1,384:29311882,39946872$1,384:29311882,39946872 k1,384:29512411,39946872:200529 x1,384:32679083,39946872 k1,384:32879612,39946872:200529 x1,384:33389475,39946872 k1,384:33590005,39946872:200530 x1,384:34390844,39946872 k1,384:34591373,39946872:200529 x1,384:35428916,39946872 k1,384:35428916,39946872:0 ) (1,384:20045308,40733304:15383608,446948,9168 x1,384:20591220,40733304 g1,384:20755060,40733304 x1,384:21555899,40733304 g1,384:21719739,40733304 x1,384:22047419,40733304 x1,384:23778210,40733304 g1,384:23942050,40733304 x1,384:26198433,40733304 x1,384:26326233,40733304 k1,384:35428916,40733304:9102683 g1,384:35428916,40733304 ) ] (1,395:21356030,42823394:14072886,446948,141880 (1,394:21356030,42823394:0,291271,0 g1,394:21356030,42823394 g1,394:20045308,42823394 g1,394:19717628,42823394 (1,394:19717628,42823394:1310722,291271,0 k1,394:21028350,42823394:1310722 (1,394:21028350,42823394:0,291271,0 k1,394:20700669,42823394:-327681 x1,394:21028350,42823394 ) ) g1,394:21356030,42823394 ) x1,394:22156878,42823394 k1,394:22367155,42823394:210277 x1,394:23058555,42823394 x1,394:23895443,42823394 k1,394:24105720,42823394:210277 x1,394:25525215,42823394 k1,394:25735492,42823394:210277 x1,394:26754555,42823394 k1,394:26964832,42823394:210277 x1,394:28130055,42823394 x1,394:28312238,42823394 x1,394:29040333,42823394 k1,394:29250610,42823394:210277 x1,394:30124194,42823394 k1,394:30334471,42823394:210277 x1,394:31571783,42823394 k1,394:31782060,42823394:210277 x1,394:32983324,42823394 k1,394:33193601,42823394:210277 x1,394:33703464,42823394 k1,394:33913741,42823394:210277 x1,394:34532397,42823394 x1,394:35005556,42823394 x1,394:35223788,42823394 x1,394:35428916,42823394 k1,395:35428916,42823394:0 ) (1,395:21356030,43609826:14072886,446948,141880 x1,395:22666732,43609826 k1,394:22932126,43609826:265394 x1,394:23732965,43609826 k1,394:23998360,43609826:265395 x1,394:24944696,43609826 x1,394:25454559,43609826 x1,394:26729213,43609826 k1,394:26994607,43609826:265394 x1,394:27649967,43609826 x1,394:28486855,43609826 k1,394:28752249,43609826:265394 x1,394:30353936,43609826 k1,394:30619330,43609826:265394 x1,394:32038825,43609826 k1,394:32304220,43609826:265395 x1,394:33323283,43609826 k1,394:33588677,43609826:265394 x1,394:34462269,43609826 x1,394:35428916,43609826 k1,394:35428916,43609826:0 ) (1,395:21356030,44396258:14072886,446948,92400 x1,395:21974686,44396258 x1,395:23230342,44396258 k1,394:23513813,44396258:283471 x1,394:24460149,44396258 k1,394:24719694,44396258:259545 x1,394:25229557,44396258 k1,394:25489103,44396258:259546 x1,394:26107759,44396258 x1,394:26835199,44396258 x1,394:27781517,44396258 k1,394:28041062,44396258:259545 x1,394:29387805,44396258 k1,394:29647350,44396258:259545 x1,394:30302710,44396258 x1,394:30848614,44396258 x1,394:31394526,44396258 x1,394:31669118,44396258 x1,394:32943772,44396258 k1,394:33203318,44396258:259546 x1,394:34368532,44396258 k1,394:34628077,44396258:259545 x1,394:35428916,44396258 k1,394:35428916,44396258:0 ) (1,395:21356030,45182690:14072886,446948,141880 x1,395:22083470,45182690 x1,395:23648462,45182690 x1,395:23966318,45182690 g1,394:24130158,45182690 x1,394:25294718,45182690 x1,394:26168292,45182690 g1,394:26332132,45182690 x1,394:27423947,45182690 x1,394:28515763,45182690 x1,394:30245233,45182690 k1,395:35428916,45182690:5183683 g1,395:35428916,45182690 ) (1,396:21356030,46723093:14072886,446948,9168 (1,395:21356030,46723093:0,291271,0 g1,395:21356030,46723093 g1,395:20045308,46723093 g1,395:19717628,46723093 (1,395:19717628,46723093:1310722,291271,0 k1,395:21028350,46723093:1310722 (1,395:21028350,46723093:0,291271,0 k1,395:20700669,46723093:-327681 x1,395:21028350,46723093 ) ) g1,395:21356030,46723093 ) x1,395:24233017,46723093 k1,395:24411415,46723093:178398 x1,395:25903655,46723093 k1,395:26082053,46723093:178398 x1,395:26700709,46723093 x1,395:27015285,46723093 x1,395:27296437,46723093 k1,395:27474835,46723093:178398 x1,395:28712147,46723093 k1,395:28890545,46723093:178398 x1,395:30091809,46723093 k1,395:30270207,46723093:178398 x1,395:30780070,46723093 k1,395:30958468,46723093:178398 x1,395:32596850,46723093 k1,395:32775248,46723093:178398 x1,395:33576087,46723093 k1,395:33754485,46723093:178398 x1,395:35428916,46723093 k1,396:35428916,46723093:0 ) (1,396:21356030,47509525:14072886,446948,141880 x1,396:23430885,47509525 k1,395:23676343,47509525:245458 x1,395:25131887,47509525 x1,395:25969412,47509525 k1,395:26214869,47509525:245457 x1,395:26724732,47509525 x1,395:27452819,47509525 x1,395:28144865,47509525 x1,395:28836911,47509525 x1,395:28958159,47509525 k1,395:29406172,47509525:448013 x1,395:30425252,47509525 k1,395:30670709,47509525:245457 x1,395:31507597,47509525 x1,395:33109275,47509525 k1,395:33354733,47509525:245458 x1,395:34154917,47509525 x1,395:35428916,47509525 k1,395:35428916,47509525:0 ) (1,396:21356030,48295957:14072886,446948,141880 x1,396:21901942,48295957 k1,395:22082149,48295957:180207 x1,395:22882988,48295957 k1,395:23063194,48295957:180206 x1,395:23573057,48295957 x1,395:24301144,48295957 x1,395:24993190,48295957 x1,395:25685236,48295957 k1,395:25865443,48295957:180207 x1,395:27284283,48295957 k1,395:27464489,48295957:180206 x1,395:28264673,48295957 x1,395:29174961,48295957 x1,395:30303480,48295957 k1,395:30483687,48295957:180207 x1,395:31612197,48295957 k1,395:31792403,48295957:180206 x1,395:32957617,48295957 k1,395:33137824,48295957:180207 x1,395:33938663,48295957 k1,395:34118869,48295957:180206 x1,395:35210684,48295957 x1,395:35428916,48295957 k1,396:35428916,48295957:0 ) ] g1,396:35428916,48295957 ) ) ] h1,396:3729360,50262037:31699556,0,0 ] ] !32891 10 !11 11 [1,440:1,50262036:35428915,50262035,0 [1,440:3729360,50262036:31699556,49329695,0 [1,440:3729360,1718773:31699556,786432,0 h1,440:3729360,1718773:31699556,0,0 ] [1,440:3729360,48295956:31699556,44938783,0 (1,440:3729360,48295956:31699556,44938783,0 (1,440:3729360,48295956:15383608,44938783,0 [1,420:3729360,48295956:15383608,44938783,0 (1,396:5040082,4012533:14072886,379448,7531 x1,396:6131898,4012533 x1,396:7861368,4012533 k1,396:19112968,4012533:11251600 g1,396:19112968,4012533 ) (1,397:5040082,5438685:14072886,446948,141880 (1,396:5040082,5438685:0,291271,0 g1,396:5040082,5438685 g1,396:3729360,5438685 g1,396:3401680,5438685 (1,396:3401680,5438685:1310722,291271,0 k1,396:4712402,5438685:1310722 (1,396:4712402,5438685:0,291271,0 k1,396:4384721,5438685:-327681 x1,396:4712402,5438685 ) ) g1,396:5040082,5438685 ) x1,396:6059162,5438685 k1,396:6259189,5438685:200027 x1,396:7132773,5438685 x1,396:8224581,5438685 k1,396:8424608,5438685:200027 x1,396:8970520,5438685 k1,396:9170547,5438685:200027 x1,396:9971386,5438685 k1,396:10171413,5438685:200027 x1,396:11627612,5438685 k1,396:11827638,5438685:200026 x1,396:12155318,5438685 x1,396:13817300,5438685 k1,396:14017327,5438685:200027 x1,396:14963663,5438685 k1,396:15163690,5438685:200027 x1,396:16619234,5438685 x1,396:17456759,5438685 k1,396:17656786,5438685:200027 x1,396:18166649,5438685 x1,396:18894736,5438685 x1,396:19112968,5438685 k1,397:19112968,5438685:0 ) (1,397:5040082,6225117:14072886,446948,141880 x1,397:5732128,6225117 x1,397:6424174,6225117 k1,396:6599506,6225117:175332 x1,396:8091746,6225117 k1,396:8267077,6225117:175331 x1,396:8776940,6225117 k1,396:8952272,6225117:175332 x1,396:10117495,6225117 x1,396:10590654,6225117 x1,396:11428197,6225117 k1,396:11603529,6225117:175332 x1,396:12404368,6225117 k1,396:12579700,6225117:175332 x1,396:13235060,6225117 x1,396:13780964,6225117 x1,396:14326876,6225117 x1,396:14601468,6225117 x1,396:15876122,6225117 k1,396:16051453,6225117:175331 x1,396:16561316,6225117 x1,396:16872612,6225117 x1,396:17408692,6225117 k1,396:17584024,6225117:175332 x1,396:18530360,6225117 x1,396:18894736,6225117 x1,396:19112968,6225117 k1,397:19112968,6225117:0 ) (1,397:5040082,7011549:14072886,446948,141880 x1,397:6568370,7011549 k1,396:6761172,7011549:192802 x1,396:7744194,7011549 k1,396:7936996,7011549:192802 x1,396:9283739,7011549 k1,396:9476542,7011549:192803 x1,396:10968782,7011549 k1,396:11161584,7011549:192802 x1,396:12071216,7011549 k1,396:12264018,7011549:192802 x1,396:12882674,7011549 k1,396:13075476,7011549:192802 x1,396:14276740,7011549 k1,396:14469543,7011549:192803 x1,396:14979406,7011549 k1,396:15172208,7011549:192802 x1,396:15754160,7011549 x1,396:16372816,7011549 x1,396:17865046,7011549 k1,396:18057848,7011549:192802 x1,396:18894736,7011549 x1,396:19112968,7011549 k1,397:19112968,7011549:0 ) (1,397:5040082,7797981:14072886,446948,141880 x1,397:6023104,7797981 g1,396:6186944,7797981 x1,396:6987783,7797981 g1,396:7151623,7797981 x1,396:7770279,7797981 x1,396:8534423,7797981 x1,396:10463126,7797981 g1,396:10626966,7797981 x1,396:11172878,7797981 g1,396:11336718,7797981 x1,396:11882630,7797981 x1,396:12901701,7797981 g1,396:13065541,7797981 x1,396:13830332,7797981 x1,396:14958196,7797981 g1,396:15122036,7797981 x1,396:16596579,7797981 k1,397:19112968,7797981:2516389 g1,397:19112968,7797981 ) (1,398:5040082,9224132:14072886,446948,141880 (1,397:5040082,9224132:0,291271,0 g1,397:5040082,9224132 g1,397:3729360,9224132 g1,397:3401680,9224132 (1,397:3401680,9224132:1310722,291271,0 k1,397:4712402,9224132:1310722 (1,397:4712402,9224132:0,291271,0 k1,397:4384721,9224132:-327681 x1,397:4712402,9224132 ) ) g1,397:5040082,9224132 ) x1,397:5404458,9224132 x1,397:5940546,9224132 k1,397:6096716,9224132:156170 x1,397:6897555,9224132 k1,397:7053725,9224132:156170 x1,397:8109501,9224132 k1,397:8265663,9224132:156162 x1,397:10559405,9224132 k1,397:10715575,9224132:156170 x1,397:11935190,9224132 k1,397:12092894,9224132:157704 x1,397:12711550,9224132 k1,397:12867719,9224132:156169 x1,397:16215917,9224132 k1,397:16372087,9224132:156170 x1,397:18083214,9224132 k1,397:18239384,9224132:156170 x1,397:19112968,9224132 k1,397:19112968,9224132:0 ) (1,398:5040082,10010564:14072886,446948,145636 x1,398:6277394,10010564 k1,397:6421631,10010564:144237 x1,397:7040287,10010564 x1,397:7351583,10010564 x1,397:9088934,10010564 k1,397:9233171,10010564:144237 x1,397:10398385,10010564 k1,397:10542622,10010564:144237 x1,397:11343461,10010564 k1,397:11487698,10010564:144237 x1,397:12616217,10010564 k1,397:12760454,10010564:144237 x1,397:13306366,10010564 k1,397:13450603,10010564:144237 x1,397:14251442,10010564 k1,397:14395679,10010564:144237 x1,397:17148167,10010564 k1,397:17292385,10010564:144218 x1,397:19112968,10010564 k1,397:19112968,10010564:0 ) (1,398:5040082,10796996:14072886,446948,7531 x1,398:7971461,10796996 k1,398:19112968,10796996:11141507 g1,398:19112968,10796996 ) (1,399:5040082,12223148:14072886,446948,141880 (1,398:5040082,12223148:0,291271,0 g1,398:5040082,12223148 g1,398:3729360,12223148 g1,398:3401680,12223148 (1,398:3401680,12223148:1310722,291271,0 k1,398:4712402,12223148:1310722 (1,398:4712402,12223148:0,291271,0 k1,398:4384721,12223148:-327681 x1,398:4712402,12223148 ) ) g1,398:5040082,12223148 ) x1,398:5877626,12223148 k1,398:6071232,12223148:193606 x1,398:6398912,12223148 x1,398:8060894,12223148 k1,398:8254531,12223148:193637 x1,398:11276998,12223148 k1,398:11470634,12223148:193636 x1,398:12089290,12223148 x1,398:12403866,12223148 x1,398:12685018,12223148 k1,398:12878655,12223148:193637 x1,398:14115967,12223148 k1,398:14309603,12223148:193636 x1,398:15583619,12223148 k1,398:15777256,12223148:193637 x1,398:17269496,12223148 k1,398:17463132,12223148:193636 x1,398:18118492,12223148 k1,398:18312129,12223148:193637 x1,398:19112968,12223148 k1,398:19112968,12223148:0 ) (1,399:5040082,13009580:14072886,446948,141880 x1,399:6095858,13009580 g1,398:6259698,13009580 x1,398:8553440,13009580 g1,398:8717280,13009580 x1,398:9773055,13009580 g1,398:9936895,13009580 x1,398:10774439,13009580 g1,398:10938279,13009580 x1,398:12322371,13009580 x1,398:13629800,13009580 g1,398:13793640,13009580 x1,398:17343021,13009580 k1,399:19112968,13009580:1769947 g1,399:19112968,13009580 ) (1,400:5040082,14435731:14072886,446948,141880 (1,399:5040082,14435731:0,291271,0 g1,399:5040082,14435731 g1,399:3729360,14435731 g1,399:3401680,14435731 (1,399:3401680,14435731:1310722,291271,0 k1,399:4712402,14435731:1310722 (1,399:4712402,14435731:0,291271,0 k1,399:4384721,14435731:-327681 x1,399:4712402,14435731 ) ) g1,399:5040082,14435731 ) x1,399:6059162,14435731 k1,399:6493084,14435731:433922 x1,399:6784060,14435731 x1,399:7101916,14435731 x1,399:8812387,14435731 k1,399:9246309,14435731:433922 x1,399:10302084,14435731 x1,399:11576091,14435731 k1,399:12010014,14435731:433923 x1,399:12300990,14435731 x1,399:12615566,14435731 x1,399:12896718,14435731 x1,399:14024582,14435731 k1,399:14458504,14435731:433922 x1,399:15914703,14435731 k1,399:16348625,14435731:433922 x1,399:17294961,14435731 x1,399:18241935,14435731 k1,399:18675857,14435731:433922 x1,399:19112968,14435731 k1,399:19112968,14435731:0 ) (1,400:5040082,15222163:14072886,533465,141880 x1,400:6860010,15222163 k1,399:7065416,15222163:205406 $1,399:7065416,15222163x1,399:8230504,15222163k1,399:8406925,15222163:176421x1,399:8916651,15222163k1,399:9093072,15222163:176421x1,399:9748434,15222163(1,399:9748434,14984338:555238,295640,0x1,399:10270904,14984338)$1,399:10303672,15222163 k1,399:10509078,15222163:205406 x1,399:11018941,15222163 $1,399:11018941,15222163(1,399:11018941,14984338:703604,295640,54614x1,399:11689777,14984338)$1,399:11722545,15222163 x1,399:11886385,15222163 k1,399:12102182,15222163:215797 x1,399:13267405,15222163 x1,399:13449588,15222163 x1,399:14869083,15222163 k1,399:15074489,15222163:205406 x1,399:15729849,15222163 k1,399:15935255,15222163:205406 x1,399:17391454,15222163 k1,399:17596860,15222163:205406 x1,399:18106723,15222163 k1,399:18312129,15222163:205406 x1,399:19112968,15222163 k1,399:19112968,15222163:0 ) (1,400:5040082,16008595:14072886,446948,141880 x1,400:8243457,16008595 g1,399:8407297,16008595 x1,399:9353633,16008595 g1,399:9517473,16008595 x1,399:10063377,16008595 x1,399:10381233,16008595 x1,399:12438383,16008595 k1,400:19112968,16008595:6674585 g1,400:19112968,16008595 ) (1,401:5040082,17434747:14072886,446948,141880 (1,400:5040082,17434747:0,291271,0 g1,400:5040082,17434747 g1,400:3729360,17434747 g1,400:3401680,17434747 (1,400:3401680,17434747:1310722,291271,0 k1,400:4712402,17434747:1310722 (1,400:4712402,17434747:0,291271,0 k1,400:4384721,17434747:-327681 x1,400:4712402,17434747 ) ) g1,400:5040082,17434747 ) x1,400:6641762,17434747 k1,400:6968544,17434747:326782 x1,400:7623904,17434747 k1,400:7950685,17434747:326781 x1,400:8897659,17434747 x1,400:9698498,17434747 k1,400:10025280,17434747:326782 x1,400:10643936,17434747 x1,400:10934912,17434747 x1,400:12100117,17434747 k1,400:12426899,17434747:326782 x1,400:12972803,17434747 x1,400:13337169,17434747 x1,400:14866095,17434747 k1,400:15192876,17434747:326781 x1,400:16139212,17434747 k1,400:16465994,17434747:326782 x1,400:16756970,17434747 x1,400:17074826,17434747 x1,400:18093897,17434747 x1,400:18421577,17434747 x1,400:18894736,17434747 x1,400:19112968,17434747 k1,401:19112968,17434747:0 ) (1,401:5040082,18221179:14072886,446948,141880 x1,401:6059808,18221179 k1,400:6221760,18221179:161952 x1,400:7495767,18221179 k1,400:7657720,18221179:161953 x1,400:9336608,18221179 x1,400:9912672,18221179 k1,400:10074624,18221179:161952 x1,400:10875463,18221179 k1,400:11037416,18221179:161953 x1,400:11547279,18221179 x1,400:12275366,18221179 x1,400:12967412,18221179 x1,400:13659458,18221179 k1,400:13821410,18221179:161952 x1,400:15423089,18221179 x1,400:16260632,18221179 k1,400:16422585,18221179:161953 x1,400:16932448,18221179 k1,400:17094400,18221179:161952 x1,400:17895239,18221179 k1,400:18057192,18221179:161953 x1,400:19112968,18221179 k1,400:19112968,18221179:0 ) (1,401:5040082,19007611:14072886,446948,141880 x1,401:6496281,19007611 x1,401:7333824,19007611 k1,400:7515191,19007611:181367 x1,400:8570966,19007611 k1,400:8752333,19007611:181367 x1,400:10500164,19007611 k1,400:10681531,19007611:181367 x1,400:11300187,19007611 k1,400:11481554,19007611:181367 x1,400:12027466,19007611 x1,400:12355146,19007611 k1,400:12536514,19007611:181368 x1,400:13956647,19007611 k1,400:14138014,19007611:181367 x1,400:15812445,19007611 k1,400:15993812,19007611:181367 x1,400:17122331,19007611 k1,400:17303698,19007611:181367 x1,400:18104537,19007611 k1,400:18285904,19007611:181367 x1,400:18576880,19007611 x1,400:18894736,19007611 x1,400:19112968,19007611 k1,401:19112968,19007611:0 ) (1,401:5040082,19794043:14072886,446948,141880 x1,401:6750553,19794043 g1,400:6914393,19794043 x1,400:7970168,19794043 x1,400:9244175,19794043 g1,400:9408015,19794043 x1,400:9698991,19794043 x1,400:10013567,19794043 x1,400:10294719,19794043 x1,400:11422583,19794043 g1,400:11586423,19794043 x1,400:13042622,19794043 g1,400:13206462,19794043 x1,400:14152798,19794043 x1,400:15099772,19794043 x1,400:15221020,19794043 k1,401:19112968,19794043:3891948 g1,401:19112968,19794043 ) (1,402:5040082,21220194:14072886,446948,141880 (1,401:5040082,21220194:0,291271,0 g1,401:5040082,21220194 g1,401:3729360,21220194 g1,401:3401680,21220194 (1,401:3401680,21220194:1310722,291271,0 k1,401:4712402,21220194:1310722 (1,401:4712402,21220194:0,291271,0 k1,401:4384721,21220194:-327681 x1,401:4712402,21220194 ) ) g1,401:5040082,21220194 ) x1,401:7842377,21220194 k1,401:7972679,21220194:130302 x1,401:10520675,21220194 k1,401:10650977,21220194:130302 x1,401:11524569,21220194 x1,401:12491216,21220194 k1,401:12621518,21220194:130302 x1,401:13385662,21220194 k1,401:13515964,21220194:130302 x1,401:14316803,21220194 k1,401:14447104,21220194:130301 x1,401:15502880,21220194 k1,401:15633149,21220194:130269 x1,401:17926891,21220194 k1,401:18057193,21220194:130302 x1,401:19112968,21220194 k1,401:19112968,21220194:0 ) (1,402:5040082,22006626:14072886,446948,141880 x1,402:5913666,22006626 k1,401:6081336,22006626:167670 x1,401:6882175,22006626 k1,401:7049846,22006626:167671 x1,401:7559709,22006626 k1,401:7727379,22006626:167670 x1,401:8346035,22006626 k1,401:8513705,22006626:167670 x1,401:9751663,22006626 k1,401:9919334,22006626:167671 x1,401:11976484,22006626 k1,401:12145111,22006626:168627 x1,401:12472791,22006626 x1,401:12969550,22006626 k1,401:13137221,22006626:167671 x1,401:13574332,22006626 k1,401:13742002,22006626:167670 x1,401:14032978,22006626 x1,401:16061305,22006626 k1,401:16228975,22006626:167670 x1,401:16738838,22006626 k1,401:16906509,22006626:167671 x1,401:17525165,22006626 k1,401:17692835,22006626:167670 x1,401:19112968,22006626 k1,401:19112968,22006626:0 ) (1,402:5040082,22793058:14072886,446948,141880 x1,402:7114274,22793058 g1,401:7278114,22793058 x1,401:7824026,22793058 g1,401:7987866,22793058 x1,401:8788705,22793058 g1,401:8952545,22793058 x1,401:9462408,22793058 x1,401:9919192,22793058 g1,401:10083032,22793058 x1,401:12412814,22793058 g1,401:12576654,22793058 x1,401:12867630,22793058 x1,401:13182206,22793058 x1,401:14591222,22793058 g1,401:14755062,22793058 x1,401:16211261,22793058 g1,401:16375101,22793058 x1,401:18268411,22793058 x1,401:18389659,22793058 k1,402:19112968,22793058:723309 g1,402:19112968,22793058 ) (1,407:3729360,24481354:15383608,541848,14544 k1,407:8777594,24481354:5048234 (1,407:8777594,24481354:0,0,0 g1,407:8777594,24481354 ) x1,407:14064734,24481354 g1,407:14064734,24481354 k1,407:19112968,24481354:5048234 ) (1,410:5039682,25551990:14073286,403728,127692 (1,409:5039682,25551990:0,402253,92009 g1,409:5039682,25551990 g1,409:4057040,25551990 g1,409:3729360,25551990 (1,409:3729360,25551990:982642,402253,92009 k1,409:4024272,25551990:294912 x1,409:4712002,25551990 ) g1,409:5039682,25551990 ) x1,409:5711485,25551990 k1,409:5896839,25551990:185354 x1,409:6470146,25551990 k1,409:6655499,25551990:185353 x1,409:8442047,25551990 k1,409:8636875,25551990:194828 x1,409:9210174,25551990 k1,409:9395528,25551990:185354 x1,409:10149317,25551990 k1,409:10344145,25551990:194828 x1,409:12048116,25551990 k1,409:12233469,25551990:185353 x1,409:12724790,25551990 k1,409:12910144,25551990:185354 x1,409:14908444,25551990 k1,409:15093797,25551990:185353 x1,409:16535906,25551990 k1,409:16721260,25551990:185354 x1,409:19112968,25551990 k1,409:19112968,25551990:0 ) (1,410:5039682,26272886:14073286,402253,127692 x1,410:8479486,26272886 k1,409:8630152,26272886:150666 x1,409:9055995,26272886 k1,409:9206662,26272886:150667 x1,409:10681797,26272886 x1,409:11006789,26272886 x1,409:11377790,26272886 x1,409:11721654,26272886 k1,409:11872320,26272886:150666 x1,409:12396667,26272886 x1,409:12768258,26272886 x1,409:13875358,26272886 k1,409:14026827,26272886:151469 x1,409:14911555,26272886 k1,409:15062221,26272886:150666 x1,409:17470438,26272886 k1,409:17621907,26272886:151469 x1,409:19112968,26272886 k1,409:19112968,26272886:0 ) (1,410:5039682,26993782:14073286,402253,127692 x1,410:6366786,26993782 g1,409:6514242,26993782 x1,409:9152508,26993782 g1,409:9299964,26993782 x1,409:9742332,26993782 g1,409:9889788,26993782 x1,409:10970933,26993782 x1,409:12003125,26993782 k1,410:19112968,26993782:7109843 g1,410:19112968,26993782 ) (1,411:5039682,28034538:14073286,403728,127692 (1,410:5039682,28034538:0,402253,92009 g1,410:5039682,28034538 g1,410:4057040,28034538 g1,410:3729360,28034538 (1,410:3729360,28034538:982642,402253,92009 k1,410:4024272,28034538:294912 x1,410:4712002,28034538 ) g1,410:5039682,28034538 ) x1,410:5612989,28034538 k1,410:5733845,28034538:120856 x1,410:6749520,28034538 x1,410:7450820,28034538 k1,410:7576996,28034538:126176 x1,410:8150295,28034538 k1,410:8271151,28034538:120856 x1,410:9024940,28034538 k1,410:9151116,28034538:126176 x1,410:11149999,28034538 k1,410:11270854,28034538:120855 x1,410:11762175,28034538 k1,410:11883031,28034538:120856 x1,410:14077740,28034538 x1,410:15716258,28034538 x1,410:17616071,28034538 k1,410:17736927,28034538:120856 x1,410:19112968,28034538 k1,410:19112968,28034538:0 ) (1,411:5039682,28755434:14073286,402253,127692 x1,411:7431390,28755434 g1,410:7578846,28755434 x1,410:8004689,28755434 g1,410:8152145,28755434 x1,410:9593656,28755434 x1,410:9964657,28755434 x1,410:10308521,28755434 g1,410:10455977,28755434 x1,410:13781389,28755434 g1,410:13928845,28755434 x1,410:15517221,28755434 g1,410:15664677,28755434 x1,410:16516379,28755434 x1,410:16587752,28755434 g1,410:16735208,28755434 x1,410:17161059,28755434 x1,410:18570149,28755434 k1,411:19112968,28755434:542819 g1,411:19112968,28755434 ) (1,412:5039682,29796189:14073286,403728,127692 (1,411:5039682,29796189:0,402253,92009 g1,411:5039682,29796189 g1,411:4057040,29796189 g1,411:3729360,29796189 (1,411:3729360,29796189:982642,402253,92009 k1,411:4024272,29796189:294912 x1,411:4712002,29796189 ) g1,411:5039682,29796189 ) x1,411:5612989,29796189 k1,411:5844512,29796189:231523 x1,411:6860187,29796189 x1,411:7561487,29796189 k1,411:7814026,29796189:252539 x1,411:8387325,29796189 k1,411:8618848,29796189:231523 x1,411:9372637,29796189 k1,411:9625176,29796189:252539 x1,411:10968781,29796189 x1,411:11624059,29796189 k1,411:11855581,29796189:231522 x1,411:12281424,29796189 k1,411:12512947,29796189:231523 x1,411:13397668,29796189 x1,411:13954458,29796189 x1,411:14325459,29796189 k1,411:14556897,29796189:231438 x1,411:15048218,29796189 k1,411:15279741,29796189:231523 x1,411:17474450,29796189 x1,411:19112968,29796189 k1,412:19112968,29796189:0 ) (1,412:5039682,30517085:14073286,402253,127692 x1,412:6939495,30517085 k1,411:7107482,30517085:167987 x1,411:8483523,30517085 k1,411:8651509,30517085:167986 x1,411:9372272,30517085 x1,411:10387939,30517085 x1,411:11043217,30517085 k1,411:11211204,30517085:167987 x1,411:11898934,30517085 k1,411:12066921,30517085:167987 x1,411:12689771,30517085 x1,411:13606935,30517085 k1,411:13774921,30517085:167986 x1,411:15216439,30517085 x1,411:15970228,30517085 k1,411:16138215,30517085:167987 x1,411:17514847,30517085 x1,411:17924181,30517085 k1,411:18097300,30517085:173119 x1,411:18916559,30517085 x1,411:19112968,30517085 k1,412:19112968,30517085:0 ) (1,412:5039682,31237981:14073286,402253,127692 x1,412:6120228,31237981 x1,412:7103452,31237981 k1,411:7254662,31237981:151210 x1,411:7745983,31237981 k1,411:7897193,31237981:151210 x1,411:8421540,31237981 x1,411:8793131,31237981 x1,411:9900231,31237981 k1,411:10052380,31237981:152149 x1,411:10740109,31237981 x1,411:11151214,31237981 k1,411:11302424,31237981:151210 x1,411:11728275,31237981 x1,411:13284208,31237981 k1,411:13436357,31237981:152149 x1,411:15910051,31237981 k1,411:16062200,31237981:152149 x1,411:17241840,31237981 k1,411:17393046,31237981:151206 x1,411:19112968,31237981 k1,411:19112968,31237981:0 ) (1,412:5039682,31958877:14073286,402253,127692 x1,412:5482050,31958877 g1,411:5629506,31958877 x1,411:7005563,31958877 x1,411:7300475,31958877 k1,412:19112968,31958877:11812493 g1,412:19112968,31958877 ) (1,413:5039682,32999633:14073286,403728,127692 (1,412:5039682,32999633:0,402253,92009 g1,412:5039682,32999633 g1,412:4057040,32999633 g1,412:3729360,32999633 (1,412:3729360,32999633:982642,402253,92009 k1,412:4024272,32999633:294912 x1,412:4712002,32999633 ) g1,412:5039682,32999633 ) x1,412:5711485,32999633 k1,412:5945260,32999633:233775 x1,412:6518567,32999633 k1,412:6752343,32999633:233776 x1,412:7080281,32999633 x1,412:8414446,32999633 x1,412:8538314,32999633 k1,412:8793669,32999633:255355 x1,412:9366968,32999633 k1,412:9600743,32999633:233775 x1,412:10354532,32999633 k1,412:10609887,32999633:255355 x1,412:11788936,32999633 k1,412:12022712,32999633:233776 x1,412:12711032,32999633 x1,412:13005944,32999633 x1,412:13277268,32999633 x1,412:13530304,32999633 x1,412:14709937,32999633 k1,412:14943712,32999633:233775 x1,412:15795414,32999633 k1,412:16029190,32999633:233776 x1,412:16749953,32999633 x1,412:17011831,32999633 x1,412:17470118,32999633 x1,412:18387872,32999633 k1,412:18621647,32999633:233775 x1,412:19112968,32999633 k1,412:19112968,32999633:0 ) (1,413:5039682,33720529:14073286,402253,127692 x1,413:5760437,33720529 k1,412:6030090,33720529:269653 x1,412:6718410,33720529 x1,412:7799539,33720529 k1,412:8069193,33720529:269654 x1,412:9117295,33720529 x1,412:10067493,33720529 k1,412:10337146,33720529:269653 x1,412:11123961,33720529 x1,412:12139038,33720529 k1,412:12408692,33720529:269654 x1,412:13555889,33720529 k1,412:13825542,33720529:269653 x1,412:14316855,33720529 k1,412:14586508,33720529:269653 x1,412:14848386,33720529 k1,412:15118040,33720529:269654 x1,412:15478421,33720529 x1,412:16092414,33720529 k1,412:16362067,33720529:269653 x1,412:17049796,33720529 x1,412:17737517,33720529 x1,412:17999395,33720529 x1,412:18457682,33720529 x1,412:19112968,33720529 k1,412:19112968,33720529:0 ) (1,413:5039682,34441425:14073286,402253,127692 x1,413:6120812,34441425 x1,413:6677602,34441425 x1,413:7245012,34441425 k1,412:7416862,34441425:171850 x1,412:8104592,34441425 k1,412:8276467,34441425:171875 x1,412:9717985,34441425 x1,412:10471774,34441425 k1,412:10643648,34441425:171874 x1,412:12020280,34441425 k1,412:12192155,34441425:171875 x1,412:12847441,34441425 x1,412:14353831,34441425 k1,412:14525705,34441425:171874 x1,412:15377407,34441425 k1,412:15549282,34441425:171875 x1,412:15975133,34441425 x1,412:16237011,34441425 x1,412:16517177,34441425 x1,412:16934178,34441425 x1,412:17524002,34441425 x1,412:18703634,34441425 x1,412:19112968,34441425 k1,412:19112968,34441425:0 ) (1,413:5039682,35162321:14073286,402253,127692 x1,413:5367620,35162321 x1,412:5739211,35162321 x1,412:6846311,35162321 g1,412:6993767,35162321 x1,412:7419618,35162321 x1,412:7910938,35162321 x1,412:8762640,35162321 x1,412:9057552,35162321 x1,412:9506997,35162321 x1,412:9630865,35162321 g1,412:9778321,35162321 x1,412:10712593,35162321 g1,412:10860049,35162321 x1,412:12187153,35162321 g1,412:12334609,35162321 x1,412:15103815,35162321 g1,412:15251271,35162321 x1,412:15693639,35162321 g1,412:15841095,35162321 x1,412:17217152,35162321 x1,412:18544256,35162321 k1,413:19112968,35162321:568712 g1,413:19112968,35162321 ) (1,414:5039682,36203077:14073286,403728,127692 (1,413:5039682,36203077:0,402253,92009 g1,413:5039682,36203077 g1,413:4057040,36203077 g1,413:3729360,36203077 (1,413:3729360,36203077:982642,402253,92009 k1,413:4024272,36203077:294912 x1,413:4712002,36203077 ) g1,413:5039682,36203077 ) x1,413:5711485,36203077 k1,413:5928914,36203077:217429 x1,413:6502221,36203077 k1,413:6719650,36203077:217429 x1,413:8506198,36203077 k1,413:8741119,36203077:234921 x1,413:9314418,36203077 k1,413:9531847,36203077:217429 x1,413:10285636,36203077 k1,413:10520558,36203077:234922 x1,413:11830539,36203077 x1,413:12813173,36203077 x1,413:13992805,36203077 k1,413:14210233,36203077:217428 x1,413:14570614,36203077 x1,413:15347997,36203077 x1,413:16888011,36203077 k1,413:17105440,36203077:217429 x1,413:17564317,36203077 k1,413:17781746,36203077:217429 x1,413:18731944,36203077 x1,413:18928353,36203077 x1,413:19112968,36203077 k1,414:19112968,36203077:0 ) (1,414:5039682,36923973:14073286,402253,127692 x1,414:5563438,36923973 x1,414:5989281,36923973 x1,414:6873411,36923973 k1,413:7083286,36923973:209875 x1,413:7574607,36923973 x1,413:8229885,36923973 x1,413:8852727,36923973 x1,413:9016692,36923973 x1,413:9835935,36923973 k1,413:10045810,36923973:209875 x1,413:11782233,36923973 k1,413:11992108,36923973:209875 x1,413:12581932,36923973 k1,413:12791807,36923973:209875 x1,413:13839909,36923973 x1,413:14790107,36923973 k1,413:14999982,36923973:209875 x1,413:16671526,36923973 x1,413:17080860,36923973 k1,413:17306339,36923973:225479 x1,413:17634277,36923973 x1,413:18005868,36923973 x1,413:19112968,36923973 k1,413:19112968,36923973:0 ) (1,414:5039682,37644869:14073286,402253,127692 x1,414:5727411,37644869 x1,413:6138516,37644869 g1,413:6285972,37644869 x1,413:6711823,37644869 x1,413:7580632,37644869 g1,413:7728088,37644869 x1,413:8579790,37644869 x1,413:8651163,37644869 g1,413:8798619,37644869 x1,413:10125723,37644869 g1,413:10273179,37644869 x1,413:12550480,37644869 g1,413:12697936,37644869 x1,413:13140304,37644869 g1,413:13287760,37644869 x1,413:14663817,37644869 x1,413:15990921,37644869 k1,414:19112968,37644869:3122047 g1,414:19112968,37644869 ) (1,415:5039682,38685625:14073286,403728,127692 (1,414:5039682,38685625:0,402253,92009 g1,414:5039682,38685625 g1,414:4057040,38685625 g1,414:3729360,38685625 (1,414:3729360,38685625:982642,402253,92009 k1,414:4024272,38685625:294912 x1,414:4712002,38685625 ) g1,414:5039682,38685625 ) x1,414:5711485,38685625 k1,414:5861510,38685625:150025 x1,414:6434817,38685625 k1,414:6584842,38685625:150025 x1,414:8371390,38685625 k1,414:8522057,38685625:150667 x1,414:9095356,38685625 k1,414:9245381,38685625:150025 x1,414:9999170,38685625 k1,414:10149837,38685625:150667 x1,414:10772096,38685625 x1,414:13089493,38685625 k1,414:13239518,38685625:150025 x1,414:13927248,38685625 k1,414:14077273,38685625:150025 x1,414:15780670,38685625 x1,414:16945556,38685625 k1,414:17095581,38685625:150025 x1,414:17586902,38685625 k1,414:17736927,38685625:150025 x1,414:19112968,38685625 k1,414:19112968,38685625:0 ) (1,415:5039682,39406521:14073286,402253,127692 x1,415:7431390,39406521 k1,414:7618264,39406521:186874 x1,414:8077141,39406521 k1,414:8264015,39406521:186874 x1,414:8984770,39406521 k1,414:9171645,39406521:186875 x1,414:11267851,39406521 k1,414:11454725,39406521:186874 x1,414:11946046,39406521 k1,414:12132920,39406521:186874 x1,414:12689710,39406521 k1,414:12876584,39406521:186874 x1,414:14874884,39406521 k1,414:15061758,39406521:186874 x1,414:16503867,39406521 x1,414:16913201,39406521 k1,414:17109930,39406521:196729 x1,414:17634277,39406521 x1,414:18005868,39406521 x1,414:19112968,39406521 k1,414:19112968,39406521:0 ) (1,415:5039682,40127417:14073286,402253,127692 x1,415:5465533,40127417 x1,414:6646944,40127417 g1,414:6794400,40127417 x1,414:8284871,40127417 g1,414:8432327,40127417 x1,414:9759431,40127417 g1,414:9906887,40127417 x1,414:12577596,40127417 g1,414:12725052,40127417 x1,414:13167420,40127417 g1,414:13314876,40127417 x1,414:14690933,40127417 x1,414:16018037,40127417 k1,415:19112968,40127417:3094931 g1,415:19112968,40127417 ) (1,416:5039682,41168172:14073286,403728,127692 (1,415:5039682,41168172:0,394002,92009 g1,415:5039682,41168172 g1,415:4057040,41168172 g1,415:3729360,41168172 (1,415:3729360,41168172:982642,394002,92009 k1,415:4024272,41168172:294912 x1,415:4712002,41168172 ) g1,415:5039682,41168172 ) x1,415:5612989,41168172 k1,415:5795798,41168172:182809 x1,415:6156179,41168172 x1,415:6259988,41168172 k1,415:6442796,41168172:182808 x1,415:7753368,41168172 x1,415:7877236,41168172 k1,415:8068883,41168172:191647 x1,415:8642182,41168172 k1,415:8824990,41168172:182808 x1,415:9578779,41168172 k1,415:9770426,41168172:191647 x1,415:10458155,41168172 x1,415:10720033,41168172 x1,415:11000199,41168172 x1,415:11417200,41168172 x1,415:12007024,41168172 x1,415:13186656,41168172 k1,415:13369465,41168172:182809 x1,415:13860786,41168172 k1,415:14043594,41168172:182808 x1,415:14600384,41168172 k1,415:14783193,41168172:182809 x1,415:15405453,41168172 x1,415:16355651,41168172 k1,415:16538459,41168172:182808 x1,415:17914500,41168172 k1,415:18097309,41168172:182809 x1,415:19112968,41168172 k1,415:19112968,41168172:0 ) (1,416:5039682,41889068:14073286,402253,127692 x1,416:6153853,41889068 x1,416:6317818,41889068 x1,416:6973104,41889068 k1,415:7131214,41889068:158110 x1,415:8179907,41889068 k1,415:8338016,41889068:158109 x1,415:9124831,41889068 x1,415:9845586,41889068 k1,415:10003696,41889068:158110 x1,415:10331634,41889068 x1,415:11043540,41889068 k1,415:11201650,41889068:158110 x1,415:12249753,41889068 x1,415:13232387,41889068 x1,415:14412019,41889068 k1,415:14570129,41889068:158110 x1,415:15225415,41889068 x1,415:15782205,41889068 x1,415:16405047,41889068 x1,415:17027889,41889068 k1,415:17185998,41889068:158109 x1,415:17873728,41889068 k1,415:18031838,41889068:158110 x1,415:18916559,41889068 x1,415:19112968,41889068 k1,416:19112968,41889068:0 ) (1,416:5039682,42609964:14073286,402253,127692 x1,416:5596472,42609964 x1,416:5967473,42609964 x1,416:6311337,42609964 g1,415:6458793,42609964 x1,415:6786731,42609964 x1,415:7158322,42609964 x1,415:8265422,42609964 g1,415:8412878,42609964 x1,415:9100607,42609964 x1,415:9511712,42609964 g1,415:9659168,42609964 x1,415:10085019,42609964 x1,415:10953828,42609964 g1,415:11101284,42609964 x1,415:11952986,42609964 x1,415:12024359,42609964 g1,415:12171815,42609964 x1,415:13498919,42609964 g1,415:13646375,42609964 x1,415:15956703,42609964 g1,415:16104159,42609964 x1,415:16546527,42609964 g1,415:16693983,42609964 x1,415:17775128,42609964 x1,415:18807320,42609964 k1,416:19112968,42609964:305648 g1,416:19112968,42609964 ) (1,417:5039682,43650720:14073286,403728,127692 (1,416:5039682,43650720:0,402253,92009 g1,416:5039682,43650720 g1,416:4057040,43650720 g1,416:3729360,43650720 (1,416:3729360,43650720:982642,402253,92009 k1,416:4024272,43650720:294912 x1,416:4712002,43650720 ) g1,416:5039682,43650720 ) x1,416:5612989,43650720 k1,416:5775244,43650720:162255 x1,416:6283081,43650720 k1,416:6445335,43650720:162254 x1,416:8656561,43650720 k1,416:8822515,43650720:165954 x1,416:9395814,43650720 k1,416:9558069,43650720:162255 x1,416:10311858,43650720 k1,416:10477812,43650720:165954 x1,416:10739690,43650720 x1,416:11413271,43650720 x1,416:12036113,43650720 x1,416:12559869,43650720 x1,416:13477623,43650720 k1,416:13639877,43650720:162254 x1,416:14131198,43650720 k1,416:14293453,43650720:162255 x1,416:14981773,43650720 x1,416:15538563,43650720 x1,416:16455727,43650720 k1,416:16617981,43650720:162254 x1,416:16978362,43650720 x1,416:17803524,43650720 k1,416:17965779,43650720:162255 x1,416:19112968,43650720 k1,416:19112968,43650720:0 ) (1,417:5039682,44371616:14073286,402253,8251 x1,417:6087785,44371616 x1,417:7070419,44371616 x1,417:8479486,44371616 k1,416:8750158,44371616:270672 x1,416:9209035,44371616 k1,416:9479706,44371616:270671 x1,416:10200461,44371616 k1,416:10471133,44371616:270672 x1,416:11421331,44371616 x1,416:12141496,44371616 x1,416:12567339,44371616 x1,416:12927713,44371616 x1,416:13451469,44371616 x1,416:14369223,44371616 k1,416:14639895,44371616:270672 x1,416:15131216,44371616 k1,416:15401887,44371616:270671 x1,416:16449989,44371616 x1,416:17400187,44371616 k1,416:17670859,44371616:270672 x1,416:19112968,44371616 k1,416:19112968,44371616:0 ) (1,417:5039682,45092512:14073286,402253,127692 x1,417:5956854,45092512 x1,416:6826836,45092512 k1,416:6974650,45092512:147814 x1,416:7383984,45092512 k1,416:7531887,45092512:147903 x1,416:7859825,45092512 x1,416:8231416,45092512 x1,416:9338516,45092512 k1,416:9486419,45092512:147903 x1,416:10174148,45092512 x1,416:10585253,45092512 k1,416:10733066,45092512:147813 x1,416:11158917,45092512 x1,416:12027726,45092512 k1,416:12175630,45092512:147904 x1,416:13027332,45092512 x1,416:13098705,45092512 k1,416:13246518,45092512:147813 x1,416:14573622,45092512 k1,416:14721525,45092512:147903 x1,416:16998826,45092512 k1,416:17146729,45092512:147903 x1,416:17589097,45092512 k1,416:17736911,45092512:147814 x1,416:19112968,45092512 k1,417:19112968,45092512:0 ) (1,417:5039682,45813408:14073286,402253,8251 x1,417:6366786,45813408 k1,417:19112968,45813408:12746182 g1,417:19112968,45813408 ) (1,418:5039682,46854164:14073286,403728,127692 (1,417:5039682,46854164:0,402253,92009 g1,417:5039682,46854164 g1,417:4057040,46854164 g1,417:3729360,46854164 (1,417:3729360,46854164:982642,402253,92009 k1,417:4024272,46854164:294912 x1,417:4712002,46854164 ) g1,417:5039682,46854164 ) x1,417:5612989,46854164 k1,417:5746091,46854164:133102 x1,417:6253928,46854164 k1,417:6387029,46854164:133101 x1,417:8598255,46854164 k1,417:8734228,46854164:135973 x1,417:9307527,46854164 k1,417:9440628,46854164:133101 x1,417:10194417,46854164 k1,417:10330390,46854164:135973 x1,417:11247554,46854164 x1,417:12001343,46854164 k1,417:12134445,46854164:133102 x1,417:12822765,46854164 x1,417:13379555,46854164 x1,417:14296719,46854164 k1,417:14429820,46854164:133101 x1,417:14790201,46854164 x1,417:15615363,46854164 k1,417:15748465,46854164:133102 x1,417:17125089,46854164 k1,417:17258190,46854164:133101 x1,417:17717067,46854164 k1,417:17850169,46854164:133102 x1,417:18505462,46854164 x1,417:18916559,46854164 x1,417:19112968,46854164 k1,418:19112968,46854164:0 ) (1,418:5039682,47575060:14073286,402253,127692 x1,418:5334594,47575060 x1,418:6022315,47575060 k1,417:6154170,47575060:131855 x1,417:7202272,47575060 x1,417:8152470,47575060 k1,417:8284326,47575060:131856 x1,417:9726435,47575060 k1,417:9858290,47575060:131855 x1,417:10218671,47575060 x1,417:11513899,47575060 k1,417:11645754,47575060:131855 x1,417:12235578,47575060 k1,417:12367434,47575060:131856 x1,417:13743475,47575060 k1,417:13875330,47575060:131855 x1,417:14596093,47575060 x1,417:15611760,47575060 x1,417:16267038,47575060 k1,417:16398894,47575060:131856 x1,417:16824737,47575060 k1,417:16956592,47575060:131855 x1,417:17841313,47575060 x1,417:18398103,47575060 x1,417:18769104,47575060 x1,417:19112968,47575060 k1,417:19112968,47575060:0 ) (1,418:5039682,48295956:14073286,402253,83160 x1,418:8365094,48295956 g1,417:8512550,48295956 x1,417:10100926,48295956 g1,417:10248382,48295956 x1,417:11100084,48295956 x1,417:11171457,48295956 g1,417:11318913,48295956 x1,417:13072450,48295956 k1,418:19112968,48295956:6040518 g1,418:19112968,48295956 ) ] g1,440:19112968,48295956 ) k1,440:19579138,48295956:466170 r1,440:19579138,48295956:0,44938783,0 k1,440:20045308,48295956:466170 (1,440:20045308,48295956:15383608,44938783,0 [1,440:20045308,48295956:15383608,44938783,0 (1,419:21355630,4012533:14073286,403728,127692 (1,418:21355630,4012533:0,402253,92009 g1,418:21355630,4012533 g1,418:20372988,4012533 g1,418:20045308,4012533 (1,418:20045308,4012533:982642,402253,92009 g1,418:20045308,4012533 x1,418:21027950,4012533 ) g1,418:21355630,4012533 ) x1,418:21732521,4012533 k1,418:21949042,4012533:216521 x1,418:22522349,4012533 k1,418:22738870,4012533:216521 x1,418:23656617,4012533 x1,418:24377372,4012533 x1,418:25376530,4012533 k1,418:25610317,4012533:233787 x1,418:26183616,4012533 k1,418:26400137,4012533:216521 x1,418:27153926,4012533 k1,418:27387714,4012533:233788 x1,418:29680920,4012533 k1,418:29897441,4012533:216521 x1,418:30356318,4012533 k1,418:30572839,4012533:216521 x1,418:31654566,4012533 x1,418:32309844,4012533 k1,418:32526365,4012533:216521 x1,418:33017686,4012533 k1,418:33234207,4012533:216521 x1,418:35428916,4012533 k1,419:35428916,4012533:0 ) (1,419:21355630,4733429:14073286,402253,127692 x1,419:22862618,4733429 x1,419:24729987,4733429 k1,418:24903807,4733429:173820 x1,418:25624570,4733429 x1,418:26443813,4733429 k1,418:26617634,4733429:173821 x1,418:27436885,4733429 x1,418:27895762,4733429 x1,418:29239360,4733429 k1,418:29413180,4733429:173820 x1,418:29839023,4733429 k1,418:30012844,4733429:173821 x1,418:30897565,4733429 x1,418:31454355,4733429 x1,418:31825356,4733429 x1,418:32169220,4733429 k1,418:32343040,4733429:173820 x1,418:32867387,4733429 x1,418:33238978,4733429 x1,418:34346078,4733429 k1,418:34526490,4733429:180412 x1,418:34952341,4733429 x1,418:35232507,4733429 x1,418:35428916,4733429 k1,419:35428916,4733429:0 ) (1,419:21355630,5454325:14073286,402253,127692 x1,419:22256875,5454325 g1,418:22404331,5454325 x1,418:23894802,5454325 g1,418:24042258,5454325 x1,418:25369362,5454325 g1,418:25516818,5454325 x1,418:28089031,5454325 g1,418:28236487,5454325 x1,418:28678855,5454325 g1,418:28826311,5454325 x1,418:30202368,5454325 x1,418:31529472,5454325 k1,419:35428916,5454325:3899444 g1,419:35428916,5454325 ) (1,420:21355630,6437365:14073286,403728,92009 (1,419:21355630,6437365:0,402253,92009 g1,419:21355630,6437365 g1,419:20372988,6437365 g1,419:20045308,6437365 (1,419:20045308,6437365:982642,402253,92009 g1,419:20045308,6437365 x1,419:21027950,6437365 ) g1,419:21355630,6437365 ) x1,419:21732521,6437365 k1,419:21911176,6437365:178655 x1,419:22484483,6437365 k1,419:22663138,6437365:178655 x1,419:25300798,6437365 k1,419:25487253,6437365:186455 x1,419:26060552,6437365 k1,419:26239207,6437365:178655 x1,419:26992996,6437365 k1,419:27179451,6437365:186455 x1,419:29178334,6437365 k1,419:29356989,6437365:178655 x1,419:29848310,6437365 k1,419:30026966,6437365:178656 x1,419:30747721,6437365 k1,419:30926376,6437365:178655 x1,419:31286757,6437365 x1,419:32581985,6437365 k1,419:32760640,6437365:178655 x1,419:33251961,6437365 k1,419:33430616,6437365:178655 x1,419:35428916,6437365 k1,419:35428916,6437365:0 ) (1,420:21355630,7158261:14073286,402253,127692 x1,420:22797739,7158261 k1,419:23002858,7158261:205119 x1,419:23330796,7158261 x1,419:25451784,7158261 k1,419:25656902,7158261:205118 x1,419:26246726,7158261 k1,419:26451845,7158261:205119 x1,419:27827886,7158261 k1,419:28033004,7158261:205118 x1,419:30424712,7158261 k1,419:30629831,7158261:205119 x1,419:31055674,7158261 k1,419:31260792,7158261:205118 x1,419:32702303,7158261 x1,419:33073304,7158261 x1,419:33417168,7158261 k1,419:33622287,7158261:205119 x1,419:33950225,7158261 x1,419:34321816,7158261 x1,419:35428916,7158261 k1,419:35428916,7158261:0 ) (1,420:21355630,7879157:14073286,402253,127692 x1,420:21781481,7879157 x1,419:23419415,7879157 x1,419:23868860,7879157 x1,419:23992728,7879157 g1,419:24140184,7879157 x1,419:25074456,7879157 g1,419:25221912,7879157 x1,419:26549016,7879157 g1,419:26696472,7879157 x1,419:29794206,7879157 g1,419:29941662,7879157 x1,419:30384030,7879157 g1,419:30531486,7879157 x1,419:31907543,7879157 x1,419:33234647,7879157 k1,420:35428916,7879157:2194269 g1,420:35428916,7879157 ) (1,422:21355630,8862197:14073286,402253,92009 (1,420:21355630,8862197:0,402253,92009 g1,420:21355630,8862197 g1,420:20372988,8862197 g1,420:20045308,8862197 (1,420:20045308,8862197:982642,402253,92009 g1,420:20045308,8862197 x1,420:21027950,8862197 ) g1,420:21355630,8862197 ) x1,420:22027433,8862197 k1,420:22167188,8862197:139755 x1,420:22740495,8862197 k1,420:22880251,8862197:139756 x1,420:23503101,8862197 x1,420:24862633,8862197 k1,420:25003928,8862197:141295 x1,420:25577235,8862197 k1,420:25716991,8862197:139756 x1,420:26257855,8862197 k1,420:26397610,8862197:139755 x1,420:27478746,8862197 x1,420:28395910,8862197 x1,420:29362625,8862197 k1,420:29503920,8862197:141295 x1,420:30732513,8862197 k1,420:30872269,8862197:139756 x1,420:32182842,8862197 x1,420:32920122,8862197 k1,420:33061417,8862197:141295 x1,420:34273500,8862197 k1,420:34413248,8862197:139748 x1,420:35232507,8862197 x1,420:35428916,8862197 k1,422:35428916,8862197:0 ) (1,422:21355630,9583093:14073286,402253,127692 x1,422:22436176,9583093 x1,422:23419400,9583093 k1,420:23561483,9583093:142083 x1,420:24347716,9583093 x1,420:25084996,9583093 k1,420:25228155,9583093:143159 x1,420:25670523,9583093 k1,420:25812606,9583093:142083 x1,420:26861307,9583093 k1,420:27005181,9583093:143874 x1,420:27676984,9583093 k1,421:27819068,9583093:142084 x1,421:28392375,9583093 k1,421:28534458,9583093:142083 x1,421:29157308,9583093 x1,421:30516840,9583093 k1,421:30659998,9583093:143158 x1,421:31331801,9583093 k1,421:31473885,9583093:142084 x1,421:31834266,9583093 x1,421:31938075,9583093 k1,421:32080158,9583093:142083 x1,421:32408096,9583093 x1,421:32508370,9583093 k1,421:32650454,9583093:142084 x1,421:33142357,9583093 x1,421:33733944,9583093 k1,421:33877102,9583093:143158 x1,421:35286767,9583093 x1,421:35428916,9583093 k1,422:35428916,9583093:0 ) (1,422:21355630,10303989:14073286,402253,104688 x1,422:22174889,10303989 g1,421:22322345,10303989 x1,421:22764713,10303989 g1,421:22912169,10303989 x1,421:24681641,10303989 g1,421:24829097,10303989 x1,421:26549019,10303989 k1,422:35428916,10303989:8879897 g1,422:35428916,10303989 ) (1,424:21355630,11287029:14073286,402253,127692 (1,422:21355630,11287029:0,402253,92009 g1,422:21355630,11287029 g1,422:20372988,11287029 g1,422:20045308,11287029 (1,422:20045308,11287029:982642,402253,92009 g1,422:20045308,11287029 x1,422:21027950,11287029 ) g1,422:21355630,11287029 ) x1,422:21928937,11287029 k1,422:22227329,11287029:298392 x1,422:24144233,11287029 k1,422:24480359,11287029:336126 x1,422:24808297,11287029 x1,422:24908571,11287029 k1,422:25206963,11287029:298392 x1,422:26190186,11287029 x1,422:26910941,11287029 x1,422:28565385,11287029 k1,422:28901511,11287029:336126 x1,422:30211491,11287029 x1,422:31161689,11287029 k1,422:31460081,11287029:298392 x1,422:32902190,11287029 k1,422:33200582,11287029:298392 x1,422:34708153,11287029 x1,422:35428916,11287029 k1,422:35428916,11287029:0 ) (1,424:21355630,12007925:14073286,402253,127692 x1,424:22306410,12007925 x1,424:22601322,12007925 x1,424:23027165,12007925 x1,424:24174354,12007925 k1,422:24458899,12007925:284545 x1,422:24983246,12007925 x1,422:25737035,12007925 k1,422:26021579,12007925:284544 x1,422:28413302,12007925 x1,422:28822636,12007925 k1,422:29107181,12007925:284545 x1,422:32432593,12007925 k1,422:32751410,12007925:318817 x1,422:34143385,12007925 k1,422:34462201,12007925:318816 x1,422:35428916,12007925 k1,422:35428916,12007925:0 ) (1,424:21355630,12728821:14073286,402253,127692 x1,424:21797998,12728821 k1,422:21990230,12728821:192232 x1,422:23071375,12728821 x1,422:24120076,12728821 k1,422:24334697,12728821:214621 x1,422:24908004,12728821 k1,423:25100236,12728821:192232 x1,423:25788556,12728821 x1,423:27148678,12728821 k1,423:27352104,12728821:203426 x1,423:27680042,12728821 x1,423:27780316,12728821 k1,423:27972548,12728821:192232 x1,423:28955771,12728821 x1,423:29676526,12728821 x1,423:31330970,12728821 k1,423:31534397,12728821:203427 x1,423:32844377,12728821 x1,423:33794575,12728821 k1,423:33986807,12728821:192232 x1,423:35428916,12728821 k1,423:35428916,12728821:0 ) (1,424:21355630,13449717:14073286,402253,127692 x1,424:22306410,13449717 x1,424:22601322,13449717 x1,424:23027165,13449717 x1,424:24338319,13449717 k1,423:24789829,13449717:451510 x1,423:26165870,13449717 k1,423:26447659,13449717:281789 x1,423:26938980,13449717 x1,423:27594258,13449717 x1,423:28217100,13449717 x1,423:28381065,13449717 x1,423:29200308,13449717 k1,423:29482097,13449717:281789 x1,423:30333799,13449717 k1,423:30615588,13449717:281789 x1,423:31172378,13449717 x1,423:31681986,13449717 x1,423:32134375,13449717 x1,423:33412503,13449717 x1,423:33821837,13449717 k1,423:34137209,13449717:315372 x1,423:34760059,13449717 x1,423:35428916,13449717 k1,423:35428916,13449717:0 ) (1,424:21355630,14170613:14073286,402253,104688 x1,424:22010916,14170613 x1,423:22291082,14170613 x1,423:22400205,14170613 g1,423:22547661,14170613 x1,423:23235980,14170613 g1,423:23383436,14170613 x1,423:24071165,14170613 x1,423:24906342,14170613 g1,423:25053798,14170613 x1,423:26659274,14170613 g1,423:26806730,14170613 x1,423:27249098,14170613 g1,423:27396554,14170613 x1,423:29166026,14170613 g1,423:29313482,14170613 x1,423:30606954,14170613 g1,423:30754410,14170613 x1,423:32277923,14170613 k1,424:35428916,14170613:3150993 g1,424:35428916,14170613 ) (1,425:21355630,15153653:14073286,402253,127692 (1,424:21355630,15153653:0,402253,92009 g1,424:21355630,15153653 g1,424:20372988,15153653 g1,424:20045308,15153653 (1,424:20045308,15153653:982642,402253,92009 g1,424:20045308,15153653 x1,424:21027950,15153653 ) g1,424:21355630,15153653 ) x1,424:21928937,15153653 k1,424:22152000,15153653:223063 x1,424:23167675,15153653 x1,424:23868975,15153653 k1,424:24110939,15153653:241964 x1,424:24684246,15153653 k1,424:24907309,15153653:223063 x1,424:25792037,15153653 x1,424:26493337,15153653 k1,424:26735301,15153653:241964 x1,424:29388903,15153653 k1,424:29611890,15153653:222987 x1,424:31283441,15153653 k1,424:31506504,15153653:223063 x1,424:33177465,15153653 x1,424:34354151,15153653 k1,424:34577214,15153653:223063 x1,424:35428916,15153653 k1,424:35428916,15153653:0 ) (1,425:21355630,15874549:14073286,402253,127692 x1,425:23747330,15874549 k1,424:23982219,15874549:234889 x1,424:26571508,15874549 x1,424:26980842,15874549 k1,424:27237589,15874549:256747 x1,424:30563001,15874549 k1,424:30819748,15874549:256747 x1,424:32408124,15874549 k1,424:32664871,15874549:256747 x1,424:33516573,15874549 x1,424:33587946,15874549 k1,424:33822835,15874549:234889 x1,424:35428916,15874549 k1,424:35428916,15874549:0 ) (1,425:21355630,16595445:14073286,402253,104688 x1,425:22928096,16595445 k1,425:35428916,16595445:12500820 g1,425:35428916,16595445 ) (1,426:21355630,17578485:14073286,402253,127692 (1,425:21355630,17578485:0,402253,92009 g1,425:21355630,17578485 g1,425:20372988,17578485 g1,425:20045308,17578485 (1,425:20045308,17578485:982642,402253,92009 g1,425:20045308,17578485 x1,425:21027950,17578485 ) g1,425:21355630,17578485 ) x1,425:21928937,17578485 k1,425:22152000,17578485:223063 x1,425:23167675,17578485 x1,425:23868975,17578485 k1,425:24110939,17578485:241964 x1,425:24684246,17578485 k1,425:24907309,17578485:223063 x1,425:25792037,17578485 x1,425:26493337,17578485 k1,425:26735301,17578485:241964 x1,425:29388903,17578485 k1,425:29611890,17578485:222987 x1,425:31283441,17578485 k1,425:31506504,17578485:223063 x1,425:33177465,17578485 x1,425:34354151,17578485 k1,425:34577214,17578485:223063 x1,425:35428916,17578485 k1,425:35428916,17578485:0 ) (1,426:21355630,18299381:14073286,402253,127692 x1,426:23747330,18299381 g1,425:23894786,18299381 x1,425:26713510,18299381 x1,425:27122844,18299381 g1,425:27270300,18299381 x1,425:30595712,18299381 g1,425:30743168,18299381 x1,425:31971760,18299381 g1,425:32119216,18299381 x1,425:33200345,18299381 g1,425:33347801,18299381 x1,425:34920267,18299381 k1,426:35428916,18299381:508649 g1,426:35428916,18299381 ) (1,427:21355630,19282421:14073286,402253,127692 (1,426:21355630,19282421:0,402253,92009 g1,426:21355630,19282421 g1,426:20372988,19282421 g1,426:20045308,19282421 (1,426:20045308,19282421:982642,402253,92009 g1,426:20045308,19282421 x1,426:21027950,19282421 ) g1,426:21355630,19282421 ) x1,426:21928937,19282421 k1,426:22147844,19282421:218907 x1,426:24294184,19282421 k1,426:24530953,19282421:236769 x1,426:25710002,19282421 k1,426:25928909,19282421:218907 x1,426:27927209,19282421 k1,426:28146116,19282421:218907 x1,426:29588225,19282421 k1,426:29807132,19282421:218907 x1,426:32199415,19282421 k1,426:32418322,19282421:218907 x1,426:32877199,19282421 k1,426:33096106,19282421:218907 x1,426:33816861,19282421 k1,426:34035768,19282421:218907 x1,426:35019582,19282421 x1,426:35428916,19282421 k1,426:35428916,19282421:0 ) (1,427:21355630,20003317:14073286,402253,127692 x1,427:24681042,20003317 g1,426:24828498,20003317 x1,426:26549002,20003317 g1,426:26696458,20003317 x1,426:28269506,20003317 g1,426:28416962,20003317 x1,426:30218855,20003317 x1,426:31594912,20003317 x1,426:32627104,20003317 k1,427:35428916,20003317:2801812 g1,427:35428916,20003317 ) (1,428:21355630,20986357:14073286,402253,127692 (1,427:21355630,20986357:0,402253,92009 g1,427:21355630,20986357 g1,427:20372988,20986357 g1,427:20045308,20986357 (1,427:20045308,20986357:982642,402253,92009 g1,427:20045308,20986357 x1,427:21027950,20986357 ) g1,427:21355630,20986357 ) x1,427:21863467,20986357 k1,427:22284761,20986357:421294 x1,427:22841551,20986357 x1,427:23793527,20986357 k1,427:24283279,20986357:489752 x1,427:24955082,20986357 k1,427:25376376,20986357:421294 x1,427:25868279,20986357 x1,427:26459866,20986357 k1,427:26949618,20986357:489752 x1,427:27571877,20986357 x1,427:28304433,20986357 x1,427:29353134,20986357 k1,427:29774428,20986357:421294 x1,427:30265749,20986357 k1,427:30687043,20986357:421294 x1,427:31735145,20986357 x1,427:32685343,20986357 k1,427:33106636,20986357:421293 x1,427:34548745,20986357 k1,427:34970039,20986357:421294 x1,427:35428916,20986357 k1,427:35428916,20986357:0 ) (1,428:21355630,21707253:14073286,402253,127692 x1,428:23911304,21707253 x1,427:24282305,21707253 k1,427:24961722,21707253:679417 x1,427:27746263,21707253 k1,427:28426217,21707253:679954 x1,427:29277919,21707253 k1,427:29957872,21707253:679953 x1,427:30547688,21707253 x1,427:30833758,21707253 x1,427:32308302,21707253 k1,427:32988255,21707253:679953 x1,427:34069392,21707253 x1,427:35019582,21707253 x1,427:35428916,21707253 k1,427:35428916,21707253:0 ) (1,428:21355630,22428149:14073286,402253,83160 x1,428:24681042,22428149 g1,427:24828498,22428149 x1,427:26416874,22428149 g1,427:26564330,22428149 x1,427:27416032,22428149 x1,427:27487405,22428149 g1,427:27634861,22428149 x1,427:29600734,22428149 k1,428:35428916,22428149:5828182 g1,428:35428916,22428149 ) (1,429:21355630,23411189:14073286,402253,127692 (1,428:21355630,23411189:0,402253,92009 g1,428:21355630,23411189 g1,428:20372988,23411189 g1,428:20045308,23411189 (1,428:20045308,23411189:982642,402253,92009 g1,428:20045308,23411189 x1,428:21027950,23411189 ) g1,428:21355630,23411189 ) x1,428:21896494,23411189 k1,428:22163158,23411189:266664 x1,428:22834961,23411189 k1,428:23101625,23411189:266664 x1,428:24756053,23411189 k1,428:25052519,23411189:296466 x1,428:26362499,23411189 x1,428:27312697,23411189 k1,428:27579361,23411189:266664 x1,428:29021470,23411189 k1,428:29288135,23411189:266665 x1,428:30008898,23411189 x1,428:31024565,23411189 x1,428:31679843,23411189 k1,428:31946507,23411189:266664 x1,428:32405384,23411189 k1,428:32672048,23411189:266664 x1,428:33392803,23411189 k1,428:33659467,23411189:266664 x1,428:34347787,23411189 x1,428:35428916,23411189 k1,428:35428916,23411189:0 ) (1,429:21355630,24132085:14073286,402253,127692 x1,429:22403732,24132085 x1,429:23353930,24132085 k1,428:23646282,24132085:292352 x1,428:24433097,24132085 x1,428:25448174,24132085 k1,428:25740527,24132085:292353 x1,428:26887724,24132085 k1,428:27180076,24132085:292352 x1,428:27540457,24132085 x1,428:28154450,24132085 k1,428:28446803,24132085:292353 x1,428:29134532,24132085 x1,428:29822253,24132085 x1,428:30084131,24132085 x1,428:30542418,24132085 x1,428:31197704,24132085 k1,428:31490056,24132085:292352 x1,428:32046846,24132085 x1,428:32482135,24132085 x1,428:33399299,24132085 x1,428:33808633,24132085 k1,428:34137209,24132085:328576 x1,428:34760059,24132085 x1,428:35428916,24132085 k1,428:35428916,24132085:0 ) (1,429:21355630,24852981:14073286,402253,127692 x1,429:22010916,24852981 x1,428:22291082,24852981 x1,428:22400205,24852981 g1,428:22547661,24852981 x1,428:23235980,24852981 g1,428:23383436,24852981 x1,428:24071165,24852981 x1,428:24906342,24852981 g1,428:25053798,24852981 x1,428:26659274,24852981 g1,428:26806730,24852981 x1,428:27544010,24852981 g1,428:27691466,24852981 x1,428:28936568,24852981 g1,428:29084024,24852981 x1,428:30411128,24852981 g1,428:30558584,24852981 x1,428:32475512,24852981 k1,429:35428916,24852981:2953404 g1,429:35428916,24852981 ) (1,430:21355630,25836021:14073286,403728,92009 (1,429:21355630,25836021:0,402253,92009 g1,429:21355630,25836021 g1,429:20372988,25836021 g1,429:20045308,25836021 (1,429:20045308,25836021:982642,402253,92009 g1,429:20045308,25836021 x1,429:21027950,25836021 ) g1,429:21355630,25836021 ) x1,429:21928937,25836021 k1,429:22121182,25836021:192245 x1,429:22956950,25836021 k1,429:23160393,25836021:203443 x1,429:23733692,25836021 k1,429:23925937,25836021:192245 x1,429:24679726,25836021 k1,429:24883168,25836021:203442 x1,429:26193148,25836021 x1,429:27143346,25836021 k1,429:27335591,25836021:192245 x1,429:28777700,25836021 k1,429:28969945,25836021:192245 x1,429:29330326,25836021 x1,429:30625554,25836021 k1,429:30817800,25836021:192246 x1,429:31276677,25836021 k1,429:31468922,25836021:192245 x1,429:32157242,25836021 x1,429:33238371,25836021 k1,429:33430616,25836021:192245 x1,429:34478718,25836021 x1,429:35428916,25836021 k1,429:35428916,25836021:0 ) (1,430:21355630,26556917:14073286,402253,127692 x1,430:22142445,26556917 x1,430:23157522,26556917 k1,429:23292542,26556917:135020 x1,429:24439739,26556917 k1,429:24574760,26556917:135021 x1,429:24935141,26556917 x1,429:25549134,26556917 k1,429:25684154,26556917:135020 x1,429:26371883,26556917 x1,429:27059604,26556917 x1,429:27321482,26556917 x1,429:27779769,26556917 x1,429:28435055,26556917 k1,429:28570076,26556917:135021 x1,429:29421778,26556917 k1,429:29556798,26556917:135020 x1,429:30048119,26556917 x1,429:30473962,26556917 x1,429:31227161,26556917 x1,429:32570758,26556917 k1,429:32705779,26556917:135021 x1,429:33525037,26556917 x1,429:34245202,26556917 k1,429:34380222,26556917:135020 x1,429:35232507,26556917 x1,429:35428916,26556917 k1,430:35428916,26556917:0 ) (1,430:21355630,27277813:14073286,402253,127692 x1,430:22436759,27277813 k1,429:22555979,27277813:119220 x1,429:23997497,27277813 x1,429:24751286,27277813 k1,429:24870507,27277813:119221 x1,429:26017704,27277813 x1,429:26427038,27277813 k1,429:26551905,27277813:124867 x1,429:27518620,27277813 k1,429:27637841,27277813:119221 x1,429:28014732,27277813 k1,429:28133952,27277813:119220 x1,429:29461048,27277813 k1,429:29580269,27277813:119221 x1,429:30203119,27277813 x1,429:30871976,27277813 k1,429:30991196,27277813:119220 x1,429:31728476,27277813 k1,429:31853344,27277813:124868 x1,429:33229393,27277813 x1,429:33982592,27277813 k1,429:34101812,27277813:119220 x1,429:35428916,27277813 k1,429:35428916,27277813:0 ) (1,430:21355630,27998709:14073286,402253,8251 x1,430:23272558,27998709 k1,430:35428916,27998709:12156358 g1,430:35428916,27998709 ) (1,432:21355630,28981749:14073286,402253,92009 (1,430:21355630,28981749:0,402253,92009 g1,430:21355630,28981749 g1,430:20372988,28981749 g1,430:20045308,28981749 (1,430:20045308,28981749:982642,402253,92009 g1,430:20045308,28981749 x1,430:21027950,28981749 ) g1,430:21355630,28981749 ) x1,430:21928937,28981749 k1,430:22215527,28981749:286590 x1,430:23706588,28981749 k1,430:24027962,28981749:321374 x1,430:24617778,28981749 x1,430:26279283,28981749 x1,430:26934569,28981749 x1,430:27917785,28981749 k1,430:28204375,28981749:286590 x1,430:29155155,28981749 x1,430:29450067,28981749 x1,430:29875910,28981749 x1,430:30793664,28981749 k1,430:31080255,28981749:286591 x1,430:31571576,28981749 k1,430:31858166,28981749:286590 x1,430:33430616,28981749 x1,430:35428916,28981749 k1,430:35428916,28981749:0 ) (1,432:21355630,29702645:14073286,402253,127692 x1,432:22797739,29702645 k1,430:23045858,29702645:248119 x1,430:23537179,29702645 x1,430:23963022,29702645 x1,430:24683187,29702645 x1,430:25830376,29702645 x1,430:26239710,29702645 k1,430:26512995,29702645:273285 x1,430:26840933,29702645 x1,430:27212524,29702645 x1,430:28319624,29702645 k1,430:28592909,29702645:273285 x1,430:30706230,29702645 k1,430:30979514,29702645:273284 x1,430:31683760,29702645 k1,430:31931879,29702645:248119 x1,430:32259817,29702645 x1,430:32341803,29702645 k1,430:32589923,29702645:248120 x1,430:33245216,29702645 x1,430:33998407,29702645 k1,430:34246526,29702645:248119 x1,430:34705403,29702645 k1,430:34953522,29702645:248119 x1,430:35428916,29702645 k1,430:35428916,29702645:0 ) (1,432:21355630,30423541:14073286,402253,127692 x1,432:21912420,30423541 x1,432:22422028,30423541 x1,432:23175227,30423541 x1,432:23299095,30423541 g1,430:23446551,30423541 x1,430:24478735,30423541 g1,430:24626191,30423541 x1,430:26346113,30423541 g1,430:26493569,30423541 x1,430:26935937,30423541 g1,431:27083393,30423541 x1,431:28459450,30423541 x1,431:29786554,30423541 k1,432:35428916,30423541:5642362 g1,432:35428916,30423541 ) (1,434:21355630,31406581:14073286,402253,92009 (1,432:21355630,31406581:0,402253,92009 g1,432:21355630,31406581 g1,432:20372988,31406581 g1,432:20045308,31406581 (1,432:20045308,31406581:982642,402253,92009 g1,432:20045308,31406581 x1,432:21027950,31406581 ) g1,432:21355630,31406581 ) x1,432:22027433,31406581 k1,432:22156084,31406581:128651 x1,432:22516465,31406581 x1,432:22620274,31406581 k1,432:22748926,31406581:128652 x1,432:23076864,31406581 x1,432:23177138,31406581 k1,432:23305789,31406581:128651 x1,432:23797692,31406581 x1,432:24389279,31406581 k1,432:24521692,31406581:132413 x1,432:26716992,31406581 k1,432:26845644,31406581:128652 x1,432:27271495,31406581 k1,432:27400128,31406581:128633 x1,432:28087857,31406581 x1,432:28498962,31406581 k1,432:28627613,31406581:128651 x1,432:29774212,31406581 x1,432:32133469,31406581 k1,432:32262121,31406581:128652 x1,432:32590059,31406581 x1,432:32843095,31406581 x1,432:33465347,31406581 x1,432:34055155,31406581 k1,432:34183806,31406581:128651 x1,432:35428916,31406581 k1,432:35428916,31406581:0 ) (1,434:21355630,32127477:14073286,402253,127692 x1,434:21814507,32127477 k1,432:21992334,32127477:177827 x1,432:22943114,32127477 x1,432:23238026,32127477 x1,432:24089712,32127477 k1,432:24267538,32127477:177826 x1,432:24988293,32127477 k1,432:25166120,32127477:177827 x1,432:26214222,32127477 x1,432:27164420,32127477 k1,432:27342247,32127477:177827 x1,432:28784356,32127477 k1,432:28962183,32127477:177827 x1,432:30469754,32127477 x1,432:31059578,32127477 k1,432:31237404,32127477:177826 x1,432:32089106,32127477 k1,432:32266933,32127477:177827 x1,432:32758254,32127477 x1,432:33413532,32127477 x1,432:34036374,32127477 x1,432:34200339,32127477 x1,432:35019582,32127477 x1,432:35428916,32127477 k1,432:35428916,32127477:0 ) (1,434:21355630,32848373:14073286,402253,127692 x1,434:21683568,32848373 x1,433:22055159,32848373 x1,433:23162259,32848373 k1,433:23449504,32848373:287245 x1,433:24039320,32848373 x1,433:24334232,32848373 x1,433:24617350,32848373 x1,433:26664619,32848373 k1,433:26951864,32848373:287245 x1,433:28049518,32848373 k1,433:28308806,32848373:259288 x1,433:29635910,32848373 k1,433:29923155,32848373:287245 x1,433:30709387,32848373 x1,433:31037325,32848373 x1,433:32769046,32848373 k1,433:33056291,32848373:287245 x1,433:33793571,32848373 k1,433:34052859,32848373:259288 x1,433:35428916,32848373 k1,434:35428916,32848373:0 ) (1,434:21355630,33569269:14073286,402253,8251 x1,434:22535278,33569269 k1,434:35428916,33569269:12893638 g1,434:35428916,33569269 ) (1,435:21355630,34552309:14073286,403728,92009 (1,434:21355630,34552309:0,402253,92009 g1,434:21355630,34552309 g1,434:20372988,34552309 g1,434:20045308,34552309 (1,434:20045308,34552309:982642,402253,92009 g1,434:20045308,34552309 x1,434:21027950,34552309 ) g1,434:21355630,34552309 ) x1,434:21928937,34552309 k1,434:22278760,34552309:349823 x1,434:22852067,34552309 k1,434:23201890,34552309:349823 x1,434:24119652,34552309 x1,434:25118810,34552309 k1,434:25519224,34552309:400414 x1,434:26092523,34552309 k1,434:26442346,34552309:349823 x1,434:27196135,34552309 k1,434:27596550,34552309:400415 x1,434:28809208,34552309 x1,434:29104120,34552309 x1,434:29529963,34552309 x1,434:30447717,34552309 k1,434:30797540,34552309:349823 x1,434:31288861,34552309 k1,434:31638684,34552309:349823 x1,434:32686786,34552309 x1,434:33636984,34552309 k1,434:33986807,34552309:349823 x1,434:35428916,34552309 k1,434:35428916,34552309:0 ) (1,435:21355630,35273205:14073286,402253,127692 x1,435:21846951,35273205 x1,435:23255443,35273205 k1,434:23619988,35273205:364545 x1,434:24996029,35273205 k1,434:25360573,35273205:364544 x1,434:26081336,35273205 x1,434:27097003,35273205 x1,434:27752281,35273205 k1,434:28116826,35273205:364545 x1,434:28804556,35273205 k1,434:29169101,35273205:364545 x1,434:30053822,35273205 x1,434:30610612,35273205 x1,434:30981613,35273205 x1,434:31325477,35273205 k1,434:31690021,35273205:364544 x1,434:32509280,35273205 x1,434:33589826,35273205 x1,434:34573050,35273205 k1,434:34937595,35273205:364545 x1,434:35428916,35273205 k1,434:35428916,35273205:0 ) (1,435:21355630,35994101:14073286,402253,127692 x1,435:21879977,35994101 x1,434:22251568,35994101 x1,434:23358668,35994101 g1,434:23506124,35994101 x1,434:24390852,35994101 g1,434:24538308,35994101 x1,434:25128124,35994101 x1,434:25914349,35994101 x1,434:26946525,35994101 g1,434:27093981,35994101 x1,434:28437586,35994101 g1,434:28585042,35994101 x1,434:30304964,35994101 g1,434:30452420,35994101 x1,434:30894788,35994101 g1,434:31042244,35994101 x1,434:32123389,35994101 x1,434:32713213,35994101 k1,435:35428916,35994101:2715703 g1,435:35428916,35994101 ) (1,436:21355630,36977141:14073286,403728,127692 (1,435:21355630,36977141:0,402253,92009 g1,435:21355630,36977141 g1,435:20372988,36977141 g1,435:20045308,36977141 (1,435:20045308,36977141:982642,402253,92009 g1,435:20045308,36977141 x1,435:21027950,36977141 ) g1,435:21355630,36977141 ) x1,435:21732521,36977141 k1,435:21971221,36977141:238700 x1,435:22397072,36977141 x1,435:22468445,36977141 k1,435:22707144,36977141:238699 x1,435:23690376,36977141 x1,435:24116227,36977141 x1,435:24667119,36977141 x1,435:24776242,36977141 k1,435:25037753,36977141:261511 x1,435:25611052,36977141 k1,435:25849751,36977141:238699 x1,435:26603540,36977141 k1,435:26865051,36977141:261511 x1,435:28044100,36977141 k1,435:28282799,36977141:238699 x1,435:29266031,36977141 x1,435:30084699,36977141 x1,435:30969419,36977141 k1,435:31208119,36977141:238700 x1,435:31895848,36977141 x1,435:32879072,36977141 k1,435:33117772,36977141:238700 x1,435:33609093,36977141 k1,435:33847792,36977141:238699 x1,435:34109670,36977141 k1,435:34348370,36977141:238700 x1,435:34774221,36977141 x1,435:35232507,36977141 x1,435:35428916,36977141 k1,436:35428916,36977141:0 ) (1,436:21355630,37698037:14073286,402253,127692 x1,436:22404331,37698037 k1,435:22517093,37698037:112762 x1,435:23369378,37698037 x1,435:24254098,37698037 k1,435:24366860,37698037:112762 x1,435:25054590,37698037 k1,435:25167352,37698037:112762 x1,435:25888107,37698037 k1,435:26000869,37698037:112762 x1,435:26951067,37698037 k1,435:27063795,37698037:112728 x1,435:28505313,37698037 x1,435:29259102,37698037 k1,435:29371864,37698037:112762 x1,435:30748496,37698037 k1,435:30861258,37698037:112762 x1,435:31352579,37698037 x1,435:31909369,37698037 x1,435:32695602,37698037 x1,435:33317854,37698037 x1,435:34071643,37698037 k1,435:34184405,37698037:112762 x1,435:35232507,37698037 x1,435:35428916,37698037 k1,436:35428916,37698037:0 ) (1,436:21355630,38418933:14073286,402253,127692 x1,436:22305828,38418933 k1,435:22426371,38418933:120543 x1,435:23868480,38418933 k1,435:23989022,38418933:120542 x1,435:24841299,38418933 x1,435:25005264,38418933 x1,435:25300176,38418933 x1,435:25559108,38418933 x1,435:26476862,38418933 k1,435:26597405,38418933:120543 x1,435:26957786,38418933 x1,435:27735169,38418933 x1,435:29275183,38418933 x1,435:29684517,38418933 k1,435:29810443,38418933:125926 x1,435:30629702,38418933 x1,435:31710248,38418933 x1,435:32693472,38418933 k1,435:32814014,38418933:120542 x1,435:33305335,38418933 k1,435:33425878,38418933:120543 x1,435:33950225,38418933 x1,435:34321816,38418933 x1,435:35428916,38418933 k1,435:35428916,38418933:0 ) (1,436:21355630,39139829:14073286,402253,127692 x1,436:22043359,39139829 x1,435:22454464,39139829 g1,435:22601920,39139829 x1,435:23027771,39139829 x1,435:24583704,39139829 g1,435:24731160,39139829 x1,435:27204854,39139829 g1,435:27352310,39139829 x1,435:28531950,39139829 g1,435:28679406,39139829 x1,435:30399328,39139829 g1,435:30546784,39139829 x1,435:30989152,39139829 g1,435:31136608,39139829 x1,435:32512665,39139829 x1,435:32807577,39139829 k1,436:35428916,39139829:2621339 g1,436:35428916,39139829 ) (1,437:21355630,40122869:14073286,403728,127692 (1,436:21355630,40122869:0,402253,92009 g1,436:21355630,40122869 g1,436:20372988,40122869 g1,436:20045308,40122869 (1,436:20045308,40122869:982642,402253,92009 g1,436:20045308,40122869 x1,436:21027950,40122869 ) g1,436:21355630,40122869 ) x1,436:22027433,40122869 k1,436:22198320,40122869:170887 x1,436:22558701,40122869 x1,436:22662510,40122869 k1,436:22833397,40122869:170887 x1,436:23161335,40122869 x1,436:23261609,40122869 k1,436:23432497,40122869:170888 x1,436:23924400,40122869 x1,436:24515987,40122869 k1,436:24692731,40122869:176744 x1,436:25266030,40122869 k1,436:25436918,40122869:170888 x1,436:26190707,40122869 k1,436:26367451,40122869:176744 x1,436:31019928,40122869 k1,436:31190816,40122869:170888 x1,436:31878546,40122869 k1,436:32049433,40122869:170887 x1,436:33360587,40122869 x1,436:34766708,40122869 k1,436:34937595,40122869:170887 x1,436:35428916,40122869 k1,436:35428916,40122869:0 ) (1,437:21355630,40843765:14073286,402253,127692 x1,437:22076385,40843765 k1,436:22277176,40843765:200791 x1,436:24275476,40843765 k1,436:24476266,40843765:200790 x1,436:25918375,40843765 k1,436:26119166,40843765:200791 x1,436:28511449,40843765 k1,436:28712240,40843765:200791 x1,436:29171117,40843765 k1,436:29371907,40843765:200790 x1,436:30092662,40843765 k1,436:30293453,40843765:200791 x1,436:31243651,40843765 x1,436:31652985,40843765 k1,436:31853775,40843765:200790 x1,436:34736804,40843765 k1,436:34937595,40843765:200791 x1,436:35428916,40843765 k1,436:35428916,40843765:0 ) (1,437:21355630,41564661:14073286,402253,127692 x1,437:21879977,41564661 x1,436:22251568,41564661 x1,436:23358668,41564661 g1,436:23506124,41564661 x1,436:24390852,41564661 g1,436:24538308,41564661 x1,436:26946525,41564661 g1,436:27093981,41564661 x1,436:28437586,41564661 g1,436:28585042,41564661 x1,436:30304964,41564661 g1,436:30452420,41564661 x1,436:30894788,41564661 g1,436:31042244,41564661 x1,436:32418301,41564661 x1,436:32713213,41564661 k1,437:35428916,41564661:2715703 g1,437:35428916,41564661 ) (1,438:21355630,42547701:14073286,403728,127692 (1,437:21355630,42547701:0,402253,92009 g1,437:21355630,42547701 g1,437:20372988,42547701 g1,437:20045308,42547701 (1,437:20045308,42547701:982642,402253,92009 g1,437:20045308,42547701 x1,437:21027950,42547701 ) g1,437:21355630,42547701 ) x1,437:21732521,42547701 k1,437:21916919,42547701:184398 x1,437:22490226,42547701 k1,437:22674624,42547701:184398 x1,437:25312284,42547701 k1,437:25505917,42547701:193633 x1,437:26079216,42547701 k1,437:26263615,42547701:184399 x1,437:27017404,42547701 k1,437:27211037,42547701:193633 x1,437:27964236,42547701 x1,437:28235560,42547701 x1,437:29340873,42547701 x1,437:30517559,42547701 k1,437:30701957,42547701:184398 x1,437:31619711,42547701 k1,437:31804109,42547701:184398 x1,437:33802409,42547701 k1,437:33986807,42547701:184398 x1,437:35428916,42547701 k1,437:35428916,42547701:0 ) (1,438:21355630,43268597:14073286,402253,127692 x1,438:21716011,43268597 x1,438:23011239,43268597 k1,437:23196404,43268597:185165 x1,437:23655281,43268597 k1,437:23840445,43268597:185164 x1,437:24561200,43268597 k1,437:24746365,43268597:185165 x1,437:25696563,43268597 k1,437:25881690,43268597:185127 x1,437:28076997,43268597 k1,437:28262161,43268597:185164 x1,437:29409358,43268597 k1,437:29594523,43268597:185165 x1,437:31265476,43268597 x1,437:31674810,43268597 k1,437:31869401,43268597:194591 x1,437:34752430,43268597 k1,437:34937595,43268597:185165 x1,437:35428916,43268597 k1,437:35428916,43268597:0 ) (1,438:21355630,43989493:14073286,402253,127692 x1,438:21879977,43989493 x1,437:22251568,43989493 x1,437:23358668,43989493 g1,437:23506124,43989493 x1,437:24193853,43989493 x1,437:24604958,43989493 g1,437:24752414,43989493 x1,437:26734198,43989493 g1,437:26881654,43989493 x1,437:29355348,43989493 g1,437:29502804,43989493 x1,437:30682444,43989493 g1,437:30829900,43989493 x1,437:32549822,43989493 g1,437:32697278,43989493 x1,437:33139646,43989493 g1,437:33287102,43989493 x1,437:34663159,43989493 x1,437:34958071,43989493 k1,438:35428916,43989493:470845 g1,438:35428916,43989493 ) ] g1,440:35428916,48295956 ) ) ] h1,440:3729360,50262036:31699556,0,0 ] ] !67028 11 Input:31:/Users/gd/Documents/Research/Linear Collider/International Linear Collider/ILC-Americas/CESR-TA/Electron cloud/ECLOUD12/Beam dynamics paper/./ECLOUD12_CesrtTA_Dugan.aux !189 Postamble: Count:9480 !27 Post scriptum:	arxiv-papers	2013-10-01T12:02:06	2024-09-04T02:49:51.802401	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "G. Dugan (Cornell U., CLASSE), M. Billing (Cornell U., CLASSE), K.\n Butler (Cornell U., CLASSE), J. Chu (Cornell U., CLASSE), J. Crittenden\n (Cornell U., CLASSE), M. Forster (Cornell U., CLASSE), D. Kreinick (Cornell\n U., CLASSE), R. Meller (Cornell U., CLASSE), M. Palmer (Cornell U., CLASSE),\n G. Ramirez (Cornell U., CLASSE), M. Rendina (Cornell U., CLASSE), N. Rider\n (Cornell U., CLASSE), K. Sonnad (Cornell U., CLASSE), H. Williams (Cornell\n U., CLASSE), R. Campbell (Cal. Poly.), R. Holtzapple (Cal. Poly.), M.\n Randazzo (Cal. Poly.), J. Flanagan (KEK, Tsukuba), K.Ohmi (KEK, Tsukuba), M.\n Furman (LBNL, Berkeley), M. Venturini (LBNL, Berkeley), M.Pivi (SLAC)", "submitter": "Scientific Information Service CERN", "url": "https://arxiv.org/abs/1310.0261" }

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